diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/-3lwzxWH7r8_postprocess.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/-3lwzxWH7r8_postprocess.srt
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+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/-3lwzxWH7r8_postprocess.srt
@@ -0,0 +1,1544 @@
+1
+00:00:21,600 --> 00:00:29,560
+ال .. في المحاضرة السابقة بدأنا التعرف على cushy
+
+2
+00:00:29,560 --> 00:00:35,020
+sequences فأخدنا تعريف ال cushy sequence و أثبتنا
+
+3
+00:00:35,020 --> 00:00:41,000
+أنه كل convergence sequence is cushy و أعتقد كمان
+
+4
+00:00:41,000 --> 00:00:48,510
+أثبتنا أنه كل cushy sequence is bounded صحيح؟اليوم
+
+5
+00:00:48,510 --> 00:00:54,970
+هنثبت العكس و هو ان كل كوشي sequence is convergent
+
+6
+00:00:54,970 --> 00:01:00,630
+فنستعيد بس نستذكر مع بعض تعريف الكوشي sequence
+
+7
+00:01:00,630 --> 00:01:07,450
+definition a
+
+8
+00:01:07,450 --> 00:01:14,010
+sequence of real numbers xn is
+
+9
+00:01:14,010 --> 00:01:14,710
+cauchy
+
+10
+00:01:18,570 --> 00:01:26,170
+إذا تحقق الشرط التالي لكل epsilon أكبر من الصفر
+
+11
+00:01:26,170 --> 00:01:32,270
+يوجد capital N depends on epsilon natural number
+
+12
+00:01:32,270 --> 00:01:41,660
+such that لو كان N و N bigger than or equal Nthis
+
+13
+00:01:41,660 --> 00:01:48,840
+implies أن absolute value ل xn minus xm أصغر من
+
+14
+00:01:48,840 --> 00:01:53,960
+إيصال وشوفنا
+
+15
+00:01:53,960 --> 00:02:03,260
+المرة اللي فاتت أو برهننا limit 2 و 21 every
+
+16
+00:02:03,260 --> 00:02:06,820
+convergent
+
+17
+00:02:11,450 --> 00:02:17,190
+sequence is cauchy
+
+18
+00:02:17,190 --> 00:02:27,510
+ثم برهنة another لمبة لمبة اتنين و عشرين بتقول
+
+19
+00:02:27,510 --> 00:02:34,250
+اللمبة هذه ان every cauchy
+
+20
+00:02:34,250 --> 00:02:35,010
+sequence
+
+21
+00:02:40,290 --> 00:02:49,630
+is bounded اليوم
+
+22
+00:02:49,630 --> 00:02:59,890
+هنثبت نظرية مهمة نظرية اتنين تلاتة وعشرين وهذه
+
+23
+00:02:59,890 --> 00:03:07,310
+النظرية هي كوشي كوشي
+
+24
+00:03:07,310 --> 00:03:08,150
+criterion
+
+25
+00:03:11,820 --> 00:03:18,680
+أو معيار كوشي معيار
+
+26
+00:03:18,680 --> 00:03:25,580
+كوشي للتقارب النظرية
+
+27
+00:03:25,580 --> 00:03:35,800
+بتنص على أن a sequence x in contained in R is
+
+28
+00:03:35,800 --> 00:03:36,700
+convergent
+
+29
+00:03:39,150 --> 00:03:55,130
+is convergent if and only if it is cauchy any
+
+30
+00:03:55,130 --> 00:04:00,610
+sequence of real numbers بتكون convergent if and
+
+31
+00:04:00,610 --> 00:04:04,750
+only if it is cauchy البرهان
+
+32
+00:04:09,110 --> 00:04:15,430
+this part اللي هو ال only if part هذا هو نفسه لمّة
+
+33
+00:04:15,430 --> 00:04:29,890
+واحدة عشرين if x in is convergent then
+
+34
+00:04:29,890 --> 00:04:32,990
+by
+
+35
+00:04:32,990 --> 00:04:37,210
+لمّة واحدة عشرين
+
+36
+00:04:40,710 --> 00:04:46,370
+it is cushy it is cushy
+
+37
+00:04:46,370 --> 00:04:51,850
+لأن هذا جزء برهناه في المحاضرة السابقة على صورة
+
+38
+00:04:51,850 --> 00:05:00,710
+لمة واحد وعشرين ال .. ال if part هنبرهنه اليوم
+
+39
+00:05:00,710 --> 00:05:09,520
+هنشوف مع بعض assume العكسassume أن الـ sequence x
+
+40
+00:05:09,520 --> 00:05:16,100
+in is Cauchy وبدنا
+
+41
+00:05:16,100 --> 00:05:25,280
+نثبت إنها convergent طيب بما إنها Cauchy then
+
+42
+00:05:25,280 --> 00:05:31,540
+by لمّا اتنين و عشرين تطلع bounded
+
+43
+00:05:36,760 --> 00:05:43,280
+إذا by لمبة إتنين و عشرين ال sequence x in is
+
+44
+00:05:43,280 --> 00:05:52,560
+bounded بستخدام
+
+45
+00:05:52,560 --> 00:05:55,600
+Bolzano-Weierstrass theorem
+
+46
+00:06:05,480 --> 00:06:09,240
+اللي أخدناها المحاضرة السابقة أو اللي قبلها هذا
+
+47
+00:06:09,240 --> 00:06:15,960
+اختصار بولزانو ويرشتراس بولزانو ويرشتراس هنا بتقول
+
+48
+00:06:15,960 --> 00:06:18,900
+انه كل bounded sequence has a convergent
+
+49
+00:06:18,900 --> 00:06:27,720
+subsequence فهي عندي bounded sequence sequence x
+
+50
+00:06:27,720 --> 00:06:33,480
+in has a
+
+51
+00:06:33,480 --> 00:06:34,560
+convergent
+
+52
+00:06:39,950 --> 00:06:44,370
+sub-sequence x
+
+53
+00:06:44,370 --> 00:06:57,970
+in k وها دي converges to x star تنتمي إلى R طبعا؟
+
+54
+00:06:57,970 --> 00:07:03,550
+إذن هذه sub-sequence من x in وconvergent to some x
+
+55
+00:07:03,550 --> 00:07:05,350
+star تنتمي إلى R
+
+56
+00:07:08,910 --> 00:07:14,610
+طيب احنا عايزين نثبت claim عايزين
+
+57
+00:07:14,610 --> 00:07:24,210
+احنا نثبت ان ال sequence xn converges الى العدد
+
+58
+00:07:24,210 --> 00:07:32,530
+x star وبالتالي هيك بنكمل برهان انظرية صح؟ فلبرهان
+
+59
+00:07:32,530 --> 00:07:33,090
+ذلك
+
+60
+00:07:36,470 --> 00:07:44,330
+نستخدم تعريف epsilon capital N للـ limit فبنبدأ بـ
+
+61
+00:07:44,330 --> 00:07:47,790
+epsilon أكبر من السفر عشوائية let epsilon أكبر من
+
+62
+00:07:47,790 --> 00:07:57,240
+السفر be givenنحتاج أن نشهر أن هناك كابتل N كمية
+
+63
+00:07:57,240 --> 00:07:59,060
+عامة تعتمد على إبسلون كمية عامة تعتمد على إبسلون
+
+64
+00:07:59,060 --> 00:08:00,120
+كمية عامة تعتمد على إبسلون كمية عامة تعتمد على
+
+65
+00:08:00,120 --> 00:08:02,780
+إبسلون كمية عامة تعتمد على إبسلون كمية عامة تعتمد
+
+66
+00:08:02,780 --> 00:08:05,280
+على إبسلون كمية عامة تعتمد على إبسلون كمية عامة
+
+67
+00:08:05,280 --> 00:08:08,080
+تعتمد على إبسلون كمية عامة تعتمد على إبسلون كمية
+
+68
+00:08:08,080 --> 00:08:09,960
+عامة تعتمد على إبسلون كمية عامة تعتمد على إبسلون
+
+69
+00:08:09,960 --> 00:08:14,020
+كمية عامة تعتمد على إبسلون
+
+70
+00:08:14,020 --> 00:08:21,080
+كمية عامة تعتمد على إبسلون
+
+71
+00:08:23,720 --> 00:08:27,960
+وهي إبسلون given، إذا by definition of Cauchy
+
+72
+00:08:27,960 --> 00:08:33,920
+sequence there exists capital N depends on إبسلون
+
+73
+00:08:33,920 --> 00:08:44,200
+natural number such that لكل N و M أكبر من أو ساوي
+
+74
+00:08:44,200 --> 00:08:50,400
+capital N، this implies an absolute X N minus X M
+
+75
+00:08:52,000 --> 00:09:00,760
+less than epsilon at null نسمي
+
+76
+00:09:00,760 --> 00:09:06,380
+ال implication هيا دي star طيب
+
+77
+00:09:06,380 --> 00:09:12,900
+احنا حصلنا على انه ال sequence x أو ال subsequence
+
+78
+00:09:12,900 --> 00:09:18,080
+x in k converges to x star
+
+79
+00:09:20,890 --> 00:09:25,870
+إذا لنفس الـ Epsilon و Epsilon هي نفس الـ Epsilon
+
+80
+00:09:25,870 --> 00:09:34,130
+given فمن تعريف ال convergence for
+
+81
+00:09:34,130 --> 00:09:39,950
+same Epsilon أكبر من الصفر نفس الـ Epsilon اللي
+
+82
+00:09:39,950 --> 00:09:47,070
+هناك نقدر نلاقي يوجد capital K عدد طبيعي capital K
+
+83
+00:09:49,980 --> 00:09:55,920
+وهذا العدد .. هذا عبارة عن عدد طبيعي وهذا العدد
+
+84
+00:09:55,920 --> 00:10:01,240
+الطبيعي هو واحد من مؤشرات الـ subsequence اللي هم
+
+85
+00:10:01,240 --> 00:10:10,300
+n واحد, n اتنين, n تلاتة و
+
+86
+00:10:10,300 --> 00:10:17,500
+هكذاإذا يوجد كابتل K اللي هو واحد عدد طبيعي وهذا
+
+87
+00:10:17,500 --> 00:10:23,080
+واحد من مؤشرات ال subsequence ممكن أختاره هذا
+
+88
+00:10:23,080 --> 00:10:32,400
+كابتل K أكبر من أو ساوي كابتل N بحيث
+
+89
+00:10:32,400 --> 00:10:41,800
+أن ال absolute value ل Xcapital K minus X star
+
+90
+00:10:41,800 --> 00:10:50,000
+أصغر من إبسلون على اتنين كمان
+
+91
+00:10:50,000 --> 00:10:54,280
+مرة السب سيكوينس هي هتconverge ل X star إذا في
+
+92
+00:10:54,280 --> 00:11:02,780
+capital K natural number و هو واحد من large واحد
+
+93
+00:11:02,780 --> 00:11:10,220
+من ال indices و طبعا كبيرهو ممكن نختاره أكبر من أو
+
+94
+00:11:10,220 --> 00:11:15,720
+ساوي capital N بحيث المسافة بين X كابتل K و X أصلا
+
+95
+00:11:15,720 --> 00:11:21,480
+أصغر من ي على 2 هو ممكن أن أنا يعني هذا أحط هنا K
+
+96
+00:11:21,480 --> 00:11:26,160
+و أقول أن هذا أصغر من ي على 2 لكل K أكبر من أو
+
+97
+00:11:26,160 --> 00:11:33,030
+ساوي كابتل Kصح؟ مش هيك تعريف ال convergence لكن
+
+98
+00:11:33,030 --> 00:11:39,730
+انا بدي اخد K بساوي كابتل K وبالتالي اخد بس X
+
+99
+00:11:39,730 --> 00:11:45,270
+كابتل K المسافة بينها و بين X star أصغر من Y على 2
+
+100
+00:11:45,270 --> 00:11:53,290
+نسمي المتباينة هذه double star الان
+
+101
+00:11:53,290 --> 00:11:53,950
+now
+
+102
+00:11:59,240 --> 00:12:08,140
+أنا عندي كابتل كأكبر من أو ساوي كابتل N so
+
+103
+00:12:08,140 --> 00:12:14,320
+by star by
+
+104
+00:12:14,320 --> 00:12:25,600
+star with M بساوي كابتل K we
+
+105
+00:12:25,600 --> 00:12:26,720
+have لدينا
+
+106
+00:12:30,820 --> 00:12:40,660
+absolute xn minus x capital k أصغر من y ع 2 نسمي
+
+107
+00:12:40,660 --> 00:12:49,900
+هذه المتباينة triple triple star كمان مرة ال k هذه
+
+108
+00:12:49,900 --> 00:12:59,980
+اختارناها أكبر منها و يساوي n و من starإذا كانت
+
+109
+00:12:59,980 --> 00:13:05,100
+الـ K .. إذا خدت M بساوي كابتال K و هذه أكبر من أو
+
+110
+00:13:05,100 --> 00:13:11,400
+ساوي N فبتصير المتباينة هذه absolute XN minus XK
+
+111
+00:13:11,400 --> 00:13:17,260
+أزرع من إبسط على اتنين و الـ N هذه لازم تكون أكبر
+
+112
+00:13:17,260 --> 00:13:22,780
+من أو ساوي M، إذن هذا صحيح لكل small M أكبر من أو
+
+113
+00:13:22,780 --> 00:13:24,260
+ساوي كابتال M
+
+114
+00:13:29,670 --> 00:13:35,670
+تمام hence by
+
+115
+00:13:35,670 --> 00:13:44,050
+double star الآن من double star and triple star
+
+116
+00:13:48,930 --> 00:13:57,270
+لدينا we have لو كان n أكبر من أو ساوي capital N
+
+117
+00:13:57,270 --> 00:14:11,330
+فهذا بيقدي أنه absolute xn minus x star طبعا
+
+118
+00:14:11,330 --> 00:14:18,530
+هنا هترح x capital K و هرجعها
+
+119
+00:14:28,690 --> 00:14:38,730
+إذا I subtracted XK and get it back باخد هدوع
+
+120
+00:14:38,730 --> 00:14:43,640
+الأثنين مع بعض و التحدين هدوع مع بعضالـ absolute
+
+121
+00:14:43,640 --> 00:14:49,100
+value بالترانجل inequality بالترانجل الانيقواليتي
+
+122
+00:14:49,100 --> 00:14:54,380
+هذا أصغر من أسابع absolute الحد الأول اللي هو xn
+
+123
+00:14:54,380 --> 00:15:01,400
+minus xk زائد absolute الحد التاني اللي هو xk
+
+124
+00:15:01,400 --> 00:15:08,500
+minus x star الآن
+
+125
+00:15:08,500 --> 00:15:16,750
+باستخدام triple starمن المتباينة هذه هاي عندي انا
+
+126
+00:15:16,750 --> 00:15:23,170
+x اول شي ال n small n أكبر من أو ساوي capital N
+
+127
+00:15:23,170 --> 00:15:28,590
+هاي small n أكبر من أو ساوي capital N وبالتالي ال
+
+128
+00:15:28,590 --> 00:15:34,870
+absolute value هذه أصغر من epsilon على اتنين زاد و
+
+129
+00:15:34,870 --> 00:15:42,360
+من double star من double starهي عندي absolute x K
+
+130
+00:15:42,360 --> 00:15:47,640
+minus x star أصغر من إبسمن على اتنين المجموع بتطلع
+
+131
+00:15:47,640 --> 00:15:53,460
+إبسمن since
+
+132
+00:15:53,460 --> 00:16:00,000
+إبسمن أكبر من السفر was arbitrarily
+
+133
+00:16:03,850 --> 00:16:09,870
+نحن لدينا من مفهوم الـ convergence أنه هيك منكون
+
+134
+00:16:09,870 --> 00:16:16,690
+أثبتنا أنه limit xn as n tends to infinity equals
+
+135
+00:16:16,690 --> 00:16:26,790
+x star وهذا بكمل برهان ال claim و النظرية تمام؟
+
+136
+00:16:26,790 --> 00:16:32,160
+هاي لاحظوا أن احنابنثبت أننا ندعي أن الـSequence
+
+137
+00:16:32,160 --> 00:16:35,660
+Xn هي الـConversion لـX الصار حسب تعريف epsilon
+
+138
+00:16:35,660 --> 00:16:40,360
+capital N للـLimits بدأنا بـepsilon given عشوائية
+
+139
+00:16:40,360 --> 00:16:46,360
+عدد موجب أثبتنا هي يوجد capital N يعتمد على
+
+140
+00:16:46,360 --> 00:16:51,660
+epsilon أشرر numberبحيث انه لكل N أكبر من او ساوي
+
+141
+00:16:51,660 --> 00:16:59,400
+capital N طلع absolute xn minus x star less than
+
+142
+00:16:59,400 --> 00:17:07,300
+epsilon لما ان هذا الكلام صحيح لكل epsilonإذا by
+
+143
+00:17:07,300 --> 00:17:10,880
+definition limit xn ساوي x أسطورة، إذا ال sequence
+
+144
+00:17:10,880 --> 00:17:14,240
+convergent إذا هذا بيكمل البرهان إن لو كانت ال
+
+145
+00:17:14,240 --> 00:17:18,800
+sequence كوشي then it is convergent تمام واضح
+
+146
+00:17:18,800 --> 00:17:23,960
+البرهان؟ okay حلو إذا نعم
+
+147
+00:17:28,410 --> 00:17:32,690
+مش احنا حاكينا ان x and is bounded؟ صحيح طيب الحين
+
+148
+00:17:32,690 --> 00:17:37,530
+في عند قلب بالنزام و بالسترس في عند x and في عند
+
+149
+00:17:37,530 --> 00:17:41,470
+conversion subsequence صح هدا هي صح conversion ل x
+
+150
+00:17:41,470 --> 00:17:46,590
+and to some x star احنا أخدنا نظرية إذا كانت ال
+
+151
+00:17:46,590 --> 00:17:51,110
+conversion subsequence converge to x و x to r ف x
+
+152
+00:17:51,110 --> 00:17:55,890
+and تكون converge ل x لا مااخدنا نظرية زيك انت مش
+
+153
+00:17:55,890 --> 00:17:57,050
+خارق النظرية صح
+
+154
+00:18:00,740 --> 00:18:05,020
+لأ النظرية مابتحكيش هيك معلش النظرية هذه بتقول لو
+
+155
+00:18:05,020 --> 00:18:09,700
+أنا في عندي bounded sequence و لو كل convergent
+
+156
+00:18:09,700 --> 00:18:13,940
+sequence من ال sequence هذه convergent لعدد X
+
+157
+00:18:13,940 --> 00:18:19,160
+فلازم ال sequence نفسها تكون convergent ل X أنا
+
+158
+00:18:19,160 --> 00:18:22,900
+عندي بس sequence sub sequence واحدة converged ل X
+
+159
+00:18:22,900 --> 00:18:26,860
+أصلا و ليس every convergent subsequence converged
+
+160
+00:18:26,860 --> 00:18:31,790
+ل X أصلافالفرض التاني تبع النظرية اللي بتحكي عنها
+
+161
+00:18:31,790 --> 00:18:38,270
+مش متحقق وبالتالي لا استطيع تطبيق النظرية تمام؟ في
+
+162
+00:18:38,270 --> 00:18:45,290
+اي سؤال تاني؟ okay ده سؤال كتير يعني مهم و .. و ..
+
+163
+00:18:45,290 --> 00:18:51,790
+و جيد و ياريت يعني اي حد عنده تساؤل زي هذا يعني
+
+164
+00:18:51,790 --> 00:18:58,410
+يسأله هل في اي شي في القرآن مش واضح؟ واضح اكتر من
+
+165
+00:18:58,410 --> 00:19:02,990
+هيك؟Okay أعتقد أن البرهان واضح يعني لو قرأته
+
+166
+00:19:02,990 --> 00:19:10,710
+بتماعه هتجد أنه يعني سهل و بسيط طيب ناخد أمثلة على
+
+167
+00:19:10,710 --> 00:19:17,430
+كيف نستخدم تعريف ال koshi sequence في إثبات أنه
+
+168
+00:19:17,430 --> 00:19:25,730
+given sequence is koshi باستخدام التعريف مباشرة و
+
+169
+00:19:25,730 --> 00:19:28,410
+ليس باستخدام اللي هو koshi criterion
+
+170
+00:19:31,950 --> 00:19:47,490
+إذا ناخد هنا بعض الأمثلة examples
+
+171
+00:19:47,490 --> 00:19:52,930
+الأمثلة
+
+172
+00:19:52,930 --> 00:20:00,250
+دي أنا أعطيها الرقم 224 أول مثال show
+
+173
+00:20:04,710 --> 00:20:13,310
+directly show direct that
+
+174
+00:20:13,310 --> 00:20:20,390
+ال sequence ال
+
+175
+00:20:20,390 --> 00:20:25,130
+sequence واحد على ان is Cauchy
+
+176
+00:20:35,150 --> 00:20:39,630
+لما اقول show directly ان ال sequence معينة is
+
+177
+00:20:39,630 --> 00:20:44,450
+Cauchy معناها ده بدي استخدم التعريف بدي استخدم
+
+178
+00:20:44,450 --> 00:20:50,450
+التعريف تبع Cauchy sequence فنشوف
+
+179
+00:20:50,450 --> 00:20:56,210
+مع بعض طبعا
+
+180
+00:20:56,210 --> 00:21:02,230
+البرهانباستخدام التعريف هنبدأ بأبسلون أكبر من
+
+181
+00:21:02,230 --> 00:21:07,510
+السفر ونرد عليها بcapital N بتخلي ال implication
+
+182
+00:21:07,510 --> 00:21:13,890
+هي دي تتحقق بالظبط زي .. يعني قريب يعني بالظبط زي
+
+183
+00:21:13,890 --> 00:21:18,370
+ما عملنا في اثبات ان ال sequence is convergent و
+
+184
+00:21:18,370 --> 00:21:27,670
+هنستخدم الarchimedean property نشوف مع بعض let
+
+185
+00:21:29,570 --> 00:21:37,110
+بالمناسبة .. بالمناسبة يعني احنا كيف نحدد ال
+
+186
+00:21:37,110 --> 00:21:40,010
+capital N for any given epsilon؟
+
+187
+00:21:48,010 --> 00:21:54,270
+أنا يعني هي عندي absolute xn minus xm لو في عندي
+
+188
+00:21:54,270 --> 00:21:59,310
+epsilon given epsilon موجة given فمن الآخر أنا
+
+189
+00:21:59,310 --> 00:22:05,190
+عايز أثبت أنه هذا أصغر من إمسنان، مظبوط؟ طب ما هذا
+
+190
+00:22:05,190 --> 00:22:11,530
+عبارة عن absolute واحد على n minus واحد على m وهذا
+
+191
+00:22:11,530 --> 00:22:16,250
+أصغر من أو ساوي absolute واحد على n زائد absolute
+
+192
+00:22:16,250 --> 00:22:23,490
+واحد على mبصبوط وهذه أعداد موجبة فهذا واحد على M
+
+193
+00:22:23,490 --> 00:22:28,290
+زائد واحد على M طيب
+
+194
+00:22:28,290 --> 00:22:33,950
+أنا عايز أجيب capital M بحيث
+
+195
+00:22:33,950 --> 00:22:39,050
+أنه لو كانت ال N و ال M أكبر من أو ساوي capital N
+
+196
+00:22:39,050 --> 00:22:44,670
+فبدنا هذا يؤدي إلى ال absolute value هذه أصغر من
+
+197
+00:22:44,670 --> 00:22:49,460
+إبسلونإذن ال N و ال M هدول لازم يكونوا أكبر من
+
+198
+00:22:49,460 --> 00:22:53,880
+capital N اللي أنا مش عارف إيش هي، بدي أجيبها، إذن
+
+199
+00:22:53,880 --> 00:23:02,480
+و بالتالي من هنا هذا بيقدي إن واحد على N و كذلك
+
+200
+00:23:02,480 --> 00:23:10,060
+واحد على M أصغر من أوسع واحد على capital N، صح؟إذا
+
+201
+00:23:10,060 --> 00:23:14,720
+كانت n أكبر من أو يساوي capital N ف1 على n هتصير
+
+202
+00:23:14,720 --> 00:23:19,100
+أصغر من أو يساوي 1 على capital N وكذلك بالنسبة ل
+
+203
+00:23:19,100 --> 00:23:25,620
+M، مظبوط؟ إذاً هذا هيصير أصغر من أو يساوي 1 على
+
+204
+00:23:25,620 --> 00:23:30,620
+capital N وهذا أصغر من 1 على capital N بساوي 2 على
+
+205
+00:23:30,620 --> 00:23:34,980
+capital M الآن بدي أخلي هذا، متى بيكون هذا أصغر من
+
+206
+00:23:34,980 --> 00:23:45,300
+epsilon؟أه، إذا هاخد n أصغر من epsilon على 2 أو 1
+
+207
+00:23:45,300 --> 00:23:51,080
+على n أصغر من epsilon على 2 إذا هذا أصغر من
+
+208
+00:23:51,080 --> 00:23:56,720
+epsilon عندما 1 على n أصغر من epsilon على 2 طيب،
+
+209
+00:23:56,720 --> 00:24:03,640
+أنا لو بدأت بepsilon عدد موجب فepsilon على 2 بطلع
+
+210
+00:24:03,640 --> 00:24:08,540
+عدد موجب و by Archimedean propertyلأي عدد موجب زي
+
+211
+00:24:08,540 --> 00:24:13,740
+هذا يوجد capital N عدد طبيعي بحيث مقلوب و أصغر من
+
+212
+00:24:13,740 --> 00:24:18,440
+epsilon ع اتنين اذا capital N اللي بتعتمد ع ال
+
+213
+00:24:18,440 --> 00:24:22,380
+given epsilon لازم تكون مقلوبها أصغر من epsilon ع
+
+214
+00:24:22,380 --> 00:24:26,800
+اتنين عشان يطلع هذا أصغر من epsilon شوفتوا كيف
+
+215
+00:24:26,800 --> 00:24:31,240
+منطلق ال capital N و ال Archimedean property طبعا
+
+216
+00:24:31,240 --> 00:24:38,930
+تضمنلي وجود مثل هالعدد capital Nتمام؟ إذا باجي
+
+217
+00:24:38,930 --> 00:24:42,790
+بقول هنا let epsilon الكلام هذا طبعا بعمله في
+
+218
+00:24:42,790 --> 00:24:47,690
+الهامش بعدين باجي برتبه بقول let epsilon أكبر من
+
+219
+00:24:47,690 --> 00:24:56,610
+السفر be given إذا
+
+220
+00:24:56,610 --> 00:25:00,930
+it choose by
+
+221
+00:25:00,930 --> 00:25:04,110
+Archimedean property
+
+222
+00:25:08,750 --> 00:25:19,250
+نختار capital N عدد طبيعي بحيث انه مقلوب ال N أصغر
+
+223
+00:25:19,250 --> 00:25:24,910
+من إبسلون على اتنين إذا هان أثبتت يوجد capital N
+
+224
+00:25:24,910 --> 00:25:29,190
+وهي اعتمد على إبسلون هي مرتبطة بإبسلون
+
+225
+00:25:35,760 --> 00:25:41,740
+هذا هيعطينا ال implication تبع الكوشي sequence
+
+226
+00:25:41,740 --> 00:25:46,240
+then
+
+227
+00:25:46,240 --> 00:25:54,280
+لو أخدت N و M أكبر من أوسع ال capital N هذه
+
+228
+00:25:54,280 --> 00:26:04,420
+فبالتأكيد هذا هيقدر ال 1 على N و كذلكواحد على M
+
+229
+00:26:04,420 --> 00:26:09,300
+كلهما أصغر من أو ساوي واحد على capital N وهذا
+
+230
+00:26:09,300 --> 00:26:15,120
+بدوره بيقدي أنه absolute واحد على M minus واحد على
+
+231
+00:26:15,120 --> 00:26:26,020
+M طبعا هذه XM وهذه XM فشوفنا أن هذا أصغر من أو
+
+232
+00:26:26,020 --> 00:26:29,940
+ساوي absolute واحد على Mباستخدام ال triangle
+
+233
+00:26:29,940 --> 00:26:36,140
+inequality زائد absolute سالب واحد على M اللي هو
+
+234
+00:26:36,140 --> 00:26:44,520
+absolute واحد على M طيب هذا بساوي واحد على M زائد
+
+235
+00:26:44,520 --> 00:26:49,800
+واحد على M لإن رد عداد موجبة وقول إن هذا أصغر من
+
+236
+00:26:49,800 --> 00:26:55,830
+أو يساوي واحد على Mزايد واحد على N وهذا بيساوي
+
+237
+00:26:55,830 --> 00:27:05,510
+اتنين على N وهذا من الاختيار تبعنا ل capital N by
+
+238
+00:27:05,510 --> 00:27:15,990
+star اتنين على N أصغر من epsilon طب
+
+239
+00:27:15,990 --> 00:27:22,830
+ما هذه .. هذا هو شرط Koshi صح؟ هذا هو شرط Koshiإذا
+
+240
+00:27:22,830 --> 00:27:28,310
+by definition بما أن هذا صحيح لكل epsilon since
+
+241
+00:27:28,310 --> 00:27:39,990
+epsilon أكبر من الصفر was arbitrary by
+
+242
+00:27:39,990 --> 00:27:45,830
+definition of Cauchy sequence ال sequence xn is
+
+243
+00:27:45,830 --> 00:27:50,990
+اللي هي واحد على n اللي الحد العام تبعها واحد على
+
+244
+00:27:50,990 --> 00:27:57,610
+nis Cauchy تمام
+
+245
+00:27:57,610 --> 00:28:04,230
+هنا أثبتنا إن ال sequence Cauchy مباشرة باستخدام
+
+246
+00:28:04,230 --> 00:28:12,050
+التعريف طبعا في برهان تاني ممكن نستخدم Cauchy
+
+247
+00:28:12,050 --> 00:28:18,290
+criterion احنا ممكن نثبت إن ال sequence هذي
+
+248
+00:28:18,290 --> 00:28:24,620
+convergentو أثبتنا هذا الكلام جبليك صح؟ و حسب
+
+249
+00:28:24,620 --> 00:28:28,640
+cushy criterion بما أنه ال sequence convergent
+
+250
+00:28:28,640 --> 00:28:32,760
+then it is cushy صح؟ هذا برهان تاني لكن إذا كنا
+
+251
+00:28:32,760 --> 00:28:38,260
+لكم برهنيها directly يعني استخدم التعريف لازم
+
+252
+00:28:38,260 --> 00:28:45,600
+البرهان هذا هو اللي إيه تكتبوه واضح تمام؟ في أي
+
+253
+00:28:45,600 --> 00:28:46,400
+استفسار؟
+
+254
+00:28:50,060 --> 00:28:51,700
+ناخد مثال تاني
+
+255
+00:29:18,730 --> 00:29:27,750
+مثال تقم اتنين consider .. consider
+
+256
+00:29:27,750 --> 00:29:36,370
+ال sequence defined
+
+257
+00:29:36,370 --> 00:29:38,710
+inductively
+
+258
+00:29:48,750 --> 00:29:52,770
+إذا في عندي sequence معرفة بطريقة استقرائية
+
+259
+00:29:52,770 --> 00:30:03,070
+كالتالي كما هي ليه هناخد x1 بساوي واحد و x2 بساوي
+
+260
+00:30:03,070 --> 00:30:12,710
+اتنين طب و xn-n أكبر من أو ساوي تلاتة هناخده بساوي
+
+261
+00:30:12,710 --> 00:30:24,340
+نص فيxn سالب اتنين زائد xn negative one طبعا هذا
+
+262
+00:30:24,340 --> 00:30:30,740
+لكل n أدب طبيعي أكبر من أو ساوى تلاتة اذا هنا في
+
+263
+00:30:30,740 --> 00:30:35,160
+اندي سيكوانس معرفة بطريقة استقرائية اول حدين اللي
+
+264
+00:30:35,160 --> 00:30:41,200
+هم قيم معينة الحد التالت وانت طالع معرف بدلالة
+
+265
+00:30:41,200 --> 00:30:47,520
+الحد اللي حدين اللي جابله مباشرةهذا طبعا بيعطينا
+
+266
+00:30:47,520 --> 00:30:53,720
+sequence المطلوب عايزين نثبت show ان ال sequence x
+
+267
+00:30:53,720 --> 00:31:04,020
+in is convergent و converges to the number 5 over
+
+268
+00:31:04,020 --> 00:31:12,220
+3 البرهان
+
+269
+00:31:17,020 --> 00:31:24,000
+هنثبت we first show
+
+270
+00:31:24,000 --> 00:31:37,700
+that sequence xn converges by
+
+271
+00:31:37,700 --> 00:31:41,700
+showing
+
+272
+00:31:41,700 --> 00:31:46,540
+بإثبات أنه
+
+273
+00:31:51,510 --> 00:32:00,170
+إنها كوشي thanks
+
+274
+00:32:00,170 --> 00:32:07,610
+to koshi criterion
+
+275
+00:32:07,610 --> 00:32:17,390
+طبعا هذا بفضل معيار كوشي أو كوشي criterion هنثبت
+
+276
+00:32:17,390 --> 00:32:23,510
+الأول أن ال sequence هي to convergentبإثبات إنه
+
+277
+00:32:23,510 --> 00:32:28,970
+كوشي وهذا طبعا حسب كوشي criterion إذا أثبتنا إن ال
+
+278
+00:32:28,970 --> 00:32:35,730
+sequence كوشي بتكون convergent تمام فنشوف كيف ممكن
+
+279
+00:32:35,730 --> 00:32:40,150
+نثبت الكلام هذا فأول شيء بدي أثبت إن ال sequence
+
+280
+00:32:40,150 --> 00:32:44,750
+bounded إذن هنا الإدعاء
+
+281
+00:32:44,750 --> 00:32:51,710
+الأول أو claim number oneالسيكونس xn الحد العام
+
+282
+00:32:51,710 --> 00:32:56,890
+تبعها أكبر من أو ساوي الواحد أصغر من أو ساوي اتنين
+
+283
+00:32:56,890 --> 00:33:05,050
+لكل n في n لبرهان
+
+284
+00:33:05,050 --> 00:33:11,810
+ذلك to see this use
+
+285
+00:33:11,810 --> 00:33:14,310
+induction
+
+286
+00:33:19,650 --> 00:33:27,010
+on n so I will leave it for you to prove claim one
+
+287
+00:33:27,010 --> 00:33:33,250
+by induction on n فالحالة
+
+288
+00:33:33,250 --> 00:33:38,010
+لو بنشوف بقرا ال statement هذا when n equals one
+
+289
+00:33:38,010 --> 00:33:44,090
+هذا معناه ان المتباين هذه هتكون x one أكبر من أو
+
+290
+00:33:44,090 --> 00:33:50,360
+ساوى الواحد أصغر من أو ساوى اتنين وهذا trueو هذه
+
+291
+00:33:50,360 --> 00:33:56,880
+صحيحة لأن هاي x واحد بساوي واحد و الواحد أكبر من
+
+292
+00:33:56,880 --> 00:34:01,620
+أو ساوي الواحد هو less than or equal to لذن ال
+
+293
+00:34:01,620 --> 00:34:06,000
+statement هذا is true for n بساوي one assume it is
+
+294
+00:34:06,000 --> 00:34:09,620
+true for n بساوي k و prove it for n بساوي k زاد
+
+295
+00:34:09,620 --> 00:34:13,500
+واحد فيعني
+
+296
+00:34:13,500 --> 00:34:16,200
+هسيبكم أنتم تكملوا البرهان البرهان سهل
+
+297
+00:34:19,520 --> 00:34:28,600
+So this is claim one الان by claim one
+
+298
+00:34:28,600 --> 00:34:36,400
+By claim one The
+
+299
+00:34:36,400 --> 00:34:43,020
+sequence x in is bounded حسب
+
+300
+00:34:43,020 --> 00:34:50,140
+claim one لأن claim oneأثبتنا فيه أو هتثبتوا فيه
+
+301
+00:34:50,140 --> 00:34:53,880
+ان الـ x in ال sequence x كل حدود ال sequence
+
+302
+00:34:53,880 --> 00:34:57,740
+محصورة بين واحد واتنين وبالتالي bounded below by
+
+303
+00:34:57,740 --> 00:35:02,680
+one bound above by two وبالتالي bounded okay إذا
+
+304
+00:35:02,680 --> 00:35:15,440
+ال sequence bounded الآن لو كتبنا writing
+
+305
+00:35:15,440 --> 00:35:16,220
+out
+
+306
+00:35:21,120 --> 00:35:29,260
+الأول مرات .. المرات
+
+307
+00:35:29,260 --> 00:35:32,100
+الأول مرات .. المرات الأول مرات .. المرات الأول
+
+308
+00:35:32,100 --> 00:35:32,120
+المرات الأول مرات .. المرات الأول مرات الأول مرات
+
+309
+00:35:32,120 --> 00:35:33,160
+مرات الأول مرات الأول مرات الأول مرات الأول مرات
+
+310
+00:35:33,160 --> 00:35:33,480
+الأول مرات الأول مرات الأول مرات الأول مرات الأول
+
+311
+00:35:33,480 --> 00:35:34,040
+مرات الأول مرات الأول مرات الأول مرات الأول مرات
+
+312
+00:35:34,040 --> 00:35:34,600
+الأول مرات الأول مرات الأول مرات الأول مرات الأول
+
+313
+00:35:34,600 --> 00:35:35,980
+مرات الأول مرات الأول مرات الأول مرات الأول مرات
+
+314
+00:35:35,980 --> 00:35:39,620
+الأول مرات الأول
+
+315
+00:35:39,620 --> 00:35:46,440
+مرات الأول
+
+316
+00:35:46,440 --> 00:35:47,720
+مرات
+
+317
+00:35:49,600 --> 00:35:56,440
+is not monotone لو
+
+318
+00:35:56,440 --> 00:36:02,300
+كتبنا أول أربع خمس ست حدود من ال sequence هذه
+
+319
+00:36:02,300 --> 00:36:08,060
+فبلاحظ أنها ليست monotone ليست increasing neither
+
+320
+00:36:08,060 --> 00:36:14,100
+increasing nor decreasingوبالتالي نقدر نستخدم الـ
+
+321
+00:36:14,100 --> 00:36:23,600
+monotone convergence theorem so we can't we can't
+
+322
+00:36:23,600 --> 00:36:31,120
+use ال monotone convergence theorem we
+
+323
+00:36:31,120 --> 00:36:31,940
+can't
+
+324
+00:36:35,030 --> 00:36:42,910
+we can't use monotone convergence theorem الـ
+
+325
+00:36:42,910 --> 00:36:46,570
+sequence bounded عشان استخدم الـ monotone
+
+326
+00:36:46,570 --> 00:36:49,530
+convergence theorem لازم تكون monotone increasing
+
+327
+00:36:49,530 --> 00:36:53,730
+او monotone decreasing ف it is not monotone
+
+328
+00:36:53,730 --> 00:36:56,610
+فماقدرش استخدم ال monotone convergence theorem
+
+329
+00:36:56,610 --> 00:37:03,210
+عشان افحص ال convergenceالـ sequence لازم ابحث عن
+
+330
+00:37:03,210 --> 00:37:09,890
+طريقه تانية غير الـ monotone convergence فيها طيب
+
+331
+00:37:09,890 --> 00:37:16,530
+هنثبت claim 2 claim
+
+332
+00:37:16,530 --> 00:37:24,230
+2 ادعاء تاني وهو ان ال sequence xn بتحقق المعادلة
+
+333
+00:37:24,230 --> 00:37:30,290
+absolute xn minus xn زيادة واحدبساوي واحد على
+
+334
+00:37:30,290 --> 00:37:37,450
+اتنين قص ان نجاتف وان وهذا الكلام صحيح for every n
+
+335
+00:37:37,450 --> 00:37:41,950
+في n to
+
+336
+00:37:41,950 --> 00:37:50,510
+see
+
+337
+00:37:50,510 --> 00:37:54,790
+this لبرهان ذلك use induction
+
+338
+00:37:57,680 --> 00:38:03,880
+use induction on n برضه ممكن برهان المعادلة هذه by
+
+339
+00:38:03,880 --> 00:38:09,460
+induction on n هينبرهن
+
+340
+00:38:09,460 --> 00:38:13,540
+البرهان if
+
+341
+00:38:13,540 --> 00:38:24,670
+n بسبب واحد ف absolute x واحد minus xأتنين بساوي
+
+342
+00:38:24,670 --> 00:38:30,010
+absolute واحد سالي اتنين بساوي absolute واحد بساوي
+
+343
+00:38:30,010 --> 00:38:35,810
+واحد هذا الطرف اليمين و الطرف اليسار واحد على
+
+344
+00:38:35,810 --> 00:38:41,370
+اتنين plus n minus واحد بساوي واحد على اتنين plus
+
+345
+00:38:41,370 --> 00:38:49,110
+سفر بساوي واحدة واحد بساوي واحد اذا
+
+346
+00:38:49,110 --> 00:38:54,930
+المعادلة true for n بساوي واحدطيب assume ال
+
+347
+00:38:54,930 --> 00:39:06,670
+induction hypothesis الفرض طبع ال induction assume
+
+348
+00:39:06,670 --> 00:39:12,710
+أنه ال ..
+
+349
+00:39:12,710 --> 00:39:25,640
+ال claim is true for n بساوة kو k طبعا أكبر من أول
+
+350
+00:39:25,640 --> 00:39:30,920
+سالة من واحد هذا
+
+351
+00:39:30,920 --> 00:39:37,460
+معناه أن absolute xk minus xk زيادة واحد بسالة
+
+352
+00:39:37,460 --> 00:39:44,020
+واحد على اتنين أس كسالب واحد، صح؟ هذه الأدارة
+
+353
+00:39:44,020 --> 00:39:45,600
+صحيحة and k
+
+354
+00:39:49,840 --> 00:39:54,580
+الان تعالى نثبت صحة العبارة عند n بساوي k زايد
+
+355
+00:39:54,580 --> 00:39:59,420
+واحد ناخد الطرف الشمال عندما n بساوي k زايد واحد
+
+356
+00:39:59,420 --> 00:40:06,600
+هذا عبارة عن x k زايد واحد سالد x k زايد اتنين
+
+357
+00:40:06,600 --> 00:40:14,020
+بدنا نثبت ان هذا بساوي واحد على اتنين اص k صح؟ طب
+
+358
+00:40:14,020 --> 00:40:21,460
+تعالى نشوفهي absolute xk plus one minus الان xk
+
+359
+00:40:21,460 --> 00:40:26,760
+زائد اتنين من ال definition تبع ال sequence بدل n
+
+360
+00:40:26,760 --> 00:40:38,320
+بدل n بk زائد اتنين فبطلع نص في xk زائد xk زائد
+
+361
+00:40:38,320 --> 00:40:38,760
+واحد
+
+362
+00:40:49,170 --> 00:41:04,590
+وهذا بيساوي و هذا بيساوي نص في absolute x x
+
+363
+00:41:04,590 --> 00:41:09,430
+k negative x k plus one
+
+364
+00:41:16,590 --> 00:41:19,730
+بعد ما نطرح بطلع عنده نص عامل مشترك و absolute
+
+365
+00:41:19,730 --> 00:41:26,890
+الان by induction hypothesis من الفرض تبع ال
+
+366
+00:41:26,890 --> 00:41:33,130
+induction ال absolute value هذه أيها ايش بيساوي
+
+367
+00:41:33,130 --> 00:41:39,210
+عوض عنها اي نص ضرب one over two to k negative one
+
+368
+00:41:39,210 --> 00:41:43,550
+ويساوي واحد على
+
+369
+00:42:09,140 --> 00:42:09,700
+اثنين اثنين اثنين اثنين اثنين اثنين اثنين اثنين
+
+370
+00:42:09,700 --> 00:42:09,720
+اثنين اثنين اثنين اثنين اثنين اثنين اثنين اثنين
+
+371
+00:42:09,720 --> 00:42:09,820
+اثنين اثنين اثنين اثنين اثنين اثنين اثنين اثنين
+
+372
+00:42:09,820 --> 00:42:10,040
+اثنين اثنين اثنين اثنين اثنين اثنين اثنين اثنين
+
+373
+00:42:10,040 --> 00:42:10,480
+اثنين اثنين اثنين اثنين اثنين اثنين اثنين اثنين
+
+374
+00:42:10,480 --> 00:42:10,960
+اثنين اثنين اثنين اثنين اثنين اثنين اثنين اثنين
+
+375
+00:42:15,820 --> 00:42:23,080
+الان باستخدام ال claim التاني ممكن نثبت شغلة مهمة
+
+376
+00:42:23,080 --> 00:42:36,380
+في البرهان اذا
+
+377
+00:42:36,380 --> 00:42:43,650
+خليها هادى للمرة الجاية بس بدى اكتبهاخليكم أنتوا
+
+378
+00:42:43,650 --> 00:42:53,390
+تفكروا فيها .. خليكم أنتوا تفكروا فيها Now
+
+379
+00:42:53,390 --> 00:43:11,210
+using a claim to verify .. verify that ..
+
+380
+00:43:14,770 --> 00:43:23,190
+F M أكبر من N فهذا
+
+381
+00:43:23,190 --> 00:43:33,530
+بيقدي أن absolute X N minus X M أصغر من واحد على
+
+382
+00:43:33,530 --> 00:43:39,170
+اتنين قص M نجاتي باتنين
+
+383
+00:43:45,950 --> 00:43:54,290
+إذاً هذا ممكن إثباته by ال triangle ال equality و
+
+384
+00:43:54,290 --> 00:44:06,850
+claim اثنين فبنوقف
+
+385
+00:44:06,850 --> 00:44:14,460
+هنا و بنكمل ال .. بنكمل ان شاء اللهالبرهان في
+
+386
+00:44:14,460 --> 00:44:19,680
+المحاضرة الجاية، في حد عنده أي سؤال أو استفسار؟
+
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/-3lwzxWH7r8_raw.json b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/-3lwzxWH7r8_raw.json
new file mode 100644
index 0000000000000000000000000000000000000000..8db3380f41eea0bc4c4ec266fbbf22a4657c50da
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/-3lwzxWH7r8_raw.json
@@ -0,0 +1 @@
+{"segments": [{"id": 1, "seek": 4630, "start": 21.6, "end": 46.3, "text": "ال .. في المحاضرة السابقة بدأنا التعرف على cushy sequences فأخدنا تعريف ال cushy sequence و أثبتنا أنه كل convergence sequence is cushy و أعتقد كمان أثبتنا أنه كل cushy sequence is bounded صحيح؟", "tokens": [6027, 4386, 8978, 9673, 5016, 46958, 25720, 21136, 16758, 28671, 47525, 10721, 8315, 16712, 3615, 28480, 15844, 18422, 88, 22978, 6156, 10721, 9778, 3215, 8315, 37279, 16572, 5172, 2423, 18422, 88, 8310, 4032, 5551, 12984, 3555, 2655, 8315, 14739, 3224, 28242, 32181, 8310, 307, 18422, 88, 4032, 5551, 34268, 28543, 9122, 2304, 7649, 5551, 12984, 3555, 2655, 8315, 14739, 3224, 28242, 18422, 88, 8310, 307, 37498, 20328, 5016, 1829, 5016, 22807], "avg_logprob": -0.11143662966787815, "compression_ratio": 1.6449704142011834, "no_speech_prob": 0.0, "words": [{"start": 21.6, "end": 21.98, "word": "ال", "probability": 0.935546875}, {"start": 21.98, "end": 22.76, "word": " ..", "probability": 0.77978515625}, {"start": 22.76, "end": 23.2, "word": " في", "probability": 0.7607421875}, {"start": 23.2, "end": 23.8, "word": " المحاضرة", "probability": 0.9891357421875}, {"start": 23.8, "end": 24.46, "word": " السابقة", "probability": 0.9825846354166666}, {"start": 24.46, "end": 26.62, "word": " بدأنا", "probability": 0.717529296875}, {"start": 26.62, "end": 28.88, "word": " التعرف", "probability": 0.9425455729166666}, {"start": 28.88, "end": 29.12, "word": " على", "probability": 0.8720703125}, {"start": 29.12, "end": 29.56, "word": " cushy", "probability": 0.59625244140625}, {"start": 29.56, "end": 30.2, "word": " sequences", "probability": 0.9658203125}, {"start": 30.2, "end": 32.78, "word": " فأخدنا", "probability": 0.92060546875}, {"start": 32.78, "end": 33.2, "word": " تعريف", "probability": 0.9892578125}, {"start": 33.2, "end": 33.34, "word": " ال", "probability": 0.85986328125}, {"start": 33.34, "end": 33.64, "word": " cushy", "probability": 0.91748046875}, {"start": 33.64, "end": 34.18, "word": " sequence", "probability": 0.92431640625}, {"start": 34.18, "end": 34.34, "word": " و", "probability": 0.81787109375}, {"start": 34.34, "end": 35.02, "word": " أثبتنا", "probability": 0.98505859375}, {"start": 35.02, "end": 35.94, "word": " أنه", "probability": 0.7216796875}, {"start": 35.94, "end": 36.4, "word": " كل", "probability": 0.98974609375}, {"start": 36.4, "end": 36.94, "word": " convergence", "probability": 0.953125}, {"start": 36.94, "end": 37.62, "word": " sequence", "probability": 0.98779296875}, {"start": 37.62, "end": 37.88, "word": " is", "probability": 0.9345703125}, {"start": 37.88, "end": 39.28, "word": " cushy", "probability": 0.9345703125}, {"start": 39.28, "end": 39.92, "word": " و", "probability": 0.80126953125}, {"start": 39.92, "end": 40.44, "word": " أعتقد", "probability": 0.908203125}, {"start": 40.44, "end": 41.0, "word": " كمان", "probability": 0.97265625}, {"start": 41.0, "end": 41.82, "word": " أثبتنا", "probability": 0.99365234375}, {"start": 41.82, "end": 43.3, "word": " أنه", "probability": 0.86376953125}, {"start": 43.3, "end": 43.62, "word": " كل", "probability": 0.98583984375}, {"start": 43.62, "end": 44.54, "word": " cushy", "probability": 0.951416015625}, {"start": 44.54, "end": 45.04, "word": " sequence", "probability": 0.9697265625}, {"start": 45.04, "end": 45.26, "word": " is", "probability": 0.96630859375}, {"start": 45.26, "end": 45.6, "word": " bounded", "probability": 0.9228515625}, {"start": 45.6, "end": 46.3, "word": " صحيح؟", "probability": 0.89365234375}], "temperature": 1.0}, {"id": 2, "seek": 7470, "start": 47.47, "end": 74.71, "text": "اليوم هنثبت العكس و هو ان كل كوشي sequence is convergent فنستعيد بس نستذكر مع بعض تعريف الكوشي sequence definition a sequence of real numbers xn is cauchy", "tokens": [6027, 1829, 20498, 8032, 1863, 12984, 3555, 2655, 18863, 4117, 3794, 4032, 31439, 16472, 28242, 9122, 2407, 8592, 1829, 8310, 307, 9652, 6930, 6156, 1863, 14851, 3615, 25708, 4724, 3794, 8717, 14851, 8848, 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is", "probability": 0.91064453125}, {"start": 54.17, "end": 54.97, "word": " convergent", "probability": 0.953857421875}, {"start": 54.97, "end": 57.05, "word": " فنستعيد", "probability": 0.9310546875}, {"start": 57.05, "end": 57.29, "word": " بس", "probability": 0.976318359375}, {"start": 57.29, "end": 57.91, "word": " نستذكر", "probability": 0.969482421875}, {"start": 57.91, "end": 58.11, "word": " مع", "probability": 0.9970703125}, {"start": 58.11, "end": 58.57, "word": " بعض", "probability": 0.986572265625}, {"start": 58.57, "end": 59.57, "word": " تعريف", "probability": 0.9734700520833334}, {"start": 59.57, "end": 60.09, "word": " الكوشي", "probability": 0.8509521484375}, {"start": 60.09, "end": 60.63, "word": " sequence", "probability": 0.9853515625}, {"start": 60.63, "end": 62.37, "word": " definition", "probability": 0.72998046875}, {"start": 62.37, "end": 67.45, "word": " a", "probability": 0.744140625}, {"start": 67.45, "end": 68.01, "word": " sequence", "probability": 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0.508331298828125}, {"start": 109.36, "end": 113.96, "word": " وشوفنا", "probability": 0.83935546875}, {"start": 113.96, "end": 114.44, "word": " المرة", "probability": 0.9619140625}, {"start": 114.44, "end": 114.62, "word": " اللي", "probability": 0.866455078125}, {"start": 114.62, "end": 115.26, "word": " فاتت", "probability": 0.9630533854166666}, {"start": 115.26, "end": 116.44, "word": " أو", "probability": 0.78173828125}, {"start": 116.44, "end": 117.28, "word": " برهننا", "probability": 0.74560546875}, {"start": 117.28, "end": 118.72, "word": " limit", "probability": 0.673828125}, {"start": 118.72, "end": 119.34, "word": " 2", "probability": 0.35791015625}, {"start": 119.34, "end": 120.12, "word": " و", "probability": 0.54150390625}, {"start": 120.12, "end": 120.72, "word": " 21", "probability": 0.60888671875}, {"start": 120.72, "end": 123.26, "word": " every", "probability": 0.71533203125}, {"start": 123.26, "end": 126.82, "word": " convergent", "probability": 0.954833984375}], "temperature": 1.0}, {"id": 5, "seek": 15501, "start": 131.45, "end": 155.01, "text": "sequence is cauchy ثم برهنة another لمبة لمبة اتنين و عشرين بتقول اللمبة هذه ان every cauchy sequence", "tokens": [11834, 655, 307, 1335, 625, 88, 38637, 2304, 4724, 2288, 3224, 1863, 3660, 1071, 32767, 49401, 32767, 49401, 1975, 2655, 1863, 9957, 4032, 6225, 46309, 9957, 39894, 39648, 13672, 2304, 49401, 29538, 16472, 633, 1335, 625, 88, 8310], "avg_logprob": -0.28826120725044835, "compression_ratio": 1.2654867256637168, "no_speech_prob": 0.0, "words": [{"start": 131.45, "end": 132.63, "word": "sequence", "probability": 0.5546875}, {"start": 132.63, "end": 133.31, "word": " is", "probability": 0.9189453125}, {"start": 133.31, "end": 137.19, "word": " cauchy", "probability": 0.7333170572916666}, {"start": 137.19, "end": 139.23, "word": " ثم", "probability": 0.962890625}, {"start": 139.23, "end": 140.41, "word": " برهنة", "probability": 0.766162109375}, {"start": 140.41, "end": 141.57, "word": " another", "probability": 0.90771484375}, {"start": 141.57, "end": 142.21, "word": " لمبة", "probability": 0.59814453125}, {"start": 142.21, "end": 144.03, "word": " لمبة", "probability": 0.8740234375}, {"start": 144.03, "end": 144.35, "word": " اتنين", "probability": 0.8677978515625}, {"start": 144.35, "end": 144.45, "word": " و", "probability": 0.9189453125}, {"start": 144.45, "end": 145.03, "word": " عشرين", "probability": 0.8141276041666666}, {"start": 145.03, "end": 147.51, "word": " بتقول", "probability": 0.94677734375}, {"start": 147.51, "end": 147.99, "word": " اللمبة", "probability": 0.7184244791666666}, {"start": 147.99, "end": 148.37, "word": " هذه", "probability": 0.64453125}, {"start": 148.37, "end": 148.67, "word": " ان", "probability": 0.77978515625}, {"start": 148.67, "end": 150.95, "word": " every", "probability": 0.71484375}, {"start": 150.95, "end": 154.25, "word": " cauchy", "probability": 0.78466796875}, {"start": 154.25, "end": 155.01, "word": " sequence", "probability": 0.98681640625}], "temperature": 1.0}, {"id": 6, "seek": 18815, "start": 160.29, "end": 188.15, "text": "is bounded اليوم هنثبت نظرية مهمة نظرية اتنين تلاتة وعشرين وهذه النظرية هي كوشي كوشي criterion", "tokens": [271, 37498, 45595, 20498, 8032, 1863, 12984, 3555, 2655, 8717, 19913, 2288, 10632, 3714, 16095, 3660, 8717, 19913, 2288, 10632, 1975, 2655, 1863, 9957, 6055, 1211, 9307, 3660, 4032, 3615, 46309, 9957, 37037, 24192, 28239, 19913, 2288, 10632, 39896, 9122, 2407, 8592, 1829, 9122, 2407, 8592, 1829, 46691], "avg_logprob": -0.2256058624812535, "compression_ratio": 1.3963963963963963, "no_speech_prob": 0.0, "words": [{"start": 160.29, "end": 160.79, "word": "is", "probability": 0.7607421875}, {"start": 160.79, "end": 161.43, "word": " bounded", "probability": 0.95263671875}, {"start": 161.43, "end": 169.63, "word": " اليوم", "probability": 0.91650390625}, {"start": 169.63, "end": 170.49, "word": " هنثبت", "probability": 0.90009765625}, {"start": 170.49, "end": 171.51, "word": " نظرية", "probability": 0.935302734375}, {"start": 171.51, "end": 173.01, "word": " مهمة", "probability": 0.9903971354166666}, {"start": 173.01, "end": 174.69, "word": " نظرية", "probability": 0.952880859375}, {"start": 174.69, "end": 175.37, "word": " اتنين", "probability": 0.876953125}, {"start": 175.37, "end": 176.77, "word": " تلاتة", "probability": 0.953369140625}, {"start": 176.77, "end": 177.55, "word": " وعشرين", "probability": 0.831787109375}, {"start": 177.55, "end": 179.89, "word": " وهذه", "probability": 0.609130859375}, {"start": 179.89, "end": 180.77, "word": " النظرية", "probability": 0.8465576171875}, {"start": 180.77, "end": 181.21, "word": " هي", "probability": 0.0982666015625}, {"start": 181.21, "end": 183.89, "word": " كوشي", "probability": 0.80377197265625}, {"start": 183.89, "end": 187.31, "word": " كوشي", "probability": 0.75506591796875}, {"start": 187.31, "end": 188.15, "word": " criterion", "probability": 0.5283203125}], "temperature": 1.0}, {"id": 7, "seek": 21670, "start": 191.82, "end": 216.7, "text": "أو معيار كوشي معيار كوشي للتقارب النظرية بتنص على أن a sequence x in contained in R is convergent", "tokens": [10721, 2407, 20449, 1829, 9640, 9122, 2407, 8592, 1829, 20449, 1829, 9640, 9122, 2407, 8592, 1829, 24976, 2655, 4587, 9640, 3555, 28239, 19913, 2288, 10632, 39894, 1863, 9381, 15844, 14739, 257, 8310, 2031, 294, 16212, 294, 497, 307, 9652, 6930], "avg_logprob": -0.15377286003857124, "compression_ratio": 1.2280701754385965, "no_speech_prob": 0.0, "words": [{"start": 191.82, "end": 192.3, "word": "أو", "probability": 0.874267578125}, {"start": 192.3, "end": 193.02, "word": " معيار", "probability": 0.97119140625}, {"start": 193.02, "end": 193.66, "word": " كوشي", "probability": 0.9071044921875}, {"start": 193.66, "end": 198.68, "word": " معيار", "probability": 0.951171875}, {"start": 198.68, "end": 199.28, "word": " كوشي", "probability": 0.9801025390625}, {"start": 199.28, "end": 200.68, "word": " للتقارب", "probability": 0.9904296875}, {"start": 200.68, "end": 205.58, "word": " النظرية", "probability": 0.9322509765625}, {"start": 205.58, "end": 206.28, "word": " بتنص", "probability": 0.8566080729166666}, {"start": 206.28, "end": 206.52, "word": " على", "probability": 0.71484375}, {"start": 206.52, "end": 206.76, "word": " أن", "probability": 0.41845703125}, {"start": 206.76, "end": 207.08, "word": " a", "probability": 0.321044921875}, {"start": 207.08, "end": 207.8, "word": " sequence", "probability": 0.96484375}, {"start": 207.8, "end": 209.54, "word": " x", "probability": 0.4814453125}, {"start": 209.54, "end": 210.06, "word": " in", "probability": 0.60009765625}, {"start": 210.06, "end": 211.1, "word": " contained", "probability": 0.548828125}, {"start": 211.1, "end": 211.36, "word": " in", "probability": 0.94189453125}, {"start": 211.36, "end": 211.94, "word": " R", "probability": 0.91162109375}, {"start": 211.94, "end": 215.8, "word": " is", "probability": 0.9013671875}, {"start": 215.8, "end": 216.7, "word": " convergent", "probability": 0.957763671875}], "temperature": 1.0}, {"id": 8, "seek": 24475, "start": 219.15, "end": 244.75, "text": "is convergent if and only if it is cauchy any sequence of real numbers بتكون convergent if and only if it is cauchy البرهان", "tokens": [271, 9652, 6930, 498, 293, 787, 498, 309, 307, 1335, 625, 88, 604, 8310, 295, 957, 3547, 39894, 30544, 9652, 6930, 498, 293, 787, 498, 309, 307, 1335, 625, 88, 2423, 26890, 3224, 7649], "avg_logprob": -0.2127232108797346, "compression_ratio": 1.40625, "no_speech_prob": 0.0, "words": [{"start": 219.15, "end": 219.61, "word": "is", "probability": 0.439208984375}, {"start": 219.61, "end": 220.65, "word": " convergent", "probability": 0.94970703125}, {"start": 220.65, "end": 222.35, "word": " if", "probability": 0.91650390625}, {"start": 222.35, "end": 222.73, "word": " and", "probability": 0.943359375}, {"start": 222.73, "end": 223.07, "word": " only", "probability": 0.89990234375}, {"start": 223.07, "end": 223.53, "word": " if", "probability": 0.98193359375}, {"start": 223.53, "end": 225.67, "word": " it", "probability": 0.921875}, {"start": 225.67, "end": 226.05, "word": " is", "probability": 0.9482421875}, {"start": 226.05, "end": 227.69, "word": " cauchy", "probability": 0.7891438802083334}, {"start": 227.69, "end": 235.13, "word": " any", "probability": 0.07940673828125}, {"start": 235.13, "end": 237.37, "word": " sequence", "probability": 0.9765625}, {"start": 237.37, "end": 237.55, "word": " of", "probability": 0.97021484375}, {"start": 237.55, "end": 237.79, "word": " real", "probability": 0.95556640625}, {"start": 237.79, "end": 238.25, "word": " numbers", "probability": 0.876953125}, {"start": 238.25, "end": 238.63, "word": " بتكون", "probability": 0.79541015625}, {"start": 238.63, "end": 239.45, "word": " convergent", "probability": 0.827880859375}, {"start": 239.45, "end": 240.39, "word": " if", "probability": 0.9541015625}, {"start": 240.39, "end": 240.61, "word": " and", "probability": 0.9541015625}, {"start": 240.61, "end": 240.87, "word": " only", "probability": 0.90576171875}, {"start": 240.87, "end": 241.25, "word": " if", "probability": 0.97509765625}, {"start": 241.25, "end": 241.53, "word": " it", "probability": 0.96923828125}, {"start": 241.53, "end": 241.85, "word": " is", "probability": 0.95654296875}, {"start": 241.85, "end": 242.47, "word": " cauchy", "probability": 0.8824869791666666}, {"start": 242.47, "end": 244.75, "word": " البرهان", "probability": 0.9132080078125}], "temperature": 1.0}, {"id": 9, "seek": 27720, "start": 249.11, "end": 277.21, "text": "this part اللي هو ال only if part هذا هو نفسه لمّة واحدة عشرين if x in is convergent then by لمّة واحدة عشرين", "tokens": [11176, 644, 13672, 1829, 31439, 2423, 787, 498, 644, 23758, 31439, 8717, 36178, 3224, 32767, 11703, 3660, 36764, 24401, 3660, 6225, 46309, 9957, 498, 2031, 294, 307, 9652, 6930, 550, 538, 32767, 11703, 3660, 36764, 24401, 3660, 6225, 46309, 9957], "avg_logprob": -0.1970274419319339, "compression_ratio": 1.3162393162393162, "no_speech_prob": 0.0, "words": [{"start": 249.11, "end": 249.53, "word": "this", "probability": 0.505859375}, {"start": 249.53, "end": 249.99, "word": " part", "probability": 0.93994140625}, {"start": 249.99, "end": 250.19, "word": " اللي", "probability": 0.83544921875}, {"start": 250.19, "end": 250.35, "word": " هو", "probability": 0.99169921875}, {"start": 250.35, "end": 250.53, "word": " ال", "probability": 0.447021484375}, {"start": 250.53, "end": 250.83, "word": " only", "probability": 0.77197265625}, {"start": 250.83, "end": 251.13, "word": " if", "probability": 0.80615234375}, {"start": 251.13, "end": 251.75, "word": " part", "probability": 0.95361328125}, {"start": 251.75, "end": 254.21, "word": " هذا", "probability": 0.8193359375}, {"start": 254.21, "end": 254.47, "word": " هو", "probability": 0.97314453125}, {"start": 254.47, "end": 255.01, "word": " نفسه", "probability": 0.9378255208333334}, {"start": 255.01, "end": 255.43, "word": " لمّة", "probability": 0.6962890625}, {"start": 255.43, "end": 256.87, "word": " واحدة", "probability": 0.6767578125}, {"start": 256.87, "end": 257.25, "word": " عشرين", "probability": 0.80419921875}, {"start": 257.25, "end": 259.75, "word": " if", "probability": 0.84228515625}, {"start": 259.75, "end": 262.11, "word": " x", "probability": 0.83251953125}, {"start": 262.11, "end": 262.69, "word": " in", "probability": 0.36474609375}, {"start": 262.69, "end": 263.71, "word": " is", "probability": 0.95166015625}, {"start": 263.71, "end": 264.75, "word": " convergent", "probability": 0.97216796875}, {"start": 264.75, "end": 269.89, "word": " then", "probability": 0.818359375}, {"start": 269.89, "end": 272.99, "word": " by", "probability": 0.92138671875}, {"start": 272.99, "end": 273.71, "word": " لمّة", "probability": 0.9422200520833334}, {"start": 273.71, "end": 276.53, "word": " واحدة", "probability": 0.9606119791666666}, {"start": 276.53, "end": 277.21, "word": " عشرين", "probability": 0.9812825520833334}], "temperature": 1.0}, {"id": 10, "seek": 30429, "start": 280.71, "end": 304.29, "text": "it is cushy it is cushy لأن هذا جزء برهناه في المحاضرة السابقة على صورة لمة واحد وعشرين ال .. ال if part هنبرهنه اليوم هنشوف مع بعض assume العكس", "tokens": [270, 307, 18422, 88, 309, 307, 18422, 88, 5296, 33456, 23758, 10874, 11622, 38207, 4724, 2288, 3224, 8315, 3224, 8978, 9673, 5016, 46958, 25720, 21136, 16758, 28671, 15844, 20328, 13063, 3660, 32767, 3660, 36764, 24401, 4032, 3615, 46309, 9957, 2423, 4386, 2423, 498, 644, 8032, 1863, 26890, 3224, 1863, 3224, 45595, 20498, 8032, 1863, 8592, 38688, 20449, 45030, 11242, 6552, 18863, 4117, 3794], "avg_logprob": -0.2176513730082661, "compression_ratio": 1.392638036809816, "no_speech_prob": 0.0, "words": [{"start": 280.71, "end": 281.07, "word": "it", "probability": 0.73095703125}, {"start": 281.07, "end": 281.37, "word": " is", "probability": 0.9501953125}, {"start": 281.37, "end": 282.45, "word": " cushy", "probability": 0.77392578125}, {"start": 282.45, "end": 282.81, "word": " it", "probability": 0.42138671875}, {"start": 282.81, "end": 283.29, "word": " is", "probability": 0.97509765625}, {"start": 283.29, "end": 286.37, "word": " cushy", "probability": 0.910400390625}, {"start": 286.37, "end": 286.91, "word": " لأن", "probability": 0.33209228515625}, {"start": 286.91, "end": 287.17, "word": " هذا", "probability": 0.80859375}, {"start": 287.17, "end": 287.53, "word": " جزء", "probability": 0.9000651041666666}, {"start": 287.53, "end": 288.25, "word": " برهناه", "probability": 0.702685546875}, {"start": 288.25, "end": 289.05, "word": " في", "probability": 0.92529296875}, {"start": 289.05, "end": 289.59, "word": " المحاضرة", "probability": 0.984375}, {"start": 289.59, "end": 290.17, "word": " السابقة", "probability": 0.9830729166666666}, {"start": 290.17, "end": 290.43, "word": " على", "probability": 0.9189453125}, {"start": 290.43, "end": 291.85, "word": " صورة", "probability": 0.92333984375}, {"start": 291.85, "end": 292.87, "word": " لمة", "probability": 0.499267578125}, {"start": 292.87, "end": 293.43, "word": " واحد", "probability": 0.911865234375}, {"start": 293.43, "end": 293.99, "word": " وعشرين", "probability": 0.67254638671875}, {"start": 293.99, "end": 295.61, "word": " ال", "probability": 0.8154296875}, {"start": 295.61, "end": 295.91, "word": " ..", "probability": 0.48193359375}, {"start": 295.91, "end": 296.91, "word": " ال", "probability": 0.92822265625}, {"start": 296.91, "end": 297.19, "word": " if", "probability": 0.73046875}, {"start": 297.19, "end": 297.77, "word": " part", "probability": 0.96240234375}, {"start": 297.77, "end": 300.27, "word": " هنبرهنه", "probability": 0.9192708333333334}, {"start": 300.27, "end": 300.71, "word": " اليوم", "probability": 0.95947265625}, {"start": 300.71, "end": 302.29, "word": " هنشوف", "probability": 0.902099609375}, {"start": 302.29, "end": 302.49, "word": " مع", "probability": 0.98388671875}, {"start": 302.49, "end": 302.95, "word": " بعض", "probability": 0.99365234375}, {"start": 302.95, "end": 303.65, "word": " assume", "probability": 0.88916015625}, {"start": 303.65, "end": 304.29, "word": " العكس", "probability": 0.97607421875}], "temperature": 1.0}, {"id": 11, "seek": 33154, "start": 305.96, "end": 331.54, "text": "assume أن الـ sequence x in is Cauchy وبدنا نثبت إنها convergent طيب بما إنها Cauchy then by لمّا اتنين و عشرين تطلع bounded", "tokens": [640, 2540, 14739, 2423, 39184, 8310, 2031, 294, 307, 7544, 625, 88, 4032, 44510, 8315, 8717, 12984, 3555, 2655, 36145, 11296, 9652, 6930, 23032, 1829, 3555, 4724, 15042, 36145, 11296, 7544, 625, 88, 550, 538, 32767, 11703, 995, 1975, 2655, 1863, 9957, 4032, 6225, 46309, 9957, 6055, 9566, 1211, 3615, 37498], "avg_logprob": -0.2854567301961092, "compression_ratio": 1.1712328767123288, "no_speech_prob": 0.0, "words": [{"start": 305.96, "end": 306.84, "word": "assume", "probability": 0.7529296875}, {"start": 306.84, "end": 308.34, 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"probability": 0.812744140625}, {"start": 320.08, "end": 320.44, "word": " إنها", "probability": 0.932861328125}, {"start": 320.44, "end": 321.64, "word": " Cauchy", "probability": 0.935546875}, {"start": 321.64, "end": 325.28, "word": " then", "probability": 0.71630859375}, {"start": 325.28, "end": 325.9, "word": " by", "probability": 0.8427734375}, {"start": 325.9, "end": 327.48, "word": " لمّا", "probability": 0.7898763020833334}, {"start": 327.48, "end": 329.88, "word": " اتنين", "probability": 0.78375244140625}, {"start": 329.88, "end": 330.0, "word": " و", "probability": 0.875}, {"start": 330.0, "end": 330.58, "word": " عشرين", "probability": 0.7405598958333334}, {"start": 330.58, "end": 331.14, "word": " تطلع", "probability": 0.95263671875}, {"start": 331.14, "end": 331.54, "word": " bounded", "probability": 0.91015625}], "temperature": 1.0}, {"id": 12, "seek": 35560, "start": 336.76, "end": 355.6, "text": "إذا by لمبة إتنين و عشرين ال sequence x in is bounded بستخدام 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star", "probability": 0.0726318359375}, {"start": 703.85, "end": 704.37, "word": " أصغر", "probability": 0.9322509765625}, {"start": 704.37, "end": 704.49, "word": " من", "probability": 0.9931640625}, {"start": 704.49, "end": 704.73, "word": " Y", "probability": 0.477294921875}, {"start": 704.73, "end": 705.03, "word": " على", "probability": 0.48388671875}, {"start": 705.03, "end": 705.27, "word": " 2", "probability": 0.87744140625}, {"start": 705.27, "end": 706.59, "word": " نسمي", "probability": 0.87353515625}, {"start": 706.59, "end": 707.35, "word": " المتباينة", "probability": 0.972900390625}, {"start": 707.35, "end": 707.83, "word": " هذه", "probability": 0.5703125}, {"start": 707.83, "end": 708.43, "word": " double", "probability": 0.615234375}, {"start": 708.43, "end": 709.01, "word": " star", "probability": 0.869140625}, {"start": 709.01, "end": 713.29, "word": " الان", "probability": 0.6416015625}, {"start": 713.29, "end": 713.95, "word": " now", "probability": 0.69140625}], "temperature": 1.0}, {"id": 26, "seek": 74672, "start": 719.24, "end": 746.72, "text": "أنا عندي كابتل كأكبر من أو ساوي كابتل N so by star by star with M بساوي كابتل K we have لدينا", "tokens": [10721, 8315, 18871, 16254, 9122, 16758, 2655, 1211, 9122, 10721, 4117, 26890, 9154, 34051, 8608, 995, 45865, 9122, 16758, 2655, 1211, 426, 370, 538, 3543, 538, 3543, 365, 376, 4724, 3794, 995, 45865, 9122, 16758, 2655, 1211, 591, 321, 362, 5296, 16254, 8315], "avg_logprob": -0.286221596327695, "compression_ratio": 1.3529411764705883, "no_speech_prob": 0.0, "words": [{"start": 719.24, "end": 719.58, "word": "أنا", "probability": 0.75927734375}, {"start": 719.58, "end": 719.98, "word": " عندي", "probability": 0.88232421875}, {"start": 719.98, "end": 720.48, "word": " كابتل", "probability": 0.6629638671875}, {"start": 720.48, "end": 721.44, "word": " كأكبر", "probability": 0.736328125}, {"start": 721.44, "end": 721.58, "word": " من", "probability": 0.96484375}, {"start": 721.58, "end": 721.76, "word": " أو", "probability": 0.63916015625}, {"start": 721.76, "end": 722.1, "word": " ساوي", "probability": 0.7750651041666666}, {"start": 722.1, "end": 722.64, "word": " كابتل", "probability": 0.8704833984375}, {"start": 722.64, "end": 723.0, "word": " N", "probability": 0.416748046875}, {"start": 723.0, "end": 728.14, "word": " so", "probability": 0.2724609375}, {"start": 728.14, "end": 729.4, "word": " by", "probability": 0.315185546875}, {"start": 729.4, "end": 730.0, "word": " star", "probability": 0.765625}, {"start": 730.0, "end": 734.32, "word": " by", "probability": 0.310791015625}, {"start": 734.32, "end": 735.06, "word": " star", "probability": 0.93994140625}, {"start": 735.06, "end": 737.8, "word": " with", "probability": 0.8759765625}, {"start": 737.8, "end": 738.16, "word": " M", "probability": 0.51708984375}, {"start": 738.16, "end": 738.82, "word": " بساوي", "probability": 0.936279296875}, {"start": 738.82, "end": 739.46, "word": " كابتل", "probability": 0.9620361328125}, {"start": 739.46, "end": 739.9, "word": " K", "probability": 0.912109375}, {"start": 739.9, "end": 745.6, "word": " we", "probability": 0.783203125}, {"start": 745.6, "end": 746.1, "word": " have", "probability": 0.9716796875}, {"start": 746.1, "end": 746.72, "word": " لدينا", "probability": 0.9181315104166666}], "temperature": 1.0}, {"id": 27, "seek": 77796, "start": 750.82, "end": 777.96, "text": "absolute xn minus x capital k أصغر من y ع 2 نسمي هذه المتباينة triple triple star كمان مرة ال k هذه اختارناها أكبر منها و يساوي n و من star", "tokens": [17243, 401, 1169, 2031, 77, 3175, 2031, 4238, 350, 5551, 9381, 17082, 2288, 9154, 288, 6225, 568, 8717, 38251, 1829, 29538, 9673, 2655, 3555, 995, 9957, 3660, 15508, 15508, 3543, 9122, 2304, 7649, 3714, 25720, 2423, 350, 29538, 1975, 46456, 9640, 8315, 11296, 5551, 4117, 26890, 9154, 11296, 4032, 7251, 3794, 995, 45865, 297, 4032, 9154, 3543], "avg_logprob": -0.25363685242060957, "compression_ratio": 1.3157894736842106, 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{"start": 760.66, "end": 761.06, "word": " هذه", "probability": 0.91455078125}, {"start": 761.06, "end": 762.34, "word": " المتباينة", "probability": 0.9003092447916666}, {"start": 762.34, "end": 762.76, "word": " triple", "probability": 0.80517578125}, {"start": 762.76, "end": 765.34, "word": " triple", "probability": 0.5458984375}, {"start": 765.34, "end": 765.82, "word": " star", "probability": 0.7353515625}, {"start": 765.82, "end": 768.38, "word": " كمان", "probability": 0.9501953125}, {"start": 768.38, "end": 768.72, "word": " مرة", "probability": 0.96142578125}, {"start": 768.72, "end": 768.96, "word": " ال", "probability": 0.81396484375}, {"start": 768.96, "end": 769.3, "word": " k", "probability": 0.51123046875}, {"start": 769.3, "end": 769.9, "word": " هذه", "probability": 0.77734375}, {"start": 769.9, "end": 771.44, "word": " اختارناها", "probability": 0.8732421875}, {"start": 771.44, "end": 772.34, "word": " أكبر", "probability": 0.8673502604166666}, {"start": 772.34, "end": 772.78, "word": " منها", "probability": 0.913330078125}, {"start": 772.78, "end": 772.98, "word": " و", "probability": 0.57177734375}, {"start": 772.98, "end": 773.4, "word": " يساوي", "probability": 0.73309326171875}, {"start": 773.4, "end": 773.74, "word": " n", "probability": 0.67333984375}, {"start": 773.74, "end": 775.88, "word": " و", "probability": 0.96435546875}, {"start": 775.88, "end": 776.64, "word": " من", "probability": 0.904296875}, {"start": 776.64, "end": 777.96, "word": " star", "probability": 0.69775390625}], "temperature": 1.0}, {"id": 28, "seek": 80426, "start": 779.2, "end": 804.26, "text": "إذا كانت الـ K .. إذا خدت M بساوي كابتال K و هذه أكبر من أو ساوي N فبتصير المتباينة هذه absolute XN minus XK أزرع من إبسط على اتنين و الـ N هذه لازم تكون أكبر من أو ساوي M، إذن هذا صحيح لكل small M أكبر من أو ساوي كابتال M", "tokens": [28814, 15730, 25961, 2655, 2423, 39184, 591, 4386, 11933, 15730, 16490, 3215, 2655, 376, 4724, 3794, 995, 45865, 9122, 16758, 2655, 6027, 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"probability": 0.7415364583333334}, {"start": 798.86, "end": 799.1, "word": " هذا", "probability": 0.91943359375}, {"start": 799.1, "end": 799.6, "word": " صحيح", "probability": 0.99755859375}, {"start": 799.6, "end": 800.18, "word": " لكل", "probability": 0.980224609375}, {"start": 800.18, "end": 800.66, "word": " small", "probability": 0.763671875}, {"start": 800.66, "end": 801.1, "word": " M", "probability": 0.44482421875}, {"start": 801.1, "end": 802.34, "word": " أكبر", "probability": 0.9820963541666666}, {"start": 802.34, "end": 802.58, "word": " من", "probability": 0.99658203125}, {"start": 802.58, "end": 802.78, "word": " أو", "probability": 0.994140625}, {"start": 802.78, "end": 803.42, "word": " ساوي", "probability": 0.9705403645833334}, {"start": 803.42, "end": 804.02, "word": " كابتال", "probability": 0.83087158203125}, {"start": 804.02, "end": 804.26, "word": " M", "probability": 0.7490234375}], "temperature": 1.0}, {"id": 29, "seek": 82405, "start": 809.67, "end": 824.05, "text": "تمام hence by double star الآن من double star and triple star", "tokens": [39237, 10943, 16678, 538, 3834, 3543, 6024, 48506, 9154, 3834, 3543, 293, 15508, 3543], "avg_logprob": -0.26666667858759563, "compression_ratio": 1.1451612903225807, "no_speech_prob": 0.0, "words": [{"start": 809.6700000000001, "end": 811.07, "word": "تمام", "probability": 0.73974609375}, {"start": 811.07, "end": 812.47, "word": " hence", "probability": 0.350830078125}, {"start": 812.47, "end": 815.67, "word": " by", "probability": 0.9033203125}, {"start": 815.67, "end": 816.49, "word": " double", "probability": 0.8359375}, {"start": 816.49, "end": 817.09, "word": " star", "probability": 0.94580078125}, {"start": 817.09, "end": 817.65, "word": " الآن", "probability": 0.58349609375}, {"start": 817.65, "end": 817.97, "word": " من", "probability": 0.98486328125}, {"start": 817.97, "end": 818.35, "word": " double", "probability": 0.93017578125}, {"start": 818.35, "end": 818.99, "word": " star", "probability": 0.93310546875}, {"start": 818.99, "end": 820.69, "word": " and", "probability": 0.9541015625}, {"start": 820.69, "end": 822.85, "word": " triple", "probability": 0.9521484375}, {"start": 822.85, "end": 824.05, "word": " star", "probability": 0.96337890625}], "temperature": 1.0}, {"id": 30, "seek": 85853, "start": 828.93, "end": 858.53, "text": "لدينا we have لو كان n أكبر من أو ساوي capital N فهذا بيقدي أنه absolute xn minus x star طبعا هنا هترح x capital K و هرجعها", "tokens": [1211, 16254, 8315, 321, 362, 45164, 25961, 297, 5551, 4117, 26890, 9154, 34051, 8608, 995, 45865, 4238, 426, 6156, 3224, 15730, 4724, 1829, 4587, 16254, 14739, 3224, 8236, 2031, 77, 3175, 2031, 3543, 23032, 3555, 3615, 995, 34105, 8032, 2655, 2288, 5016, 2031, 4238, 591, 4032, 8032, 47341, 3615, 11296], "avg_logprob": -0.21706495214911067, "compression_ratio": 1.1904761904761905, "no_speech_prob": 0.0, "words": [{"start": 828.93, "end": 829.87, "word": "لدينا", "probability": 0.896484375}, 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"word": " بيقدي", "probability": 0.70599365234375}, {"start": 839.71, "end": 840.33, "word": " أنه", "probability": 0.43798828125}, {"start": 840.33, "end": 841.01, "word": " absolute", "probability": 0.88720703125}, {"start": 841.01, "end": 843.27, "word": " xn", "probability": 0.657470703125}, {"start": 843.27, "end": 843.99, "word": " minus", "probability": 0.78125}, {"start": 843.99, "end": 844.47, "word": " x", "probability": 0.99267578125}, {"start": 844.47, "end": 845.15, "word": " star", "probability": 0.33056640625}, {"start": 845.15, "end": 851.33, "word": " طبعا", "probability": 0.9764404296875}, {"start": 851.33, "end": 851.65, "word": " هنا", "probability": 0.94775390625}, {"start": 851.65, "end": 853.55, "word": " هترح", "probability": 0.8350830078125}, {"start": 853.55, "end": 855.67, "word": " x", "probability": 0.8837890625}, {"start": 855.67, "end": 856.21, "word": " capital", "probability": 0.73583984375}, {"start": 856.21, "end": 856.71, "word": " K", "probability": 0.74853515625}, {"start": 856.71, "end": 857.71, "word": " و", "probability": 0.6435546875}, {"start": 857.71, "end": 858.53, "word": " هرجعها", "probability": 0.90771484375}], "temperature": 1.0}, {"id": 31, "seek": 88163, "start": 868.69, "end": 881.63, "text": "إذا I subtracted XK and get it back باخد هدوع الأثنين مع بعض و التحدين هدوع مع بعض", "tokens": [28814, 15730, 286, 16390, 292, 1783, 42, 293, 483, 309, 646, 4724, 47283, 3215, 8032, 3215, 2407, 3615, 16247, 12984, 1863, 9957, 20449, 45030, 11242, 4032, 16712, 5016, 16254, 1863, 8032, 3215, 2407, 3615, 20449, 45030, 11242], "avg_logprob": -0.32298521070103897, "compression_ratio": 1.1844660194174756, "no_speech_prob": 0.0, "words": [{"start": 868.69, "end": 869.07, "word": "إذا", "probability": 0.5443115234375}, {"start": 869.07, "end": 870.33, "word": " I", "probability": 0.279052734375}, {"start": 870.33, "end": 871.95, "word": " subtracted", "probability": 0.749755859375}, {"start": 871.95, "end": 873.45, "word": " XK", 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اللي", "probability": 0.974365234375}, {"start": 899.4, "end": 899.62, "word": " هو", "probability": 0.986328125}, {"start": 899.62, "end": 901.4, "word": " xk", "probability": 0.969482421875}, {"start": 901.4, "end": 902.22, "word": " minus", "probability": 0.98974609375}, {"start": 902.22, "end": 902.66, "word": " x", "probability": 0.99462890625}, {"start": 902.66, "end": 903.22, "word": " star", "probability": 0.1741943359375}, {"start": 903.22, "end": 908.5, "word": " الآن", "probability": 0.6878662109375}, {"start": 908.5, "end": 909.3, "word": " باستخدام", "probability": 0.9916178385416666}, {"start": 909.3, "end": 909.78, "word": " triple", "probability": 0.4912109375}, {"start": 909.78, "end": 911.16, "word": " star", "probability": 0.919921875}], "temperature": 1.0}, {"id": 33, "seek": 93767, "start": 912.65, "end": 937.67, "text": "من المتباينة هذه هاي عندي انا x اول شي ال n small n أكبر من أو ساوي capital N هاي small n أكبر من أو ساوي capital N وبالتالي ال absolute value هذه أصغر من epsilon على اتنين زاد و من double star من double star", "tokens": [27842, 9673, 2655, 3555, 995, 9957, 3660, 29538, 8032, 47302, 18871, 16254, 1975, 8315, 2031, 1975, 12610, 44049, 2423, 297, 1359, 297, 5551, 4117, 26890, 9154, 34051, 8608, 995, 45865, 4238, 426, 8032, 47302, 1359, 297, 5551, 4117, 26890, 9154, 34051, 8608, 995, 45865, 4238, 426, 46599, 6027, 2655, 6027, 1829, 2423, 8236, 2158, 29538, 5551, 9381, 17082, 2288, 9154, 17889, 15844, 1975, 2655, 1863, 9957, 30767, 18513, 4032, 9154, 3834, 3543, 9154, 3834, 3543], "avg_logprob": -0.23087993460266212, "compression_ratio": 1.7298850574712643, "no_speech_prob": 0.0, "words": [{"start": 912.65, "end": 912.95, "word": "من", "probability": 0.88623046875}, {"start": 912.95, "end": 913.79, "word": " المتباينة", "probability": 0.9475911458333334}, {"start": 913.79, "end": 914.23, "word": " هذه", "probability": 0.81884765625}, {"start": 914.23, "end": 916.09, "word": " هاي", "probability": 0.466552734375}, {"start": 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"probability": 0.7247721354166666}, {"start": 922.39, "end": 922.77, "word": " capital", "probability": 0.71826171875}, {"start": 922.77, "end": 923.17, "word": " N", "probability": 0.71923828125}, {"start": 923.17, "end": 924.35, "word": " هاي", "probability": 0.76611328125}, {"start": 924.35, "end": 924.77, "word": " small", "probability": 0.9052734375}, {"start": 924.77, "end": 925.13, "word": " n", "probability": 0.95068359375}, {"start": 925.13, "end": 925.69, "word": " أكبر", "probability": 0.9542643229166666}, {"start": 925.69, "end": 925.83, "word": " من", "probability": 0.99365234375}, {"start": 925.83, "end": 926.01, "word": " أو", "probability": 0.97509765625}, {"start": 926.01, "end": 926.29, "word": " ساوي", "probability": 0.9775390625}, {"start": 926.29, "end": 926.69, "word": " capital", "probability": 0.8525390625}, {"start": 926.69, "end": 927.07, "word": " N", "probability": 0.97119140625}, {"start": 927.07, "end": 928.43, "word": " وبالتالي", "probability": 0.900390625}, {"start": 928.43, "end": 928.59, "word": " ال", "probability": 0.95458984375}, {"start": 928.59, "end": 928.95, "word": " absolute", "probability": 0.9208984375}, {"start": 928.95, "end": 929.51, "word": " value", "probability": 0.9970703125}, {"start": 929.51, "end": 929.87, "word": " هذه", "probability": 0.6064453125}, {"start": 929.87, "end": 930.51, "word": " أصغر", "probability": 0.964599609375}, {"start": 930.51, "end": 931.93, "word": " من", "probability": 0.9765625}, {"start": 931.93, "end": 932.29, "word": " epsilon", "probability": 0.68212890625}, {"start": 932.29, "end": 932.57, "word": " على", "probability": 0.271728515625}, {"start": 932.57, "end": 933.01, "word": " اتنين", "probability": 0.882080078125}, {"start": 933.01, "end": 933.77, "word": " زاد", "probability": 0.740966796875}, {"start": 933.77, "end": 934.87, "word": " و", "probability": 0.96240234375}, {"start": 934.87, "end": 935.19, "word": " من", "probability": 0.6748046875}, {"start": 935.19, "end": 935.61, "word": " double", "probability": 0.82373046875}, {"start": 935.61, "end": 936.23, "word": " star", "probability": 0.560546875}, {"start": 936.23, "end": 936.79, "word": " من", "probability": 0.80126953125}, {"start": 936.79, "end": 937.15, "word": " double", "probability": 0.9287109375}, {"start": 937.15, "end": 937.67, "word": " star", "probability": 0.9658203125}], "temperature": 1.0}, {"id": 34, "seek": 96000, "start": 939.8, "end": 960.0, "text": "هي عندي absolute x K minus x star أصغر من إبسمن على اتنين المجموع بتطلع إبسمن since إبسمن أكبر من السفر was arbitrarily", "tokens": [3224, 1829, 18871, 16254, 8236, 2031, 591, 3175, 2031, 3543, 5551, 9381, 17082, 2288, 9154, 11933, 3555, 3794, 27842, 15844, 1975, 2655, 1863, 9957, 9673, 7435, 2304, 45367, 39894, 9566, 1211, 3615, 11933, 3555, 3794, 27842, 1670, 11933, 3555, 3794, 27842, 5551, 4117, 26890, 9154, 21136, 5172, 2288, 390, 19071, 3289], "avg_logprob": -0.2656250097430669, "compression_ratio": 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987.89, "text": "نحن لدينا من مفهوم الـ convergence أنه هيك منكون أثبتنا أنه limit xn as n tends to infinity equals x star وهذا بكمل برهان ال claim و النظرية تمام؟ هاي لاحظوا أن احنا", "tokens": [1863, 5016, 1863, 5296, 16254, 8315, 9154, 3714, 5172, 3224, 20498, 2423, 39184, 32181, 14739, 3224, 39896, 4117, 9154, 30544, 5551, 12984, 3555, 2655, 8315, 14739, 3224, 4948, 2031, 77, 382, 297, 12258, 281, 13202, 6915, 2031, 3543, 37037, 15730, 4724, 24793, 1211, 4724, 2288, 3224, 7649, 2423, 3932, 4032, 28239, 19913, 2288, 10632, 46811, 10943, 22807, 8032, 47302, 20193, 5016, 19913, 14407, 14739, 1975, 5016, 8315], "avg_logprob": -0.4177389780388159, "compression_ratio": 1.3591160220994476, "no_speech_prob": 0.0, "words": [{"start": 963.85, "end": 964.29, "word": "نحن", "probability": 0.50927734375}, {"start": 964.29, "end": 965.47, "word": " لدينا", "probability": 0.7731119791666666}, {"start": 965.47, "end": 965.99, "word": " من", "probability": 0.87646484375}, {"start": 965.99, "end": 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tends", "probability": 0.488037109375}, {"start": 975.19, "end": 975.37, "word": " to", "probability": 0.83544921875}, {"start": 975.37, "end": 975.95, "word": " infinity", "probability": 0.8974609375}, {"start": 975.95, "end": 976.69, "word": " equals", "probability": 0.830078125}, {"start": 976.69, "end": 977.15, "word": " x", "probability": 0.9609375}, {"start": 977.15, "end": 977.69, "word": " star", "probability": 0.10491943359375}, {"start": 977.69, "end": 980.63, "word": " وهذا", "probability": 0.837646484375}, {"start": 980.63, "end": 981.23, "word": " بكمل", "probability": 0.8505859375}, {"start": 981.23, "end": 981.85, "word": " برهان", "probability": 0.924072265625}, {"start": 981.85, "end": 982.73, "word": " ال", "probability": 0.81787109375}, {"start": 982.73, "end": 983.17, "word": " claim", "probability": 0.51416015625}, {"start": 983.17, "end": 983.41, "word": " و", "probability": 0.55517578125}, {"start": 983.41, "end": 984.67, "word": " النظرية", "probability": 0.9525146484375}, {"start": 984.67, "end": 986.79, "word": " تمام؟", "probability": 0.6493326822916666}, {"start": 986.79, "end": 987.09, "word": " هاي", "probability": 0.4625244140625}, {"start": 987.09, "end": 987.51, "word": " لاحظوا", "probability": 0.82470703125}, {"start": 987.51, "end": 987.61, "word": " أن", "probability": 0.73046875}, {"start": 987.61, "end": 987.89, "word": " احنا", "probability": 0.8435872395833334}], "temperature": 1.0}, {"id": 36, "seek": 100766, "start": 988.84, "end": 1007.66, "text": "بنثبت أننا ندعي أن الـSequence Xn هي الـConversion لـX الصار حسب تعريف epsilon capital N للـLimits بدأنا بـepsilon given عشوائية عدد موجب أثبتنا هي يوجد capital N يعتمد على epsilon أشرر number", "tokens": [3555, 1863, 12984, 3555, 2655, 14739, 8315, 8717, 3215, 3615, 1829, 14739, 2423, 39184, 10637, 358, 655, 1783, 77, 39896, 2423, 39184, 9838, 29153, 5296, 39184, 55, 31767, 9640, 11331, 35457, 37279, 16572, 5172, 17889, 4238, 426, 24976, 39184, 43, 332, 1208, 47525, 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"probability": 0.505615234375}, {"start": 992.98, "end": 993.28, "word": " لـX", "probability": 0.5950520833333334}, {"start": 993.28, "end": 993.72, "word": " الصار", "probability": 0.516357421875}, {"start": 993.72, "end": 994.82, "word": " حسب", "probability": 0.810302734375}, {"start": 994.82, "end": 995.26, "word": " تعريف", "probability": 0.8606770833333334}, {"start": 995.26, "end": 995.66, "word": " epsilon", "probability": 0.1988525390625}, {"start": 995.66, "end": 996.06, "word": " capital", "probability": 0.2449951171875}, {"start": 996.06, "end": 996.36, "word": " N", "probability": 0.89208984375}, {"start": 996.36, "end": 997.06, "word": " للـLimits", "probability": 0.751611328125}, {"start": 997.06, "end": 998.5, "word": " بدأنا", "probability": 0.7568359375}, {"start": 998.5, "end": 999.06, "word": " بـepsilon", "probability": 0.7489013671875}, {"start": 999.06, "end": 999.46, "word": " given", "probability": 0.48388671875}, {"start": 999.46, "end": 1000.36, "word": " عشوائية", "probability": 0.97529296875}, {"start": 1000.36, "end": 1000.86, "word": " عدد", "probability": 0.8904622395833334}, {"start": 1000.86, "end": 1001.36, "word": " موجب", "probability": 0.859375}, {"start": 1001.36, "end": 1003.4, "word": " أثبتنا", "probability": 0.9666015625}, {"start": 1003.4, "end": 1003.66, "word": " هي", "probability": 0.525390625}, {"start": 1003.66, "end": 1004.44, "word": " يوجد", "probability": 0.84765625}, {"start": 1004.44, "end": 1005.16, "word": " capital", "probability": 0.5029296875}, {"start": 1005.16, "end": 1005.6, "word": " N", "probability": 0.943359375}, {"start": 1005.6, "end": 1006.2, "word": " يعتمد", "probability": 0.9130859375}, {"start": 1006.2, "end": 1006.36, "word": " على", "probability": 0.82080078125}, {"start": 1006.36, "end": 1006.82, "word": " epsilon", "probability": 0.79296875}, {"start": 1006.82, "end": 1007.34, "word": " أشرر", "probability": 0.4326985677083333}, {"start": 1007.34, "end": 1007.66, "word": " number", "probability": 0.384521484375}], "temperature": 1.0}, {"id": 37, "seek": 102508, "start": 1008.38, "end": 1025.08, "text": "بحيث انه لكل N أكبر من او ساوي capital N طلع absolute xn minus x star less than epsilon لما ان هذا الكلام صحيح لكل epsilon", "tokens": [49628, 1829, 12984, 16472, 3224, 5296, 28820, 426, 5551, 4117, 26890, 9154, 1975, 2407, 8608, 995, 45865, 4238, 426, 23032, 1211, 3615, 8236, 2031, 77, 3175, 2031, 3543, 1570, 813, 17889, 5296, 15042, 16472, 23758, 2423, 28820, 10943, 20328, 5016, 1829, 5016, 5296, 28820, 17889], "avg_logprob": -0.29228941253993823, "compression_ratio": 1.2444444444444445, "no_speech_prob": 0.0, "words": [{"start": 1008.38, "end": 1009.08, "word": "بحيث", "probability": 0.83544921875}, {"start": 1009.08, "end": 1009.52, "word": " انه", "probability": 0.6846923828125}, {"start": 1009.52, "end": 1010.08, "word": " لكل", "probability": 0.978515625}, {"start": 1010.08, "end": 1010.4, "word": " N", "probability": 0.50732421875}, {"start": 1010.4, 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sequence كوشي then it is convergent تمام واضح البرهان؟ okay حلو إذا نعم", "tokens": [28814, 15730, 538, 7123, 4948, 2031, 77, 8608, 995, 45865, 2031, 5551, 3794, 9566, 13063, 3660, 12399, 11933, 15730, 2423, 8310, 9652, 6930, 11933, 15730, 23758, 4724, 1829, 24793, 1211, 2423, 26890, 3224, 7649, 36145, 45164, 25961, 2655, 2423, 8310, 9122, 2407, 8592, 1829, 550, 309, 307, 9652, 6930, 46811, 10943, 4032, 46958, 5016, 2423, 26890, 3224, 7649, 22807, 1392, 11331, 1211, 2407, 11933, 15730, 8717, 25957], "avg_logprob": -0.3550091977505123, "compression_ratio": 1.4736842105263157, "no_speech_prob": 0.0, "words": [{"start": 1026.56, "end": 1027.02, "word": "إذا", "probability": 0.4913330078125}, {"start": 1027.02, "end": 1027.3, "word": " by", "probability": 0.3603515625}, {"start": 1027.3, "end": 1027.82, "word": " definition", "probability": 0.9267578125}, {"start": 1027.82, "end": 1028.34, "word": " limit", "probability": 0.92529296875}, {"start": 1028.34, "end": 1029.02, "word": " xn", 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في", "probability": 0.92236328125}, {"start": 1156.35, "end": 1156.95, "word": " إثبات", "probability": 0.86700439453125}, {"start": 1156.95, "end": 1157.43, "word": " أنه", "probability": 0.6025390625}, {"start": 1157.43, "end": 1157.75, "word": " given", "probability": 0.908203125}, {"start": 1157.75, "end": 1159.37, "word": " sequence", "probability": 0.97998046875}, {"start": 1159.37, "end": 1159.79, "word": " is", "probability": 0.96728515625}, {"start": 1159.79, "end": 1160.25, "word": " koshi", "probability": 0.763671875}, {"start": 1160.25, "end": 1162.21, "word": " باستخدام", "probability": 0.9765625}, {"start": 1162.21, "end": 1162.73, "word": " التعريف", "probability": 0.9949951171875}, {"start": 1162.73, "end": 1163.45, "word": " مباشرة", "probability": 0.9935302734375}, {"start": 1163.45, "end": 1165.73, "word": " و", "probability": 0.98486328125}, {"start": 1165.73, "end": 1166.13, "word": " ليس", "probability": 0.79931640625}, {"start": 1166.13, "end": 1166.91, "word": " باستخدام", "probability": 0.9942220052083334}, {"start": 1166.91, "end": 1167.25, "word": " اللي", "probability": 0.946044921875}, {"start": 1167.25, "end": 1167.51, "word": " هو", "probability": 0.9892578125}, {"start": 1167.51, "end": 1167.89, "word": " koshi", "probability": 0.828125}, {"start": 1167.89, "end": 1168.41, "word": " criterion", "probability": 0.97509765625}], "temperature": 1.0}, {"id": 43, "seek": 120025, "start": 1171.95, "end": 1200.25, "text": "إذا ناخد هنا بعض الأمثلة examples الأمثلة دي أنا أعطيها الرقم 224 أول مثال show", "tokens": [28814, 15730, 8717, 47283, 3215, 34105, 45030, 11242, 16247, 2304, 12984, 37977, 5110, 16247, 2304, 12984, 37977, 11778, 1829, 41850, 5551, 3615, 9566, 1829, 11296, 34892, 4587, 2304, 5853, 19, 5551, 12610, 50113, 6027, 855], "avg_logprob": -0.15603297907445166, "compression_ratio": 1.1944444444444444, "no_speech_prob": 0.0, "words": [{"start": 1171.95, "end": 1172.29, "word": "إذا", "probability": 0.710693359375}, {"start": 1172.29, "end": 1172.77, "word": " ناخد", "probability": 0.9775390625}, {"start": 1172.77, "end": 1173.09, "word": " هنا", "probability": 0.927734375}, {"start": 1173.09, "end": 1173.59, "word": " بعض", "probability": 0.9560546875}, {"start": 1173.59, "end": 1174.19, "word": " الأمثلة", "probability": 0.984619140625}, {"start": 1174.19, "end": 1187.49, "word": " examples", "probability": 0.630859375}, {"start": 1187.49, "end": 1192.93, "word": " الأمثلة", "probability": 0.86065673828125}, {"start": 1192.93, "end": 1193.15, "word": " دي", "probability": 0.848876953125}, {"start": 1193.15, "end": 1193.25, "word": " أنا", "probability": 0.455322265625}, {"start": 1193.25, "end": 1193.59, "word": " أعطيها", "probability": 0.9236328125}, {"start": 1193.59, "end": 1194.01, "word": " الرقم", "probability": 0.8541666666666666}, {"start": 1194.01, "end": 1195.79, "word": " 224", "probability": 0.8212890625}, {"start": 1195.79, "end": 1197.97, "word": " أول", "probability": 0.9296875}, {"start": 1197.97, "end": 1198.61, "word": " مثال", "probability": 0.798095703125}, {"start": 1198.61, "end": 1200.25, "word": " show", "probability": 0.91455078125}], "temperature": 1.0}, {"id": 44, "seek": 122513, "start": 1204.71, "end": 1225.13, "text": "directly show direct that ال sequence ال sequence واحد على ان is Cauchy", "tokens": [44868, 356, 855, 2047, 300, 2423, 8310, 2423, 8310, 36764, 24401, 15844, 16472, 307, 7544, 625, 88], "avg_logprob": -0.4366319527228673, "compression_ratio": 1.12, "no_speech_prob": 0.0, "words": [{"start": 1204.7099999999998, "end": 1206.11, "word": "directly", "probability": 0.6871337890625}, {"start": 1206.11, "end": 1207.51, "word": " show", "probability": 0.615234375}, {"start": 1207.51, "end": 1208.35, "word": " direct", "probability": 0.7919921875}, {"start": 1208.35, "end": 1213.31, "word": " that", "probability": 0.9052734375}, {"start": 1213.31, "end": 1215.85, "word": " ال", "probability": 0.2607421875}, {"start": 1215.85, "end": 1216.47, "word": " sequence", "probability": 0.89501953125}, {"start": 1216.47, "end": 1220.39, "word": " ال", "probability": 0.451416015625}, {"start": 1220.39, "end": 1221.27, "word": " sequence", "probability": 0.98291015625}, {"start": 1221.27, "end": 1221.83, "word": " واحد", "probability": 0.865966796875}, {"start": 1221.83, "end": 1222.03, "word": " على", "probability": 0.54052734375}, {"start": 1222.03, "end": 1222.41, "word": " ان", "probability": 0.400146484375}, {"start": 1222.41, "end": 1224.41, "word": " is", "probability": 0.81494140625}, {"start": 1224.41, "end": 1225.13, "word": " Cauchy", "probability": 0.712158203125}], "temperature": 1.0}, {"id": 45, "seek": 125757, "start": 1235.15, "end": 1257.57, "text": "لما اقول show directly ان ال sequence معينة is Cauchy معناها ده بدي استخدم التعريف بدي استخدم التعريف تبع Cauchy sequence فنشوف مع بعض طبعا البرهان", "tokens": [1211, 15042, 1975, 39648, 855, 3838, 16472, 2423, 8310, 20449, 9957, 3660, 307, 7544, 625, 88, 20449, 8315, 11296, 11778, 3224, 4724, 16254, 44713, 9778, 40448, 16712, 3615, 16572, 5172, 4724, 16254, 44713, 9778, 40448, 16712, 3615, 16572, 5172, 6055, 3555, 3615, 7544, 625, 88, 8310, 6156, 1863, 8592, 38688, 20449, 45030, 11242, 23032, 3555, 3615, 995, 2423, 26890, 3224, 7649], "avg_logprob": -0.16872479333992926, "compression_ratio": 1.523489932885906, "no_speech_prob": 0.0, "words": [{"start": 1235.15, "end": 1235.57, "word": "لما", "probability": 0.6851806640625}, {"start": 1235.57, "end": 1235.99, "word": " اقول", "probability": 0.6522216796875}, {"start": 1235.99, "end": 1236.27, "word": " show", "probability": 0.95263671875}, {"start": 1236.27, "end": 1236.95, "word": " directly", "probability": 0.8984375}, {"start": 1236.95, "end": 1237.31, "word": " ان", "probability": 0.62744140625}, {"start": 1237.31, "end": 1237.49, "word": " ال", "probability": 0.73681640625}, {"start": 1237.49, "end": 1238.05, "word": " sequence", "probability": 0.80224609375}, {"start": 1238.05, "end": 1239.07, "word": " معينة", "probability": 0.93359375}, {"start": 1239.07, "end": 1239.63, "word": " is", "probability": 0.794921875}, {"start": 1239.63, "end": 1240.03, "word": " Cauchy", "probability": 0.720947265625}, {"start": 1240.03, "end": 1240.51, "word": " معناها", "probability": 0.7747395833333334}, {"start": 1240.51, "end": 1240.75, "word": " ده", "probability": 0.740234375}, {"start": 1240.75, "end": 1241.09, "word": " بدي", "probability": 0.754150390625}, {"start": 1241.09, "end": 1241.97, "word": " استخدم", "probability": 0.91796875}, {"start": 1241.97, "end": 1243.57, "word": " التعريف", "probability": 0.986083984375}, {"start": 1243.57, "end": 1243.83, "word": " بدي", "probability": 0.6170654296875}, {"start": 1243.83, "end": 1244.45, "word": " استخدم", "probability": 0.9615885416666666}, {"start": 1244.45, "end": 1245.47, "word": " التعريف", "probability": 0.9945068359375}, {"start": 1245.47, "end": 1246.21, "word": " تبع", "probability": 0.9705403645833334}, {"start": 1246.21, "end": 1246.61, "word": " Cauchy", "probability": 0.8922526041666666}, {"start": 1246.61, "end": 1247.15, "word": " sequence", "probability": 0.83447265625}, {"start": 1247.15, "end": 1250.45, "word": " فنشوف", "probability": 0.947509765625}, {"start": 1250.45, "end": 1250.65, "word": " مع", "probability": 0.99658203125}, {"start": 1250.65, "end": 1251.15, "word": " بعض", "probability": 0.987548828125}, {"start": 1251.15, "end": 1256.21, "word": " طبعا", "probability": 0.9619140625}, {"start": 1256.21, "end": 1257.57, "word": " البرهان", "probability": 0.9344482421875}], "temperature": 1.0}, {"id": 46, "seek": 128767, "start": 1258.87, "end": 1287.67, "text": "باستخدام التعريف هنبدأ بأبسلون أكبر من السفر ونرد عليها بcapital N بتخلي ال implication هي دي تتحقق بالظبط زي .. يعني قريب يعني بالظبط زي ما عملنا في اثبات ان ال sequence is convergent و هنستخدم الarchimedean property نشوف مع بعض let", "tokens": [3555, 995, 14851, 9778, 3215, 10943, 16712, 3615, 16572, 5172, 8032, 1863, 44510, 10721, 4724, 10721, 3555, 3794, 1211, 11536, 5551, 4117, 26890, 9154, 21136, 5172, 2288, 4032, 1863, 2288, 3215, 25894, 11296, 4724, 9485, 1686, 426, 39894, 9778, 20292, 2423, 37814, 39896, 11778, 1829, 6055, 2655, 5016, 4587, 4587, 20666, 19913, 3555, 9566, 30767, 1829, 4386, 37495, 22653, 12174, 16572, 3555, 37495, 22653, 20666, 19913, 3555, 9566, 30767, 1829, 19446, 6225, 42213, 8315, 8978, 1975, 12984, 3555, 9307, 16472, 2423, 8310, 307, 9652, 6930, 4032, 8032, 1863, 14851, 9778, 40448, 2423, 1178, 332, 4858, 282, 4707, 8717, 8592, 38688, 20449, 45030, 11242, 718], "avg_logprob": -0.2120535759698777, "compression_ratio": 1.5020746887966805, "no_speech_prob": 0.0, "words": [{"start": 1258.87, "end": 1259.69, "word": "باستخدام", "probability": 0.9510091145833334}, {"start": 1259.69, "end": 1260.53, "word": " التعريف", "probability": 0.9873046875}, {"start": 1260.53, "end": 1261.17, "word": " هنبدأ", "probability": 0.86328125}, {"start": 1261.17, "end": 1261.71, "word": " بأبسلون", "probability": 0.6169026692708334}, {"start": 1261.71, "end": 1262.01, "word": " أكبر", "probability": 0.966796875}, {"start": 1262.01, "end": 1262.23, "word": " من", "probability": 0.974609375}, {"start": 1262.23, "end": 1262.75, "word": " السفر", "probability": 0.7836100260416666}, {"start": 1262.75, "end": 1264.15, "word": " ونرد", "probability": 0.813720703125}, {"start": 1264.15, "end": 1264.63, "word": " عليها", "probability": 0.9853515625}, {"start": 1264.63, "end": 1265.27, "word": " بcapital", "probability": 0.6790364583333334}, {"start": 1265.27, "end": 1265.57, "word": " N", "probability": 0.712890625}, {"start": 1265.57, "end": 1266.97, "word": " بتخلي", "probability": 0.7236328125}, {"start": 1266.97, "end": 1267.07, "word": " ال", "probability": 0.96142578125}, {"start": 1267.07, "end": 1267.51, "word": " implication", "probability": 0.82861328125}, {"start": 1267.51, "end": 1267.79, "word": " هي", "probability": 0.400634765625}, {"start": 1267.79, "end": 1267.95, "word": " دي", "probability": 0.599365234375}, {"start": 1267.95, "end": 1268.57, "word": " تتحقق", "probability": 0.97587890625}, {"start": 1268.57, "end": 1269.11, "word": " بالظبط", "probability": 0.84478759765625}, {"start": 1269.11, "end": 1269.35, "word": " زي", "probability": 0.81396484375}, {"start": 1269.35, "end": 1269.39, "word": " ..", "probability": 0.312255859375}, {"start": 1269.39, "end": 1269.59, "word": " يعني", "probability": 0.874267578125}, {"start": 1269.59, "end": 1270.09, "word": " قريب", "probability": 0.9822591145833334}, {"start": 1270.09, "end": 1271.33, "word": " يعني", "probability": 0.6920166015625}, {"start": 1271.33, "end": 1272.05, "word": " بالظبط", "probability": 0.994873046875}, {"start": 1272.05, "end": 1273.89, "word": " زي", "probability": 0.95068359375}, {"start": 1273.89, "end": 1274.07, "word": " ما", "probability": 0.97216796875}, {"start": 1274.07, "end": 1274.61, "word": " عملنا", "probability": 0.9602864583333334}, {"start": 1274.61, "end": 1274.95, "word": " في", "probability": 0.9248046875}, {"start": 1274.95, "end": 1275.41, "word": " اثبات", "probability": 0.8251953125}, {"start": 1275.41, "end": 1275.57, "word": " ان", "probability": 0.45654296875}, {"start": 1275.57, "end": 1275.73, "word": " ال", "probability": 0.9599609375}, {"start": 1275.73, "end": 1276.05, "word": " sequence", "probability": 0.935546875}, {"start": 1276.05, "end": 1276.79, "word": " is", "probability": 0.94287109375}, {"start": 1276.79, "end": 1277.55, "word": " convergent", "probability": 0.930419921875}, {"start": 1277.55, "end": 1278.37, "word": " و", "probability": 0.6298828125}, {"start": 1278.37, "end": 1279.05, "word": " هنستخدم", "probability": 0.9044921875}, {"start": 1279.05, "end": 1280.05, "word": " الarchimedean", "probability": 0.759326171875}, {"start": 1280.05, "end": 1280.75, "word": " property", "probability": 0.9697265625}, {"start": 1280.75, "end": 1282.27, "word": " نشوف", "probability": 0.8445638020833334}, {"start": 1282.27, "end": 1282.47, "word": " مع", "probability": 0.99609375}, {"start": 1282.47, "end": 1283.31, "word": " بعض", "probability": 0.994384765625}, {"start": 1283.31, "end": 1287.67, "word": " let", "probability": 0.84521484375}], "temperature": 1.0}, {"id": 47, "seek": 130001, "start": 1289.57, "end": 1300.01, "text": "بالمناسبة .. بالمناسبة يعني احنا كيف نحدد ال capital N for any given epsilon؟", "tokens": [3555, 45340, 8315, 35457, 3660, 4386, 20666, 2304, 8315, 35457, 3660, 37495, 22653, 1975, 5016, 8315, 9122, 33911, 8717, 24401, 3215, 2423, 4238, 426, 337, 604, 2212, 17889, 22807], "avg_logprob": -0.1499999940395355, "compression_ratio": 1.118811881188119, "no_speech_prob": 0.0, "words": [{"start": 1289.57, "end": 1290.51, "word": "بالمناسبة", "probability": 0.9466796875}, {"start": 1290.51, "end": 1290.53, "word": " ..", "probability": 0.459716796875}, {"start": 1290.53, "end": 1291.21, "word": " بالمناسبة", "probability": 0.94140625}, {"start": 1291.21, "end": 1291.39, "word": " يعني", "probability": 0.787109375}, {"start": 1291.39, "end": 1292.51, "word": " احنا", "probability": 0.9357096354166666}, {"start": 1292.51, "end": 1295.31, "word": " كيف", "probability": 0.83935546875}, {"start": 1295.31, "end": 1296.89, "word": " نحدد", "probability": 0.96923828125}, {"start": 1296.89, "end": 1297.11, "word": " ال", "probability": 0.84423828125}, {"start": 1297.11, "end": 1297.47, "word": " capital", "probability": 0.50341796875}, {"start": 1297.47, "end": 1297.87, "word": " N", "probability": 0.82080078125}, {"start": 1297.87, "end": 1298.55, "word": " for", "probability": 0.9326171875}, {"start": 1298.55, "end": 1298.87, "word": " any", "probability": 0.92724609375}, {"start": 1298.87, "end": 1299.21, "word": " given", "probability": 0.90673828125}, {"start": 1299.21, "end": 1300.01, "word": " epsilon؟", "probability": 0.898681640625}], "temperature": 1.0}, {"id": 48, "seek": 133755, "start": 1308.01, "end": 1337.55, "text": "أنا يعني هي عندي absolute xn minus xm لو في عندي epsilon given epsilon موجة given فمن الآخر أنا عايز أثبت أنه هذا أصغر من إمسنان، مظبوط؟ طب ما هذا عبارة عن absolute واحد على n minus واحد على m وهذا أصغر من أو ساوي absolute واحد على n زائد absolute واحد على m", "tokens": [10721, 8315, 37495, 22653, 39896, 18871, 16254, 8236, 2031, 77, 3175, 2031, 76, 45164, 8978, 18871, 16254, 17889, 2212, 17889, 3714, 29245, 3660, 2212, 6156, 27842, 6024, 95, 34740, 41850, 6225, 47302, 11622, 5551, 12984, 3555, 2655, 14739, 3224, 23758, 5551, 9381, 17082, 2288, 9154, 11933, 2304, 3794, 1863, 7649, 12399, 3714, 19913, 3555, 2407, 9566, 22807, 23032, 3555, 19446, 23758, 6225, 3555, 9640, 3660, 18871, 8236, 36764, 24401, 15844, 297, 3175, 36764, 24401, 15844, 275, 37037, 15730, 5551, 9381, 17082, 2288, 9154, 34051, 8608, 995, 45865, 8236, 36764, 24401, 15844, 297, 30767, 16373, 3215, 8236, 36764, 24401, 15844, 275], "avg_logprob": -0.26794553983329544, "compression_ratio": 1.8701923076923077, "no_speech_prob": 0.0, "words": 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M، مظبوط؟ إذاً هذا هيصير أصغر من أو يساوي 1 على capital N وهذا أصغر من 1 على capital N بساوي 2 على capital M الآن بدي أخلي هذا، متى بيكون هذا أصغر من epsilon؟", "tokens": [28814, 15730, 25961, 2655, 297, 5551, 4117, 26890, 9154, 34051, 7251, 3794, 995, 45865, 4238, 426, 6156, 16, 15844, 297, 8032, 2655, 9381, 13546, 5551, 9381, 17082, 2288, 9154, 34051, 7251, 3794, 995, 45865, 502, 15844, 4238, 426, 4032, 4117, 8848, 23275, 20666, 1863, 35457, 3660, 5296, 376, 12399, 3714, 19913, 3555, 2407, 9566, 22807, 11933, 15730, 14111, 23758, 8032, 1829, 9381, 13546, 5551, 9381, 17082, 2288, 9154, 34051, 7251, 3794, 995, 45865, 502, 15844, 4238, 426, 37037, 15730, 5551, 9381, 17082, 2288, 9154, 502, 15844, 4238, 426, 4724, 3794, 995, 45865, 568, 15844, 4238, 376, 6024, 48506, 4724, 16254, 5551, 9778, 20292, 23758, 12399, 44650, 7578, 4724, 1829, 30544, 23758, 5551, 9381, 17082, 2288, 9154, 17889, 22807], "avg_logprob": -0.18448003625669399, "compression_ratio": 2.076530612244898, 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1446.98, "end": 1475.36, "text": "لأي عدد موجب زي هذا يوجد capital N عدد طبيعي بحيث مقلوب و أصغر من epsilon ع اتنين اذا capital N اللي بتعتمد ع ال given epsilon لازم تكون مقلوبها أصغر من epsilon ع اتنين عشان يطلع هذا أصغر من epsilon شوفتوا كيف منطلق ال capital N و ال Archimedean property طبعا تضمنلي وجود مثل هالعدد capital N", "tokens": [1211, 10721, 1829, 6225, 3215, 3215, 3714, 29245, 3555, 30767, 1829, 23758, 7251, 29245, 3215, 4238, 426, 6225, 3215, 3215, 23032, 21292, 3615, 1829, 4724, 5016, 1829, 12984, 3714, 4587, 1211, 37746, 4032, 5551, 9381, 17082, 2288, 9154, 17889, 6225, 1975, 2655, 1863, 9957, 1975, 15730, 4238, 426, 13672, 1829, 39894, 34268, 2304, 3215, 6225, 2423, 2212, 17889, 5296, 31377, 2304, 6055, 30544, 3714, 4587, 1211, 37746, 11296, 5551, 9381, 17082, 2288, 9154, 17889, 6225, 1975, 2655, 1863, 9957, 6225, 8592, 7649, 7251, 9566, 1211, 3615, 23758, 5551, 9381, 17082, 2288, 9154, 17889, 13412, 38688, 2655, 14407, 9122, 33911, 9154, 9566, 1211, 4587, 2423, 4238, 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1483.23, "end": 1483.61, "word": " بعدين", "probability": 0.6861572265625}, {"start": 1483.61, "end": 1483.97, "word": " باجي", "probability": 0.96533203125}, {"start": 1483.97, "end": 1484.73, "word": " برتبه", "probability": 0.935302734375}, {"start": 1484.73, "end": 1485.25, "word": " بقول", "probability": 0.963134765625}, {"start": 1485.25, "end": 1485.55, "word": " let", "probability": 0.96826171875}, {"start": 1485.55, "end": 1486.19, "word": " epsilon", "probability": 0.94189453125}, {"start": 1486.19, "end": 1487.47, "word": " أكبر", "probability": 0.93017578125}, {"start": 1487.47, "end": 1487.69, "word": " من", "probability": 0.99609375}, {"start": 1487.69, "end": 1488.27, "word": " السفر", "probability": 0.82275390625}, {"start": 1488.27, "end": 1488.51, "word": " be", "probability": 0.759765625}, {"start": 1488.51, "end": 1489.01, "word": " given", "probability": 0.8974609375}, {"start": 1489.01, "end": 1496.61, "word": " إذا", "probability": 0.712158203125}, {"start": 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0.91162109375}, {"start": 1518.09, "end": 1518.45, "word": " N", "probability": 0.53466796875}, {"start": 1518.45, "end": 1519.25, "word": " أصغر", "probability": 0.973388671875}, {"start": 1519.25, "end": 1519.51, "word": " من", "probability": 0.9921875}, {"start": 1519.51, "end": 1520.33, "word": " إبسلون", "probability": 0.8156494140625}, {"start": 1520.33, "end": 1520.67, "word": " على", "probability": 0.264892578125}, {"start": 1520.67, "end": 1521.11, "word": " اتنين", "probability": 0.8531494140625}, {"start": 1521.11, "end": 1522.87, "word": " إذا", "probability": 0.35784912109375}, {"start": 1522.87, "end": 1523.11, "word": " هان", "probability": 0.4383544921875}, {"start": 1523.11, "end": 1523.71, "word": " أثبتت", "probability": 0.88349609375}, {"start": 1523.71, "end": 1524.13, "word": " يوجد", "probability": 0.9814453125}, {"start": 1524.13, "end": 1524.57, "word": " capital", "probability": 0.75146484375}, {"start": 1524.57, "end": 1524.91, "word": " N", "probability": 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1549.02, "word": " لو", "probability": 0.86083984375}, {"start": 1549.02, "end": 1549.54, "word": " أخدت", "probability": 0.9141845703125}, {"start": 1549.54, "end": 1549.88, "word": " N", "probability": 0.4189453125}, {"start": 1549.88, "end": 1550.14, "word": " و", "probability": 0.9599609375}, {"start": 1550.14, "end": 1550.6, "word": " M", "probability": 0.81640625}, {"start": 1550.6, "end": 1552.22, "word": " أكبر", "probability": 0.94775390625}, {"start": 1552.22, "end": 1552.4, "word": " من", "probability": 0.95166015625}, {"start": 1552.4, "end": 1552.92, "word": " أوسع", "probability": 0.7867838541666666}, {"start": 1552.92, "end": 1553.04, "word": " ال", "probability": 0.4013671875}, {"start": 1553.04, "end": 1553.42, "word": " capital", "probability": 0.638671875}, {"start": 1553.42, "end": 1553.78, "word": " N", "probability": 0.93896484375}, {"start": 1553.78, "end": 1554.28, "word": " هذه", "probability": 0.537109375}, {"start": 1554.28, "end": 1556.48, "word": " فبالتأكيد", "probability": 0.9709123883928571}, {"start": 1556.48, "end": 1556.82, "word": " هذا", "probability": 0.85009765625}, {"start": 1556.82, "end": 1557.62, "word": " هيقدر", "probability": 0.8277994791666666}, {"start": 1557.62, "end": 1557.96, "word": " ال", "probability": 0.6142578125}, {"start": 1557.96, "end": 1558.42, "word": " 1", "probability": 0.6083984375}, {"start": 1558.42, "end": 1558.8, "word": " على", "probability": 0.6826171875}, {"start": 1558.8, "end": 1559.2, "word": " N", "probability": 0.94091796875}, {"start": 1559.2, "end": 1560.88, "word": " و", "probability": 0.9599609375}, {"start": 1560.88, "end": 1561.66, "word": " كذلك", "probability": 0.8863932291666666}], "temperature": 1.0}, {"id": 57, "seek": 158814, "start": 1563.4, "end": 1588.14, "text": "واحد على M كلهما أصغر من أو ساوي واحد على capital N وهذا بدوره بيقدي أنه absolute واحد على M minus واحد على M طبعا هذه XM وهذه XM فشوفنا أن هذا أصغر من أو ساوي absolute واحد على M", "tokens": [14407, 24401, 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"end": 1572.24, "word": " واحد", "probability": 0.967041015625}, {"start": 1572.24, "end": 1572.48, "word": " على", "probability": 0.90869140625}, {"start": 1572.48, "end": 1572.88, "word": " M", "probability": 0.49560546875}, {"start": 1572.88, "end": 1574.16, "word": " minus", "probability": 0.72314453125}, {"start": 1574.16, "end": 1574.94, "word": " واحد", "probability": 0.965087890625}, {"start": 1574.94, "end": 1575.12, "word": " على", "probability": 0.90380859375}, {"start": 1575.12, "end": 1575.58, "word": " M", "probability": 0.826171875}, {"start": 1575.58, "end": 1577.1, "word": " طبعا", "probability": 0.9405517578125}, {"start": 1577.1, "end": 1577.42, "word": " هذه", "probability": 0.393798828125}, {"start": 1577.42, "end": 1578.26, "word": " XM", "probability": 0.625244140625}, {"start": 1578.26, "end": 1580.58, "word": " وهذه", "probability": 0.875732421875}, {"start": 1580.58, "end": 1582.7, "word": " XM", "probability": 0.943359375}, {"start": 1582.7, "end": 1584.58, 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absolute سالب واحد على M اللي هو absolute واحد على M طيب هذا بساوي واحد على M زائد واحد على M لإن رد عداد موجبة وقول إن هذا أصغر من أو يساوي واحد على M", "tokens": [3555, 995, 14851, 9778, 3215, 10943, 2423, 13369, 16970, 30767, 16373, 3215, 8236, 8608, 6027, 3555, 36764, 24401, 15844, 376, 13672, 1829, 31439, 8236, 36764, 24401, 15844, 376, 23032, 1829, 3555, 23758, 4724, 3794, 995, 45865, 36764, 24401, 15844, 376, 30767, 16373, 3215, 36764, 24401, 15844, 376, 5296, 28814, 1863, 12602, 3215, 6225, 3215, 18513, 3714, 29245, 49401, 4032, 39648, 36145, 23758, 5551, 9381, 17082, 2288, 9154, 34051, 7251, 3794, 995, 45865, 36764, 24401, 15844, 376], "avg_logprob": -0.22240260204711518, "compression_ratio": 1.7529411764705882, "no_speech_prob": 0.0, "words": [{"start": 1588.8, "end": 1589.46, "word": "باستخدام", "probability": 0.927734375}, {"start": 1589.46, "end": 1589.6, "word": " ال", "probability": 0.83056640625}, {"start": 1589.6, "end": 1589.94, "word": " triangle", "probability": 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0.904296875}, {"start": 1611.74, "end": 1612.14, "word": " M", "probability": 0.990234375}], "temperature": 1.0}, {"id": 59, "seek": 164063, "start": 1612.91, "end": 1640.63, "text": "زايد واحد على N وهذا بيساوي اتنين على N وهذا من الاختيار تبعنا ل capital N by star اتنين على N أصغر من epsilon طب ما هذه .. هذا هو شرط Koshi صح؟ هذا هو شرط Koshi", "tokens": [11622, 995, 25708, 36764, 24401, 15844, 426, 37037, 15730, 4724, 1829, 3794, 995, 45865, 1975, 2655, 1863, 9957, 15844, 426, 37037, 15730, 9154, 2423, 47283, 31371, 9640, 6055, 3555, 3615, 8315, 5296, 4238, 426, 538, 3543, 1975, 2655, 1863, 9957, 15844, 426, 5551, 9381, 17082, 2288, 9154, 17889, 23032, 3555, 19446, 29538, 4386, 23758, 31439, 13412, 2288, 9566, 591, 17392, 20328, 5016, 22807, 23758, 31439, 13412, 2288, 9566, 591, 17392], "avg_logprob": -0.21698943745922034, "compression_ratio": 1.5625, "no_speech_prob": 0.0, "words": [{"start": 1612.91, "end": 1613.57, "word": "زايد", "probability": 0.7544759114583334}, {"start": 1613.57, "end": 1614.01, "word": " واحد", "probability": 0.935546875}, {"start": 1614.01, "end": 1614.23, "word": " على", "probability": 0.77197265625}, {"start": 1614.23, "end": 1614.55, "word": " N", "probability": 0.50732421875}, {"start": 1614.55, "end": 1615.19, "word": " وهذا", "probability": 0.758056640625}, {"start": 1615.19, "end": 1615.83, "word": " بيساوي", "probability": 0.89833984375}, {"start": 1615.83, "end": 1616.41, "word": " اتنين", "probability": 0.980712890625}, {"start": 1616.41, "end": 1616.65, "word": " على", "probability": 0.90869140625}, {"start": 1616.65, "end": 1617.01, "word": " N", "probability": 0.95068359375}, {"start": 1617.01, "end": 1618.39, "word": " وهذا", "probability": 0.887939453125}, {"start": 1618.39, "end": 1618.85, "word": " من", "probability": 0.9931640625}, {"start": 1618.85, "end": 1620.95, "word": " الاختيار", "probability": 0.83734130859375}, {"start": 1620.95, "end": 1621.99, "word": " تبعنا", "probability": 0.91064453125}, {"start": 1621.99, "end": 1623.19, "word": " ل", "probability": 0.97216796875}, {"start": 1623.19, "end": 1623.63, "word": " capital", "probability": 0.49365234375}, {"start": 1623.63, "end": 1624.07, "word": " N", "probability": 0.71435546875}, {"start": 1624.07, "end": 1625.51, "word": " by", "probability": 0.5576171875}, {"start": 1625.51, "end": 1626.19, "word": " star", "probability": 0.8212890625}, {"start": 1626.19, "end": 1628.09, "word": " اتنين", "probability": 0.964599609375}, {"start": 1628.09, "end": 1628.31, "word": " على", "probability": 0.92236328125}, {"start": 1628.31, "end": 1628.85, "word": " N", "probability": 0.947265625}, {"start": 1628.85, "end": 1630.63, "word": " أصغر", "probability": 0.8719482421875}, {"start": 1630.63, "end": 1631.17, "word": " من", "probability": 0.994140625}, {"start": 1631.17, "end": 1632.61, "word": " epsilon", "probability": 0.50439453125}, {"start": 1632.61, "end": 1635.99, "word": " طب", "probability": 0.836181640625}, {"start": 1635.99, "end": 1636.15, "word": " ما", "probability": 0.65771484375}, {"start": 1636.15, "end": 1636.41, "word": " هذه", "probability": 0.259521484375}, {"start": 1636.41, "end": 1636.49, "word": " ..", "probability": 0.63330078125}, {"start": 1636.49, "end": 1636.67, "word": " هذا", "probability": 0.55419921875}, {"start": 1636.67, "end": 1636.93, "word": " هو", "probability": 0.97265625}, {"start": 1636.93, "end": 1637.27, "word": " شرط", "probability": 0.9108072916666666}, {"start": 1637.27, "end": 1637.65, "word": " Koshi", "probability": 0.45172119140625}, {"start": 1637.65, "end": 1638.85, "word": " صح؟", "probability": 0.8766276041666666}, {"start": 1638.85, "end": 1639.15, "word": " هذا", "probability": 0.90673828125}, {"start": 1639.15, "end": 1639.41, "word": " هو", "probability": 0.99267578125}, {"start": 1639.41, "end": 1639.89, "word": " شرط", "probability": 0.9853515625}, {"start": 1639.89, "end": 1640.63, "word": " Koshi", "probability": 0.927490234375}], "temperature": 1.0}, {"id": 60, "seek": 167131, "start": 1642.37, "end": 1671.31, "text": "إذا by definition بما أن هذا صحيح لكل epsilon since epsilon أكبر من الصفر was arbitrary by definition of Cauchy sequence ال sequence xn is اللي هي واحد على n اللي الحد العام تبعها واحد على n", "tokens": [28814, 15730, 538, 7123, 4724, 15042, 14739, 23758, 20328, 5016, 1829, 5016, 5296, 28820, 17889, 1670, 17889, 5551, 4117, 26890, 9154, 31767, 5172, 2288, 390, 23211, 538, 7123, 295, 7544, 625, 88, 8310, 2423, 8310, 2031, 77, 307, 13672, 1829, 39896, 36764, 24401, 15844, 297, 13672, 1829, 21542, 3215, 18863, 10943, 6055, 3555, 3615, 11296, 36764, 24401, 15844, 297], "avg_logprob": -0.28020832389593125, "compression_ratio": 1.505813953488372, "no_speech_prob": 0.0, "words": [{"start": 1642.37, "end": 1642.83, "word": "إذا", "probability": 0.5155029296875}, {"start": 1642.83, "end": 1643.45, "word": " by", "probability": 0.61083984375}, {"start": 1643.45, "end": 1644.11, "word": " definition", "probability": 0.93896484375}, {"start": 1644.11, "end": 1645.87, "word": " بما", "probability": 0.5791015625}, {"start": 1645.87, "end": 1646.03, "word": " أن", "probability": 0.6689453125}, {"start": 1646.03, "end": 1646.35, "word": " هذا", "probability": 0.86376953125}, {"start": 1646.35, "end": 1646.91, "word": " صحيح", "probability": 0.9967041015625}, {"start": 1646.91, "end": 1647.33, "word": " لكل", "probability": 0.982666015625}, {"start": 1647.33, "end": 1647.75, "word": " epsilon", "probability": 0.29248046875}, {"start": 1647.75, "end": 1648.31, "word": " since", "probability": 0.55712890625}, {"start": 1648.31, "end": 1651.25, "word": " epsilon", "probability": 0.65625}, {"start": 1651.25, "end": 1651.77, "word": " أكبر", "probability": 0.94287109375}, {"start": 1651.77, "end": 1652.03, "word": " من", "probability": 0.99365234375}, {"start": 1652.03, "end": 1652.63, "word": " الصفر", "probability": 0.8313802083333334}, {"start": 1652.63, "end": 1653.25, "word": " was", "probability": 0.88037109375}, {"start": 1653.25, "end": 1654.07, "word": " arbitrary", "probability": 0.83740234375}, {"start": 1654.07, "end": 1659.99, "word": " by", "probability": 0.89111328125}, {"start": 1659.99, "end": 1660.67, "word": " definition", "probability": 0.87890625}, {"start": 1660.67, "end": 1661.17, "word": " of", "probability": 0.9697265625}, {"start": 1661.17, "end": 1661.59, "word": " Cauchy", "probability": 0.869140625}, {"start": 1661.59, "end": 1662.25, "word": " sequence", "probability": 0.8857421875}, {"start": 1662.25, "end": 1663.51, "word": " ال", "probability": 0.491455078125}, {"start": 1663.51, "end": 1663.99, "word": " sequence", "probability": 0.90087890625}, {"start": 1663.99, "end": 1664.99, "word": " xn", "probability": 0.519775390625}, {"start": 1664.99, "end": 1665.83, "word": " is", "probability": 0.362060546875}, {"start": 1665.83, "end": 1666.05, "word": " اللي", "probability": 0.68994140625}, {"start": 1666.05, "end": 1666.31, "word": " هي", "probability": 0.921875}, {"start": 1666.31, "end": 1666.75, "word": " واحد", "probability": 0.963134765625}, {"start": 1666.75, "end": 1666.95, "word": " على", "probability": 0.70849609375}, {"start": 1666.95, "end": 1667.31, "word": " n", "probability": 0.415283203125}, {"start": 1667.31, "end": 1668.91, "word": " اللي", "probability": 0.833251953125}, {"start": 1668.91, "end": 1669.33, "word": " الحد", "probability": 0.9091796875}, {"start": 1669.33, "end": 1669.63, "word": " العام", "probability": 0.6649169921875}, {"start": 1669.63, "end": 1670.29, "word": " تبعها", "probability": 0.8441162109375}, {"start": 1670.29, "end": 1670.83, "word": " واحد", "probability": 0.96044921875}, {"start": 1670.83, "end": 1670.99, "word": " على", "probability": 0.796875}, {"start": 1670.99, "end": 1671.31, "word": " n", "probability": 0.611328125}], "temperature": 1.0}, {"id": 61, "seek": 169901, "start": 1672.61, "end": 1699.01, "text": "is Cauchy تمام هنا أثبتنا إن ال sequence Cauchy مباشرة باستخدام التعريف طبعا في برهان تاني ممكن نستخدم Cauchy criterion احنا ممكن نثبت إن ال sequence هذي convergent", "tokens": [271, 7544, 625, 88, 46811, 10943, 34105, 5551, 12984, 3555, 2655, 8315, 36145, 2423, 8310, 7544, 625, 88, 3714, 3555, 33599, 25720, 4724, 995, 14851, 9778, 3215, 10943, 16712, 3615, 16572, 5172, 23032, 3555, 3615, 995, 8978, 4724, 2288, 3224, 7649, 6055, 7649, 1829, 3714, 43020, 8717, 14851, 9778, 40448, 7544, 625, 88, 46691, 1975, 5016, 8315, 3714, 43020, 8717, 12984, 3555, 2655, 36145, 2423, 8310, 8032, 8848, 1829, 9652, 6930], "avg_logprob": -0.15896267050670254, "compression_ratio": 1.4819277108433735, "no_speech_prob": 0.0, "words": [{"start": 1672.61, "end": 1673.17, "word": "is", "probability": 0.44580078125}, {"start": 1673.17, "end": 1673.85, "word": " Cauchy", "probability": 0.7613932291666666}, {"start": 1673.85, "end": 1677.61, "word": " تمام", "probability": 0.6146240234375}, {"start": 1677.61, "end": 1680.03, "word": " هنا", "probability": 0.348876953125}, {"start": 1680.03, "end": 1680.61, "word": " أثبتنا", "probability": 0.97685546875}, {"start": 1680.61, "end": 1680.85, "word": " إن", "probability": 0.458740234375}, {"start": 1680.85, "end": 1680.97, "word": " ال", "probability": 0.68359375}, {"start": 1680.97, "end": 1681.29, "word": " sequence", "probability": 0.603515625}, {"start": 1681.29, "end": 1682.05, "word": " Cauchy", "probability": 0.8702799479166666}, {"start": 1682.05, "end": 1683.41, "word": " مباشرة", "probability": 0.974853515625}, {"start": 1683.41, "end": 1684.23, "word": " باستخدام", "probability": 0.976318359375}, {"start": 1684.23, "end": 1684.97, "word": " التعريف", "probability": 0.97802734375}, {"start": 1684.97, "end": 1687.01, "word": " طبعا", "probability": 0.972900390625}, {"start": 1687.01, "end": 1687.47, "word": " في", "probability": 0.515625}, {"start": 1687.47, "end": 1688.13, "word": " برهان", "probability": 0.95458984375}, {"start": 1688.13, "end": 1688.91, "word": " تاني", "probability": 0.9759114583333334}, {"start": 1688.91, "end": 1690.91, "word": " ممكن", "probability": 0.9716796875}, {"start": 1690.91, "end": 1691.53, "word": " نستخدم", "probability": 0.9906005859375}, {"start": 1691.53, "end": 1692.05, "word": " Cauchy", "probability": 0.9373372395833334}, {"start": 1692.05, "end": 1692.79, "word": " criterion", "probability": 0.91357421875}, {"start": 1692.79, "end": 1695.79, "word": " احنا", "probability": 0.8611653645833334}, {"start": 1695.79, "end": 1696.45, "word": " ممكن", "probability": 0.992919921875}, {"start": 1696.45, "end": 1697.07, "word": " نثبت", "probability": 0.99658203125}, {"start": 1697.07, "end": 1697.29, "word": " إن", "probability": 0.732421875}, {"start": 1697.29, "end": 1697.43, "word": " ال", "probability": 0.96630859375}, {"start": 1697.43, "end": 1697.87, "word": " sequence", "probability": 0.96923828125}, {"start": 1697.87, "end": 1698.29, "word": " هذي", "probability": 0.6671549479166666}, {"start": 1698.29, "end": 1699.01, "word": " convergent", "probability": 0.73681640625}], "temperature": 1.0}, {"id": 62, "seek": 172640, "start": 1699.88, "end": 1726.4, "text": "و أثبتنا هذا الكلام جبليك صح؟ و حسب cushy criterion بما أنه ال sequence convergent then it is cushy صح؟ هذا برهان تاني لكن إذا كنا لكم برهنيها directly يعني استخدم التعريف لازم البرهان هذا هو اللي إيه تكتبوه واضح تمام؟ في أي استفسار؟", "tokens": [2407, 5551, 12984, 3555, 2655, 8315, 23758, 2423, 28820, 10943, 10874, 3555, 20292, 4117, 20328, 5016, 22807, 4032, 11331, 35457, 18422, 88, 46691, 4724, 15042, 14739, 3224, 2423, 8310, 9652, 6930, 550, 309, 307, 18422, 88, 20328, 5016, 22807, 23758, 4724, 2288, 3224, 7649, 6055, 7649, 1829, 44381, 11933, 15730, 9122, 8315, 5296, 24793, 4724, 2288, 3224, 22653, 11296, 3838, 37495, 22653, 44713, 9778, 40448, 16712, 3615, 16572, 5172, 5296, 31377, 2304, 2423, 26890, 3224, 7649, 23758, 31439, 13672, 1829, 11933, 1829, 3224, 6055, 4117, 2655, 3555, 2407, 3224, 4032, 46958, 5016, 46811, 10943, 22807, 8978, 36632, 44713, 36178, 9640, 22807], "avg_logprob": -0.19071691118034662, "compression_ratio": 1.6017316017316017, "no_speech_prob": 0.0, "words": [{"start": 1699.88, "end": 1700.14, "word": "و", "probability": 0.984375}, {"start": 1700.14, "end": 1700.6, "word": " أثبتنا", "probability": 0.93330078125}, {"start": 1700.6, "end": 1700.8, "word": " هذا", "probability": 0.9775390625}, {"start": 1700.8, "end": 1701.16, "word": " الكلام", "probability": 0.97216796875}, {"start": 1701.16, "end": 1701.82, "word": " جبليك", "probability": 0.52154541015625}, {"start": 1701.82, "end": 1702.82, "word": " صح؟", "probability": 0.7996419270833334}, {"start": 1702.82, "end": 1703.92, "word": " و", "probability": 0.9345703125}, {"start": 1703.92, "end": 1704.62, "word": " حسب", "probability": 0.843505859375}, {"start": 1704.62, "end": 1705.06, "word": " cushy", "probability": 0.552642822265625}, {"start": 1705.06, "end": 1705.74, "word": " criterion", "probability": 0.9873046875}, {"start": 1705.74, "end": 1707.16, "word": " بما", "probability": 0.8740234375}, {"start": 1707.16, "end": 1707.4, "word": " أنه", "probability": 0.69580078125}, {"start": 1707.4, "end": 1707.46, "word": " ال", "probability": 0.451904296875}, {"start": 1707.46, "end": 1707.72, "word": " sequence", "probability": 0.97607421875}, {"start": 1707.72, "end": 1708.64, "word": " convergent", "probability": 0.79541015625}, {"start": 1708.64, "end": 1709.02, "word": " then", "probability": 0.65576171875}, {"start": 1709.02, "end": 1709.32, "word": " it", "probability": 0.97314453125}, {"start": 1709.32, "end": 1709.62, "word": " is", "probability": 0.9599609375}, {"start": 1709.62, "end": 1710.16, "word": " cushy", "probability": 0.801513671875}, {"start": 1710.16, "end": 1711.16, "word": " صح؟", "probability": 0.9236653645833334}, {"start": 1711.16, "end": 1711.4, "word": " هذا", "probability": 0.8876953125}, {"start": 1711.4, "end": 1711.8, "word": " برهان", "probability": 0.9493408203125}, {"start": 1711.8, "end": 1712.16, "word": " تاني", "probability": 0.9658203125}, {"start": 1712.16, "end": 1712.4, "word": " لكن", "probability": 0.55029296875}, {"start": 1712.4, "end": 1712.56, "word": " إذا", "probability": 0.859375}, {"start": 1712.56, "end": 1712.76, "word": " كنا", "probability": 0.607666015625}, {"start": 1712.76, "end": 1713.1, "word": " لكم", "probability": 0.741455078125}, {"start": 1713.1, "end": 1714.74, "word": " برهنيها", "probability": 0.799951171875}, {"start": 1714.74, "end": 1715.34, "word": " directly", "probability": 0.9697265625}, {"start": 1715.34, "end": 1715.72, "word": " يعني", "probability": 0.966064453125}, {"start": 1715.72, "end": 1716.16, "word": " استخدم", "probability": 0.92333984375}, {"start": 1716.16, "end": 1716.92, "word": " التعريف", "probability": 0.9747314453125}, {"start": 1716.92, "end": 1718.26, "word": " لازم", "probability": 0.9679361979166666}, {"start": 1718.26, "end": 1718.8, "word": " البرهان", "probability": 0.86651611328125}, {"start": 1718.8, "end": 1719.16, "word": " هذا", "probability": 0.91943359375}, {"start": 1719.16, "end": 1719.42, "word": " هو", "probability": 0.99609375}, {"start": 1719.42, "end": 1719.7, "word": " اللي", "probability": 0.9814453125}, {"start": 1719.7, "end": 1720.46, "word": " إيه", "probability": 0.8179524739583334}, {"start": 1720.46, "end": 1721.42, "word": " تكتبوه", "probability": 0.976806640625}, {"start": 1721.42, "end": 1722.16, "word": " واضح", "probability": 0.9664713541666666}, {"start": 1722.16, "end": 1723.9, "word": " تمام؟", "probability": 0.789794921875}, {"start": 1723.9, "end": 1725.46, "word": " في", "probability": 0.94091796875}, {"start": 1725.46, "end": 1725.6, "word": " أي", "probability": 0.8017578125}, {"start": 1725.6, "end": 1726.4, "word": " استفسار؟", "probability": 0.81695556640625}], "temperature": 1.0}, {"id": 63, "seek": 173170, "start": 1730.06, "end": 1731.7, "text": "ناخد مثال تاني", "tokens": [1863, 47283, 3215, 50113, 6027, 6055, 7649, 1829], "avg_logprob": -0.0759548611111111, "compression_ratio": 0.7647058823529411, "no_speech_prob": 0.0, "words": [{"start": 1730.06, "end": 1730.64, "word": "ناخد", "probability": 0.96728515625}, {"start": 1730.64, "end": 1731.1, "word": " مثال", "probability": 0.97021484375}, {"start": 1731.1, "end": 1731.7, "word": " تاني", "probability": 0.8798828125}], "temperature": 1.0}, {"id": 64, "seek": 177871, "start": 1758.73, "end": 1778.71, "text": "مثال تقم اتنين consider .. consider ال sequence defined inductively", "tokens": [2304, 12984, 6027, 6055, 4587, 2304, 1975, 2655, 1863, 9957, 1949, 4386, 1949, 2423, 8310, 7642, 31612, 3413], "avg_logprob": -0.3055098746952258, "compression_ratio": 1.0384615384615385, "no_speech_prob": 0.0, "words": [{"start": 1758.73, "end": 1759.59, "word": "مثال", "probability": 0.7496744791666666}, {"start": 1759.59, "end": 1760.99, "word": " تقم", "probability": 0.6710611979166666}, {"start": 1760.99, "end": 1761.55, "word": " اتنين", "probability": 0.8878173828125}, {"start": 1761.55, "end": 1764.09, "word": " consider", "probability": 0.73388671875}, {"start": 1764.09, "end": 1764.45, "word": " ..", "probability": 0.271484375}, {"start": 1764.45, "end": 1767.75, "word": " consider", "probability": 0.9033203125}, {"start": 1767.75, "end": 1768.53, "word": " ال", "probability": 0.88720703125}, {"start": 1768.53, "end": 1769.15, "word": " sequence", "probability": 0.8642578125}, {"start": 1769.15, "end": 1776.37, "word": " defined", "probability": 0.92138671875}, {"start": 1776.37, "end": 1778.71, "word": " inductively", "probability": 0.880859375}], "temperature": 1.0}, {"id": 65, "seek": 181525, "start": 1788.75, "end": 1815.25, "text": "إذا في عندي sequence معرفة بطريقة استقرائية كالتالي كما هي ليه هناخد x1 بساوي واحد و x2 بساوي اتنين طب و xn-n أكبر من أو ساوي تلاتة هناخده بساوي نص في", "tokens": [28814, 15730, 8978, 18871, 16254, 8310, 20449, 28480, 3660, 4724, 9566, 16572, 28671, 44713, 4587, 2288, 16373, 10632, 9122, 6027, 2655, 6027, 1829, 9122, 15042, 39896, 32239, 3224, 8032, 1863, 47283, 3215, 2031, 16, 4724, 3794, 995, 45865, 36764, 24401, 4032, 2031, 17, 4724, 3794, 995, 45865, 1975, 2655, 1863, 9957, 23032, 3555, 4032, 2031, 77, 12, 77, 5551, 4117, 26890, 9154, 34051, 8608, 995, 45865, 6055, 1211, 9307, 3660, 8032, 1863, 47283, 3215, 3224, 4724, 3794, 995, 45865, 8717, 9381, 8978], "avg_logprob": -0.1864646109471838, "compression_ratio": 1.548780487804878, "no_speech_prob": 0.0, "words": [{"start": 1788.75, "end": 1789.11, "word": "إذا", "probability": 0.4149169921875}, {"start": 1789.11, "end": 1789.69, "word": " في", "probability": 0.3603515625}, {"start": 1789.69, "end": 1790.01, "word": " عندي", "probability": 0.7391357421875}, {"start": 1790.01, "end": 1790.49, "word": " sequence", "probability": 0.7763671875}, {"start": 1790.49, "end": 1791.23, "word": " معرفة", "probability": 0.900390625}, {"start": 1791.23, "end": 1791.87, "word": " بطريقة", "probability": 0.9862060546875}, {"start": 1791.87, "end": 1792.77, "word": " استقرائية", "probability": 0.97099609375}, {"start": 1792.77, "end": 1793.55, "word": " كالتالي", "probability": 0.8935546875}, {"start": 1793.55, "end": 1794.21, "word": " كما", "probability": 0.853759765625}, {"start": 1794.21, "end": 1794.33, "word": " هي", "probability": 0.5263671875}, {"start": 1794.33, "end": 1794.91, "word": " ليه", "probability": 0.883544921875}, {"start": 1794.91, "end": 1796.39, "word": " هناخد", "probability": 0.8155517578125}, {"start": 1796.39, "end": 1797.01, "word": " x1", "probability": 0.629638671875}, {"start": 1797.01, "end": 1797.93, "word": " بساوي", "probability": 0.7864990234375}, {"start": 1797.93, "end": 1798.53, "word": " واحد", "probability": 0.927734375}, {"start": 1798.53, "end": 1800.81, "word": " و", "probability": 0.89990234375}, {"start": 1800.81, "end": 1801.75, "word": " x2", "probability": 0.8232421875}, {"start": 1801.75, "end": 1803.07, "word": " بساوي", "probability": 0.9493408203125}, {"start": 1803.07, "end": 1804.71, "word": " اتنين", "probability": 0.9737548828125}, {"start": 1804.71, "end": 1805.19, "word": " طب", "probability": 0.895263671875}, {"start": 1805.19, "end": 1805.31, "word": " و", "probability": 0.6015625}, {"start": 1805.31, "end": 1806.13, "word": " xn", "probability": 0.90283203125}, {"start": 1806.13, "end": 1808.47, "word": "-n", "probability": 0.4296875}, {"start": 1808.47, "end": 1808.95, "word": " أكبر", "probability": 0.9265950520833334}, {"start": 1808.95, "end": 1809.15, "word": " من", "probability": 0.95703125}, {"start": 1809.15, "end": 1809.33, "word": " أو", "probability": 0.548828125}, {"start": 1809.33, "end": 1809.69, "word": " ساوي", "probability": 0.8251953125}, {"start": 1809.69, "end": 1810.39, "word": " تلاتة", "probability": 0.944091796875}, {"start": 1810.39, "end": 1812.01, "word": " هناخده", "probability": 0.944140625}, {"start": 1812.01, "end": 1812.71, "word": " بساوي", "probability": 0.9678955078125}, {"start": 1812.71, "end": 1813.25, "word": " نص", "probability": 0.9921875}, {"start": 1813.25, "end": 1815.25, "word": " في", "probability": 0.93505859375}], "temperature": 1.0}, {"id": 66, "seek": 184450, "start": 1815.68, "end": 1844.5, "text": "xn سالب اتنين زائد xn negative one طبعا هذا لكل n أدب طبيعي أكبر من أو ساوى تلاتة اذا هنا في اندي سيكوانس معرفة بطريقة استقرائية اول حدين اللي هم قيم معينة الحد التالت وانت طالع معرف بدلالة الحد اللي حدين اللي جابله مباشرة", "tokens": [87, 77, 8608, 6027, 3555, 1975, 2655, 1863, 9957, 30767, 16373, 3215, 2031, 77, 3671, 472, 23032, 3555, 3615, 995, 23758, 5296, 28820, 297, 5551, 3215, 3555, 23032, 21292, 3615, 1829, 5551, 4117, 26890, 9154, 34051, 8608, 995, 2407, 7578, 6055, 1211, 9307, 3660, 1975, 15730, 34105, 8978, 16472, 16254, 8608, 1829, 4117, 2407, 7649, 3794, 20449, 28480, 3660, 4724, 9566, 16572, 28671, 44713, 4587, 2288, 16373, 10632, 1975, 12610, 11331, 3215, 9957, 13672, 1829, 8032, 2304, 12174, 32640, 20449, 9957, 3660, 21542, 3215, 16712, 6027, 2655, 4032, 7649, 2655, 23032, 6027, 3615, 20449, 28480, 4724, 3215, 1211, 6027, 3660, 21542, 3215, 13672, 1829, 11331, 3215, 9957, 13672, 1829, 10874, 16758, 43761, 3714, 3555, 33599, 25720], "avg_logprob": -0.23424144891592172, "compression_ratio": 1.742081447963801, "no_speech_prob": 5.960464477539063e-08, "words": [{"start": 1815.68, "end": 1817.26, "word": "xn", "probability": 0.3876953125}, {"start": 1817.26, "end": 1818.02, "word": " سالب", "probability": 0.5388081868489584}, {"start": 1818.02, "end": 1818.82, "word": " اتنين", "probability": 0.9366455078125}, {"start": 1818.82, "end": 1820.36, "word": " زائد", "probability": 0.5478922526041666}, {"start": 1820.36, "end": 1821.48, "word": " xn", "probability": 0.8603515625}, {"start": 1821.48, "end": 1822.02, "word": " negative", "probability": 0.7353515625}, {"start": 1822.02, "end": 1822.6, "word": " one", "probability": 0.95703125}, {"start": 1822.6, "end": 1824.1, "word": " طبعا", "probability": 0.85040283203125}, {"start": 1824.1, "end": 1824.34, "word": " هذا", "probability": 0.62158203125}, {"start": 1824.34, "end": 1825.08, "word": " لكل", "probability": 0.960693359375}, {"start": 1825.08, "end": 1825.48, "word": " n", "probability": 0.8125}, {"start": 1825.48, "end": 1826.4, "word": " أدب", "probability": 0.6211751302083334}, {"start": 1826.4, "end": 1826.92, "word": " طبيعي", "probability": 0.947265625}, {"start": 1826.92, "end": 1827.46, "word": " أكبر", "probability": 0.88623046875}, {"start": 1827.46, "end": 1827.7, "word": " من", "probability": 0.962890625}, {"start": 1827.7, "end": 1827.96, "word": " أو", "probability": 0.88623046875}, {"start": 1827.96, "end": 1828.38, "word": " ساوى", "probability": 0.69012451171875}, {"start": 1828.38, "end": 1828.94, "word": " تلاتة", "probability": 0.9215087890625}, {"start": 1828.94, "end": 1830.24, "word": " اذا", "probability": 0.477569580078125}, {"start": 1830.24, "end": 1830.52, "word": " هنا", "probability": 0.966796875}, {"start": 1830.52, "end": 1830.74, "word": " في", "probability": 0.55810546875}, {"start": 1830.74, "end": 1831.0, "word": " اندي", "probability": 0.4024658203125}, {"start": 1831.0, "end": 1831.58, "word": " سيكوانس", "probability": 0.827392578125}, {"start": 1831.58, "end": 1832.18, "word": " معرفة", "probability": 0.8761393229166666}, {"start": 1832.18, "end": 1832.66, "word": " بطريقة", "probability": 0.977783203125}, {"start": 1832.66, "end": 1833.58, "word": " استقرائية", "probability": 0.95859375}, {"start": 1833.58, "end": 1834.5, "word": " اول", "probability": 0.771240234375}, {"start": 1834.5, "end": 1834.98, "word": " حدين", "probability": 0.9518229166666666}, {"start": 1834.98, "end": 1835.16, "word": " اللي", "probability": 0.71337890625}, {"start": 1835.16, "end": 1835.32, "word": " هم", "probability": 0.6529541015625}, {"start": 1835.32, "end": 1835.66, "word": " قيم", "probability": 0.97314453125}, {"start": 1835.66, "end": 1836.5, "word": " معينة", "probability": 0.9905598958333334}, {"start": 1836.5, "end": 1837.38, "word": " الحد", "probability": 0.902099609375}, {"start": 1837.38, "end": 1837.92, "word": " التالت", "probability": 0.9871419270833334}, {"start": 1837.92, "end": 1838.2, "word": " وانت", "probability": 0.8356119791666666}, {"start": 1838.2, "end": 1838.66, "word": " طالع", "probability": 0.9158528645833334}, {"start": 1838.66, "end": 1840.02, "word": " معرف", "probability": 0.94091796875}, {"start": 1840.02, "end": 1841.2, "word": " بدلالة", "probability": 0.80205078125}, {"start": 1841.2, "end": 1841.78, "word": " الحد", "probability": 0.956787109375}, {"start": 1841.78, "end": 1842.74, "word": " اللي", "probability": 0.665771484375}, {"start": 1842.74, "end": 1843.34, "word": " حدين", "probability": 0.8517252604166666}, {"start": 1843.34, "end": 1843.54, "word": " اللي", "probability": 0.98095703125}, {"start": 1843.54, "end": 1843.9, "word": " جابله", "probability": 0.8738606770833334}, {"start": 1843.9, "end": 1844.5, "word": " مباشرة", "probability": 0.9923095703125}], "temperature": 1.0}, {"id": 67, "seek": 187222, "start": 1846.38, "end": 1872.22, "text": "هذا طبعا بيعطينا sequence المطلوب عايزين نثبت show ان ال sequence x in is convergent و converges to the number 5 over 3 البرهان", "tokens": [3224, 15730, 23032, 3555, 3615, 995, 4724, 40228, 9566, 1829, 8315, 8310, 9673, 9566, 1211, 37746, 6225, 47302, 11622, 9957, 8717, 12984, 3555, 2655, 855, 16472, 2423, 8310, 2031, 294, 307, 9652, 6930, 4032, 9652, 2880, 281, 264, 1230, 1025, 670, 805, 2423, 26890, 3224, 7649], "avg_logprob": -0.185505322319396, "compression_ratio": 1.297709923664122, "no_speech_prob": 0.0, "words": [{"start": 1846.38, "end": 1846.72, "word": "هذا", "probability": 0.82470703125}, {"start": 1846.72, "end": 1847.0, "word": " طبعا", "probability": 0.927490234375}, {"start": 1847.0, "end": 1847.52, "word": " بيعطينا", "probability": 0.80322265625}, {"start": 1847.52, "end": 1847.86, "word": " sequence", "probability": 0.487548828125}, {"start": 1847.86, "end": 1848.76, "word": " المطلوب", "probability": 0.9652099609375}, {"start": 1848.76, "end": 1849.78, "word": " عايزين", "probability": 0.8895263671875}, {"start": 1849.78, "end": 1850.34, "word": " نثبت", "probability": 0.9884033203125}, {"start": 1850.34, "end": 1850.8, "word": " show", "probability": 0.84033203125}, {"start": 1850.8, "end": 1852.44, "word": " ان", "probability": 0.50634765625}, {"start": 1852.44, "end": 1852.64, "word": " ال", "probability": 0.595703125}, {"start": 1852.64, "end": 1853.2, "word": " sequence", "probability": 0.91748046875}, {"start": 1853.2, "end": 1853.72, "word": " x", "probability": 0.7421875}, {"start": 1853.72, "end": 1854.2, "word": " in", "probability": 0.3115234375}, {"start": 1854.2, "end": 1855.7, "word": " is", "probability": 0.8642578125}, {"start": 1855.7, "end": 1856.9, "word": " convergent", "probability": 0.9287109375}, {"start": 1856.9, "end": 1858.72, "word": " و", "probability": 0.8564453125}, {"start": 1858.72, "end": 1859.78, "word": " converges", "probability": 0.803466796875}, {"start": 1859.78, "end": 1860.22, "word": " to", "probability": 0.96923828125}, {"start": 1860.22, "end": 1862.38, "word": " the", "probability": 0.63720703125}, {"start": 1862.38, "end": 1862.86, "word": " number", "probability": 0.96630859375}, {"start": 1862.86, "end": 1863.64, "word": " 5", "probability": 0.5078125}, {"start": 1863.64, "end": 1864.02, "word": " over", "probability": 0.72802734375}, {"start": 1864.02, "end": 1865.16, "word": " 3", "probability": 0.97998046875}, {"start": 1865.16, "end": 1872.22, "word": " البرهان", "probability": 0.8778076171875}], "temperature": 1.0}, {"id": 68, "seek": 190654, "start": 1877.02, "end": 1906.54, "text": "هنثبت we first show that sequence xn converges by showing بإثبات أنه", "tokens": [3224, 1863, 12984, 3555, 2655, 321, 700, 855, 300, 8310, 2031, 77, 9652, 2880, 538, 4099, 4724, 28814, 12984, 3555, 9307, 14739, 3224], "avg_logprob": -0.3629557229578495, "compression_ratio": 1.0, "no_speech_prob": 0.0, "words": [{"start": 1877.02, "end": 1878.06, "word": "هنثبت", "probability": 0.825537109375}, {"start": 1878.06, "end": 1879.38, "word": " we", "probability": 0.0870361328125}, {"start": 1879.38, "end": 1880.38, "word": " first", "probability": 0.91552734375}, {"start": 1880.38, "end": 1884.0, "word": " show", "probability": 0.34326171875}, {"start": 1884.0, "end": 1886.88, "word": " that", "probability": 0.8603515625}, {"start": 1886.88, "end": 1888.68, "word": " sequence", "probability": 0.73583984375}, {"start": 1888.68, "end": 1889.36, "word": " xn", "probability": 0.545654296875}, {"start": 1889.36, "end": 1890.34, "word": " converges", "probability": 0.92041015625}, {"start": 1890.34, "end": 1897.7, "word": " by", "probability": 0.57666015625}, {"start": 1897.7, "end": 1901.7, "word": " showing", "probability": 0.892578125}, {"start": 1901.7, "end": 1904.4, "word": " بإثبات", "probability": 0.88544921875}, {"start": 1904.4, "end": 1906.54, "word": " أنه", "probability": 0.768798828125}], "temperature": 1.0}, {"id": 69, "seek": 194105, "start": 1911.51, "end": 1941.05, "text": "إنها كوشي thanks to koshi criterion طبعا هذا بفضل معيار كوشي أو كوشي criterion هنثبت الأول أن ال sequence هي to convergent", "tokens": [28814, 1863, 11296, 9122, 2407, 8592, 1829, 3231, 281, 350, 17392, 46691, 23032, 3555, 3615, 995, 23758, 4724, 5172, 11242, 1211, 20449, 1829, 9640, 9122, 2407, 8592, 1829, 34051, 9122, 2407, 8592, 1829, 46691, 8032, 1863, 12984, 3555, 2655, 16247, 12610, 14739, 2423, 8310, 39896, 281, 9652, 6930], "avg_logprob": -0.2552614893232073, "compression_ratio": 1.303030303030303, "no_speech_prob": 0.0, "words": [{"start": 1911.51, "end": 1912.01, "word": "إنها", "probability": 0.729248046875}, {"start": 1912.01, "end": 1912.57, "word": " كوشي", "probability": 0.81475830078125}, {"start": 1912.57, "end": 1920.17, "word": " thanks", "probability": 0.6484375}, {"start": 1920.17, "end": 1922.13, "word": " to", "probability": 0.95263671875}, {"start": 1922.13, "end": 1923.07, "word": " koshi", "probability": 0.59619140625}, {"start": 1923.07, "end": 1927.61, "word": " criterion", "probability": 0.923828125}, {"start": 1927.61, "end": 1929.73, "word": " طبعا", "probability": 0.96826171875}, {"start": 1929.73, "end": 1930.05, "word": " هذا", "probability": 0.6845703125}, {"start": 1930.05, "end": 1931.43, "word": " بفضل", "probability": 0.9949951171875}, {"start": 1931.43, "end": 1932.95, "word": " معيار", "probability": 0.8262532552083334}, {"start": 1932.95, "end": 1933.43, "word": " كوشي", "probability": 0.897705078125}, {"start": 1933.43, "end": 1933.57, "word": " أو", "probability": 0.6923828125}, {"start": 1933.57, "end": 1933.87, "word": " كوشي", "probability": 0.83880615234375}, {"start": 1933.87, "end": 1934.29, "word": " criterion", "probability": 0.8671875}, {"start": 1934.29, "end": 1937.39, "word": " هنثبت", "probability": 0.870361328125}, {"start": 1937.39, "end": 1938.35, "word": " الأول", "probability": 0.9267578125}, {"start": 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2135.22, "end": 2135.22, "word": " الأول", "probability": 0.85302734375}, {"start": 2135.22, "end": 2135.22, "word": " مرات", "probability": 0.9777018229166666}, {"start": 2135.22, "end": 2135.22, "word": " الأول", "probability": 0.85009765625}, {"start": 2135.22, "end": 2135.74, "word": " مرات", "probability": 0.9775390625}, {"start": 2135.74, "end": 2135.96, "word": " الأول", "probability": 0.8486328125}, {"start": 2135.96, "end": 2135.98, "word": " مرات", "probability": 0.97705078125}, {"start": 2135.98, "end": 2135.98, "word": " الأول", "probability": 0.850341796875}, {"start": 2135.98, "end": 2136.0, "word": " مرات", "probability": 0.9768880208333334}, {"start": 2136.0, "end": 2139.62, "word": " الأول", "probability": 0.85107421875}, {"start": 2139.62, "end": 2141.06, "word": " مرات", "probability": 0.9767252604166666}, {"start": 2141.06, "end": 2146.44, "word": " الأول", "probability": 0.84814453125}, {"start": 2146.44, "end": 2147.72, "word": " مرات", "probability": 0.9736328125}], "temperature": 1.0}, {"id": 77, "seek": 216967, "start": 2149.6, "end": 2169.68, "text": "is not monotone لو كتبنا أول أربع خمس ست حدود من ال sequence هذه فبلاحظ أنها ليست monotone ليست increasing neither increasing nor decreasing", "tokens": [271, 406, 1108, 310, 546, 45164, 9122, 2655, 3555, 8315, 5551, 12610, 5551, 25513, 3615, 16490, 2304, 3794, 8608, 2655, 11331, 3215, 23328, 9154, 2423, 8310, 29538, 6156, 3555, 15040, 5016, 19913, 14739, 11296, 32239, 14851, 1108, 310, 546, 32239, 14851, 5662, 9662, 5662, 6051, 23223], "avg_logprob": -0.17486702698342343, "compression_ratio": 1.3428571428571427, "no_speech_prob": 0.0, "words": [{"start": 2149.6, "end": 2150.0, "word": "is", "probability": 0.390625}, {"start": 2150.0, "end": 2150.44, "word": " not", "probability": 0.96044921875}, {"start": 2150.44, "end": 2151.46, "word": " monotone", "probability": 0.8779296875}, {"start": 2151.46, "end": 2156.44, "word": " لو", "probability": 0.49462890625}, {"start": 2156.44, "end": 2156.92, "word": " كتبنا", "probability": 0.8570556640625}, {"start": 2156.92, "end": 2157.36, "word": " أول", "probability": 0.860595703125}, {"start": 2157.36, "end": 2157.92, "word": " أربع", "probability": 0.88232421875}, {"start": 2157.92, "end": 2158.42, "word": " خمس", "probability": 0.89501953125}, {"start": 2158.42, "end": 2159.3, "word": " ست", "probability": 0.936767578125}, {"start": 2159.3, "end": 2159.92, "word": " حدود", "probability": 0.8650716145833334}, {"start": 2159.92, "end": 2161.12, "word": " من", "probability": 0.9892578125}, {"start": 2161.12, "end": 2161.3, "word": " ال", "probability": 0.82421875}, {"start": 2161.3, "end": 2161.8, "word": " sequence", "probability": 0.767578125}, {"start": 2161.8, "end": 2162.3, "word": " هذه", "probability": 0.483642578125}, {"start": 2162.3, "end": 2164.36, "word": " فبلاحظ", "probability": 0.86943359375}, {"start": 2164.36, "end": 2164.76, "word": " أنها", "probability": 0.7900390625}, {"start": 2164.76, "end": 2165.22, "word": " ليست", "probability": 0.984130859375}, {"start": 2165.22, "end": 2165.72, "word": " monotone", "probability": 0.8564453125}, {"start": 2165.72, "end": 2166.2, "word": " ليست", "probability": 0.895751953125}, {"start": 2166.2, "end": 2166.8, "word": " increasing", "probability": 0.96875}, {"start": 2166.8, "end": 2168.06, "word": " neither", "probability": 0.748046875}, {"start": 2168.06, "end": 2168.74, "word": " increasing", "probability": 0.95556640625}, {"start": 2168.74, "end": 2169.14, "word": " nor", "probability": 0.96337890625}, {"start": 2169.14, "end": 2169.68, "word": " decreasing", "probability": 0.94287109375}], "temperature": 1.0}, {"id": 78, "seek": 219194, "start": 2171.54, "end": 2191.94, "text": "وبالتالي نقدر نستخدم الـ monotone convergence theorem so we can't we can't use ال monotone convergence theorem we can't", "tokens": [37746, 6027, 2655, 6027, 1829, 8717, 28543, 2288, 8717, 14851, 9778, 40448, 2423, 39184, 1108, 310, 546, 32181, 20904, 370, 321, 393, 380, 321, 393, 380, 764, 2423, 1108, 310, 546, 32181, 20904, 321, 393, 380], "avg_logprob": -0.2641469594594595, "compression_ratio": 1.4489795918367347, "no_speech_prob": 0.0, "words": [{"start": 2171.54, "end": 2172.5, "word": "وبالتالي", "probability": 0.9375}, {"start": 2172.5, "end": 2173.24, "word": " نقدر", "probability": 0.6307779947916666}, {"start": 2173.24, "end": 2173.88, "word": " نستخدم", "probability": 0.8775634765625}, {"start": 2173.88, "end": 2174.1, "word": " الـ", "probability": 0.5823974609375}, {"start": 2174.1, "end": 2174.48, "word": " monotone", "probability": 0.8922526041666666}, {"start": 2174.48, "end": 2175.1, "word": " convergence", "probability": 0.96044921875}, {"start": 2175.1, "end": 2175.56, "word": " theorem", "probability": 0.92236328125}, {"start": 2175.56, "end": 2176.86, "word": " so", "probability": 0.13232421875}, {"start": 2176.86, "end": 2178.8, "word": " we", "probability": 0.806640625}, {"start": 2178.8, "end": 2180.16, "word": " can't", "probability": 0.919921875}, {"start": 2180.16, "end": 2182.48, "word": " we", "probability": 0.18701171875}, {"start": 2182.48, "end": 2183.6, "word": " can't", "probability": 0.98095703125}, {"start": 2183.6, "end": 2184.96, "word": " use", "probability": 0.9150390625}, {"start": 2184.96, "end": 2185.84, "word": " ال", "probability": 0.7119140625}, {"start": 2185.84, "end": 2186.42, "word": " monotone", "probability": 0.8938802083333334}, {"start": 2186.42, "end": 2187.1, "word": " convergence", "probability": 0.962890625}, {"start": 2187.1, "end": 2187.74, "word": " theorem", "probability": 0.9052734375}, {"start": 2187.74, "end": 2191.12, "word": " we", "probability": 0.84130859375}, {"start": 2191.12, "end": 2191.94, "word": " can't", "probability": 0.98046875}], "temperature": 1.0}, {"id": 79, "seek": 221898, "start": 2195.03, "end": 2218.99, "text": "we can't use monotone convergence theorem الـ sequence bounded عشان استخدم الـ monotone convergence theorem لازم تكون monotone increasing او monotone decreasing ف it is not monotone فماقدرش استخدم ال monotone convergence theorem عشان افحص ال convergence", "tokens": [826, 393, 380, 764, 1108, 310, 546, 32181, 20904, 2423, 39184, 8310, 37498, 6225, 8592, 7649, 44713, 9778, 40448, 2423, 39184, 1108, 310, 546, 32181, 20904, 5296, 31377, 2304, 6055, 30544, 1108, 310, 546, 5662, 1975, 2407, 1108, 310, 546, 23223, 6156, 309, 307, 406, 1108, 310, 546, 6156, 15042, 28543, 2288, 8592, 44713, 9778, 40448, 2423, 1108, 310, 546, 32181, 20904, 6225, 8592, 7649, 1975, 5172, 5016, 9381, 2423, 32181], "avg_logprob": -0.21940103442304665, "compression_ratio": 1.8944099378881987, "no_speech_prob": 0.0, "words": [{"start": 2195.03, "end": 2195.35, "word": "we", "probability": 0.05596923828125}, {"start": 2195.35, "end": 2196.17, "word": " can't", "probability": 0.84033203125}, {"start": 2196.17, "end": 2196.79, "word": " use", "probability": 0.921875}, {"start": 2196.79, "end": 2198.71, "word": " monotone", "probability": 0.9150390625}, {"start": 2198.71, "end": 2199.61, "word": " convergence", "probability": 0.947265625}, {"start": 2199.61, "end": 2200.15, "word": " theorem", "probability": 0.83056640625}, {"start": 2200.15, "end": 2202.91, "word": " الـ", "probability": 0.33331298828125}, {"start": 2202.91, "end": 2203.29, "word": " sequence", "probability": 0.91259765625}, {"start": 2203.29, "end": 2204.03, "word": " bounded", "probability": 0.58349609375}, {"start": 2204.03, "end": 2205.57, "word": " عشان", "probability": 0.94189453125}, {"start": 2205.57, "end": 2206.07, "word": " استخدم", "probability": 0.833984375}, {"start": 2206.07, "end": 2206.29, "word": " الـ", "probability": 0.71044921875}, {"start": 2206.29, "end": 2206.57, "word": " monotone", "probability": 0.9698893229166666}, {"start": 2206.57, "end": 2207.15, "word": " convergence", "probability": 0.986328125}, {"start": 2207.15, "end": 2207.57, "word": " theorem", "probability": 0.8818359375}, {"start": 2207.57, "end": 2207.91, "word": " لازم", "probability": 0.8580729166666666}, {"start": 2207.91, "end": 2208.25, "word": " تكون", "probability": 0.94873046875}, {"start": 2208.25, "end": 2208.93, "word": " monotone", "probability": 0.89208984375}, {"start": 2208.93, "end": 2209.53, "word": " increasing", "probability": 0.96142578125}, {"start": 2209.53, "end": 2209.89, "word": " او", "probability": 0.7347412109375}, {"start": 2209.89, "end": 2210.47, "word": " monotone", "probability": 0.927734375}, {"start": 2210.47, "end": 2211.03, "word": " decreasing", "probability": 0.9560546875}, {"start": 2211.03, "end": 2212.49, "word": " ف", "probability": 0.8291015625}, {"start": 2212.49, "end": 2212.75, "word": " it", "probability": 0.70849609375}, {"start": 2212.75, "end": 2212.93, "word": " is", "probability": 0.88525390625}, {"start": 2212.93, "end": 2213.15, "word": " not", "probability": 0.96826171875}, {"start": 2213.15, "end": 2213.73, "word": " monotone", "probability": 0.9788411458333334}, {"start": 2213.73, "end": 2214.43, "word": " فماقدرش", "probability": 0.843359375}, {"start": 2214.43, "end": 2214.89, "word": " استخدم", "probability": 0.916015625}, {"start": 2214.89, "end": 2215.07, "word": " ال", "probability": 0.78759765625}, {"start": 2215.07, "end": 2215.51, "word": " monotone", "probability": 0.8426106770833334}, {"start": 2215.51, "end": 2216.13, "word": " convergence", "probability": 0.97216796875}, {"start": 2216.13, "end": 2216.61, "word": " theorem", "probability": 0.88330078125}, {"start": 2216.61, "end": 2218.03, "word": " عشان", "probability": 0.880859375}, {"start": 2218.03, "end": 2218.37, "word": " افحص", "probability": 0.88232421875}, {"start": 2218.37, "end": 2218.51, "word": " ال", "probability": 0.9765625}, {"start": 2218.51, "end": 2218.99, "word": " convergence", "probability": 0.98291015625}], "temperature": 1.0}, {"id": 80, "seek": 224813, "start": 2220.41, "end": 2248.13, "text": "الـ sequence لازم ابحث عن طريقه تانية غير الـ monotone convergence فيها طيب هنثبت claim 2 claim 2 ادعاء تاني وهو ان ال sequence xn بتحقق المعادلة absolute xn minus xn زيادة واحد", "tokens": [6027, 39184, 8310, 5296, 31377, 2304, 48127, 5016, 12984, 18871, 23032, 16572, 4587, 3224, 6055, 7649, 10632, 32771, 13546, 2423, 39184, 1108, 310, 546, 32181, 8978, 11296, 23032, 1829, 3555, 8032, 1863, 12984, 3555, 2655, 3932, 568, 3932, 568, 1975, 3215, 3615, 16606, 6055, 7649, 1829, 37037, 2407, 16472, 2423, 8310, 2031, 77, 39894, 5016, 4587, 4587, 9673, 3615, 18513, 37977, 8236, 2031, 77, 3175, 2031, 77, 30767, 1829, 18513, 3660, 36764, 24401], "avg_logprob": -0.22423986204572627, "compression_ratio": 1.4143646408839778, "no_speech_prob": 0.0, "words": [{"start": 2220.41, "end": 2220.91, "word": "الـ", "probability": 0.5501708984375}, {"start": 2220.91, "end": 2222.05, "word": " sequence", "probability": 0.70849609375}, {"start": 2222.05, "end": 2222.59, "word": " لازم", "probability": 0.9415690104166666}, {"start": 2222.59, "end": 2223.05, "word": " ابحث", "probability": 0.7925618489583334}, {"start": 2223.05, "end": 2223.21, "word": " عن", "probability": 0.98291015625}, {"start": 2223.21, "end": 2223.69, "word": " طريقه", "probability": 0.8682861328125}, {"start": 2223.69, "end": 2224.19, "word": " تانية", "probability": 0.9075520833333334}, {"start": 2224.19, "end": 2225.05, "word": " غير", "probability": 0.9375}, {"start": 2225.05, "end": 2225.33, "word": " الـ", "probability": 0.749755859375}, {"start": 2225.33, "end": 2225.79, "word": " monotone", "probability": 0.8855794270833334}, {"start": 2225.79, "end": 2226.61, "word": " convergence", "probability": 0.95703125}, {"start": 2226.61, "end": 2227.05, "word": " فيها", "probability": 0.3258056640625}, {"start": 2227.05, "end": 2229.89, "word": " طيب", "probability": 0.9498697916666666}, {"start": 2229.89, "end": 2230.51, "word": " هنثبت", "probability": 0.8439453125}, {"start": 2230.51, "end": 2230.87, "word": " claim", "probability": 0.91943359375}, {"start": 2230.87, "end": 2231.45, "word": " 2", "probability": 0.57373046875}, {"start": 2231.45, "end": 2236.53, "word": " claim", "probability": 0.435791015625}, {"start": 2236.53, "end": 2237.11, "word": " 2", "probability": 0.9775390625}, {"start": 2237.11, "end": 2237.73, "word": " ادعاء", "probability": 0.7462158203125}, {"start": 2237.73, "end": 2238.97, "word": " تاني", "probability": 0.9690755208333334}, {"start": 2238.97, "end": 2239.75, "word": " وهو", "probability": 0.7313232421875}, {"start": 2239.75, "end": 2240.03, "word": " ان", "probability": 0.818359375}, {"start": 2240.03, "end": 2240.61, "word": " ال", "probability": 0.80517578125}, {"start": 2240.61, "end": 2241.11, "word": " sequence", "probability": 0.79931640625}, {"start": 2241.11, "end": 2241.67, "word": " xn", "probability": 0.67578125}, {"start": 2241.67, "end": 2243.01, "word": " بتحقق", "probability": 0.9654541015625}, {"start": 2243.01, "end": 2244.23, "word": " المعادلة", "probability": 0.9847412109375}, {"start": 2244.23, "end": 2244.79, "word": " absolute", "probability": 0.82666015625}, {"start": 2244.79, "end": 2245.69, "word": " xn", "probability": 0.964599609375}, {"start": 2245.69, "end": 2246.09, "word": " minus", "probability": 0.63134765625}, {"start": 2246.09, "end": 2247.07, "word": " xn", "probability": 0.9716796875}, {"start": 2247.07, "end": 2247.71, "word": " زيادة", "probability": 0.6766357421875}, {"start": 2247.71, "end": 2248.13, "word": " واحد", "probability": 0.945068359375}], "temperature": 1.0}, {"id": 81, "seek": 227479, "start": 2248.95, "end": 2274.79, "text": "بساوي واحد على اتنين قص ان نجاتف وان وهذا الكلام صحيح for every n في n to see this لبرهان ذلك use induction", "tokens": [3555, 3794, 995, 45865, 36764, 24401, 15844, 1975, 2655, 1863, 9957, 12174, 9381, 16472, 8717, 7435, 9307, 5172, 4032, 7649, 37037, 15730, 2423, 28820, 10943, 20328, 5016, 1829, 5016, 337, 633, 297, 8978, 297, 281, 536, 341, 5296, 26890, 3224, 7649, 29910, 23275, 764, 33371], "avg_logprob": -0.2997622217821038, "compression_ratio": 1.2196969696969697, "no_speech_prob": 0.0, "words": [{"start": 2248.95, "end": 2249.59, "word": "بساوي", "probability": 0.679443359375}, {"start": 2249.59, "end": 2250.05, "word": " واحد", "probability": 0.89013671875}, {"start": 2250.05, "end": 2250.29, "word": " على", "probability": 0.603515625}, {"start": 2250.29, "end": 2250.95, "word": " اتنين", "probability": 0.96142578125}, {"start": 2250.95, "end": 2251.59, "word": " قص", "probability": 0.5322265625}, {"start": 2251.59, "end": 2252.19, "word": " ان", "probability": 0.468017578125}, {"start": 2252.19, "end": 2253.21, "word": " نجاتف", "probability": 0.61029052734375}, {"start": 2253.21, "end": 2253.55, "word": " وان", "probability": 0.591796875}, {"start": 2253.55, "end": 2255.13, "word": " وهذا", "probability": 0.73681640625}, {"start": 2255.13, "end": 2255.59, "word": " الكلام", "probability": 0.8819986979166666}, {"start": 2255.59, "end": 2256.23, "word": " صحيح", 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"probability": 0.962890625}], "temperature": 1.0}, {"id": 82, "seek": 230164, "start": 2277.68, "end": 2301.64, "text": "use induction on n برضه ممكن برهان المعادلة هذه by induction on n هينبرهن البرهان if n بسبب واحد ف absolute x واحد minus x", "tokens": [84, 405, 33371, 322, 297, 4724, 43042, 3224, 3714, 43020, 4724, 2288, 3224, 7649, 9673, 3615, 18513, 37977, 29538, 538, 33371, 322, 297, 8032, 9957, 26890, 3224, 1863, 2423, 26890, 3224, 7649, 498, 297, 4724, 35457, 3555, 36764, 24401, 6156, 8236, 2031, 36764, 24401, 3175, 2031], "avg_logprob": -0.20146276532335483, "compression_ratio": 1.373015873015873, "no_speech_prob": 0.0, "words": [{"start": 2277.68, "end": 2278.22, "word": "use", "probability": 0.528778076171875}, {"start": 2278.22, "end": 2278.92, "word": " induction", "probability": 0.94775390625}, {"start": 2278.92, "end": 2279.32, "word": " on", "probability": 0.93017578125}, {"start": 2279.32, "end": 2279.58, "word": " n", "probability": 0.693359375}, {"start": 2279.58, 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2561.5, "text": "الان باستخدام ال claim التاني ممكن نثبت شغلة مهمة في البرهان اذا خليها هادى للمرة الجاية بس بدى اكتبها", "tokens": [6027, 7649, 4724, 995, 14851, 9778, 3215, 10943, 2423, 3932, 16712, 7649, 1829, 3714, 43020, 8717, 12984, 3555, 2655, 13412, 17082, 37977, 3714, 16095, 3660, 8978, 2423, 26890, 3224, 7649, 1975, 15730, 16490, 20292, 11296, 8032, 18513, 7578, 5296, 19528, 25720, 25724, 995, 10632, 4724, 3794, 47525, 7578, 1975, 4117, 2655, 3555, 11296], "avg_logprob": -0.212673613594638, "compression_ratio": 1.4365079365079365, "no_speech_prob": 0.0, "words": [{"start": 2535.82, "end": 2536.38, "word": "الان", "probability": 0.778076171875}, {"start": 2536.38, "end": 2537.1, "word": " باستخدام", "probability": 0.9768880208333334}, {"start": 2537.1, "end": 2537.28, "word": " ال", "probability": 0.8310546875}, {"start": 2537.28, "end": 2537.6, "word": " claim", "probability": 0.1490478515625}, {"start": 2537.6, "end": 2538.34, "word": " التاني", "probability": 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"word": " بس", "probability": 0.9541015625}, {"start": 2560.76, "end": 2561.04, "word": " بدى", "probability": 0.578125}, {"start": 2561.04, "end": 2561.5, "word": " اكتبها", "probability": 0.9619140625}], "temperature": 1.0}, {"id": 92, "seek": 259121, "start": 2562.79, "end": 2591.21, "text": "خليكم أنتوا تفكروا فيها .. خليكم أنتوا تفكروا فيها Now using a claim to verify .. verify that ..", "tokens": [9778, 20292, 24793, 14739, 2655, 14407, 6055, 5172, 37983, 14407, 8978, 11296, 4386, 16490, 20292, 24793, 14739, 2655, 14407, 6055, 5172, 37983, 14407, 8978, 11296, 823, 1228, 257, 3932, 281, 16888, 4386, 16888, 300, 4386], "avg_logprob": -0.1588541662527455, "compression_ratio": 1.511111111111111, "no_speech_prob": 0.0, "words": [{"start": 2562.79, "end": 2563.33, "word": "خليكم", "probability": 0.7964680989583334}, {"start": 2563.33, "end": 2563.65, "word": " أنتوا", "probability": 0.8331705729166666}, {"start": 2563.65, "end": 2564.19, "word": " تفكروا", "probability": 0.963623046875}, {"start": 2564.19, "end": 2564.57, "word": " فيها", "probability": 0.982421875}, {"start": 2564.57, "end": 2564.99, "word": " ..", "probability": 0.69677734375}, {"start": 2564.99, "end": 2566.13, "word": " خليكم", "probability": 0.8092447916666666}, {"start": 2566.13, "end": 2566.47, "word": " أنتوا", "probability": 0.9737955729166666}, {"start": 2566.47, "end": 2566.97, "word": " تفكروا", "probability": 0.9903564453125}, {"start": 2566.97, "end": 2567.45, "word": " فيها", "probability": 0.99609375}, {"start": 2567.45, "end": 2573.39, "word": " Now", "probability": 0.51220703125}, {"start": 2573.39, "end": 2576.29, "word": " using", "probability": 0.66845703125}, {"start": 2576.29, "end": 2578.23, "word": " a", "probability": 0.69189453125}, {"start": 2578.23, "end": 2578.73, "word": " claim", "probability": 0.7197265625}, {"start": 2578.73, "end": 2581.11, "word": " to", "probability": 0.966796875}, {"start": 2581.11, "end": 2583.91, "word": " verify", "probability": 0.9599609375}, {"start": 2583.91, "end": 2585.93, "word": " ..", "probability": 0.83837890625}, {"start": 2585.93, "end": 2586.55, "word": " verify", "probability": 0.86376953125}, {"start": 2586.55, "end": 2588.83, "word": " that", "probability": 0.95263671875}, {"start": 2588.83, "end": 2591.21, "word": " ..", "probability": 0.89013671875}], "temperature": 1.0}, {"id": 93, "seek": 261917, "start": 2594.77, "end": 2619.17, "text": "F M أكبر من N فهذا بيقدي أن absolute X N minus X M أصغر من واحد على اتنين قص M نجاتي باتنين", "tokens": [37, 376, 5551, 4117, 26890, 9154, 426, 6156, 3224, 15730, 4724, 1829, 4587, 16254, 14739, 8236, 1783, 426, 3175, 1783, 376, 5551, 9381, 17082, 2288, 9154, 36764, 24401, 15844, 1975, 2655, 1863, 9957, 12174, 9381, 376, 8717, 7435, 9307, 1829, 4724, 9307, 1863, 9957], "avg_logprob": -0.29357640213436553, "compression_ratio": 1.1680672268907564, "no_speech_prob": 0.0, "words": [{"start": 2594.77, "end": 2596.17, "word": "F", "probability": 0.366455078125}, {"start": 2596.17, "end": 2597.57, "word": " M", "probability": 0.38427734375}, {"start": 2597.57, "end": 2598.33, "word": " أكبر", "probability": 0.9163411458333334}, {"start": 2598.33, "end": 2598.65, "word": " من", "probability": 0.99072265625}, {"start": 2598.65, "end": 2599.05, "word": " N", "probability": 0.919921875}, {"start": 2599.05, "end": 2603.19, "word": " فهذا", "probability": 0.9596354166666666}, {"start": 2603.19, "end": 2603.93, "word": " بيقدي", "probability": 0.7342529296875}, {"start": 2603.93, "end": 2604.99, "word": " أن", "probability": 0.5830078125}, {"start": 2604.99, "end": 2605.63, "word": " absolute", "probability": 0.20263671875}, {"start": 2605.63, "end": 2606.55, "word": " X", "probability": 0.662109375}, {"start": 2606.55, "end": 2607.17, "word": " N", "probability": 0.270263671875}, {"start": 2607.17, "end": 2607.93, "word": " minus", "probability": 0.767578125}, {"start": 2607.93, "end": 2608.59, "word": " X", "probability": 0.9755859375}, {"start": 2608.59, "end": 2609.45, "word": " M", "probability": 0.95068359375}, {"start": 2609.45, "end": 2611.99, "word": " أصغر", "probability": 0.980224609375}, {"start": 2611.99, "end": 2612.39, "word": " من", "probability": 0.99755859375}, {"start": 2612.39, "end": 2613.23, "word": " واحد", "probability": 0.865478515625}, {"start": 2613.23, "end": 2613.53, "word": " على", "probability": 0.7705078125}, {"start": 2613.53, "end": 2614.47, "word": " اتنين", "probability": 0.9671630859375}, {"start": 2614.47, "end": 2616.81, "word": " قص", "probability": 0.51885986328125}, {"start": 2616.81, "end": 2617.47, "word": " M", "probability": 0.68115234375}, {"start": 2617.47, "end": 2618.65, "word": " نجاتي", "probability": 0.6954345703125}, {"start": 2618.65, "end": 2619.17, "word": " باتنين", "probability": 0.775390625}], "temperature": 1.0}, {"id": 94, "seek": 265269, "start": 2625.95, "end": 2652.69, "text": "إذاً هذا ممكن إثباته by ال triangle ال equality و claim اثنين فبنوقف هنا و بنكمل ال .. بنكمل ان شاء الله", "tokens": [28814, 15730, 14111, 23758, 3714, 43020, 11933, 12984, 3555, 9307, 3224, 538, 2423, 13369, 2423, 14949, 4032, 3932, 1975, 12984, 1863, 9957, 6156, 3555, 1863, 30543, 5172, 34105, 4032, 44945, 24793, 1211, 2423, 4386, 44945, 24793, 1211, 16472, 13412, 16606, 21984], "avg_logprob": -0.2966889888048172, "compression_ratio": 1.255813953488372, "no_speech_prob": 0.0, "words": [{"start": 2625.95, "end": 2626.93, "word": "إذاً", "probability": 0.4600830078125}, {"start": 2626.93, "end": 2628.17, "word": " هذا", "probability": 0.76025390625}, {"start": 2628.17, "end": 2628.53, "word": " ممكن", "probability": 0.927001953125}, {"start": 2628.53, "end": 2629.29, "word": " إثباته", "probability": 0.90400390625}, {"start": 2629.29, "end": 2629.75, "word": " by", "probability": 0.28857421875}, {"start": 2629.75, "end": 2631.81, "word": " ال", "probability": 0.38818359375}, {"start": 2631.81, "end": 2632.43, "word": " triangle", "probability": 0.7255859375}, {"start": 2632.43, "end": 2633.47, "word": " ال", "probability": 0.25537109375}, {"start": 2633.47, "end": 2633.97, "word": " equality", "probability": 0.66015625}, {"start": 2633.97, "end": 2634.29, "word": " و", "probability": 0.9443359375}, {"start": 2634.29, "end": 2634.73, "word": " claim", "probability": 0.71875}, {"start": 2634.73, "end": 2635.47, "word": " اثنين", "probability": 0.8363037109375}, {"start": 2635.47, "end": 2646.85, "word": " فبنوقف", "probability": 0.93623046875}, {"start": 2646.85, "end": 2647.15, "word": " هنا", "probability": 0.99462890625}, {"start": 2647.15, "end": 2647.45, "word": " و", "probability": 0.83642578125}, {"start": 2647.45, "end": 2648.37, "word": " بنكمل", "probability": 0.931640625}, {"start": 2648.37, "end": 2649.41, "word": " ال", "probability": 0.61328125}, {"start": 2649.41, "end": 2650.99, "word": " ..", "probability": 0.67041015625}, {"start": 2650.99, "end": 2651.95, "word": " بنكمل", "probability": 0.81494140625}, {"start": 2651.95, "end": 2652.31, "word": " ان", "probability": 0.60546875}, {"start": 2652.31, "end": 2652.57, "word": " شاء", "probability": 0.988037109375}, {"start": 2652.57, "end": 2652.69, "word": " الله", "probability": 0.94091796875}], "temperature": 1.0}, {"id": 95, "seek": 265968, "start": 2653.76, "end": 2659.68, "text": "البرهان في المحاضرة الجاية، في حد عنده أي سؤال أو استفسار؟", "tokens": [6027, 26890, 3224, 7649, 8978, 9673, 5016, 46958, 25720, 25724, 995, 10632, 12399, 8978, 11331, 3215, 43242, 3224, 36632, 8608, 33604, 6027, 34051, 44713, 36178, 9640, 22807], "avg_logprob": -0.159598208963871, "compression_ratio": 1.2045454545454546, "no_speech_prob": 0.0, "words": [{"start": 2653.76, "end": 2654.28, "word": "البرهان", "probability": 0.9632568359375}, {"start": 2654.28, "end": 2654.46, "word": " في", "probability": 0.955078125}, {"start": 2654.46, "end": 2655.14, "word": " المحاضرة", "probability": 0.9833984375}, {"start": 2655.14, "end": 2657.2, "word": " الجاية،", "probability": 0.7115478515625}, {"start": 2657.2, "end": 2657.4, "word": " في", "probability": 0.75634765625}, {"start": 2657.4, "end": 2657.66, "word": " حد", "probability": 0.975341796875}, {"start": 2657.66, "end": 2657.92, "word": " عنده", "probability": 0.958984375}, {"start": 2657.92, "end": 2658.08, "word": " أي", "probability": 0.78076171875}, {"start": 2658.08, "end": 2658.44, "word": " سؤال", "probability": 0.9783528645833334}, {"start": 2658.44, "end": 2658.6, "word": " أو", "probability": 0.8115234375}, {"start": 2658.6, "end": 2659.68, "word": " استفسار؟", "probability": 0.785308837890625}], "temperature": 1.0}], "language": "ar", "language_probability": 1.0, "duration": 2660.58875, "duration_after_vad": 2402.93749999999}
\ No newline at end of file
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/1Uemtyp4-IM_raw.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/1Uemtyp4-IM_raw.srt
new file mode 100644
index 0000000000000000000000000000000000000000..bac53d461716a551e5b01bd0d078feb8f384c370
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/1Uemtyp4-IM_raw.srt
@@ -0,0 +1,1400 @@
+1
+00:00:23,230 --> 00:00:28,870
+بسم الله الرحمن الرحيم في الساعة هذه طبعا هيكون
+
+2
+00:00:28,870 --> 00:00:34,770
+فيانا مناقشة نشوف
+
+3
+00:00:34,770 --> 00:00:39,790
+ال section الأخيرة في chapter تلاتة نبدأ section
+
+4
+00:00:39,790 --> 00:00:43,610
+تلاتة ستة فيانكم أي سؤال في section تلاتة ستة؟
+
+5
+00:00:51,050 --> 00:00:56,790
+التالي هذا نقشناه المرة اللي فاتت طيب
+
+6
+00:00:56,790 --> 00:01:02,630
+في section تلاتة سبعة في عندكم أي أسئلة في section
+
+7
+00:01:02,630 --> 00:01:08,970
+تلاتة سبعة سؤال رقم احداشر نعم رقم احداشر
+
+8
+00:01:22,550 --> 00:01:35,490
+بس الرقم 11 تلاتة سابعة if
+
+9
+00:01:35,490 --> 00:01:42,950
+the series sigma a n with
+
+10
+00:01:42,950 --> 00:01:46,270
+a
+
+11
+00:01:46,270 --> 00:01:51,070
+n أكبر من الصفر is convergent
+
+12
+00:01:54,210 --> 00:02:01,230
+is convergent then
+
+13
+00:02:01,230 --> 00:02:14,890
+is the series sigma للجدر التربيهي ولا
+
+14
+00:02:14,890 --> 00:02:15,410
+لأ؟
+
+15
+00:02:24,900 --> 00:02:29,340
+is the series and
+
+16
+00:02:29,340 --> 00:02:39,240
+if and
+
+17
+00:02:39,240 --> 00:02:52,300
+if BN BN بساوي A واحد زائد إلى AN كل هذا مجسوم على
+
+18
+00:02:52,300 --> 00:02:52,720
+N
+
+19
+00:02:55,990 --> 00:03:03,350
+مع الـ n يشبه الـ n ثم
+
+20
+00:03:03,350 --> 00:03:08,310
+اظهر .. اظهر
+
+21
+00:03:08,310 --> 00:03:15,590
+ان السيريز سيجما bn دائما
+
+22
+00:03:15,590 --> 00:03:19,510
+.. دائما
+
+23
+00:03:19,510 --> 00:03:21,290
+متحرر
+
+24
+00:03:33,740 --> 00:03:34,160
+Okay
+
+25
+00:03:51,610 --> 00:03:56,550
+بنثبت ان لو كانت ال series هذه حدودها كلها موجبة و
+
+26
+00:03:56,550 --> 00:04:02,670
+convergent وعرفنا Pn على ان ال average لمجموعة او
+
+27
+00:04:02,670 --> 00:04:09,750
+ال average لأول n من حدود ال series An فبنثبت ان
+
+28
+00:04:09,750 --> 00:04:12,790
+ال series هذه بتطلع دائما divergent
+
+29
+00:04:18,290 --> 00:04:21,610
+وارجي ال unbounded ال series لما تكون unbounded
+
+30
+00:04:21,610 --> 00:04:25,710
+تتطير مين هي ال unbounded؟ الأسئلة ال sequence of
+
+31
+00:04:25,710 --> 00:04:36,990
+partial sums صحيح يعني
+
+32
+00:04:36,990 --> 00:04:42,710
+أنا عندي أول شي not
+
+33
+00:04:42,710 --> 00:04:43,290
+first
+
+34
+00:04:47,630 --> 00:05:00,050
+رحزي أولا أنه لكل K ينتمي إلى N EK
+
+35
+00:05:00,050 --> 00:05:12,590
+اللي هو بيساوي A1 زايد EK على N على K هذا
+
+36
+00:05:12,590 --> 00:05:15,870
+بيكون دايما أكبر من أو يساوي
+
+37
+00:05:20,590 --> 00:05:25,570
+A1 على K لأن
+
+38
+00:05:25,570 --> 00:05:32,410
+ال .. ال bus اللي هنا أكبر من A1 لأن الأعداد هنا
+
+39
+00:05:32,410 --> 00:05:37,150
+اللي في ال bus كل أعداد موجبة فال bus اللي هنا
+
+40
+00:05:37,150 --> 00:05:40,930
+أكبر من ال bus اللي هناك وبالتالي هذا دايما صحيح
+
+41
+00:05:40,930 --> 00:05:45,650
+لكل K في N hence
+
+42
+00:05:45,650 --> 00:05:46,790
+وبالتالي
+
+43
+00:05:48,890 --> 00:05:57,350
+لو أخدت الـ SIN الانف بارشيل سام للسيريز سيجما BN
+
+44
+00:06:05,920 --> 00:06:10,120
+إذن هذا عبارة عن ال F partial sum لل series sigma
+
+45
+00:06:10,120 --> 00:06:17,480
+bn الآن عندي bk أكبر من أو يساوي هاي summation من
+
+46
+00:06:17,480 --> 00:06:24,380
+k بساوي واحد إلى n و ال bk هادي أكبر من أو يساوي a
+
+47
+00:06:24,380 --> 00:06:29,640
+واحد على k ال a واحد ثابت بالنسبة ل k ده تمليش على
+
+48
+00:06:29,640 --> 00:06:36,860
+k فبطلّه برا هاي a واحد ضربSummation من K بيسار
+
+49
+00:06:36,860 --> 00:06:44,400
+واحد إلى N لواحد على K واحنا
+
+50
+00:06:44,400 --> 00:06:50,140
+أثبتنا قبل هيك أنه ال sequence of partial sums لل
+
+51
+00:06:50,140 --> 00:06:57,620
+harmonic series is unbounded
+
+52
+00:06:57,620 --> 00:07:03,380
+في كان مثال سابق بيقول إنه
+
+53
+00:07:07,390 --> 00:07:14,970
+إن الـ sequence هذه من n بساوي واحد to infinity is
+
+54
+00:07:14,970 --> 00:07:18,590
+unbounded
+
+55
+00:07:18,590 --> 00:07:25,010
+is unbounded حسب
+
+56
+00:07:25,010 --> 00:07:31,030
+مثال سألت إذا لما أضربها ال sequence هذه لما أضرب
+
+57
+00:07:31,030 --> 00:07:35,350
+حدودها أو أضربها في ثابت موجة تبقى unbounded
+
+58
+00:07:39,070 --> 00:07:48,770
+وبالتالي إذا SM هذا بيقدي ان ال sequence SM is
+
+59
+00:07:48,770 --> 00:07:52,610
+unbounded
+
+60
+00:07:52,610 --> 00:07:59,870
+therefore ال
+
+61
+00:07:59,870 --> 00:08:08,950
+limit ل SM لما انتقل ل infinity does not existand
+
+62
+00:08:08,950 --> 00:08:16,510
+therefore the series sigma dn diverges لان احنا
+
+63
+00:08:16,510 --> 00:08:19,970
+قلنا قبلك ان اي infinite series بتكون convergent
+
+64
+00:08:19,970 --> 00:08:24,570
+if and only if the sequence of partial sums is
+
+65
+00:08:24,570 --> 00:08:32,870
+convergent لان هذا هو الحل okay تمام في
+
+66
+00:08:32,870 --> 00:08:35,730
+أي أسئلة تانية في section تلاتة سبعة
+
+67
+00:08:53,340 --> 00:08:58,320
+مفهوم الحل؟ في
+
+68
+00:08:58,320 --> 00:09:03,800
+أسئلة تانية في ال section هذا أو أي section سابق؟
+
+69
+00:09:03,800 --> 00:09:11,180
+فسؤال سبعة هذا
+
+70
+00:09:11,180 --> 00:09:16,210
+المماثل بيشبه مثال تلاتة سبعة ستةفاقرأي المثال
+
+71
+00:09:16,210 --> 00:09:22,330
+حاولي تطبقي نفس الطريقة مشروحليك مثال فحاولي
+
+72
+00:09:22,330 --> 00:09:28,710
+اتجلدي المثال في اي اسئلة تانية؟
+
+73
+00:09:28,710 --> 00:09:35,950
+مان
+
+74
+00:09:35,950 --> 00:09:37,170
+لديها سؤال؟
+
+75
+00:09:56,850 --> 00:10:11,850
+في عندكم أسرة طيب
+
+76
+00:10:11,850 --> 00:10:14,790
+لما تفكروا في أسرة بدي أنا بارهنكم koshi
+
+77
+00:10:14,790 --> 00:10:21,390
+condensation set test لأن هذا في عليه أسرة ومهم
+
+78
+00:10:38,680 --> 00:10:56,660
+سؤال اتماشي section تلاتة .. سابعة قوشيز
+
+79
+00:10:56,660 --> 00:11:00,760
+condensation
+
+80
+00:11:00,760 --> 00:11:01,180
+test
+
+81
+00:11:13,290 --> 00:11:19,130
+فال test هذا بيقول let sigma
+
+82
+00:11:19,130 --> 00:11:29,970
+an be a series .. a series of
+
+83
+00:11:29,970 --> 00:11:42,270
+monotone .. of monotone decreasing positive
+
+84
+00:11:45,320 --> 00:11:54,260
+مجموعات اثنين اثنين
+
+85
+00:11:54,260 --> 00:11:54,260
+اثنين اثنين اثنين اثنين اثنين اثنين اثنين اثنين
+
+86
+00:11:54,260 --> 00:11:54,340
+اثنين اثنين اثنين اثنين اثنين اثنين اثنين اثنين
+
+87
+00:11:54,340 --> 00:11:58,160
+اثنين اثنين اثنين اثنين اثنين اثنين اثنين اثنين
+
+88
+00:11:58,160 --> 00:11:59,760
+اثنين اثنين اثنين اثنين اثنين اثنين اثنين اثنين
+
+89
+00:11:59,760 --> 00:12:03,980
+اثنين اثنين اثنين اثنين اثنين اثنين اثنين اثنين
+
+90
+00:12:03,980 --> 00:12:05,320
+اثنين اثنين اثنين اثنين اثنين اثنين اثنين اثنين
+
+91
+00:12:05,320 --> 00:12:08,420
+اثنين اثنين اثنين
+
+92
+00:12:08,420 --> 00:12:14,380
+اثنين
+
+93
+00:12:14,380 --> 00:12:14,820
+اثن
+
+94
+00:12:42,930 --> 00:12:48,630
+وهي البرهان اولا
+
+95
+00:12:48,630 --> 00:13:02,350
+خلّينا نلاحظnote that لاحظي انه لو أخدت نص في
+
+96
+00:13:02,350 --> 00:13:12,530
+summation من k بساوي zero to infinity ل two أُس k
+
+97
+00:13:12,530 --> 00:13:18,930
+في a two to k هذا
+
+98
+00:13:18,930 --> 00:13:20,830
+بيطلع بساوي نص
+
+99
+00:13:23,540 --> 00:13:33,720
+في A1 اول حد لما كدا ساوى سفر فبطلع نص A1 الحد
+
+100
+00:13:33,720 --> 00:13:43,940
+اللي بعده هيطلع A2 اللي بعده اتنين A4 واللي بعده
+
+101
+00:13:43,940 --> 00:13:52,020
+اربعة في A8 وهكذا نستمر على هذا النمط إلى
+
+102
+00:13:55,470 --> 00:14:00,650
+أتنين خلّينا ناخد المجموعة من K بساوي سفر إلى M
+
+103
+00:14:00,650 --> 00:14:08,290
+حيث M عدد طبيعي ما فأخر حد هيكون اتنين أس M سالب
+
+104
+00:14:08,290 --> 00:14:15,670
+واحد في A اتنين أس M الآن
+
+105
+00:14:15,670 --> 00:14:21,770
+هذا المجموع أصغر من A واحد نص A واحد بالتأكيد أصغر
+
+106
+00:14:21,770 --> 00:14:32,160
+من A واحدو طبعا ال .. ال .. الأعداد هذه كلها موجبة
+
+107
+00:14:32,160 --> 00:14:38,600
+و بتكون decrease in sequence فنص a1 أصغر من a1 و
+
+108
+00:14:38,600 --> 00:14:59,570
+a2 بساوي a2 و 2 a4 أصغر من a3 زائد a4 صح؟A4 أصغر
+
+109
+00:14:59,570 --> 00:15:07,110
+من A3 لأن ال sequence A N decreasing فعندي A4 زاد
+
+110
+00:15:07,110 --> 00:15:17,350
+A4 أصغر من A3 زاد A4 و هكذا برضه عندي A8 أصغر من
+
+111
+00:15:17,350 --> 00:15:25,190
+A5 و أصغر من A6 و أصغر من A7وبالتالي هدا هيكون
+
+112
+00:15:25,190 --> 00:15:30,470
+اربعة ا تمانية اصغر من مجموعة اربعة حدود اللي هم a
+
+113
+00:15:30,470 --> 00:15:42,270
+خمسة زائد a ستة زائد a سبعة زائد a تمانية و هكذا
+
+114
+00:15:42,270 --> 00:15:48,830
+استمر على هذا النمط الى ان نصل لاخر
+
+115
+00:15:51,230 --> 00:15:58,170
+هدول الحدود هيكون اصغر من .. او لحد هذا الأخير
+
+116
+00:15:58,170 --> 00:16:04,530
+اصغر من المجموعة اللي هو a اتنين اص ام سالب واحد
+
+117
+00:16:04,530 --> 00:16:11,430
+زائد واحد زائد a
+
+118
+00:16:11,430 --> 00:16:18,920
+اتنين اص ام سالب واحد زائد اتنين زائد و هكذابقت
+
+119
+00:16:18,920 --> 00:16:24,700
+أصغر من مجموعة كل ال series لأن هذه كلها حدود
+
+120
+00:16:24,700 --> 00:16:29,680
+موجبة، أعداد موجبة وهذا
+
+121
+00:16:29,680 --> 00:16:38,540
+الكلام صحيح لكل M، لكل M عدد طبيعي أكبر
+
+122
+00:16:38,540 --> 00:16:47,240
+من أو يساوي، يعني عدد طبيعي وبالتالي
+
+123
+00:16:47,240 --> 00:16:48,120
+and so
+
+124
+00:16:50,890 --> 00:17:01,690
+وبالتالي نضرب sum من k بساوي سفر إلى m لتو أس ك
+
+125
+00:17:01,690 --> 00:17:10,490
+بإتنين أس ك ده هيطلع أصغر من أو ساوي نضرب الطرفين
+
+126
+00:17:10,490 --> 00:17:15,470
+في اتنين عشان نتخلص من النصفبصير المجموع هذا أصغر
+
+127
+00:17:15,470 --> 00:17:21,410
+من أوسعه اتنين في summation من n equals zero to
+
+128
+00:17:21,410 --> 00:17:27,750
+infinity ل a n تمام؟
+
+129
+00:17:27,750 --> 00:17:34,330
+وهذا
+
+130
+00:17:34,330 --> 00:17:39,650
+صحيح لكل m belonging to N
+
+131
+00:17:44,360 --> 00:18:02,120
+بنسمي ال quality هذه واحد طيب
+
+132
+00:18:02,120 --> 00:18:05,680
+now next
+
+133
+00:18:09,650 --> 00:18:21,350
+given any m أكبر من أو سوى الواحد choose using
+
+134
+00:18:21,350 --> 00:18:34,050
+Archimedean property choose
+
+135
+00:18:34,050 --> 00:18:44,810
+k بحيث أنهtwo to K أكبر من M لأي عدد طبيعي ممكن
+
+136
+00:18:44,810 --> 00:18:58,530
+ألاقي عدد طبيعي بحياتي two to K أكبر من M then ال
+
+137
+00:18:58,530 --> 00:19:06,690
+summation from N equals zero to Mلان هذا بيطلع
+
+138
+00:19:06,690 --> 00:19:12,630
+أصغر من a0
+
+139
+00:19:12,630 --> 00:19:19,710
+زائد a1 زائد a2
+
+140
+00:19:19,710 --> 00:19:31,150
+زائد a3 زائد a4 زائد a5 زائد a6 زائد a7 زائد a8
+
+141
+00:19:35,390 --> 00:19:47,670
+مع بعض زائد و هكذا إلى اتنين
+
+142
+00:19:47,670 --> 00:19:55,190
+أسكت زائد اتنين أسكت زائد واحد زائد و هكذا إلى
+
+143
+00:19:55,190 --> 00:19:59,330
+اتنين
+
+144
+00:19:59,330 --> 00:20:03,950
+أسكت زائد واحد سالب واحد
+
+145
+00:20:12,500 --> 00:20:17,840
+أنا عند ال M هذا ال M أصغر من اتنين أس كي في آخر
+
+146
+00:20:17,840 --> 00:20:26,460
+حد اللي هو AM هيكون أصغر من A رقم اتنين أس كي أو
+
+147
+00:20:26,460 --> 00:20:34,180
+أصغر من أو ساوي اتنين رقم A أس اتنين كي زي واحد
+
+148
+00:20:34,180 --> 00:20:35,780
+minus واحد
+
+149
+00:20:43,450 --> 00:20:52,190
+والمجموع هذا .. هذا المجموع أصغر من او يساوي a0
+
+150
+00:20:52,190 --> 00:20:57,490
+زائد a1 زائد
+
+151
+00:20:57,490 --> 00:21:06,710
+2 a2 لأن a3 أصغر من a2 صح؟ عشان ال sequence an is
+
+152
+00:21:06,710 --> 00:21:13,030
+decreasing و هذا المجموع أصغر من 4 a
+
+153
+00:21:14,740 --> 00:21:26,420
+أربعة صح وهكذا إلى المجموع هذا هيكون أصغر من اتنين
+
+154
+00:21:26,420 --> 00:21:37,280
+أث كيه هذول عدد الحدود في a اتنين أث كيه يعني هذول
+
+155
+00:21:37,280 --> 00:21:41,860
+عدد الحدود عددهم اتنين أث كيه وكل واحد منهم
+
+156
+00:21:45,050 --> 00:21:55,350
+أصغر من ات اول واحد اللي هو ات نين اص كيه وهذا
+
+157
+00:21:55,350 --> 00:22:01,830
+بدوره أصغر من ات نين اص كيه زائد summation من كيه
+
+158
+00:22:01,830 --> 00:22:09,730
+بساوي zero to infinity لاتنين اص كيه في ات نين اص
+
+159
+00:22:09,730 --> 00:22:17,540
+كيه هاي أول حد ات نين اص كيهلما ك بيساوي سفر بيطلع
+
+160
+00:22:17,540 --> 00:22:25,640
+ا واحد و بعدين اللي بعده بيطلع اتنين اتنين لما ك
+
+161
+00:22:25,640 --> 00:22:33,480
+بيساوي واحد و اللي بعده اربعة اربعة و هكذا طبعا
+
+162
+00:22:33,480 --> 00:22:37,400
+هذا بوقف المجموعة هذا finite هذا أصغر من المجموعة
+
+163
+00:22:37,400 --> 00:22:41,400
+من ك بيساوي سفر إلى ملا نهاية هذا طبعا في حدود
+
+164
+00:22:41,400 --> 00:22:41,820
+أكتر
+
+165
+00:22:44,960 --> 00:22:53,040
+تمام؟ وبالتالي إذا نستنتج and so نستنتج
+
+166
+00:22:53,040 --> 00:23:02,980
+إنه المجموعة sigma from n equal zero to infinity ل
+
+167
+00:23:02,980 --> 00:23:12,050
+a nبطلع أصغر من أو ساوي a0 زاد sigma from k equals
+
+168
+00:23:12,050 --> 00:23:20,790
+zero to infinity ل 2 أُس k a2 أُس k لأن
+
+169
+00:23:20,790 --> 00:23:26,810
+هذا صحيح لكل m أكبر من أو ساوي الواحد لأن هذا
+
+170
+00:23:26,810 --> 00:23:33,330
+عبارة عن هذا عبارة عن upper bound هذا العددأو هذا
+
+171
+00:23:33,330 --> 00:23:39,530
+العدد upper bound لل sequence of partial sums هنا
+
+172
+00:23:39,530 --> 00:23:44,190
+فما
+
+173
+00:23:44,190 --> 00:23:47,210
+هذه ال sequence of partial sums is increasing
+
+174
+00:23:47,210 --> 00:23:50,750
+متزايدة
+
+175
+00:23:50,750 --> 00:23:55,110
+و bounded above by this number إذا ال limit تبعت
+
+176
+00:23:55,110 --> 00:23:58,650
+ال sequence of partial sums exist و بالساوية
+
+177
+00:23:58,650 --> 00:24:04,990
+supremumلـ sequence of partial sums الـ supremum
+
+178
+00:24:04,990 --> 00:24:11,150
+لـ sequence of partial sums أقل من ال upper bound
+
+179
+00:24:11,150 --> 00:24:13,670
+هذا upper bound لـ sequence of partial sums ال
+
+180
+00:24:13,670 --> 00:24:17,050
+supremum أصغر upper bound وبالتالي إذا ال supremum
+
+181
+00:24:17,050 --> 00:24:21,690
+لـ sequence of partial sums هو عبارة عن limit لـ
+
+182
+00:24:21,690 --> 00:24:23,730
+sequence of partial sums اللي هو مجموعة ال
+
+183
+00:24:23,730 --> 00:24:29,190
+infinite series أصغر من أو ساوي ال upper boundby
+
+184
+00:24:29,190 --> 00:24:34,290
+monotone convergence theorem السيريز
+
+185
+00:24:34,290 --> 00:24:39,610
+هذي convergence ومجموعة بساول limit ل sequence of
+
+186
+00:24:39,610 --> 00:24:44,710
+partial sums اللي هي أصغر من أو ساول عددها okay
+
+187
+00:24:44,710 --> 00:24:54,170
+إذا نسمي المتباينة هذه اتنين إذا من المتباينة واحد
+
+188
+00:24:54,170 --> 00:24:54,870
+واتنين
+
+189
+00:25:11,870 --> 00:25:19,130
+الان بمقارنة مباشرة الاختلافات
+
+190
+00:25:19,130 --> 00:25:30,640
+المتباينات واحدة و اتنين بيقدواالسيريز sigma a n
+
+191
+00:25:30,640 --> 00:25:39,720
+converges if and only if السيريز sigma اثنين اثنين
+
+192
+00:25:39,720 --> 00:25:47,820
+a اثنين اثنين converges تعالى
+
+193
+00:25:47,820 --> 00:25:54,680
+نشوف لو كانت السيريز هذه convergent فالسيريز
+
+194
+00:25:54,680 --> 00:25:55,780
+هذه convergent
+
+195
+00:25:58,080 --> 00:26:03,500
+وبالتالي طبعا أن هذا صحيح لكل M بالمناسبة بقدر أن
+
+196
+00:26:03,500 --> 00:26:08,880
+هذه أيضا sequence of partial sums هذه ال limit
+
+197
+00:26:08,880 --> 00:26:19,460
+تبعتها exist وبالتالي ال infinite series هذه إذا
+
+198
+00:26:19,460 --> 00:26:27,250
+أن ال ممكن نقول أن هذا الكلام صحيحالان لو كانت ال
+
+199
+00:26:27,250 --> 00:26:31,430
+series هادي convergent فنضربها في ثابت اتنين تطلع
+
+200
+00:26:31,430 --> 00:26:35,270
+convergent وبالتالي ال series هادي convergent by
+
+201
+00:26:35,270 --> 00:26:40,170
+direct comparison test العكس لو كانت ال series
+
+202
+00:26:40,170 --> 00:26:41,670
+هادي convergent
+
+203
+00:26:44,460 --> 00:26:50,840
+فلما أضفلها حد عدد موجب بيبقى conversion وبالتالي
+
+204
+00:26:50,840 --> 00:26:54,080
+by direct comparison test ال series الأصغر بتطلع
+
+205
+00:26:54,080 --> 00:26:58,160
+conversion okay تمام؟ لأن هذا بثبت koshi
+
+206
+00:26:58,160 --> 00:27:03,600
+condensation test هذا ال test قوي كتير ويله فوائد
+
+207
+00:27:03,600 --> 00:27:13,000
+كتيرة فمن الفوائد تبعته يعني
+
+208
+00:27:13,000 --> 00:27:13,800
+هذه مثال
+
+209
+00:27:22,170 --> 00:27:37,410
+ممكن نستنتج ال test P-series مثال،
+
+210
+00:27:37,410 --> 00:27:46,290
+أنا موجود في أحدى التمرين التمرين 13
+
+211
+00:27:53,040 --> 00:28:05,440
+تعملين تلتاش سيكشن تلاتة سبعة ايش بيقول هذا if if
+
+212
+00:28:05,440 --> 00:28:16,600
+P أكبر من السفر is a real number discuss
+
+213
+00:28:16,600 --> 00:28:20,940
+the
+
+214
+00:28:20,940 --> 00:28:21,680
+convergence
+
+215
+00:28:42,640 --> 00:28:44,720
+تعالوا نفحص
+
+216
+00:28:49,400 --> 00:28:58,120
+Summation from n equals one to infinity لإتنين أُس
+
+217
+00:28:58,120 --> 00:29:08,700
+n في واحد على هاي أو خلّيني أقول إتنين أُس n في a
+
+218
+00:29:08,700 --> 00:29:16,120
+and a إتنين أُس m إيش بيساوي هذا طبعا هاي عندي a n
+
+219
+00:29:16,120 --> 00:29:24,230
+هذا هو عبارة عن a mالحد العام لل series فان بساوي
+
+220
+00:29:24,230 --> 00:29:30,290
+1 على n to p فبتبحث هل ال series هذي convergent او
+
+221
+00:29:30,290 --> 00:29:33,990
+متى بتكون هذي ال series convergent وبالتالي بقدر
+
+222
+00:29:33,990 --> 00:29:37,890
+اطبق اللي هو cauchy condensation test فهذه عبارة
+
+223
+00:29:37,890 --> 00:29:43,970
+عن sigma from n equals one to infinityالان ايه
+
+224
+00:29:43,970 --> 00:29:53,550
+اتنين اص ان بطلع واحد على اتنين اص ان الكل اص P
+
+225
+00:29:53,550 --> 00:30:03,810
+تمام؟ وهذا بيساوي summation from n equals one to
+
+226
+00:30:03,810 --> 00:30:18,940
+infinity لاتنين اص واحد minus Pالكل أسئلة وهدي
+
+227
+00:30:18,940 --> 00:30:27,020
+is a geometric series is a geometric series
+
+228
+00:30:27,020 --> 00:30:33,680
+وبالتالي
+
+229
+00:30:33,680 --> 00:30:38,320
+مظبوط هذا عبارة عن geometric series لو بدى أكتب
+
+230
+00:30:38,320 --> 00:30:40,620
+حدود تبعتها
+
+231
+00:30:43,100 --> 00:30:52,660
+فاول حد عبارة عن اتنين اص واحد minus P الحد التاني
+
+232
+00:30:52,660 --> 00:31:00,940
+اتنين اص واحد minus P الكل تربية و هكذا فالحد
+
+233
+00:31:00,940 --> 00:31:05,480
+الاول اتنين اص واحد minus P الحد التاني اتنين اص
+
+234
+00:31:05,480 --> 00:31:09,980
+واحد minus P و هكذا with ratio
+
+235
+00:31:14,090 --> 00:31:28,710
+with ratio with
+
+236
+00:31:28,710 --> 00:31:34,830
+ratio R
+
+237
+00:31:34,830 --> 00:31:41,790
+بساوي اتنين اص واحد minus P
+
+238
+00:31:48,590 --> 00:31:58,790
+So by geometric series test it converges if
+
+239
+00:31:58,790 --> 00:32:06,450
+and all if absolute R بيساوي اتنين أس واحد minus P
+
+240
+00:32:06,450 --> 00:32:16,670
+أصغر من واحد وهذا بتحقق اتنين أس واحد minus P أصغر
+
+241
+00:32:16,670 --> 00:32:25,590
+من واحدفنقول if واحد minus P اذا
+
+242
+00:32:25,590 --> 00:32:36,910
+كان واحد minus P أصغر من السفر سالم لأن لو كان
+
+243
+00:32:36,910 --> 00:32:41,430
+واحد minus P موجب فاتنين أس أي عدد موجب عمره ما
+
+244
+00:32:41,430 --> 00:32:47,440
+بيكون أصغر من واحدنصبوت لكن لو كان الأس سالم فبصير
+
+245
+00:32:47,440 --> 00:32:52,620
+هذا واحد على اتنين أس وموجب فبصير أصغر من واحد اذا
+
+246
+00:32:52,620 --> 00:32:57,020
+هذا صحيح if and only if الأس تابع الأتنين اللي هو
+
+247
+00:32:57,020 --> 00:33:06,240
+واحد minus P أصغر من سفر if and only if واحد أصغر
+
+248
+00:33:06,240 --> 00:33:12,920
+من P أو P أكبر من واحد okay تماموهذا هو ال P
+
+249
+00:33:12,920 --> 00:33:19,120
+Series Test لان احنا استنتجنا ال P Series Test من
+
+250
+00:33:19,120 --> 00:33:26,200
+Koshi Condensation Test فاكرين ال P Series هذي او
+
+251
+00:33:26,200 --> 00:33:29,840
+ال P Series Test اثبتنا ان Convergent if and only
+
+252
+00:33:29,840 --> 00:33:35,200
+if P أكبر من 1 وDivergent اذا كانت P أصغر منها
+
+253
+00:33:35,200 --> 00:33:35,960
+وسائل 1
+
+254
+00:33:42,110 --> 00:33:51,730
+Okay إذا ال .. هذا المعنى So by Cauchy Cauchy's
+
+255
+00:33:51,730 --> 00:34:01,910
+Condensation Test The series Sigma
+
+256
+00:34:01,910 --> 00:34:07,830
+from N equals one to infinity ال one over N to P
+
+257
+00:34:08,830 --> 00:34:16,530
+convergence if and only if P أكبر من واحد وهذا هو
+
+258
+00:34:16,530 --> 00:34:23,030
+ال P-series test إذن هذا بورجينا قوة Koshi
+
+259
+00:34:23,030 --> 00:34:29,530
+Condensation Test okay تمام؟ في طبعا أسئلة أخرى
+
+260
+00:34:29,530 --> 00:34:33,430
+على Koshi Condensation Test وانا طالب منكم تحلوها
+
+261
+00:34:33,430 --> 00:34:41,340
+زي السؤال 14 و15 صح؟ففي أي شيء في الأسئلة دي أو
+
+262
+00:34:41,340 --> 00:34:47,160
+أسئلة تانية؟
+
+263
+00:34:47,160 --> 00:34:56,500
+في
+
+264
+00:34:56,500 --> 00:34:58,180
+عندكم أي أسئلة؟
+
+265
+00:35:13,000 --> 00:35:19,560
+إذا سيكشن واحد تلاتة سبعة في أي سؤال تاني عندكم في
+
+266
+00:35:19,560 --> 00:35:25,540
+الأسئلة هذه أو
+
+267
+00:35:25,540 --> 00:35:31,860
+السيكاشن السابقة أو سيكشن أربعة واحد إذا بتحبه
+
+268
+00:35:31,860 --> 00:35:35,560
+سيكشن أربعة واحد
+
+269
+00:36:06,090 --> 00:36:13,070
+مافيش أسئلة؟ طيب ال .. مدان مافيش أسئلة نواصل ..
+
+270
+00:36:13,070 --> 00:36:16,190
+نكمل
+
+271
+00:36:16,190 --> 00:36:17,490
+المحاضرة في السابقة
+
+272
+00:36:49,090 --> 00:36:53,250
+المرة الأخرى اتحدثنا عن ال two-sided limits و عن
+
+273
+00:36:53,250 --> 00:37:00,350
+ال one-sided limits و أخدنا بعض النظريات و قلنا إن
+
+274
+00:37:00,350 --> 00:37:05,090
+جميع النظريات اللي برهنناها هو one-sided limit
+
+275
+00:37:05,090 --> 00:37:12,990
+صحيحة لل two-sided limitsأو المباريات الصحيحة لـ
+
+276
+00:37:12,990 --> 00:37:17,070
+two-sided limits بتكون أيضا صحيحة لـ one-sided
+
+277
+00:37:17,070 --> 00:37:26,650
+limit فناخد
+
+278
+00:37:26,650 --> 00:37:31,350
+أنفلة show
+
+279
+00:37:31,350 --> 00:37:31,950
+that
+
+280
+00:37:35,020 --> 00:37:55,100
+Limit لـ Signum X لإن X تقول لسفر لا يوجد فنلاحظ
+
+281
+00:37:55,100 --> 00:37:59,600
+أن Limit لأول شئ Signum X
+
+282
+00:38:03,790 --> 00:38:11,230
+بساوي x على absolute x لكل x لا يساوي سفر لما
+
+283
+00:38:11,230 --> 00:38:15,010
+أعرفنا الدالة هذه قلت لها بس هي نفسها x على
+
+284
+00:38:15,010 --> 00:38:20,690
+absolute x لو كان x بساوي سفر الآن ال limit ل
+
+285
+00:38:20,690 --> 00:38:30,890
+sigma x لما x تقول إلى سفر من اليمين بساوي ال
+
+286
+00:38:30,890 --> 00:38:31,310
+limit
+
+287
+00:38:35,810 --> 00:38:41,530
+لما x تقول إلى صفر من اليمين لما x تقول إلى صفر من
+
+288
+00:38:41,530 --> 00:38:50,190
+اليمين لما x تقول إلى صفر من اليمين لما x تقول إلى
+
+289
+00:38:50,190 --> 00:38:55,650
+صفر من اليمين لما x تقول إلى صفر من اليمين لما x
+
+290
+00:38:55,650 --> 00:38:57,330
+تقول إلى صفر من اليمين لما x تقول إلى صفر من
+
+291
+00:38:57,330 --> 00:39:02,560
+اليمين لما x تقول إلى صفر من اليمينلما x تقول أصغر
+
+292
+00:39:02,560 --> 00:39:21,640
+من اليسار لما x أصغر من صفر لما
+
+293
+00:39:21,640 --> 00:39:28,560
+x أصغر من صفر لما x أصغر من صفر لما x أصغر من صفر
+
+294
+00:39:28,560 --> 00:39:33,950
+لما x أصغر من صفرالسالب واحد بيطلع السالب واحد ان
+
+295
+00:39:33,950 --> 00:39:37,670
+انا عندي ال limit من اليامين يساوي واحد ال limit
+
+296
+00:39:37,670 --> 00:39:44,230
+من اليسار يساوي سالب واحد مش متساوي اتين so by
+
+297
+00:39:44,230 --> 00:39:50,150
+theorem حسب النظرية اللي أخدناها theorem اربعة
+
+298
+00:39:50,150 --> 00:39:55,630
+تلاتة تلاتة بيطلع
+
+299
+00:39:55,630 --> 00:40:01,080
+عندي ال limit او ال two sided limitللـ signal
+
+300
+00:40:01,080 --> 00:40:09,560
+function لما x تقول السفر does not exist تمام؟
+
+301
+00:40:09,560 --> 00:40:22,280
+طيب خلّيني انا اخد show
+
+302
+00:40:22,280 --> 00:40:27,380
+that ال
+
+303
+00:40:27,380 --> 00:40:32,350
+limit لل function e والواحد على xلما x تقول إلى
+
+304
+00:40:32,350 --> 00:40:40,550
+سفر من اليمين does not exist and
+
+305
+00:40:40,550 --> 00:40:43,910
+من
+
+306
+00:40:43,910 --> 00:40:51,170
+ال limit لنفس ال function e to واحد على x لما x
+
+307
+00:40:51,170 --> 00:40:58,710
+تقول إلى سفر من اليسار تطلع موجودة و بساوي سفر
+
+308
+00:41:23,830 --> 00:41:31,010
+طيب ال ..
+
+309
+00:41:31,010 --> 00:41:34,050
+نحاول نبرهن الجزء الأول
+
+310
+00:41:56,420 --> 00:42:03,380
+بناخد الجزء الأول let
+
+311
+00:42:03,380 --> 00:42:13,540
+z of x بساوي e to 1 على x حفة x لا تساوي 0 وبدنا
+
+312
+00:42:13,540 --> 00:42:19,260
+نثبت to
+
+313
+00:42:19,260 --> 00:42:28,130
+show ان ال limitلـ g of x لما x تقول لصفر من
+
+314
+00:42:28,130 --> 00:42:38,750
+اليمين does not exist it suffices to
+
+315
+00:42:38,750 --> 00:42:42,710
+show يكفي
+
+316
+00:42:42,710 --> 00:42:52,650
+اثبات ان ال function g of x is not bounded on
+
+317
+00:42:56,170 --> 00:43:05,850
+on a right .. on a right neighborhood
+
+318
+00:43:05,850 --> 00:43:13,670
+.. on a right neighborhood اللي هو سفر دلتا of
+
+319
+00:43:13,670 --> 00:43:15,230
+zero
+
+320
+00:43:25,230 --> 00:43:28,670
+أخذنا قبل ذلك نظرية بتقول إيه ده عشان أثبت أنه ال
+
+321
+00:43:28,670 --> 00:43:35,710
+limit ل function عن نقطة معينة مش موجودة يكفي أثبت
+
+322
+00:43:35,710 --> 00:43:43,910
+أنه أنه الدالة unbounded عند أي unbounded
+
+323
+00:43:43,910 --> 00:43:48,650
+عند أي neighborhood
+
+324
+00:43:48,650 --> 00:43:56,210
+للنقطة الآن بالنسبة لل one-sided limitعشان أقول إن
+
+325
+00:43:56,210 --> 00:44:02,230
+ال limit ل function زي هذه g of x لما x تقول إلى
+
+326
+00:44:02,230 --> 00:44:09,430
+سفر من اليمين does not exist فهي
+
+327
+00:44:09,430 --> 00:44:16,390
+السفر و ال x تقول لسفر من اليمين فبدل ما أخد delta
+
+328
+00:44:16,390 --> 00:44:20,290
+neighborhood للسفر
+
+329
+00:44:20,290 --> 00:44:29,960
+فباخد right neighborhoodright neighborhood للسفر
+
+330
+00:44:29,960 --> 00:44:35,960
+فيكفي ان ال function هذه ماهياش bounded عن كل
+
+331
+00:44:35,960 --> 00:44:41,780
+right neighborhood يعني جوار من اليمين للسفر لان
+
+332
+00:44:41,780 --> 00:44:46,000
+انا بتعامل مع نهاية من اليمين لكن لما كنت اتعامل
+
+333
+00:44:46,000 --> 00:44:51,240
+مع نهاية من الطرفين فكنت ااخد delta neighborhood
+
+334
+00:44:51,240 --> 00:44:56,840
+كاملفلو أثبتت إن الـ function هذه ماهياش bounded
+
+335
+00:44:56,840 --> 00:45:01,280
+عند أي right neighborhood للصفر على الصورة هذه
+
+336
+00:45:01,280 --> 00:45:06,220
+فحسب نظرية سابقة الدالة مش ممكن يكون لها limit من
+
+337
+00:45:06,220 --> 00:45:09,960
+اليمين عند الصفر لأن لو كان لها limit عند الصفر من
+
+338
+00:45:09,960 --> 00:45:15,820
+اليمين فلازم تكون bounded على some neighborhood ..
+
+339
+00:45:15,820 --> 00:45:25,650
+right neighborhood للصفرOkay تمام و لإثبات ذلك to
+
+340
+00:45:25,650 --> 00:45:29,270
+see
+
+341
+00:45:29,270 --> 00:45:39,290
+this we use ال inequality التالية وهي T أكبر من
+
+342
+00:45:39,290 --> 00:45:47,010
+سفر دايما أصغر من E أس T for all T أكبر من سفر هذه
+
+343
+00:45:47,010 --> 00:45:54,200
+المتباينةهذه المتباينة موجودة
+
+344
+00:45:54,200 --> 00:46:01,780
+برهانة C Chapter 8 برهانة
+
+345
+00:46:01,780 --> 00:46:07,600
+موجودة في Chapter 8 اللي هتاخدوه لاحقا فهنستخدم
+
+346
+00:46:07,600 --> 00:46:11,460
+اللي هو المتباينة هذه في اثبات ان ال function
+
+347
+00:46:11,460 --> 00:46:17,420
+ماهياش bounded على neighborhood او right
+
+348
+00:46:17,420 --> 00:46:25,130
+neighborhood للصفرOkay عشان الوجد خلص بنوقف و
+
+349
+00:46:25,130 --> 00:46:29,590
+بناخد خمس دقايق break و بعدين بنكمل ان شاء الله
+
+350
+00:46:29,590 --> 00:46:35,550
+البرهانة فحنوقف و نكمل في الجزء التالي من المحاضرة
+
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/2qaKB7theEQ_postprocess.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/2qaKB7theEQ_postprocess.srt
new file mode 100644
index 0000000000000000000000000000000000000000..b89a6e27d6d974580d69fb80e82c8077b047c6a2
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/2qaKB7theEQ_postprocess.srt
@@ -0,0 +1,1608 @@
+1
+00:00:19,940 --> 00:00:25,840
+السلام عليكم هنكمل
+
+2
+00:00:25,840 --> 00:00:33,420
+اليوم section اربعة اتنين في ال section هذا كان
+
+3
+00:00:33,420 --> 00:00:38,780
+اتبقى بس ان احنا نثبت النظرية اللى كتبتها على
+
+4
+00:00:38,780 --> 00:00:44,700
+اللوحى النظرية هذه بتنص على ان لو كان في هندي
+
+5
+00:00:44,700 --> 00:00:53,020
+function من A لRو c cluster point للset a وإذا كان
+
+6
+00:00:53,020 --> 00:00:59,120
+limit ال function عن c exist وموجبة أو
+
+7
+00:00:59,120 --> 00:01:04,880
+على التوالي إذا كانت limit f of x عن c موجودة
+
+8
+00:01:04,880 --> 00:01:09,080
+وسالبة فيوجد
+
+9
+00:01:09,080 --> 00:01:14,990
+نقدر نلاقي delta neighborhood دي delta لل cبحيث إن
+
+10
+00:01:14,990 --> 00:01:19,510
+الدالة هتكون إذا كانت ال limit موجبة فالدالة هتكون
+
+11
+00:01:19,510 --> 00:01:26,670
+موجبة على ال delta never hood ل C وإذا
+
+12
+00:01:26,670 --> 00:01:31,950
+كانت ال limit سالبة فالدالة هتكون سالبة على جوار
+
+13
+00:01:31,950 --> 00:01:38,890
+delta ل C هذه
+
+14
+00:01:38,890 --> 00:01:43,210
+نظرية تشبه نظرية سابقة بخصوص limits of sequences
+
+15
+00:01:44,920 --> 00:01:48,400
+النظرية اللي فاتت بتاعة الـ sequences المشابهة
+
+16
+00:01:48,400 --> 00:01:52,640
+بتقول لو كانت ال sequence النهاية تبعتها limit x
+
+17
+00:01:52,640 --> 00:01:58,200
+in exist وموجبة فلازم ال sequence تكون حدودها من
+
+18
+00:01:58,200 --> 00:02:02,420
+capital N وانت طالع كلها موجبة و لو كانت ال limit
+
+19
+00:02:02,420 --> 00:02:06,640
+لل sequence exist و سالبة فلازم حدود ال sequence
+
+20
+00:02:06,640 --> 00:02:10,900
+من capital N وانت طالع كلها تكون سالبة فهذه شبيهة
+
+21
+00:02:10,900 --> 00:02:18,100
+فيهاوالبرهان سهل وشبيه بالبرهان تبع النظرية
+
+22
+00:02:18,100 --> 00:02:24,320
+المشابهة في حالة ال sequences فناخد الحالة assume
+
+23
+00:02:24,320 --> 00:02:27,360
+ناخد
+
+24
+00:02:27,360 --> 00:02:32,400
+الحالة اللي فيها ال limit ل
+
+25
+00:02:32,400 --> 00:02:40,060
+f of x at c exists and equals عدد l موجب
+
+26
+00:02:53,880 --> 00:03:00,200
+فإذا كانت ال limit موجبة بنا أثبت أن يوجد delta
+
+27
+00:03:00,200 --> 00:03:07,240
+neverhood إلى آخر A فخلّينا
+
+28
+00:03:07,240 --> 00:03:17,740
+ناخد let epsilon في الحالة دي let epsilon بساوي
+
+29
+00:03:17,740 --> 00:03:26,600
+L على 2 فهذا عدد موجبالان by definition of limit
+
+30
+00:03:26,600 --> 00:03:32,640
+of function by epsilon delta definition لأي
+
+31
+00:03:32,640 --> 00:03:38,140
+epsilon موجبة زي هذه يوجد delta تعتمد على L على 2
+
+32
+00:03:38,140 --> 00:03:43,280
+اللي هي ال epsilon عدد موجب بحيث انه لو كان X
+
+33
+00:03:45,890 --> 00:03:51,090
+ينتمي إلى a و absolute x minus c أصغر من delta
+
+34
+00:03:51,090 --> 00:03:59,790
+أكبر من 0 فهذا بتضمن أن absolute f of x minus L
+
+35
+00:03:59,790 --> 00:04:04,510
+أصغر من إبسلم اللي هي عبارة عن L ع 2
+
+36
+00:04:08,170 --> 00:04:15,990
+فحل المتباينة هذه في f of x فتصير f of x minus L
+
+37
+00:04:15,990 --> 00:04:24,930
+أصغر من L على 2 أكبر من سالب L على 2 وهذا
+
+38
+00:04:24,930 --> 00:04:29,210
+بيقدي أن
+
+39
+00:04:29,210 --> 00:04:30,350
+f of x
+
+40
+00:04:34,740 --> 00:04:45,980
+من هنا F of X تطلع أكبر من L على 2 لأنه لما أخد
+
+41
+00:04:45,980 --> 00:04:50,240
+سالب L أنجلها عن ناحية التانية فتصير F of X أكبر
+
+42
+00:04:50,240 --> 00:04:55,700
+من L سالب L على 2 تطلع L على 2 و ال L موجبة إذا L
+
+43
+00:04:55,700 --> 00:05:06,860
+على 2 موجبة إذا هيك بنكون أثبتناإن ال F of X طلعت
+
+44
+00:05:06,860 --> 00:05:18,580
+أكبر من سفر لمين لكل X تنتمي إلى A ومن
+
+45
+00:05:18,580 --> 00:05:28,080
+المتباينة هذه هذا معناه X لا تساوي Cإن الـ X ينتمي
+
+46
+00:05:28,080 --> 00:05:34,480
+إلى A و لا تساوي C يعني موجودة في A و مش موجودة في
+
+47
+00:05:34,480 --> 00:05:44,280
+singleton set C و المتباينة هذه هذي معناها إن X
+
+48
+00:05:44,280 --> 00:05:46,460
+ينتمي ل V Delta
+
+49
+00:05:56,010 --> 00:06:00,790
+x-c أصغر من دلتا بكافئ
+
+50
+00:06:10,030 --> 00:06:17,770
+إن X أصغر من C زائد Delta أكبر من C سارب Delta
+
+51
+00:06:17,770 --> 00:06:21,550
+فهذا
+
+52
+00:06:21,550 --> 00:06:27,890
+معناه X تمتني لفترة مفتوحة Delta-neighborhood ل-C
+
+53
+00:06:29,570 --> 00:06:34,830
+Okay تمام اذا f of x اللي اعطاها موجبة لكل x في a
+
+54
+00:06:34,830 --> 00:06:43,250
+ومختلفة عن c وايضا من هنا ال x أيضا تنتمي ل delta
+
+55
+00:06:43,250 --> 00:06:48,850
+neighborhood ل c وبالتالي تنتمي لتقاطع المجمعتين
+
+56
+00:06:48,850 --> 00:06:55,270
+اذا هذا بثبت المظرية في حالة لما يكون ال limit
+
+57
+00:06:55,270 --> 00:06:56,490
+تبعتي موجبة
+
+58
+00:07:00,240 --> 00:07:08,880
+لو كانت ال limit سالبة فالبرهان مشابه لأن ال
+
+59
+00:07:08,880 --> 00:07:18,680
+proof of the caseلما تكون ال limit ل f of x لما x
+
+60
+00:07:18,680 --> 00:07:30,600
+تقول ل c بساوي العدد سالب is similar to
+
+61
+00:07:30,600 --> 00:07:38,200
+above case مشابه
+
+62
+00:07:38,200 --> 00:07:50,120
+للبرهان السابق علىفي الحالة هذه take start with
+
+63
+00:07:50,120 --> 00:07:55,920
+epsilon بساوي سالب ال ع اتنين وهذا بطلع عدد موجب
+
+64
+00:07:55,920 --> 00:08:01,060
+يعني ابدوا البرهان بدل ما بدنا بepsilon بساوي ال ع
+
+65
+00:08:01,060 --> 00:08:04,340
+اتنين ابدوا ابسون .. ابسون بساوي
+
+66
+00:08:14,820 --> 00:08:20,320
+البرهان البرهان البرهان البرهان البرهان البرهان
+
+67
+00:08:20,320 --> 00:08:20,800
+البرهان البرهان البرهان البرهان البرهان البرهان
+
+68
+00:08:20,800 --> 00:08:20,860
+البرهان البرهان البرهان البرهان البرهان البرهان
+
+69
+00:08:20,860 --> 00:08:24,600
+البرهان البرهان البرهان البرهان البرهان البرهان
+
+70
+00:08:24,600 --> 00:08:24,640
+البرهان البرهان البرهان البرهان البرهان البرهان
+
+71
+00:08:24,640 --> 00:08:25,080
+البرهان البرهان البرهان البرهان البرهان البرهان
+
+72
+00:08:25,080 --> 00:08:33,380
+البرهان البرهان
+
+73
+00:08:34,550 --> 00:08:40,910
+لأن ال L سالمة و هذا صحيح لكل X في جوار Delta ل C
+
+74
+00:08:40,910 --> 00:08:46,530
+و في A minus single to C okay؟ لأن حاسبكم أنتم
+
+75
+00:08:46,530 --> 00:08:50,990
+تكتبوا البرهان تبع الحالة التانية تمام؟ واضح؟ في
+
+76
+00:08:50,990 --> 00:08:54,850
+أي سؤال أو سفسار؟ تمام؟
+
+77
+00:09:00,180 --> 00:09:08,120
+Okay إذا نبدأ section جديد
+
+78
+00:09:08,120 --> 00:09:26,200
+section
+
+79
+00:09:26,200 --> 00:09:29,460
+أربعة تلاتة
+
+80
+00:09:33,950 --> 00:09:47,190
+بعض التطبيقات .. بعض التطبيقات لـ
+
+81
+00:09:47,190 --> 00:09:53,590
+limit concept
+
+82
+00:10:04,410 --> 00:10:12,090
+بعض التعاملات أو توصية بعض مفاهيم النهايات احنا
+
+83
+00:10:12,090 --> 00:10:16,470
+قبل هيك درسنا في section 4.1 و 4.2 ال limit of
+
+84
+00:10:16,470 --> 00:10:20,450
+function او ال two sided limit لل function عن نقطة
+
+85
+00:10:20,450 --> 00:10:24,810
+معينة اليوم هندرس ال one sided limit ل function عن
+
+86
+00:10:24,810 --> 00:10:30,960
+نقطة and cluster point للمجال تبعهامقصود بال one
+
+87
+00:10:30,960 --> 00:10:34,040
+sided limit اللي هو limit من اليمين أو limit من
+
+88
+00:10:34,040 --> 00:10:38,740
+اليسار ونشوف ما هي علاقة ال one sided limit بال
+
+89
+00:10:38,740 --> 00:10:43,320
+two sided limit فنعرف
+
+90
+00:10:43,320 --> 00:10:47,880
+الأول definition نعرف ال one sided limit
+
+91
+00:10:47,880 --> 00:10:53,000
+definition let
+
+92
+00:10:55,520 --> 00:11:03,780
+fb function from a to r and c be a cluster point
+
+93
+00:11:03,780 --> 00:11:06,840
+cluster
+
+94
+00:11:06,840 --> 00:11:22,340
+point of a واحد او
+
+95
+00:11:22,340 --> 00:11:34,220
+خلّيالـ cluster point of المجموعة A
+
+96
+00:11:34,220 --> 00:11:40,880
+تقاطع الفترة المفتوحة من C إلى infinity اللي هي كل
+
+97
+00:11:40,880 --> 00:11:47,240
+ال X مجموعة كل العناصر X تنتبه إلى A حيث X أكبر من
+
+98
+00:11:47,240 --> 00:11:53,520
+C نقول
+
+99
+00:11:57,140 --> 00:12:03,280
+إن العدد alien 10 إلى R is
+
+100
+00:12:03,280 --> 00:12:14,600
+a right .. is a right hand limit .. right hand
+
+101
+00:12:14,600 --> 00:12:29,490
+limit of ال function F at ..x بساوي c if الشرط
+
+102
+00:12:29,490 --> 00:12:35,570
+التالي بتحقق لكل
+
+103
+00:12:35,570 --> 00:12:40,630
+إبسلون given
+
+104
+00:12:40,630 --> 00:12:43,970
+إبسلون
+
+105
+00:12:43,970 --> 00:12:51,040
+أكبر من السفر يوجد delta تعتمد علىepsilon عدد موجة
+
+106
+00:12:51,040 --> 00:13:01,180
+بهاث انه لو كان x ينتمي ل a و x minus c أكبر من
+
+107
+00:13:01,180 --> 00:13:07,760
+سفر أصغر من delta فهذا بتضمن ان absolute f of x
+
+108
+00:13:07,760 --> 00:13:13,540
+minus l أصغر من epsilon in
+
+109
+00:13:13,540 --> 00:13:17,720
+this case in
+
+110
+00:13:17,720 --> 00:13:28,480
+this casewe write نكتب أن ال limit لل function f
+
+111
+00:13:28,480 --> 00:13:37,920
+عندما x تقول إلى c من اليمين بساوي العدد ال ..
+
+112
+00:13:37,920 --> 00:13:44,920
+تمام؟ لأن هذا تعريف ال limit from the right أو ال
+
+113
+00:13:44,920 --> 00:13:49,460
+right hand limit لل function f عند النقطة c
+
+114
+00:14:05,180 --> 00:14:13,440
+إذا أنا عندي هذه خط الأعداد وهي النقطة C وانا عندي
+
+115
+00:14:13,440 --> 00:14:21,900
+ال C هي cluster point ل
+
+116
+00:14:21,900 --> 00:14:26,180
+A .. لكل ال X موجود في A و أكبر من C
+
+117
+00:14:32,710 --> 00:14:37,650
+فبنقول إن ال limit عند x بالساوية c أو ال function
+
+118
+00:14:37,650 --> 00:14:42,150
+في إلها right-hand limit و ال right-hand limit هي
+
+119
+00:14:42,150 --> 00:14:48,410
+العدد L إذا كان لأي إبسلون أكبر من السفر بتقدر
+
+120
+00:14:48,410 --> 00:14:53,390
+نلاقي delta عدد موجة بيعتمد على إبسلون بحيث لكل x
+
+121
+00:14:53,390 --> 00:15:01,510
+في المجموعة A إذا كانت ال X هذه على يمين ال C
+
+122
+00:15:05,140 --> 00:15:12,460
+والمسافة بينها وبين الـ C أصغر من Delta فبتطلع
+
+123
+00:15:12,460 --> 00:15:19,860
+المسافة بين F و XL أصغر من Y بالمثل
+
+124
+00:15:19,860 --> 00:15:24,100
+ممكن نعرف ال limit from the right يعني أنا بدي
+
+125
+00:15:24,100 --> 00:15:27,800
+أعرف ال limit from the right أو ال right hand
+
+126
+00:15:27,800 --> 00:15:34,750
+limitهي نفس let f be function from A to R و C
+
+127
+00:15:34,750 --> 00:15:41,010
+cluster point للمجموعة A تقاطع الفترة المفتوحة من
+
+128
+00:15:41,010 --> 00:15:50,850
+سالب مالة نهاية إلى C اللي هي كل ال X في A حيث X
+
+129
+00:15:50,850 --> 00:15:58,490
+هتكون أصغر من مرة هذه أصغر من Cفنقول إن الـ real
+
+130
+00:15:58,490 --> 00:16:06,070
+number L هو بدل right hand limit هيكون left hand
+
+131
+00:16:06,070 --> 00:16:10,550
+limit of f at c if given epsilon there exists
+
+132
+00:16:10,550 --> 00:16:15,310
+delta depends on epsilon بحيث أنه لكل x ينتمي إلى
+
+133
+00:16:15,310 --> 00:16:21,900
+aلكل X هنتمي إلى A وال X طبعا موجودة في الفترة هذه
+
+134
+00:16:21,900 --> 00:16:28,600
+يعني ال X المرة هذه على يسار المرة هذه ال X هتكون
+
+135
+00:16:31,420 --> 00:16:35,260
+موجودة في A وفي الفترة المفتوحة من سالب مالنهاية
+
+136
+00:16:35,260 --> 00:16:44,720
+إلى C يعني ال X هتكون على يسار ال C وبالتالي هنا
+
+137
+00:16:44,720 --> 00:16:51,240
+ال C minus المسافة بين X و C absolute X minus C
+
+138
+00:16:51,240 --> 00:16:57,640
+هتطلع بساوي C minus X فلو كانت المسافة هذه أصغر من
+
+139
+00:16:57,640 --> 00:16:59,660
+Delta وطبعا أكبر من سفر
+
+140
+00:17:09,550 --> 00:17:18,370
+هذا الشرط سيصبح c-x أصغر من دلتا أكبر من سفر فهذا
+
+141
+00:17:18,370 --> 00:17:22,910
+لازم يضمن أن absolute of f of x minus l أصغر من
+
+142
+00:17:22,910 --> 00:17:29,610
+إبسمن في الحالة هذه بيقول إن ال limit لf of x لما
+
+143
+00:17:29,610 --> 00:17:31,850
+x تقول لc من اليسار
+
+144
+00:17:34,230 --> 00:17:39,550
+بس n بساوي LL okay انها تعريف ال lift hand limit
+
+145
+00:17:39,550 --> 00:17:44,890
+او ال limit from the lift okay تعديل
+
+146
+00:17:44,890 --> 00:17:50,770
+بسيط بس طيب
+
+147
+00:17:50,770 --> 00:17:58,170
+ال limits هذه هنشوف يعني بعد شوية ان ال one sided
+
+148
+00:17:58,170 --> 00:18:03,210
+limits ده functional نقطةممكن يعني التنتين يكونوا
+
+149
+00:18:03,210 --> 00:18:09,750
+موجودين عند النقطة ويلهم نفس القيمة أو ممكن
+
+150
+00:18:09,750 --> 00:18:15,430
+التنتين يكونوا موجودين عند نقطة لكن قيمهم مختلفة
+
+151
+00:18:15,430 --> 00:18:20,270
+زي الـ Signum function عند الصفر شوفنا أن ال limit
+
+152
+00:18:20,270 --> 00:18:23,350
+تبعتها من اليمين واحد و ال limit تبعتها من اليسار
+
+153
+00:18:23,350 --> 00:18:27,850
+ثالث واحدإذا ممكن ال two sided limits يكونوا
+
+154
+00:18:27,850 --> 00:18:33,630
+موجودات لكن they are different مختلفات، ممكن برضه
+
+155
+00:18:33,630 --> 00:18:37,790
+one sided limit تكون موجودة and the other may not
+
+156
+00:18:37,790 --> 00:18:42,090
+exist، ممكن ما تكونش موجودة من أساسه
+
+157
+00:18:44,810 --> 00:18:54,930
+ممكن ال one sided limits ولا واحدة فيهم تكون
+
+158
+00:18:54,930 --> 00:19:03,250
+موجودة فكل الحلقات هذه هنشوفها في أمثلة لاحقة لكن
+
+159
+00:19:03,250 --> 00:19:08,030
+الأول خلّينا نبرهن النظرية التالية
+
+160
+00:19:13,670 --> 00:19:18,750
+طبعا هنا بنحب ال ..
+
+161
+00:19:18,750 --> 00:19:24,870
+النوّه أن كل نظريات اللي أثبتناها في section 4.1
+
+162
+00:19:24,870 --> 00:19:31,790
+أو 4.2 بخصوص ال two sided limit هتكون صحيحة بخصوص
+
+163
+00:19:31,790 --> 00:19:38,030
+ال right limit و كذلك صحيحة بخصوص ال left hand
+
+164
+00:19:38,030 --> 00:19:38,430
+limit
+
+165
+00:19:41,100 --> 00:19:46,240
+فعلى سبيل المثال وليس الحصر احنا أخدنا sequential
+
+166
+00:19:46,240 --> 00:19:51,240
+criterion sequential criterion for two sided limit
+
+167
+00:19:51,240 --> 00:19:56,580
+الآن هنكتب برضه sequential criterion for right
+
+168
+00:19:56,580 --> 00:20:09,420
+limit sequential criterion for right
+
+169
+00:20:09,420 --> 00:20:10,100
+hand
+
+170
+00:20:25,970 --> 00:20:35,670
+limits let f from a to r be function and c be
+
+171
+00:20:35,670 --> 00:20:37,330
+cluster
+
+172
+00:20:39,230 --> 00:20:47,910
+point of A then the following statements are
+
+173
+00:20:47,910 --> 00:20:54,190
+equivalent الأبارات التالية متكافئة واحد ال limit
+
+174
+00:20:54,190 --> 00:21:01,810
+ل F of X as X tends to C from the right exist
+
+175
+00:21:01,810 --> 00:21:06,030
+وبساوي عدد L اتنين
+
+176
+00:21:13,830 --> 00:21:20,370
+for for every sequence
+
+177
+00:21:20,370 --> 00:21:36,530
+x in contained in a تقاطع c و infinity such that
+
+178
+00:21:38,210 --> 00:21:47,290
+limit x in as n tends to infinity بيساوي c we have
+
+179
+00:21:47,290 --> 00:21:51,090
+limit
+
+180
+00:21:51,090 --> 00:21:59,170
+لل image of the sequence x in بيساوي العدد L
+
+181
+00:22:10,340 --> 00:22:17,060
+البرهان شبيه بالبرهان الخاص بالـ two-sided limit
+
+182
+00:22:17,060 --> 00:22:24,100
+فمثلا لو بدنا نبرهن proof لو بدى برهن الأبعار
+
+183
+00:22:24,100 --> 00:22:30,720
+الأولى بتأدي للتانية فبنقول assume .. نبدأ ب
+
+184
+00:22:30,720 --> 00:22:38,240
+assume أن ال limit ال right limitالـ F عند الـ C
+
+185
+00:22:38,240 --> 00:22:44,240
+exist بساوي L وبدنا
+
+186
+00:22:44,240 --> 00:22:48,680
+نثبت أن الـ two بيطلع العبارة تنين بتطلع صحيحة
+
+187
+00:22:48,680 --> 00:22:55,400
+لبرهان العبارة to prove two
+
+188
+00:22:55,400 --> 00:22:56,260
+holds
+
+189
+00:22:59,500 --> 00:23:08,320
+لت نبدأ لت xn contained in a تقاطع c إلى infinity
+
+190
+00:23:08,320 --> 00:23:12,360
+بsequence
+
+191
+00:23:12,360 --> 00:23:19,040
+such that ال limit تبعتها as n tends to infinity
+
+192
+00:23:19,040 --> 00:23:24,440
+بساوي c إذا أنا باخد sequence في المجموعة a
+
+193
+00:23:24,440 --> 00:23:30,410
+وحدودها كلهم أكبر من cو بفرض أن ال limit لل
+
+194
+00:23:30,410 --> 00:23:38,870
+sequence يادي بيساوي العدد c نحتاج أن نظهر عشان
+
+195
+00:23:38,870 --> 00:23:46,250
+نثبت اتنين باقي نثبت أن ال limit نحتاج أن نظهر أن
+
+196
+00:23:46,250 --> 00:23:53,530
+ال limit لل image of the sequence xn as n tends to
+
+197
+00:23:53,530 --> 00:24:01,630
+infinity بساوي Lهيك بنكون أثبتنا أن العبارة 2
+
+198
+00:24:01,630 --> 00:24:10,150
+صحيحة، مصبوط، صح؟ طيب لبرهان ذلك to
+
+199
+00:24:10,150 --> 00:24:11,130
+see this
+
+200
+00:24:16,090 --> 00:24:19,390
+بدأ اثبت ان ال limit لل sequence هذه بساوي عدد L
+
+201
+00:24:19,390 --> 00:24:23,890
+فبستخدم تعريف epsilon capital N لل limit فلازم
+
+202
+00:24:23,890 --> 00:24:31,510
+نبدأ with epsilon أكبر من السفر ب given طيب مش
+
+203
+00:24:31,510 --> 00:24:41,970
+احنا فرضينSince الـ right limit ل F and C موجود أو
+
+204
+00:24:41,970 --> 00:24:46,770
+بالساوي L من تعريف ال right limit there exists
+
+205
+00:24:46,770 --> 00:24:52,230
+delta depends on epsilon positive number بحيث إنه
+
+206
+00:24:52,230 --> 00:25:02,200
+لو كانت ال X تنتمي إلى A و X minus C أكبر من 0أصغر
+
+207
+00:25:02,200 --> 00:25:09,760
+من دلتا هذا معناه بيقدي أنه absolute f of x minus
+
+208
+00:25:09,760 --> 00:25:22,600
+L أصغر من إبسم نسمي ال implication هذي star now
+
+209
+00:25:22,600 --> 00:25:30,880
+for the aboveالدلتا أكبر من السفر لدلتا هذه
+
+210
+00:25:30,880 --> 00:25:34,720
+الاتباع الموجبة لدلتا هذه الاتباع الموجبة لدلتا
+
+211
+00:25:34,720 --> 00:25:36,800
+هذه الاتباع الموجبة لدلتا هذه الاتباع الموجبة
+
+212
+00:25:36,800 --> 00:25:38,440
+لدلتا هذه الاتباع الموجبة لدلتا هذه الاتباع
+
+213
+00:25:38,440 --> 00:25:41,400
+الموجبة لدلتا هذه الاتباع الموجبة لدلتا هذه
+
+214
+00:25:41,400 --> 00:25:41,660
+الاتباع الموجبة لدلتا هذه الاتباع الموجبة لدلتا
+
+215
+00:25:41,660 --> 00:25:42,040
+هذه الاتباع الموجبة لدلتا هذه الاتباع الموجبة
+
+216
+00:25:42,040 --> 00:25:43,160
+لدلتا هذه الاتباع الموجبة لدلتا هذه الاتباع
+
+217
+00:25:43,160 --> 00:25:51,140
+الموجبة لدلتا هذه الاتباع الموجبة لدلتا هذه
+
+218
+00:25:51,140 --> 00:25:57,180
+الاتباع الموجبة لدلتا هذه الاتباع الموجبة لدلت
+
+219
+00:25:57,830 --> 00:26:05,110
+natural number عدد طبيعي بحيث انه لو كان ال N أكبر
+
+220
+00:26:05,110 --> 00:26:12,510
+من أو ساوي capital N فهذا بتضمن ان absolute xn
+
+221
+00:26:12,510 --> 00:26:20,090
+minus c أصغر من delta نسمي ال implication هذه
+
+222
+00:26:20,090 --> 00:26:21,050
+double star
+
+223
+00:26:30,480 --> 00:26:44,680
+hence و بالتالي star and double star imply بيؤدوا
+
+224
+00:26:44,680 --> 00:26:51,860
+إلى ما يلي انه لو كانت ال N أكبر من أو ساوي
+
+225
+00:26:51,860 --> 00:26:56,360
+capital N فمن
+
+226
+00:26:56,360 --> 00:26:57,380
+double star
+
+227
+00:26:59,860 --> 00:27:04,940
+لو كانت n أكبر من أو ساوي capital N فمن double
+
+228
+00:27:04,940 --> 00:27:21,340
+star بطلع absolute xn minus c أصغر من delta هذا
+
+229
+00:27:21,340 --> 00:27:24,800
+بيقدّي أن xn
+
+230
+00:27:26,360 --> 00:27:35,400
+minus C أكبر من سفر أصغر من Delta ليه؟ لأن ال XM
+
+231
+00:27:35,400 --> 00:27:44,420
+موجودة تنتمي لإيه؟ هو أكبر من C، لذلك هذا لأن XM
+
+232
+00:27:44,420 --> 00:27:48,400
+أكبر
+
+233
+00:27:48,400 --> 00:27:58,050
+من Cفبالتالي absolute xn-c أكبر من 0 وبالتالي
+
+234
+00:27:58,050 --> 00:28:06,790
+absolute xn-c absolute عدد موجب بساوي نفسه لأن ال
+
+235
+00:28:06,790 --> 00:28:15,750
+absolute value هنا ل xn-c بساوي xn-c لأن xn أكبر
+
+236
+00:28:15,750 --> 00:28:18,110
+من c وطبعا
+
+237
+00:28:21,830 --> 00:28:32,590
+هذا أكبر من السفر لأن xn لا تساوي c أكبر من c الان
+
+238
+00:28:32,590 --> 00:28:39,310
+من ال star هذا بيقدي by starالـ star بتقول إذا
+
+239
+00:28:39,310 --> 00:28:45,330
+كانت ال X أو هنا في الحالة تبعتنا X in ال X in هذه
+
+240
+00:28:45,330 --> 00:28:49,990
+تنتمي لإيه؟ ال X in هي تنتمي لإيه؟ و بعدين هي عندي
+
+241
+00:28:49,990 --> 00:28:56,470
+X in سالب C أكبر من سفر أصغر من Delta إذا by star
+
+242
+00:28:56,470 --> 00:29:06,950
+بتطلع absolute F of X in minus L أصغر من Y تمام؟
+
+243
+00:29:10,290 --> 00:29:18,790
+الان نلاحظ ان ابسلون was arbitrary ابسلون
+
+244
+00:29:18,790 --> 00:29:28,230
+was arbitrary since
+
+245
+00:29:28,230 --> 00:29:36,890
+ابسلون أكبر من السفر was arbitrary اذا هيك بنكون
+
+246
+00:29:36,890 --> 00:29:43,470
+احنا أثبتناإنه لأي إبسلون أو لكل إبسلون يوجد Delta
+
+247
+00:29:43,470 --> 00:29:50,890
+لأ لكل إبسلون يوجد capital N يعتمد على ال Delta
+
+248
+00:29:50,890 --> 00:29:55,070
+وبالتالي تعتمد على إبسلون لأن ال Delta تعتمد على
+
+249
+00:29:55,070 --> 00:30:01,410
+إبسلون بحيث إنه لكل N أكبر من أو سوى capital N طلع
+
+250
+00:30:01,410 --> 00:30:06,190
+عندي absolute f of xn minus L أصغر من إبسلونإذاً
+
+251
+00:30:06,190 --> 00:30:12,750
+by epsilon capital N definition of limit بيطلع هيك
+
+252
+00:30:12,750 --> 00:30:18,710
+بيكون أثبتنا أنه limit ال sequence f of x n as n
+
+253
+00:30:18,710 --> 00:30:23,710
+tends to infinity بساوي L وهذا اللي بدنا يعني هذا
+
+254
+00:30:23,710 --> 00:30:26,570
+اللي احنا ايه اللي عايزين نثبته
+
+255
+00:30:29,870 --> 00:30:35,490
+إذاً هيك منكون أثبتنا أنه إيه اتنين holds وبالتالي
+
+256
+00:30:35,490 --> 00:30:41,670
+هيك هذا بيكمل برهان واحد implies two okay تمام؟
+
+257
+00:30:41,670 --> 00:30:46,490
+بالمثل ممكن انبرهن اتنين implies one
+
+258
+00:30:55,620 --> 00:31:03,360
+the proof of اتنين implies العبارة
+
+259
+00:31:03,360 --> 00:31:11,200
+التانية implies الأولى is similar is
+
+260
+00:31:11,200 --> 00:31:18,600
+similar to is
+
+261
+00:31:18,600 --> 00:31:22,000
+similar to proof of
+
+262
+00:31:24,130 --> 00:31:34,570
+the sequential criterion for two-sided limit
+
+263
+00:31:34,570 --> 00:31:45,850
+exercises
+
+264
+00:31:45,850 --> 00:31:50,980
+يعني اتمرنوا عليهاانا ارجع لبرهان ال sequential
+
+265
+00:31:50,980 --> 00:31:55,520
+criterion for two-sided limit وشوفوا اقرأوا
+
+266
+00:31:55,520 --> 00:31:59,600
+البرهان و اعملوا التعديلات البسيطة على البرهان لان
+
+267
+00:31:59,600 --> 00:32:03,920
+هنا احنا نتعامل مع right hand limit او limit from
+
+268
+00:32:03,920 --> 00:32:07,500
+the right rather than two-sided limit زي ما عملنا
+
+269
+00:32:07,500 --> 00:32:12,920
+في البرهان تبع واحد implies اتنين okay فحاسيبكم
+
+270
+00:32:12,920 --> 00:32:17,740
+انتوا تكتبوا البرهان تبع اتنين بيقدي لواحدبنفس
+
+271
+00:32:17,740 --> 00:32:21,700
+الطريقة اللى برهنها فى حالة ال two sided limit
+
+272
+00:32:21,700 --> 00:32:30,080
+okay تمام فى اى سؤال طبعا ممكن برضه ايضا يوجد
+
+273
+00:32:30,080 --> 00:32:35,500
+ممكننا نثبت sequential criterion for left hand
+
+274
+00:32:35,500 --> 00:32:42,620
+limit او limit from the left نفس الطريقةOkay فيعني
+
+275
+00:32:42,620 --> 00:32:47,080
+احنا مش هنكتب طبعا نظرية دي هنعتبرها نظرية قائمة و
+
+276
+00:32:47,080 --> 00:32:53,010
+صحيحة و مش بدون برهان okay تمام؟إذن هذه واحدة من
+
+277
+00:32:53,010 --> 00:32:58,650
+النظريات اللي برهناها في section 4.1 و 4.2 و
+
+278
+00:32:58,650 --> 00:33:04,470
+بالمثل كل نظريات اللي برهناهم لـ two sided limit
+
+279
+00:33:04,470 --> 00:33:10,590
+في section 4.1 و 4.2 هنعتبرهم قائمين أو نعتبر
+
+280
+00:33:10,590 --> 00:33:15,330
+نظريات هذه صحيحة لـ left limit و right limit
+
+281
+00:33:22,080 --> 00:33:37,560
+في نظرية أخرى مهمة وهي التعطيل
+
+282
+00:33:37,560 --> 00:33:43,000
+العلاقة بين ال two sided limits و ال one sided
+
+283
+00:33:43,000 --> 00:33:49,100
+limits ف
+
+284
+00:33:51,870 --> 00:34:01,250
+if b function from a to r and let c be a cluster
+
+285
+00:34:01,250 --> 00:34:05,450
+point
+
+286
+00:34:05,450 --> 00:34:08,690
+of
+
+287
+00:34:08,690 --> 00:34:15,310
+المجموعة a تقاطع الفترة المفتوحة from c to
+
+288
+00:34:15,310 --> 00:34:24,070
+infinity and of a تقاطعالـ open interval from
+
+289
+00:34:24,070 --> 00:34:32,250
+negative infinity to c then
+
+290
+00:34:32,250 --> 00:34:42,450
+الـ two-sided limit للـ function f and c بتكون
+
+291
+00:34:42,450 --> 00:34:47,730
+موجودة وبتساوي
+
+292
+00:34:47,730 --> 00:34:54,760
+عدد L if and only ifالـ one-sided limit أو الـ
+
+293
+00:34:54,760 --> 00:35:02,120
+limit from the right the limit at C from the right
+
+294
+00:35:02,120 --> 00:35:12,860
+exist و بساوي L and the limit of F at C from the
+
+295
+00:35:12,860 --> 00:35:19,360
+left exist و بتساوي نفس العدد L وهذه نظرية أخذناها
+
+296
+00:35:19,360 --> 00:35:21,520
+في تفاضل ألف إذا بتذكروا
+
+297
+00:35:24,420 --> 00:35:29,460
+متى ال limit عند نقطة في مجالها او cluster point
+
+298
+00:35:29,460 --> 00:35:34,940
+لمجالها بتكون exist بالساوية عدد اذا كانت ال limit
+
+299
+00:35:34,940 --> 00:35:37,980
+من اليمين موجودة و ال limit من اليسار موجودة و
+
+300
+00:35:37,980 --> 00:35:47,600
+التنتين متساويتين و بساوي نفس العدد هناك
+
+301
+00:35:47,600 --> 00:35:50,500
+بس ماكنش البرهان المطلوب منكم المرة دي احنا
+
+302
+00:35:50,500 --> 00:35:58,420
+مطالبينبالبرهان البرهان يعني كتير سهل ينتج من
+
+303
+00:35:58,420 --> 00:36:06,780
+التعريفات proof ف .. هحاول أبرهنلكم ال F part هذا
+
+304
+00:36:06,780 --> 00:36:16,520
+مسمى ال F part يعني هفرض أنه assume أنه
+
+305
+00:36:16,520 --> 00:36:17,780
+ال one sided limits
+
+306
+00:36:24,700 --> 00:36:29,160
+the limit from the right exist وبساوي L وكذلك
+
+307
+00:36:29,160 --> 00:36:36,000
+limit from the left موجودة
+
+308
+00:36:36,000 --> 00:36:41,840
+وبساوي العدد L وعايز اثبت ان ال limit from the two
+
+309
+00:36:41,840 --> 00:36:48,940
+sides exist اذا هنا هذا الفرض المطلوب
+
+310
+00:37:02,770 --> 00:37:09,030
+أكلم الـ two-sided limit لـ الـ function f at x
+
+311
+00:37:09,030 --> 00:37:14,530
+بساوي c exist و بساوي نفس القيمة أو نفس الأعداد L
+
+312
+00:37:14,530 --> 00:37:27,010
+لبرهان ذلك to see this لبرهان ذلك بنحاول نطبق
+
+313
+00:37:27,010 --> 00:37:33,000
+تعريف epsilon deltaلـ limit of function فبنبدأ
+
+314
+00:37:33,000 --> 00:37:41,140
+بنقول let epsilon أكبر من السفر be given طيب
+
+315
+00:37:41,140 --> 00:37:47,380
+أنا من الفرض أنا فارض تعالى نستفيد من الفرض للوصول
+
+316
+00:37:47,380 --> 00:37:51,720
+إلى المطلوب هذا برهان مباشر البرهان المباشر ده
+
+317
+00:37:51,720 --> 00:37:57,420
+ناخد الفرض بنشتغل عليه بنحط عليه شوية برات و بعدين
+
+318
+00:37:57,420 --> 00:38:05,060
+بنطلع منهالمطلوب فمن الفرض فرضين احنا ان ال limit
+
+319
+00:38:05,060 --> 00:38:14,060
+ل f of x as x tends to c positive لما انه ال limit
+
+320
+00:38:14,060 --> 00:38:20,020
+من اليمين عن c بساوي L y أكبر من سفر given by
+
+321
+00:38:20,020 --> 00:38:25,190
+definitionthere exists delta واحد بالساوي delta
+
+322
+00:38:25,190 --> 00:38:32,830
+واحد تعتمد على epsilon عدد موجب بحيث أنه لو كان x
+
+323
+00:38:32,830 --> 00:38:40,650
+ينتمي إلى a و x minus c أكبر من سفر أصغر من delta
+
+324
+00:38:40,650 --> 00:38:48,350
+واحد فهذا بتضمن أن absolute f of x minus l أصغر من
+
+325
+00:38:48,350 --> 00:38:48,810
+epsilon
+
+326
+00:38:52,780 --> 00:39:00,720
+نسمي الـ implication head star also كذلك بما أن
+
+327
+00:39:00,720 --> 00:39:08,420
+احنا فرضين ان ال limit ل f of x as x tends to c
+
+328
+00:39:08,420 --> 00:39:13,900
+from the left exist وequal نفس العدد L، اذا by
+
+329
+00:39:13,900 --> 00:39:18,980
+definition of left hand limitthere exists delta
+
+330
+00:39:18,980 --> 00:39:21,880
+تانية مش صارت الـ delta هذه تكون نفس الـ delta
+
+331
+00:39:21,880 --> 00:39:27,040
+اللي فوق ماحد بيقدر يجزم ذلك فنسميها delta تانية
+
+332
+00:39:27,040 --> 00:39:32,980
+there exists delta two depends طبعا بالتأكيد تعتمد
+
+333
+00:39:32,980 --> 00:39:38,560
+على إبسلون وعدد موجة بحيث أنه حسب التعريف لكل x
+
+334
+00:39:39,250 --> 00:39:46,270
+تنتمي إلى a و c minus x أكبر من سفر أصغر من delta
+
+335
+00:39:46,270 --> 00:39:54,290
+و 2 طبعا هذا بتضمن أن absolute f of x minus n less
+
+336
+00:39:54,290 --> 00:40:00,710
+than epsilon انسمي ال implication هذه double star
+
+337
+00:40:00,710 --> 00:40:05,390
+خلينا
+
+338
+00:40:05,390 --> 00:40:11,530
+ناخد كالعادة deltaنعرف delta على إنها minimum ال
+
+339
+00:40:11,530 --> 00:40:17,530
+minimum الأصغر بين delta واحد و delta اتنين طبعا
+
+340
+00:40:17,530 --> 00:40:21,890
+هذه بالتأكيد هيطلع الصغيرة بين الاتنين هتكون واحدة
+
+341
+00:40:21,890 --> 00:40:27,770
+منهم وبالتالي تطلع عدد موجب وتعتمد على epsilon إذن
+
+342
+00:40:27,770 --> 00:40:30,930
+هيثبت أن يوجد delta تعتمد على epsilon و ال delta
+
+343
+00:40:30,930 --> 00:40:36,110
+هي عدد موجب الان for this delta تعالى نشوف
+
+344
+00:40:40,450 --> 00:40:49,310
+لو كان x ينتمي ل a و absolute x minus c أكبر من
+
+345
+00:40:49,310 --> 00:40:54,510
+صفر أصغر من دلتا الان
+
+346
+00:40:54,510 --> 00:40:57,850
+بناخد delta بساوي ال minimum ل delta واحد و delta
+
+347
+00:40:57,850 --> 00:41:02,060
+اتنينطبعا بما ان دلتا واحد ودلتا اتنين اعداد موجب
+
+348
+00:41:02,060 --> 00:41:06,080
+اذا دلتا اعداد موجب وكذلك تعتمد على epsilon لان
+
+349
+00:41:06,080 --> 00:41:10,380
+دلتا واحد ودلتا اتنين تعتمد على epsilon الان لو
+
+350
+00:41:10,380 --> 00:41:16,720
+أخدت x تنتمي لمجموعة a و ال x صارت مختلفة عن ال c
+
+351
+00:41:16,720 --> 00:41:23,320
+و المسافة بينها و بين ال c أصغر من دلتا هذا معناه
+
+352
+00:41:23,320 --> 00:41:34,510
+هذا معناه انهال X لا تساوي C وبالتالي
+
+353
+00:41:34,510 --> 00:41:48,230
+ال X ممكن تكون اصغر من C او ال X اكبر من C فهذا
+
+354
+00:41:48,230 --> 00:41:55,630
+بيقدي انه ال .. ال
+
+355
+00:41:55,630 --> 00:42:04,400
+.. ال .. اذا كانت ال Xإذا كانت الـ X أكبر من C لو
+
+356
+00:42:04,400 --> 00:42:08,980
+كانت الـ X أكبر من C فهذا بقدّي أن absolute X
+
+357
+00:42:08,980 --> 00:42:15,200
+minus C بساوي X minus C بصير الـ absolute value
+
+358
+00:42:15,200 --> 00:42:20,560
+هذه عبارة عن X minus C هو أكبر من 0 أصغر من Delta
+
+359
+00:42:20,560 --> 00:42:29,800
+ولو كانتال X أصغر من C فال absolute value هذه
+
+360
+00:42:29,800 --> 00:42:37,360
+بيصير C minus X أكبر من سفر أصغر من Delta في
+
+361
+00:42:37,360 --> 00:42:41,460
+الحالة الأولى ال Delta تبعتي هذه أصغر من أو ساوي
+
+362
+00:42:41,460 --> 00:42:47,120
+Delta واحد صح؟ ال Delta هذه هي ال minimum ل Delta
+
+363
+00:42:47,120 --> 00:42:50,760
+واحد و Delta اتنين وبالتالي أصغر من أو ساوي Delta
+
+364
+00:42:50,760 --> 00:42:58,090
+واحد وبالتالي من ال starإذا كانت x تنتمي إلى a و x
+
+365
+00:42:58,090 --> 00:43:03,990
+minus c أكبر من سفر أصغر من دلتا واحد من ال star
+
+366
+00:43:03,990 --> 00:43:11,770
+بيطلع عندي absolute f of x minus L أصغر من يوإذا
+
+367
+00:43:11,770 --> 00:43:17,510
+كانت ال X أصغر من ال C فبطلع absolute X سالب C
+
+368
+00:43:17,510 --> 00:43:22,870
+بيطلع بيساوي C سالب X أصغر من Delta وطبعا X مستويش
+
+369
+00:43:22,870 --> 00:43:29,190
+C أكبر من 0 وال Delta هذه من تعريفها أصغر من أو
+
+370
+00:43:29,190 --> 00:43:35,300
+يساوي Delta 2باستخدام double star ال implication
+
+371
+00:43:35,300 --> 00:43:41,420
+double star لما يكون ال X تنتمي ل A و C minus X
+
+372
+00:43:41,420 --> 00:43:46,640
+أكبر من 0 أصغر من Delta 2 هذا بيقدر أن absolute F
+
+373
+00:43:46,640 --> 00:43:53,680
+of X minus L أصغر من إبسن إذن في كل الأحوال هذه
+
+374
+00:43:53,680 --> 00:43:58,180
+بتقدر أن absolute F of X minus L أصغر من إبسن
+
+375
+00:43:58,180 --> 00:43:59,400
+تمام؟
+
+376
+00:44:02,170 --> 00:44:06,090
+طب ما هذا هو تعريف epsilon delta لل limit of
+
+377
+00:44:06,090 --> 00:44:12,270
+function صح؟ إذا نيجي بنقول هنا since epsilon أكبر
+
+378
+00:44:12,270 --> 00:44:15,870
+من السفر was arbitrary
+
+379
+00:44:17,410 --> 00:44:22,850
+إذا احنا بنكون أثبتنا لكل إبسلون أكبر من سفر يوجد
+
+380
+00:44:22,850 --> 00:44:27,950
+Delta تعتمد على إبسلون عدد موجب بحيث لكل X تنتمي ل
+
+381
+00:44:27,950 --> 00:44:32,210
+A و Absolute X minus C أكبر من سفر أصغر من Delta
+
+382
+00:44:32,210 --> 00:44:37,810
+طلع عندي Absolute F of X في الحالتين minus L أصغر
+
+383
+00:44:37,810 --> 00:44:41,630
+من إبسلون وبالتالي إذا هذا صحيح لكل إبسلون
+
+384
+00:44:41,630 --> 00:44:45,620
+وبالتالي by إبسلون Delta definition of limitأو
+
+385
+00:44:45,620 --> 00:44:54,600
+function we have أثبتنا أن ال limit ل f of x as x
+
+386
+00:44:54,600 --> 00:45:01,380
+tends to c بساوي العدد L okay تمام، إذا هذا بثبت
+
+387
+00:45:01,380 --> 00:45:04,840
+اللي هو لو كان ال two sided limits موجودين
+
+388
+00:45:04,840 --> 00:45:10,730
+متساويتين، لأ لو كان ال one sided limitsكلا هما
+
+389
+00:45:10,730 --> 00:45:15,530
+موجودة و بساوة قيمة مشتركة L ف ال two sided limit
+
+390
+00:45:15,530 --> 00:45:20,130
+تطلع exist و قيمتها بساوة القيمة المشتركة الان
+
+391
+00:45:20,130 --> 00:45:28,210
+برهان العكس أسهل لذن هكتب هنا ال proof of
+
+392
+00:45:28,210 --> 00:45:36,650
+the canvas is easier أسهل
+
+393
+00:45:38,760 --> 00:45:44,180
+So exercise it يعني
+
+394
+00:45:44,180 --> 00:45:49,780
+اتمرّض عليها لو كانت ال two-sided limit exist فمن
+
+395
+00:45:49,780 --> 00:45:55,840
+السهل أن نثبت أن ال right hand limit exist و ال
+
+396
+00:45:55,840 --> 00:46:00,600
+left hand limit exist و كلهم لهم نفس القيمة okay
+
+397
+00:46:00,600 --> 00:46:05,170
+تمام؟إذا هنوقف هنا و في المحاضرة الجاية ان شاء
+
+398
+00:46:05,170 --> 00:46:09,710
+الله هناخد أمثلة على one-sided limits إما في اتين
+
+399
+00:46:09,710 --> 00:46:13,350
+موجودين و متساوياتين أو اتنين موجودين و مختلفتين
+
+400
+00:46:13,350 --> 00:46:18,190
+أو واحدة موجودة و اتنين مش موجودة و هكذا، هنشوف كل
+
+401
+00:46:18,190 --> 00:46:24,670
+الأنواع و كل ال situations، تمام؟ okay شكرا لكم و
+
+402
+00:46:24,670 --> 00:46:26,550
+نشوفكم ان شاء الله المرة القادمة
+
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/2qaKB7theEQ_raw.json b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/2qaKB7theEQ_raw.json
new file mode 100644
index 0000000000000000000000000000000000000000..6fdcac040ddb211862ae976dd951b7d14e43a6a5
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/2qaKB7theEQ_raw.json
@@ -0,0 +1 @@
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0.3037109375}, {"start": 106.0, "end": 106.52, "word": " فاتت", "probability": 0.8600260416666666}, {"start": 106.52, "end": 107.02, "word": " بتاعة", "probability": 0.6787923177083334}, {"start": 107.02, "end": 107.22, "word": " الـ", "probability": 0.755859375}, {"start": 107.22, "end": 107.58, "word": " sequences", "probability": 0.49658203125}, {"start": 107.58, "end": 108.4, "word": " المشابهة", "probability": 0.899609375}, {"start": 108.4, "end": 108.72, "word": " بتقول", "probability": 0.951904296875}, {"start": 108.72, "end": 108.92, "word": " لو", "probability": 0.7841796875}, {"start": 108.92, "end": 109.36, "word": " كانت", "probability": 0.986572265625}, {"start": 109.36, "end": 110.36, "word": " ال", "probability": 0.8935546875}, {"start": 110.36, "end": 110.86, "word": " sequence", "probability": 0.8212890625}, {"start": 110.86, "end": 111.6, "word": " النهاية", "probability": 0.7596842447916666}, {"start": 111.6, "end": 112.1, "word": " تبعتها", "probability": 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"probability": 0.88134765625}, {"start": 123.7, "end": 123.92, "word": " و", "probability": 0.9375}, {"start": 123.92, "end": 124.4, "word": " سالبة", "probability": 0.796875}, {"start": 124.4, "end": 125.66, "word": " فلازم", "probability": 0.9329833984375}, {"start": 125.66, "end": 126.06, "word": " حدود", "probability": 0.9908854166666666}, {"start": 126.06, "end": 126.22, "word": " ال", "probability": 0.97412109375}, {"start": 126.22, "end": 126.64, "word": " sequence", "probability": 0.99560546875}, {"start": 126.64, "end": 126.88, "word": " من", "probability": 0.98828125}, {"start": 126.88, "end": 127.18, "word": " capital", "probability": 0.86962890625}, {"start": 127.18, "end": 127.42, "word": " N", "probability": 0.93310546875}, {"start": 127.42, "end": 127.72, "word": " وانت", "probability": 0.9256184895833334}, {"start": 127.72, "end": 128.14, "word": " طالع", "probability": 0.9275716145833334}, {"start": 128.14, "end": 129.0, "word": " كلها", "probability": 0.98779296875}, {"start": 129.0, "end": 129.32, "word": " تكون", "probability": 0.98095703125}, {"start": 129.32, "end": 129.88, "word": " سالبة", "probability": 0.9791666666666666}, {"start": 129.88, "end": 130.44, "word": " فهذه", "probability": 0.7352701822916666}, {"start": 130.44, "end": 130.9, "word": " شبيهة", "probability": 0.874267578125}, {"start": 130.9, "end": 131.2, "word": " فيها", "probability": 0.976806640625}], "temperature": 1.0}, {"id": 5, "seek": 16006, "start": 131.78, "end": 160.06, "text": "والبرهان سهل وشبيه بالبرهان تبع النظرية المشابهة في حالة ال sequences فناخد الحالة assume ناخد الحالة اللي فيها ال limit ل f of x at c exists and equals عدد l موجب", "tokens": [2407, 6027, 26890, 3224, 7649, 8608, 3224, 1211, 4032, 8592, 21292, 3224, 20666, 26890, 3224, 7649, 6055, 3555, 3615, 28239, 19913, 2288, 10632, 9673, 8592, 16758, 3224, 3660, 8978, 11331, 6027, 3660, 2423, 22978, 6156, 1863, 47283, 3215, 21542, 6027, 3660, 6552, 8717, 47283, 3215, 21542, 6027, 3660, 13672, 1829, 8978, 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140.0, "word": " ال", "probability": 0.88916015625}, {"start": 140.0, "end": 140.52, "word": " sequences", "probability": 0.40380859375}, {"start": 140.52, "end": 142.94, "word": " فناخد", "probability": 0.9122314453125}, {"start": 142.94, "end": 143.52, "word": " الحالة", "probability": 0.9910481770833334}, {"start": 143.52, "end": 144.32, "word": " assume", "probability": 0.58251953125}, {"start": 144.32, "end": 147.36, "word": " ناخد", "probability": 0.8531901041666666}, {"start": 147.36, "end": 147.96, "word": " الحالة", "probability": 0.9949544270833334}, {"start": 147.96, "end": 148.26, "word": " اللي", "probability": 0.8193359375}, {"start": 148.26, "end": 148.7, "word": " فيها", "probability": 0.983154296875}, {"start": 148.7, "end": 148.96, "word": " ال", "probability": 0.669921875}, {"start": 148.96, "end": 149.36, "word": " limit", "probability": 0.966796875}, {"start": 149.36, "end": 152.4, "word": " ل", "probability": 0.7607421875}, {"start": 152.4, "end": 152.68, "word": " f", "probability": 0.365478515625}, {"start": 152.68, "end": 152.92, "word": " of", "probability": 0.455810546875}, {"start": 152.92, "end": 153.36, "word": " x", "probability": 0.9033203125}, {"start": 153.36, "end": 154.78, "word": " at", "probability": 0.86083984375}, {"start": 154.78, "end": 155.12, "word": " c", "probability": 0.7890625}, {"start": 155.12, "end": 155.82, "word": " exists", "probability": 0.71484375}, {"start": 155.82, "end": 157.52, "word": " and", "probability": 0.94873046875}, {"start": 157.52, "end": 158.16, "word": " equals", "probability": 0.95703125}, {"start": 158.16, "end": 158.74, "word": " عدد", "probability": 0.9814453125}, {"start": 158.74, "end": 159.14, "word": " l", "probability": 0.245849609375}, {"start": 159.14, "end": 160.06, "word": " موجب", "probability": 0.9319661458333334}], "temperature": 1.0}, {"id": 6, "seek": 20080, "start": 173.88, "end": 200.8, "text": "فإذا كانت ال limit موجبة بنا أثبت أن يوجد delta neverhood إلى آخر A فخلّينا ناخد let epsilon في الحالة دي let epsilon بساوي L على 2 فهذا عدد موجب", "tokens": [5172, 28814, 15730, 25961, 2655, 2423, 4948, 3714, 29245, 49401, 4724, 8315, 5551, 12984, 3555, 2655, 14739, 7251, 29245, 3215, 8289, 1128, 3809, 30731, 19753, 34740, 316, 6156, 9778, 1211, 11703, 9957, 995, 8717, 47283, 3215, 718, 17889, 8978, 21542, 6027, 3660, 11778, 1829, 718, 17889, 4724, 3794, 995, 45865, 441, 15844, 568, 6156, 3224, 15730, 6225, 3215, 3215, 3714, 29245, 3555], "avg_logprob": -0.3554067517083789, "compression_ratio": 1.3272727272727274, "no_speech_prob": 0.0, "words": [{"start": 173.88, "end": 174.48, "word": "فإذا", "probability": 0.6401774088541666}, {"start": 174.48, "end": 174.92, "word": " كانت", "probability": 0.93017578125}, {"start": 174.92, "end": 175.02, "word": " ال", "probability": 0.51025390625}, {"start": 175.02, "end": 175.38, "word": " limit", "probability": 0.71337890625}, {"start": 175.38, "end": 176.48, "word": " موجبة", "probability": 0.9217122395833334}, {"start": 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"probability": 0.9192708333333334}], "temperature": 1.0}, {"id": 7, "seek": 22328, "start": 203.14, "end": 223.28, "text": "الان by definition of limit of function by epsilon delta definition لأي epsilon موجبة زي هذه يوجد delta تعتمد على L على 2 اللي هي ال epsilon عدد موجب بحيث انه لو كان X", "tokens": [6027, 7649, 538, 7123, 295, 4948, 295, 2445, 538, 17889, 8289, 7123, 5296, 10721, 1829, 17889, 3714, 29245, 49401, 30767, 1829, 29538, 7251, 29245, 3215, 8289, 6055, 34268, 2304, 3215, 15844, 441, 15844, 568, 13672, 1829, 39896, 2423, 17889, 6225, 3215, 3215, 3714, 29245, 3555, 4724, 5016, 1829, 12984, 16472, 3224, 45164, 25961, 1783], "avg_logprob": -0.19247158819978888, "compression_ratio": 1.421383647798742, "no_speech_prob": 0.0, "words": [{"start": 203.14, "end": 203.6, "word": "الان", "probability": 0.572998046875}, {"start": 203.6, "end": 203.92, "word": " by", "probability": 0.775390625}, {"start": 203.92, "end": 204.56, "word": " definition", "probability": 0.8974609375}, {"start": 204.56, "end": 206.18, "word": " of", "probability": 0.8828125}, {"start": 206.18, "end": 206.6, "word": " limit", "probability": 0.73876953125}, {"start": 206.6, "end": 206.78, "word": " of", "probability": 0.955078125}, {"start": 206.78, "end": 207.44, "word": " function", "probability": 0.9453125}, {"start": 207.44, "end": 208.06, "word": " by", "probability": 0.6708984375}, {"start": 208.06, "end": 208.46, "word": " epsilon", "probability": 0.73046875}, {"start": 208.46, "end": 208.8, "word": " delta", "probability": 0.73193359375}, {"start": 208.8, "end": 209.52, "word": " definition", "probability": 0.91650390625}, {"start": 209.52, "end": 212.64, "word": " لأي", "probability": 0.8128255208333334}, {"start": 212.64, "end": 213.02, "word": " epsilon", "probability": 0.81787109375}, {"start": 213.02, "end": 213.48, "word": " موجبة", "probability": 0.8260091145833334}, {"start": 213.48, "end": 213.78, "word": " زي", "probability": 0.876708984375}, {"start": 213.78, "end": 214.08, "word": " هذه", "probability": 0.88916015625}, {"start": 214.08, "end": 214.7, "word": " يوجد", "probability": 0.9720052083333334}, {"start": 214.7, "end": 215.2, "word": " delta", "probability": 0.62451171875}, {"start": 215.2, "end": 216.8, "word": " تعتمد", "probability": 0.96728515625}, {"start": 216.8, "end": 217.12, "word": " على", "probability": 0.86865234375}, {"start": 217.12, "end": 217.56, "word": " L", "probability": 0.5986328125}, {"start": 217.56, "end": 217.78, "word": " على", "probability": 0.286376953125}, {"start": 217.78, "end": 218.14, "word": " 2", "probability": 0.86181640625}, {"start": 218.14, "end": 218.34, "word": " اللي", "probability": 0.7998046875}, {"start": 218.34, "end": 218.54, "word": " هي", "probability": 0.93505859375}, {"start": 218.54, "end": 218.68, "word": " ال", "probability": 0.87939453125}, {"start": 218.68, "end": 219.12, "word": " epsilon", "probability": 0.5341796875}, {"start": 219.12, "end": 219.76, "word": " عدد", "probability": 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function f عندما x تقول إلى c من اليمين بساوي العدد ال .. تمام؟ لأن هذا تعريف ال limit from the right أو ال right hand limit لل function f عند النقطة c", "tokens": [826, 2464, 8717, 4117, 2655, 3555, 14739, 2423, 4948, 24976, 2445, 283, 43242, 15042, 2031, 6055, 39648, 30731, 269, 9154, 2423, 32640, 9957, 4724, 3794, 995, 45865, 18863, 3215, 3215, 2423, 4386, 46811, 10943, 22807, 5296, 33456, 23758, 37279, 16572, 5172, 2423, 4948, 490, 264, 558, 34051, 2423, 558, 1011, 4948, 24976, 2445, 283, 43242, 28239, 47432, 3660, 269], "avg_logprob": -0.2348958323399226, "compression_ratio": 1.4912280701754386, "no_speech_prob": 0.0, "words": [{"start": 800.92, "end": 801.34, "word": "we", "probability": 0.1475830078125}, {"start": 801.34, "end": 802.16, "word": " write", "probability": 0.9169921875}, {"start": 802.16, "end": 804.96, "word": " نكتب", "probability": 0.9244384765625}, {"start": 804.96, "end": 805.24, "word": " أن", "probability": 0.325439453125}, {"start": 805.24, "end": 805.44, 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النقطة C وانا عندي ال C هي cluster point ل A .. لكل ال X موجود في A و أكبر من C", "tokens": [28814, 15730, 41850, 18871, 16254, 29538, 16490, 9566, 16247, 22488, 18513, 37037, 1829, 28239, 47432, 3660, 383, 4032, 7649, 995, 18871, 16254, 2423, 383, 39896, 13630, 935, 5296, 316, 4386, 5296, 28820, 2423, 1783, 3714, 29245, 23328, 8978, 316, 4032, 5551, 4117, 26890, 9154, 383], "avg_logprob": -0.27683423265166907, "compression_ratio": 1.2794117647058822, "no_speech_prob": 0.0, "words": [{"start": 845.18, "end": 845.46, "word": "إذا", "probability": 0.439910888671875}, {"start": 845.46, "end": 845.72, "word": " أنا", "probability": 0.7236328125}, {"start": 845.72, "end": 846.1, "word": " عندي", "probability": 0.828125}, {"start": 846.1, "end": 846.48, "word": " هذه", "probability": 0.495361328125}, {"start": 846.48, "end": 847.16, "word": " خط", "probability": 0.96533203125}, {"start": 847.16, "end": 847.9, "word": " الأعداد", "probability": 0.8922526041666666}, {"start": 847.9, "end": 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إبسلون أكبر من السفر بتقدر نلاقي delta عدد موجة بيعتمد على إبسلون بحيث لكل x في المجموعة A إذا كانت ال X هذه على يمين ال C", "tokens": [5172, 3555, 1863, 39648, 36145, 2423, 4948, 43242, 2031, 20666, 3794, 995, 2407, 10632, 269, 34051, 2423, 2445, 8978, 11933, 1211, 11296, 558, 12, 5543, 4948, 4032, 2423, 558, 12, 5543, 4948, 39896, 18863, 3215, 3215, 441, 11933, 15730, 25961, 5296, 10721, 1829, 11933, 3555, 3794, 1211, 11536, 5551, 4117, 26890, 9154, 21136, 5172, 2288, 39894, 28543, 2288, 8717, 15040, 38436, 8289, 6225, 3215, 3215, 3714, 29245, 3660, 4724, 1829, 34268, 2304, 3215, 15844, 11933, 3555, 3794, 1211, 11536, 4724, 5016, 1829, 12984, 5296, 28820, 2031, 8978, 9673, 7435, 2304, 2407, 27884, 316, 11933, 15730, 25961, 2655, 2423, 1783, 29538, 15844, 7251, 2304, 9957, 2423, 383], "avg_logprob": -0.21509930630710639, "compression_ratio": 1.6309012875536482, "no_speech_prob": 0.0, "words": [{"start": 872.71, "end": 873.43, "word": "فبنقول", "probability": 0.8428955078125}, 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تعريف ال lift hand limit او ال limit from the lift okay تعديل بسيط بس طيب ال limits هذه هنشوف يعني بعد شوية ان ال one sided limits ده functional نقطة", "tokens": [3555, 3794, 297, 4724, 3794, 995, 45865, 441, 43, 1392, 16472, 11296, 37279, 16572, 5172, 2423, 5533, 1011, 4948, 1975, 2407, 2423, 4948, 490, 264, 5533, 1392, 37279, 16254, 1211, 4724, 3794, 1829, 9566, 4724, 3794, 23032, 1829, 3555, 2423, 10406, 29538, 8032, 1863, 8592, 38688, 37495, 22653, 39182, 13412, 2407, 10632, 16472, 2423, 472, 41651, 10406, 11778, 3224, 11745, 8717, 47432, 3660], "avg_logprob": -0.3525390671566129, "compression_ratio": 1.4545454545454546, "no_speech_prob": 0.0, "words": [{"start": 1054.23, "end": 1054.69, "word": "بس", "probability": 0.679931640625}, {"start": 1054.69, "end": 1055.03, "word": " n", "probability": 0.06793212890625}, {"start": 1055.03, "end": 1055.99, "word": " بساوي", "probability": 0.717529296875}, {"start": 1055.99, "end": 1056.53, "word": " LL", "probability": 0.4373779296875}, 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{"start": 1060.83, "end": 1061.17, "word": " lift", "probability": 0.71923828125}, {"start": 1061.17, "end": 1061.83, "word": " okay", "probability": 0.66748046875}, {"start": 1061.83, "end": 1064.89, "word": " تعديل", "probability": 0.6944986979166666}, {"start": 1064.89, "end": 1065.37, "word": " بسيط", "probability": 0.99072265625}, {"start": 1065.37, "end": 1065.89, "word": " بس", "probability": 0.91796875}, {"start": 1065.89, "end": 1070.77, "word": " طيب", "probability": 0.8956705729166666}, {"start": 1070.77, "end": 1071.13, "word": " ال", "probability": 0.97509765625}, {"start": 1071.13, "end": 1072.51, "word": " limits", "probability": 0.845703125}, {"start": 1072.51, "end": 1072.97, "word": " هذه", "probability": 0.47607421875}, {"start": 1072.97, "end": 1073.69, "word": " هنشوف", "probability": 0.961669921875}, {"start": 1073.69, "end": 1074.01, "word": " يعني", "probability": 0.933349609375}, {"start": 1074.01, "end": 1074.31, "word": " بعد", "probability": 0.89990234375}, 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ال limit تبعتها من اليمين واحد و ال limit تبعتها من اليسار ثالث واحد", "tokens": [2304, 43020, 37495, 22653, 16712, 29399, 9957, 7251, 30544, 14407, 3714, 29245, 23328, 9957, 43242, 28239, 47432, 3660, 4032, 26895, 16095, 8717, 36178, 25062, 32640, 3660, 34051, 3714, 43020, 16712, 29399, 9957, 7251, 30544, 14407, 3714, 29245, 23328, 9957, 43242, 8717, 47432, 3660, 44381, 12174, 32640, 16095, 3714, 46456, 46538, 3660, 30767, 1829, 2423, 39184, 13515, 449, 2445, 43242, 31767, 5172, 2288, 13412, 38688, 8315, 14739, 2423, 4948, 6055, 3555, 34268, 11296, 9154, 2423, 32640, 9957, 36764, 24401, 4032, 2423, 4948, 6055, 3555, 34268, 11296, 9154, 45595, 3794, 9640, 38637, 6027, 12984, 36764, 24401], "avg_logprob": -0.15789473276389274, "compression_ratio": 2.0364583333333335, "no_speech_prob": 0.0, "words": [{"start": 1081.13, "end": 1081.73, "word": "ممكن", "probability": 0.800537109375}, {"start": 1081.73, "end": 1082.05, "word": " يعني", "probability": 0.607177734375}, {"start": 1082.05, 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x in as n tends to infinity بيساوي c we have limit لل image of the sequence x in بيساوي العدد L", "tokens": [4197, 270, 2031, 294, 382, 297, 12258, 281, 13202, 4724, 1829, 3794, 995, 45865, 269, 321, 362, 4948, 24976, 3256, 295, 264, 8310, 2031, 294, 4724, 1829, 3794, 995, 45865, 18863, 3215, 3215, 441], "avg_logprob": -0.2609375136239188, "compression_ratio": 1.1214953271028036, "no_speech_prob": 0.0, "words": [{"start": 1298.21, "end": 1298.81, "word": "limit", "probability": 0.63525390625}, {"start": 1298.81, "end": 1299.21, "word": " x", "probability": 0.78759765625}, {"start": 1299.21, "end": 1299.65, "word": " in", "probability": 0.3193359375}, {"start": 1299.65, "end": 1300.13, "word": " as", "probability": 0.9130859375}, {"start": 1300.13, "end": 1300.49, "word": " n", "probability": 0.7275390625}, {"start": 1300.49, "end": 1300.77, "word": " tends", "probability": 0.81103515625}, {"start": 1300.77, "end": 1300.89, "word": " to", "probability": 0.93701171875}, {"start": 1300.89, "end": 1301.45, "word": " infinity", "probability": 0.89697265625}, {"start": 1301.45, "end": 1302.47, "word": " بيساوي", "probability": 0.755517578125}, {"start": 1302.47, "end": 1303.01, "word": " c", "probability": 0.51171875}, {"start": 1303.01, "end": 1304.53, "word": " we", "probability": 0.408447265625}, {"start": 1304.53, "end": 1307.29, "word": " have", "probability": 0.974609375}, {"start": 1307.29, "end": 1311.09, "word": " limit", "probability": 0.728515625}, {"start": 1311.09, "end": 1312.75, "word": " لل", "probability": 0.7109375}, {"start": 1312.75, "end": 1313.25, "word": " image", "probability": 0.499267578125}, {"start": 1313.25, "end": 1313.61, "word": " of", "probability": 0.97412109375}, {"start": 1313.61, "end": 1313.85, "word": " the", "probability": 0.8310546875}, {"start": 1313.85, "end": 1314.45, "word": " sequence", "probability": 0.9814453125}, {"start": 1314.45, "end": 1314.87, "word": " x", "probability": 0.98193359375}, {"start": 1314.87, "end": 1315.45, "word": " in", "probability": 0.84033203125}, {"start": 1315.45, "end": 1318.17, "word": " بيساوي", "probability": 0.9353515625}, {"start": 1318.17, "end": 1318.85, "word": " العدد", "probability": 0.9772135416666666}, {"start": 1318.85, "end": 1319.17, "word": " L", "probability": 0.64599609375}], "temperature": 1.0}, {"id": 49, "seek": 135510, "start": 1330.34, "end": 1355.1, "text": "البرهان شبيه بالبرهان الخاص بالـ two-sided limit فمثلا لو بدنا نبرهن proof لو بدى برهن الأبعار الأولى بتأدي للتانية فبنقول assume .. نبدأ ب assume أن ال limit ال right limit", "tokens": [6027, 26890, 3224, 7649, 13412, 21292, 3224, 20666, 26890, 3224, 7649, 33962, 33546, 20666, 39184, 732, 12, 30941, 4948, 6156, 2304, 12984, 15040, 45164, 47525, 8315, 8717, 26890, 3224, 1863, 8177, 45164, 47525, 7578, 4724, 2288, 3224, 1863, 16247, 3555, 3615, 9640, 16247, 12610, 7578, 39894, 10721, 16254, 24976, 2655, 7649, 10632, 6156, 3555, 1863, 39648, 6552, 4386, 8717, 44510, 10721, 4724, 6552, 14739, 2423, 4948, 2423, 558, 4948], "avg_logprob": -0.2316964247397014, "compression_ratio": 1.5952380952380953, "no_speech_prob": 0.0, "words": [{"start": 1330.34, "end": 1331.26, "word": "البرهان", "probability": 0.96484375}, {"start": 1331.26, "end": 1332.42, "word": " شبيه", "probability": 0.8653971354166666}, {"start": 1332.42, "end": 1335.06, "word": " بالبرهان", "probability": 0.9346923828125}, {"start": 1335.06, "end": 1335.72, "word": " الخاص", "probability": 0.973876953125}, {"start": 1335.72, "end": 1336.02, "word": " بالـ", "probability": 0.6444091796875}, {"start": 1336.02, "end": 1336.24, "word": " two", "probability": 0.72314453125}, {"start": 1336.24, "end": 1336.62, "word": "-sided", "probability": 0.7685546875}, {"start": 1336.62, "end": 1337.06, "word": " limit", "probability": 0.91259765625}, {"start": 1337.06, "end": 1339.08, "word": " فمثلا", "probability": 0.93115234375}, {"start": 1339.08, "end": 1339.82, "word": " لو", "probability": 0.673828125}, {"start": 1339.82, "end": 1340.18, "word": " بدنا", "probability": 0.962890625}, {"start": 1340.18, "end": 1340.9, "word": " نبرهن", "probability": 0.71844482421875}, {"start": 1340.9, "end": 1341.62, "word": " proof", "probability": 0.5625}, {"start": 1341.62, "end": 1342.86, "word": " لو", "probability": 0.818359375}, {"start": 1342.86, "end": 1343.2, "word": " بدى", "probability": 0.544189453125}, {"start": 1343.2, "end": 1343.64, "word": " برهن", "probability": 0.8782958984375}, {"start": 1343.64, "end": 1344.1, "word": " الأبعار", "probability": 0.58392333984375}, {"start": 1344.1, "end": 1344.58, "word": " الأولى", "probability": 0.9833984375}, {"start": 1344.58, "end": 1345.4, "word": " بتأدي", "probability": 0.7711588541666666}, {"start": 1345.4, "end": 1346.48, "word": " للتانية", "probability": 0.934326171875}, {"start": 1346.48, "end": 1349.18, "word": " فبنقول", "probability": 0.9326171875}, {"start": 1349.18, "end": 1350.0, "word": " assume", "probability": 0.84912109375}, {"start": 1350.0, "end": 1350.16, "word": " ..", "probability": 0.444580078125}, {"start": 1350.16, "end": 1350.62, "word": " نبدأ", "probability": 0.9462890625}, {"start": 1350.62, "end": 1350.72, "word": " ب", "probability": 0.916015625}, {"start": 1350.72, "end": 1351.46, "word": " assume", "probability": 0.473388671875}, {"start": 1351.46, "end": 1352.62, "word": " أن", "probability": 0.541015625}, {"start": 1352.62, "end": 1353.02, "word": " ال", "probability": 0.66748046875}, {"start": 1353.02, "end": 1353.46, "word": " limit", "probability": 0.9248046875}, {"start": 1353.46, "end": 1354.28, "word": " ال", "probability": 0.32763671875}, {"start": 1354.28, "end": 1354.64, "word": " right", "probability": 0.90869140625}, {"start": 1354.64, "end": 1355.1, "word": " limit", "probability": 0.98193359375}], "temperature": 1.0}, {"id": 50, "seek": 137626, "start": 1356.36, "end": 1376.26, "text": "الـ F عند الـ C exist بساوي L وبدنا نثبت أن الـ two بيطلع العبارة تنين بتطلع صحيحة لبرهان العبارة to prove two holds", "tokens": [6027, 39184, 479, 43242, 2423, 39184, 383, 2514, 4724, 3794, 995, 45865, 441, 4032, 44510, 8315, 8717, 12984, 3555, 2655, 14739, 2423, 39184, 732, 4724, 1829, 9566, 1211, 3615, 18863, 3555, 9640, 3660, 6055, 1863, 9957, 39894, 9566, 1211, 3615, 20328, 5016, 1829, 5016, 3660, 5296, 26890, 3224, 7649, 18863, 3555, 9640, 3660, 281, 7081, 732, 9190], "avg_logprob": -0.31761852369226257, "compression_ratio": 1.3455882352941178, "no_speech_prob": 0.0, "words": [{"start": 1356.36, "end": 1356.74, "word": "الـ", "probability": 0.586669921875}, {"start": 1356.74, "end": 1357.04, "word": " F", "probability": 0.49658203125}, {"start": 1357.04, "end": 1357.52, "word": " عند", "probability": 0.9228515625}, {"start": 1357.52, "end": 1357.88, "word": " الـ", "probability": 0.7705078125}, {"start": 1357.88, "end": 1358.24, "word": " C", "probability": 0.77587890625}, {"start": 1358.24, "end": 1359.76, "word": " exist", "probability": 0.3125}, {"start": 1359.76, "end": 1360.7, "word": " بساوي", "probability": 0.7476806640625}, {"start": 1360.7, "end": 1361.16, "word": " L", "probability": 0.79541015625}, {"start": 1361.16, "end": 1364.24, "word": " وبدنا", "probability": 0.53826904296875}, {"start": 1364.24, "end": 1364.6, "word": " نثبت", "probability": 0.9918212890625}, {"start": 1364.6, "end": 1364.76, "word": " أن", "probability": 0.436279296875}, {"start": 1364.76, "end": 1364.92, "word": " الـ", "probability": 0.571533203125}, {"start": 1364.92, "end": 1365.1, "word": " two", "probability": 0.58642578125}, {"start": 1365.1, "end": 1365.6, "word": " بيطلع", "probability": 0.824169921875}, {"start": 1365.6, "end": 1366.28, "word": " العبارة", "probability": 0.773468017578125}, {"start": 1366.28, "end": 1367.38, "word": " تنين", "probability": 0.54638671875}, {"start": 1367.38, "end": 1367.8, "word": " بتطلع", "probability": 0.842041015625}, {"start": 1367.8, "end": 1368.68, "word": " صحيحة", "probability": 0.98076171875}, {"start": 1368.68, "end": 1369.6, "word": " لبرهان", "probability": 0.9019775390625}, {"start": 1369.6, "end": 1370.8, "word": " العبارة", "probability": 0.9952392578125}, {"start": 1370.8, "end": 1371.26, "word": " to", "probability": 0.8251953125}, {"start": 1371.26, "end": 1371.84, "word": " prove", "probability": 0.9541015625}, {"start": 1371.84, "end": 1375.4, "word": " two", "probability": 0.919921875}, {"start": 1375.4, "end": 1376.26, "word": " holds", "probability": 0.7060546875}], "temperature": 1.0}, {"id": 51, "seek": 140684, "start": 1379.5, "end": 1406.84, "text": "لت نبدأ لت xn contained in a تقاطع c إلى infinity بsequence such that ال limit تبعتها as n tends to infinity بساوي c إذا أنا باخد sequence في المجموعة a وحدودها كلهم أكبر من c", "tokens": [1211, 2655, 8717, 44510, 10721, 5296, 2655, 2031, 77, 16212, 294, 257, 6055, 4587, 41193, 3615, 269, 30731, 13202, 4724, 11834, 655, 1270, 300, 2423, 4948, 6055, 3555, 34268, 11296, 382, 297, 12258, 281, 13202, 4724, 3794, 995, 45865, 269, 11933, 15730, 41850, 4724, 47283, 3215, 8310, 8978, 9673, 7435, 2304, 2407, 27884, 257, 4032, 24401, 23328, 11296, 28242, 16095, 5551, 4117, 26890, 9154, 269], "avg_logprob": -0.27509468974489154, "compression_ratio": 1.3519553072625698, "no_speech_prob": 0.0, "words": [{"start": 1379.5, "end": 1380.0, "word": "لت", "probability": 0.30706787109375}, {"start": 1380.0, "end": 1380.66, "word": " نبدأ", "probability": 0.9557291666666666}, {"start": 1380.66, "end": 1381.38, "word": " لت", "probability": 0.6356201171875}, {"start": 1381.38, "end": 1382.96, "word": " xn", "probability": 0.2958984375}, {"start": 1382.96, "end": 1384.74, "word": " contained", "probability": 0.52978515625}, {"start": 1384.74, "end": 1385.2, "word": " in", "probability": 0.93017578125}, {"start": 1385.2, "end": 1385.52, "word": " a", "probability": 0.37890625}, {"start": 1385.52, "end": 1386.74, "word": " تقاطع", "probability": 0.93701171875}, {"start": 1386.74, "end": 1387.3, "word": " c", "probability": 0.76806640625}, {"start": 1387.3, "end": 1387.68, "word": " إلى", "probability": 0.755859375}, {"start": 1387.68, "end": 1388.32, "word": " infinity", "probability": 0.7333984375}, {"start": 1388.32, "end": 1392.36, "word": " بsequence", "probability": 0.7347819010416666}, {"start": 1392.36, "end": 1393.62, "word": " such", "probability": 0.80126953125}, {"start": 1393.62, "end": 1394.18, "word": " that", "probability": 0.955078125}, {"start": 1394.18, "end": 1395.7, "word": " ال", "probability": 0.55029296875}, {"start": 1395.7, "end": 1396.02, "word": " limit", "probability": 0.8037109375}, {"start": 1396.02, "end": 1396.84, "word": " تبعتها", "probability": 0.90966796875}, {"start": 1396.84, "end": 1397.7, "word": " as", "probability": 0.90478515625}, {"start": 1397.7, "end": 1398.04, "word": " n", "probability": 0.6103515625}, {"start": 1398.04, "end": 1398.3, "word": " tends", "probability": 0.33251953125}, {"start": 1398.3, "end": 1398.42, "word": " to", "probability": 0.9189453125}, {"start": 1398.42, "end": 1399.04, "word": " infinity", "probability": 0.90185546875}, {"start": 1399.04, "end": 1400.5, "word": " بساوي", "probability": 0.82568359375}, {"start": 1400.5, "end": 1400.82, "word": " c", "probability": 0.86474609375}, {"start": 1400.82, "end": 1401.5, "word": " إذا", "probability": 0.42047119140625}, {"start": 1401.5, "end": 1401.7, "word": " أنا", "probability": 0.904296875}, {"start": 1401.7, "end": 1402.08, "word": " باخد", "probability": 0.9541015625}, {"start": 1402.08, "end": 1402.7, "word": " sequence", "probability": 0.95703125}, {"start": 1402.7, "end": 1403.56, "word": " في", "probability": 0.9375}, {"start": 1403.56, "end": 1404.24, "word": " المجموعة", "probability": 0.891748046875}, {"start": 1404.24, "end": 1404.44, "word": " a", "probability": 0.75146484375}, {"start": 1404.44, "end": 1405.12, "word": " وحدودها", "probability": 0.923583984375}, {"start": 1405.12, "end": 1405.72, "word": " كلهم", "probability": 0.93408203125}, {"start": 1405.72, "end": 1406.3, "word": " أكبر", "probability": 0.962890625}, {"start": 1406.3, "end": 1406.5, "word": " من", "probability": 0.99462890625}, {"start": 1406.5, "end": 1406.84, "word": " c", "probability": 0.9453125}], "temperature": 1.0}, {"id": 52, "seek": 143587, "start": 1408.17, "end": 1435.87, "text": "و بفرض أن ال limit لل sequence يادي بيساوي العدد c نحتاج أن نظهر عشان نثبت اتنين باقي نثبت أن ال limit نحتاج أن نظهر أن ال limit لل image of the sequence xn as n tends to infinity بساوي L", "tokens": [2407, 4724, 5172, 43042, 14739, 2423, 4948, 24976, 8310, 7251, 18513, 1829, 4724, 1829, 3794, 995, 45865, 18863, 3215, 3215, 269, 8717, 33753, 26108, 14739, 8717, 19913, 3224, 2288, 6225, 8592, 7649, 8717, 12984, 3555, 2655, 1975, 2655, 1863, 9957, 4724, 995, 38436, 8717, 12984, 3555, 2655, 14739, 2423, 4948, 8717, 33753, 26108, 14739, 8717, 19913, 3224, 2288, 14739, 2423, 4948, 24976, 3256, 295, 264, 8310, 2031, 77, 382, 297, 12258, 281, 13202, 4724, 3794, 995, 45865, 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"word": " c", "probability": 0.3583984375}, {"start": 1412.25, "end": 1414.95, "word": " نحتاج", "probability": 0.5382486979166666}, {"start": 1414.95, "end": 1415.01, "word": " أن", "probability": 0.38818359375}, {"start": 1415.01, "end": 1415.43, "word": " نظهر", "probability": 0.78326416015625}, {"start": 1415.43, "end": 1418.87, "word": " عشان", "probability": 0.7102864583333334}, {"start": 1418.87, "end": 1420.13, "word": " نثبت", "probability": 0.9912109375}, {"start": 1420.13, "end": 1420.69, "word": " اتنين", "probability": 0.82568359375}, {"start": 1420.69, "end": 1421.97, "word": " باقي", "probability": 0.7742513020833334}, {"start": 1421.97, "end": 1422.59, "word": " نثبت", "probability": 0.9727783203125}, {"start": 1422.59, "end": 1422.89, "word": " أن", "probability": 0.5732421875}, {"start": 1422.89, "end": 1423.29, "word": " ال", "probability": 0.5849609375}, {"start": 1423.29, "end": 1423.75, "word": " limit", "probability": 0.92431640625}, {"start": 1423.75, "end": 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"probability": 0.338623046875}, {"start": 1641.34, "end": 1642.1, "word": " بيقدّي", "probability": 0.677978515625}, {"start": 1642.1, "end": 1643.08, "word": " أن", "probability": 0.397216796875}, {"start": 1643.08, "end": 1644.8, "word": " xn", "probability": 0.849365234375}], "temperature": 1.0}, {"id": 61, "seek": 166936, "start": 1646.36, "end": 1669.36, "text": "minus C أكبر من سفر أصغر من Delta ليه؟ لأن ال XM موجودة تنتمي لإيه؟ هو أكبر من C، لذلك هذا لأن XM أكبر من C", "tokens": [2367, 301, 383, 5551, 4117, 26890, 9154, 8608, 5172, 2288, 5551, 9381, 17082, 2288, 9154, 18183, 32239, 3224, 22807, 5296, 33456, 2423, 1783, 44, 3714, 29245, 23328, 3660, 6055, 29399, 2304, 1829, 5296, 28814, 1829, 3224, 22807, 31439, 5551, 4117, 26890, 9154, 383, 12399, 5296, 8848, 23275, 23758, 5296, 33456, 1783, 44, 5551, 4117, 26890, 9154, 383], "avg_logprob": -0.2727640224941846, "compression_ratio": 1.376, "no_speech_prob": 0.0, "words": [{"start": 1646.36, "end": 1647.04, "word": "minus", "probability": 0.6212158203125}, {"start": 1647.04, "end": 1647.32, "word": " C", "probability": 0.505859375}, {"start": 1647.32, "end": 1647.9, "word": " أكبر", "probability": 0.93603515625}, {"start": 1647.9, "end": 1648.16, "word": " من", "probability": 0.9951171875}, {"start": 1648.16, "end": 1648.68, "word": " سفر", "probability": 0.752685546875}, {"start": 1648.68, "end": 1650.36, "word": " أصغر", "probability": 0.9476318359375}, {"start": 1650.36, "end": 1650.56, "word": " من", "probability": 0.998046875}, {"start": 1650.56, "end": 1650.96, "word": " Delta", "probability": 0.494140625}, {"start": 1650.96, "end": 1653.86, "word": " ليه؟", "probability": 0.728515625}, {"start": 1653.86, "end": 1654.32, "word": " لأن", "probability": 0.96240234375}, {"start": 1654.32, "end": 1654.52, "word": " ال", "probability": 0.64306640625}, {"start": 1654.52, "end": 1655.4, "word": " XM", "probability": 0.49957275390625}, {"start": 1655.4, "end": 1658.12, "word": " موجودة", "probability": 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exercises يعني اتمرنوا عليها", "tokens": [3322, 42881, 46691, 337, 732, 12, 30941, 4948, 11900, 37495, 22653, 1975, 2655, 29973, 1863, 14407, 25894, 11296], "avg_logprob": -0.32915295425214264, "compression_ratio": 0.967391304347826, "no_speech_prob": 0.0, "words": [{"start": 1884.13, "end": 1884.51, "word": "the", "probability": 0.1903076171875}, {"start": 1884.51, "end": 1885.31, "word": " sequential", "probability": 0.96630859375}, {"start": 1885.31, "end": 1886.45, "word": " criterion", "probability": 0.962890625}, {"start": 1886.45, "end": 1889.19, "word": " for", "probability": 0.951171875}, {"start": 1889.19, "end": 1890.37, "word": " two", "probability": 0.90234375}, {"start": 1890.37, "end": 1891.25, "word": "-sided", "probability": 0.70361328125}, {"start": 1891.25, "end": 1894.57, "word": " limit", "probability": 0.6982421875}, {"start": 1894.57, "end": 1905.85, "word": " exercises", "probability": 0.1864013671875}, {"start": 1905.85, "end": 1906.93, "word": " يعني", 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2152.02, "end": 2177.78, "text": "بالبرهان البرهان يعني كتير سهل ينتج من التعريفات proof ف .. هحاول أبرهنلكم ال F part هذا مسمى ال F part يعني هفرض أنه assume أنه ال one sided limits", "tokens": [3555, 6027, 26890, 3224, 7649, 2423, 26890, 3224, 7649, 37495, 22653, 9122, 2655, 13546, 8608, 3224, 1211, 7251, 29399, 7435, 9154, 16712, 3615, 16572, 5172, 9307, 8177, 6156, 4386, 8032, 5016, 995, 12610, 5551, 26890, 3224, 1863, 23275, 2304, 2423, 479, 644, 23758, 3714, 38251, 7578, 2423, 479, 644, 37495, 22653, 8032, 5172, 43042, 14739, 3224, 6552, 14739, 3224, 2423, 472, 41651, 10406], "avg_logprob": -0.19604492210783064, "compression_ratio": 1.4556962025316456, "no_speech_prob": 0.0, "words": [{"start": 2152.02, "end": 2152.84, "word": "بالبرهان", "probability": 0.8900390625}, {"start": 2152.84, "end": 2154.56, "word": " البرهان", "probability": 0.8814697265625}, {"start": 2154.56, "end": 2154.98, "word": " يعني", "probability": 0.9365234375}, {"start": 2154.98, "end": 2155.48, "word": " 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"end": 2378.56, "text": "there exists delta تانية مش صارت الـ delta هذه تكون نفس الـ delta اللي فوق ماحد بيقدر يجزم ذلك فنسميها delta تانية there exists delta two depends طبعا بالتأكيد تعتمد على إبسلون وعدد موجة بحيث أنه حسب التعريف لكل x", "tokens": [15456, 8198, 8289, 6055, 7649, 10632, 37893, 20328, 9640, 2655, 2423, 39184, 8289, 29538, 6055, 30544, 8717, 36178, 2423, 39184, 8289, 13672, 1829, 6156, 30543, 19446, 24401, 4724, 1829, 28543, 2288, 7251, 7435, 11622, 2304, 29910, 23275, 6156, 1863, 38251, 1829, 11296, 8289, 6055, 7649, 10632, 456, 8198, 8289, 732, 5946, 23032, 3555, 3615, 995, 20666, 2655, 10721, 4117, 25708, 6055, 34268, 2304, 3215, 15844, 11933, 3555, 3794, 1211, 11536, 4032, 22488, 3215, 3714, 29245, 3660, 4724, 5016, 1829, 12984, 14739, 3224, 11331, 35457, 16712, 3615, 16572, 5172, 5296, 28820, 2031], "avg_logprob": -0.2666440113731053, "compression_ratio": 1.5817307692307692, "no_speech_prob": 0.0, "words": [{"start": 2357.62, "end": 2358.04, "word": "there", 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{"start": 2772.41, "end": 2772.51, "word": " و", "probability": 0.98583984375}, {"start": 2772.51, "end": 2773.35, "word": " مختلفتين", "probability": 0.95380859375}, {"start": 2773.35, "end": 2773.97, "word": " أو", "probability": 0.8017578125}, {"start": 2773.97, "end": 2774.39, "word": " واحدة", "probability": 0.9817708333333334}, {"start": 2774.39, "end": 2774.91, "word": " موجودة", "probability": 0.9881591796875}, {"start": 2774.91, "end": 2775.03, "word": " و", "probability": 0.94970703125}, {"start": 2775.03, "end": 2775.33, "word": " اتنين", "probability": 0.63238525390625}, {"start": 2775.33, "end": 2775.53, "word": " مش", "probability": 0.96826171875}, {"start": 2775.53, "end": 2776.37, "word": " موجودة", "probability": 0.993896484375}, {"start": 2776.37, "end": 2776.97, "word": " و", "probability": 0.912109375}, {"start": 2776.97, "end": 2777.45, "word": " هكذا،", "probability": 0.773712158203125}, {"start": 2777.45, "end": 2777.93, "word": " هنشوف", "probability": 0.83404541015625}, {"start": 2777.93, "end": 2778.19, "word": " كل", "probability": 0.98486328125}, {"start": 2778.19, "end": 2779.09, "word": " الأنواع", "probability": 0.9569091796875}, {"start": 2779.09, "end": 2779.45, "word": " و", "probability": 0.99560546875}, {"start": 2779.45, "end": 2779.73, "word": " كل", "probability": 0.91162109375}, {"start": 2779.73, "end": 2779.95, "word": " ال", "probability": 0.92529296875}, {"start": 2779.95, "end": 2781.31, "word": " situations،", "probability": 0.515625}, {"start": 2781.31, "end": 2782.85, "word": " تمام؟", "probability": 0.9794921875}, {"start": 2782.85, "end": 2783.47, "word": " okay", "probability": 0.62744140625}, {"start": 2783.47, "end": 2783.93, "word": " شكرا", "probability": 0.95068359375}, {"start": 2783.93, "end": 2784.37, "word": " لكم", "probability": 0.6917724609375}, {"start": 2784.37, "end": 2784.67, "word": " و", "probability": 0.97802734375}, {"start": 2784.67, "end": 2785.25, "word": " نشوفكم", "probability": 0.9599609375}, {"start": 2785.25, "end": 2785.43, "word": " ان", "probability": 0.64208984375}, {"start": 2785.43, "end": 2785.59, "word": " شاء", "probability": 0.996337890625}, {"start": 2785.59, "end": 2785.67, "word": " الله", "probability": 0.96533203125}, {"start": 2785.67, "end": 2786.03, "word": " المرة", "probability": 0.950439453125}, {"start": 2786.03, "end": 2786.55, "word": " القادمة", "probability": 0.9900716145833334}], "temperature": 1.0}], "language": "ar", "language_probability": 1.0, "duration": 2786.859, "duration_after_vad": 2533.8606249999893}
\ No newline at end of file
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/2qaKB7theEQ_raw.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/2qaKB7theEQ_raw.srt
new file mode 100644
index 0000000000000000000000000000000000000000..a0b9234974d403d8d3e11e6edea12b3ab80aab21
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/2qaKB7theEQ_raw.srt
@@ -0,0 +1,1620 @@
+1
+00:00:19,940 --> 00:00:25,840
+السلام عليكم هنكمل
+
+2
+00:00:25,840 --> 00:00:33,420
+اليوم section اربعة اتنين في ال section هذا كان
+
+3
+00:00:33,420 --> 00:00:38,780
+اتبقى بس ان احنا نثبت النظرية اللى كتبتها على
+
+4
+00:00:38,780 --> 00:00:44,700
+اللوحى النظرية هذه بتنص على ان لو كان في هندي
+
+5
+00:00:44,700 --> 00:00:53,020
+function من A لRو c cluster point للset a وإذا كان
+
+6
+00:00:53,020 --> 00:00:59,120
+limit ال function عن c exist وموجبة أو
+
+7
+00:00:59,120 --> 00:01:04,880
+على التوالي إذا كانت limit f of x عن c موجودة
+
+8
+00:01:04,880 --> 00:01:09,080
+وسالبة فيوجد
+
+9
+00:01:09,080 --> 00:01:14,990
+نقدر نلاقي delta neighborhood دي delta لل cبحيث إن
+
+10
+00:01:14,990 --> 00:01:19,510
+الدالة هتكون إذا كانت ال limit موجبة فالدالة هتكون
+
+11
+00:01:19,510 --> 00:01:26,670
+موجبة على ال delta never hood ل C وإذا
+
+12
+00:01:26,670 --> 00:01:31,950
+كانت ال limit سالبة فالدالة هتكون سالبة على جوار
+
+13
+00:01:31,950 --> 00:01:38,890
+delta ل C هذه
+
+14
+00:01:38,890 --> 00:01:43,210
+نظرية تشبه نظرية سابقة بخصوص limits of sequences
+
+15
+00:01:44,920 --> 00:01:48,400
+النظرية اللي فاتت بتاعة الـ sequences المشابهة
+
+16
+00:01:48,400 --> 00:01:52,640
+بتقول لو كانت ال sequence النهاية تبعتها limit x
+
+17
+00:01:52,640 --> 00:01:58,200
+in exist وموجبة فلازم ال sequence تكون حدودها من
+
+18
+00:01:58,200 --> 00:02:02,420
+capital N وانت طالع كلها موجبة و لو كانت ال limit
+
+19
+00:02:02,420 --> 00:02:06,640
+لل sequence exist و سالبة فلازم حدود ال sequence
+
+20
+00:02:06,640 --> 00:02:10,900
+من capital N وانت طالع كلها تكون سالبة فهذه شبيهة
+
+21
+00:02:10,900 --> 00:02:18,100
+فيهاوالبرهان سهل وشبيه بالبرهان تبع النظرية
+
+22
+00:02:18,100 --> 00:02:24,320
+المشابهة في حالة ال sequences فناخد الحالة assume
+
+23
+00:02:24,320 --> 00:02:27,360
+ناخد
+
+24
+00:02:27,360 --> 00:02:32,400
+الحالة اللي فيها ال limit ل
+
+25
+00:02:32,400 --> 00:02:40,060
+f of x at c exists and equals عدد l موجب
+
+26
+00:02:53,880 --> 00:03:00,200
+فإذا كانت ال limit موجبة بنا أثبت أن يوجد delta
+
+27
+00:03:00,200 --> 00:03:07,240
+neverhood إلى آخر A فخلّينا
+
+28
+00:03:07,240 --> 00:03:17,740
+ناخد let epsilon في الحالة دي let epsilon بساوي
+
+29
+00:03:17,740 --> 00:03:26,600
+L على 2 فهذا عدد موجبالان by definition of limit
+
+30
+00:03:26,600 --> 00:03:32,640
+of function by epsilon delta definition لأي
+
+31
+00:03:32,640 --> 00:03:38,140
+epsilon موجبة زي هذه يوجد delta تعتمد على L على 2
+
+32
+00:03:38,140 --> 00:03:43,280
+اللي هي ال epsilon عدد موجب بحيث انه لو كان X
+
+33
+00:03:45,890 --> 00:03:51,090
+ينتمي إلى a و absolute x minus c أصغر من delta
+
+34
+00:03:51,090 --> 00:03:59,790
+أكبر من 0 فهذا بتضمن أن absolute f of x minus L
+
+35
+00:03:59,790 --> 00:04:04,510
+أصغر من إبسلم اللي هي عبارة عن L ع 2
+
+36
+00:04:08,170 --> 00:04:15,990
+فحل المتباينة هذه في f of x فتصير f of x minus L
+
+37
+00:04:15,990 --> 00:04:24,930
+أصغر من L على 2 أكبر من سالب L على 2 وهذا
+
+38
+00:04:24,930 --> 00:04:29,210
+بيقدي أن
+
+39
+00:04:29,210 --> 00:04:30,350
+f of x
+
+40
+00:04:34,740 --> 00:04:45,980
+من هنا F of X تطلع أكبر من L على 2 لأنه لما أخد
+
+41
+00:04:45,980 --> 00:04:50,240
+سالب L أنجلها عن ناحية التانية فتصير F of X أكبر
+
+42
+00:04:50,240 --> 00:04:55,700
+من L سالب L على 2 تطلع L على 2 و ال L موجبة إذا L
+
+43
+00:04:55,700 --> 00:05:06,860
+على 2 موجبة إذا هيك بنكون أثبتناإن ال F of X طلعت
+
+44
+00:05:06,860 --> 00:05:18,580
+أكبر من سفر لمين لكل X تنتمي إلى A ومن
+
+45
+00:05:18,580 --> 00:05:28,080
+المتباينة هذه هذا معناه X لا تساوي Cإن الـ X ينتمي
+
+46
+00:05:28,080 --> 00:05:34,480
+إلى A و لا تساوي C يعني موجودة في A و مش موجودة في
+
+47
+00:05:34,480 --> 00:05:44,280
+singleton set C و المتباينة هذه هذي معناها إن X
+
+48
+00:05:44,280 --> 00:05:46,460
+ينتمي ل V Delta
+
+49
+00:05:56,010 --> 00:06:00,790
+x-c أصغر من دلتا بكافئ
+
+50
+00:06:10,030 --> 00:06:17,770
+إن X أصغر من C زائد Delta أكبر من C سارب Delta
+
+51
+00:06:17,770 --> 00:06:21,550
+فهذا
+
+52
+00:06:21,550 --> 00:06:27,890
+معناه X تمتني لفترة مفتوحة Delta-neighborhood ل-C
+
+53
+00:06:29,570 --> 00:06:34,830
+Okay تمام اذا f of x اللي اعطاها موجبة لكل x في a
+
+54
+00:06:34,830 --> 00:06:43,250
+ومختلفة عن c وايضا من هنا ال x أيضا تنتمي ل delta
+
+55
+00:06:43,250 --> 00:06:48,850
+neighborhood ل c وبالتالي تنتمي لتقاطع المجمعتين
+
+56
+00:06:48,850 --> 00:06:55,270
+اذا هذا بثبت المظرية في حالة لما يكون ال limit
+
+57
+00:06:55,270 --> 00:06:56,490
+تبعتي موجبة
+
+58
+00:07:00,240 --> 00:07:08,880
+لو كانت ال limit سالبة فالبرهان مشابه لأن ال
+
+59
+00:07:08,880 --> 00:07:18,680
+proof of the caseلما تكون ال limit ل f of x لما x
+
+60
+00:07:18,680 --> 00:07:30,600
+تقول ل c بساوي العدد سالب is similar to
+
+61
+00:07:30,600 --> 00:07:38,200
+above case مشابه
+
+62
+00:07:38,200 --> 00:07:50,120
+للبرهان السابق علىفي الحالة هذه take start with
+
+63
+00:07:50,120 --> 00:07:55,920
+epsilon بساوي سالب ال ع اتنين وهذا بطلع عدد موجب
+
+64
+00:07:55,920 --> 00:08:01,060
+يعني ابدوا البرهان بدل ما بدنا بepsilon بساوي ال ع
+
+65
+00:08:01,060 --> 00:08:04,340
+اتنين ابدوا ابسون .. ابسون بساوي
+
+66
+00:08:14,820 --> 00:08:20,320
+البرهان البرهان البرهان البرهان البرهان البرهان
+
+67
+00:08:20,320 --> 00:08:20,800
+البرهان البرهان البرهان البرهان البرهان البرهان
+
+68
+00:08:20,800 --> 00:08:20,800
+البرهان البرهان البرهان البرهان البرهان البرهان
+
+69
+00:08:20,800 --> 00:08:20,800
+البرهان البرهان البرهان البرهان البرهان البرهان
+
+70
+00:08:20,800 --> 00:08:20,800
+البرهان البرهان البرهان البرهان البرهان البرهان
+
+71
+00:08:20,800 --> 00:08:20,860
+البرهان البرهان البرهان البرهان البرهان البرهان
+
+72
+00:08:20,860 --> 00:08:24,600
+البرهان البرهان البرهان البرهان البرهان البرهان
+
+73
+00:08:24,600 --> 00:08:24,640
+البرهان البرهان البرهان البرهان البرهان البرهان
+
+74
+00:08:24,640 --> 00:08:25,080
+البرهان البرهان البرهان البرهان البرهان البرهان
+
+75
+00:08:25,080 --> 00:08:33,380
+البرهان البرهان
+
+76
+00:08:34,550 --> 00:08:40,910
+لأن ال L سالمة و هذا صحيح لكل X في جوار Delta ل C
+
+77
+00:08:40,910 --> 00:08:46,530
+و في A minus single to C okay؟ لأن حاسبكم أنتم
+
+78
+00:08:46,530 --> 00:08:50,990
+تكتبوا البرهان تبع الحالة التانية تمام؟ واضح؟ في
+
+79
+00:08:50,990 --> 00:08:54,850
+أي سؤال أو سفسار؟ تمام؟
+
+80
+00:09:00,180 --> 00:09:08,120
+Okay إذا نبدأ section جديد
+
+81
+00:09:08,120 --> 00:09:26,200
+section
+
+82
+00:09:26,200 --> 00:09:29,460
+أربعة تلاتة
+
+83
+00:09:33,950 --> 00:09:47,190
+بعض التطبيقات .. بعض التطبيقات لـ
+
+84
+00:09:47,190 --> 00:09:53,590
+limit concept
+
+85
+00:10:04,410 --> 00:10:12,090
+بعض التعاملات أو توصية بعض مفاهيم النهايات احنا
+
+86
+00:10:12,090 --> 00:10:16,470
+قبل هيك درسنا في section 4.1 و 4.2 ال limit of
+
+87
+00:10:16,470 --> 00:10:20,450
+function او ال two sided limit لل function عن نقطة
+
+88
+00:10:20,450 --> 00:10:24,810
+معينة اليوم هندرس ال one sided limit ل function عن
+
+89
+00:10:24,810 --> 00:10:30,960
+نقطة and cluster point للمجال تبعهامقصود بال one
+
+90
+00:10:30,960 --> 00:10:34,040
+sided limit اللي هو limit من اليمين أو limit من
+
+91
+00:10:34,040 --> 00:10:38,740
+اليسار ونشوف ما هي علاقة ال one sided limit بال
+
+92
+00:10:38,740 --> 00:10:43,320
+two sided limit فنعرف
+
+93
+00:10:43,320 --> 00:10:47,880
+الأول definition نعرف ال one sided limit
+
+94
+00:10:47,880 --> 00:10:53,000
+definition let
+
+95
+00:10:55,520 --> 00:11:03,780
+fb function from a to r and c be a cluster point
+
+96
+00:11:03,780 --> 00:11:06,840
+cluster
+
+97
+00:11:06,840 --> 00:11:22,340
+point of a واحد او
+
+98
+00:11:22,340 --> 00:11:34,220
+خلّيالـ cluster point of المجموعة A
+
+99
+00:11:34,220 --> 00:11:40,880
+تقاطع الفترة المفتوحة من C إلى infinity اللي هي كل
+
+100
+00:11:40,880 --> 00:11:47,240
+ال X مجموعة كل العناصر X تنتبه إلى A حيث X أكبر من
+
+101
+00:11:47,240 --> 00:11:53,520
+C نقول
+
+102
+00:11:57,140 --> 00:12:03,280
+إن العدد alien 10 إلى R is
+
+103
+00:12:03,280 --> 00:12:14,600
+a right .. is a right hand limit .. right hand
+
+104
+00:12:14,600 --> 00:12:29,490
+limit of ال function F at ..x بساوي c if الشرط
+
+105
+00:12:29,490 --> 00:12:35,570
+التالي بتحقق لكل
+
+106
+00:12:35,570 --> 00:12:40,630
+إبسلون given
+
+107
+00:12:40,630 --> 00:12:43,970
+إبسلون
+
+108
+00:12:43,970 --> 00:12:51,040
+أكبر من السفر يوجد delta تعتمد علىepsilon عدد موجة
+
+109
+00:12:51,040 --> 00:13:01,180
+بهاث انه لو كان x ينتمي ل a و x minus c أكبر من
+
+110
+00:13:01,180 --> 00:13:07,760
+سفر أصغر من delta فهذا بتضمن ان absolute f of x
+
+111
+00:13:07,760 --> 00:13:13,540
+minus l أصغر من epsilon in
+
+112
+00:13:13,540 --> 00:13:17,720
+this case in
+
+113
+00:13:17,720 --> 00:13:28,480
+this casewe write نكتب أن ال limit لل function f
+
+114
+00:13:28,480 --> 00:13:37,920
+عندما x تقول إلى c من اليمين بساوي العدد ال ..
+
+115
+00:13:37,920 --> 00:13:44,920
+تمام؟ لأن هذا تعريف ال limit from the right أو ال
+
+116
+00:13:44,920 --> 00:13:49,460
+right hand limit لل function f عند النقطة c
+
+117
+00:14:05,180 --> 00:14:13,440
+إذا أنا عندي هذه خط الأعداد وهي النقطة C وانا عندي
+
+118
+00:14:13,440 --> 00:14:21,900
+ال C هي cluster point ل
+
+119
+00:14:21,900 --> 00:14:26,180
+A .. لكل ال X موجود في A و أكبر من C
+
+120
+00:14:32,710 --> 00:14:37,650
+فبنقول إن ال limit عند x بالساوية c أو ال function
+
+121
+00:14:37,650 --> 00:14:42,150
+في إلها right-hand limit و ال right-hand limit هي
+
+122
+00:14:42,150 --> 00:14:48,410
+العدد L إذا كان لأي إبسلون أكبر من السفر بتقدر
+
+123
+00:14:48,410 --> 00:14:53,390
+نلاقي delta عدد موجة بيعتمد على إبسلون بحيث لكل x
+
+124
+00:14:53,390 --> 00:15:01,510
+في المجموعة A إذا كانت ال X هذه على يمين ال C
+
+125
+00:15:05,140 --> 00:15:12,460
+والمسافة بينها وبين الـ C أصغر من Delta فبتطلع
+
+126
+00:15:12,460 --> 00:15:19,860
+المسافة بين F و XL أصغر من Y بالمثل
+
+127
+00:15:19,860 --> 00:15:24,100
+ممكن نعرف ال limit from the right يعني أنا بدي
+
+128
+00:15:24,100 --> 00:15:27,800
+أعرف ال limit from the right أو ال right hand
+
+129
+00:15:27,800 --> 00:15:34,750
+limitهي نفس let f be function from A to R و C
+
+130
+00:15:34,750 --> 00:15:41,010
+cluster point للمجموعة A تقاطع الفترة المفتوحة من
+
+131
+00:15:41,010 --> 00:15:50,850
+سالب مالة نهاية إلى C اللي هي كل ال X في A حيث X
+
+132
+00:15:50,850 --> 00:15:58,490
+هتكون أصغر من مرة هذه أصغر من Cفنقول إن الـ real
+
+133
+00:15:58,490 --> 00:16:06,070
+number L هو بدل right hand limit هيكون left hand
+
+134
+00:16:06,070 --> 00:16:10,550
+limit of f at c if given epsilon there exists
+
+135
+00:16:10,550 --> 00:16:15,310
+delta depends on epsilon بحيث أنه لكل x ينتمي إلى
+
+136
+00:16:15,310 --> 00:16:21,900
+aلكل X هنتمي إلى A وال X طبعا موجودة في الفترة هذه
+
+137
+00:16:21,900 --> 00:16:28,600
+يعني ال X المرة هذه على يسار المرة هذه ال X هتكون
+
+138
+00:16:31,420 --> 00:16:35,260
+موجودة في A وفي الفترة المفتوحة من سالب مالنهاية
+
+139
+00:16:35,260 --> 00:16:44,720
+إلى C يعني ال X هتكون على يسار ال C وبالتالي هنا
+
+140
+00:16:44,720 --> 00:16:51,240
+ال C minus المسافة بين X و C absolute X minus C
+
+141
+00:16:51,240 --> 00:16:57,640
+هتطلع بساوي C minus X فلو كانت المسافة هذه أصغر من
+
+142
+00:16:57,640 --> 00:16:59,660
+Delta وطبعا أكبر من سفر
+
+143
+00:17:09,550 --> 00:17:18,370
+هذا الشرط سيصبح c-x أصغر من دلتا أكبر من سفر فهذا
+
+144
+00:17:18,370 --> 00:17:22,910
+لازم يضمن أن absolute of f of x minus l أصغر من
+
+145
+00:17:22,910 --> 00:17:29,610
+إبسمن في الحالة هذه بيقول إن ال limit لf of x لما
+
+146
+00:17:29,610 --> 00:17:31,850
+x تقول لc من اليسار
+
+147
+00:17:34,230 --> 00:17:39,550
+بس n بساوي LL okay انها تعريف ال lift hand limit
+
+148
+00:17:39,550 --> 00:17:44,890
+او ال limit from the lift okay تعديل
+
+149
+00:17:44,890 --> 00:17:50,770
+بسيط بس طيب
+
+150
+00:17:50,770 --> 00:17:58,170
+ال limits هذه هنشوف يعني بعد شوية ان ال one sided
+
+151
+00:17:58,170 --> 00:18:03,210
+limits ده functional نقطةممكن يعني التنتين يكونوا
+
+152
+00:18:03,210 --> 00:18:09,750
+موجودين عند النقطة ويلهم نفس القيمة أو ممكن
+
+153
+00:18:09,750 --> 00:18:15,430
+التنتين يكونوا موجودين عند نقطة لكن قيمهم مختلفة
+
+154
+00:18:15,430 --> 00:18:20,270
+زي الـ Signum function عند الصفر شوفنا أن ال limit
+
+155
+00:18:20,270 --> 00:18:23,350
+تبعتها من اليمين واحد و ال limit تبعتها من اليسار
+
+156
+00:18:23,350 --> 00:18:27,850
+ثالث واحدإذا ممكن ال two sided limits يكونوا
+
+157
+00:18:27,850 --> 00:18:33,630
+موجودات لكن they are different مختلفات، ممكن برضه
+
+158
+00:18:33,630 --> 00:18:37,790
+one sided limit تكون موجودة and the other may not
+
+159
+00:18:37,790 --> 00:18:42,090
+exist، ممكن ما تكونش موجودة من أساسه
+
+160
+00:18:44,810 --> 00:18:54,930
+ممكن ال one sided limits ولا واحدة فيهم تكون
+
+161
+00:18:54,930 --> 00:19:03,250
+موجودة فكل الحلقات هذه هنشوفها في أمثلة لاحقة لكن
+
+162
+00:19:03,250 --> 00:19:08,030
+الأول خلّينا نبرهن النظرية التالية
+
+163
+00:19:13,670 --> 00:19:18,750
+طبعا هنا بنحب ال ..
+
+164
+00:19:18,750 --> 00:19:24,870
+النوّه أن كل نظريات اللي أثبتناها في section 4.1
+
+165
+00:19:24,870 --> 00:19:31,790
+أو 4.2 بخصوص ال two sided limit هتكون صحيحة بخصوص
+
+166
+00:19:31,790 --> 00:19:38,030
+ال right limit و كذلك صحيحة بخصوص ال left hand
+
+167
+00:19:38,030 --> 00:19:38,430
+limit
+
+168
+00:19:41,100 --> 00:19:46,240
+فعلى سبيل المثال وليس الحصر احنا أخدنا sequential
+
+169
+00:19:46,240 --> 00:19:51,240
+criterion sequential criterion for two sided limit
+
+170
+00:19:51,240 --> 00:19:56,580
+الآن هنكتب برضه sequential criterion for right
+
+171
+00:19:56,580 --> 00:20:09,420
+limit sequential criterion for right
+
+172
+00:20:09,420 --> 00:20:10,100
+hand
+
+173
+00:20:25,970 --> 00:20:35,670
+limits let f from a to r be function and c be
+
+174
+00:20:35,670 --> 00:20:37,330
+cluster
+
+175
+00:20:39,230 --> 00:20:47,910
+point of A then the following statements are
+
+176
+00:20:47,910 --> 00:20:54,190
+equivalent الأبارات التالية متكافئة واحد ال limit
+
+177
+00:20:54,190 --> 00:21:01,810
+ل F of X as X tends to C from the right exist
+
+178
+00:21:01,810 --> 00:21:06,030
+وبساوي عدد L اتنين
+
+179
+00:21:13,830 --> 00:21:20,370
+for for every sequence
+
+180
+00:21:20,370 --> 00:21:36,530
+x in contained in a تقاطع c و infinity such that
+
+181
+00:21:38,210 --> 00:21:47,290
+limit x in as n tends to infinity بيساوي c we have
+
+182
+00:21:47,290 --> 00:21:51,090
+limit
+
+183
+00:21:51,090 --> 00:21:59,170
+لل image of the sequence x in بيساوي العدد L
+
+184
+00:22:10,340 --> 00:22:17,060
+البرهان شبيه بالبرهان الخاص بالـ two-sided limit
+
+185
+00:22:17,060 --> 00:22:24,100
+فمثلا لو بدنا نبرهن proof لو بدى برهن الأبعار
+
+186
+00:22:24,100 --> 00:22:30,720
+الأولى بتأدي للتانية فبنقول assume .. نبدأ ب
+
+187
+00:22:30,720 --> 00:22:38,240
+assume أن ال limit ال right limitالـ F عند الـ C
+
+188
+00:22:38,240 --> 00:22:44,240
+exist بساوي L وبدنا
+
+189
+00:22:44,240 --> 00:22:48,680
+نثبت أن الـ two بيطلع العبارة تنين بتطلع صحيحة
+
+190
+00:22:48,680 --> 00:22:55,400
+لبرهان العبارة to prove two
+
+191
+00:22:55,400 --> 00:22:56,260
+holds
+
+192
+00:22:59,500 --> 00:23:08,320
+لت نبدأ لت xn contained in a تقاطع c إلى infinity
+
+193
+00:23:08,320 --> 00:23:12,360
+بsequence
+
+194
+00:23:12,360 --> 00:23:19,040
+such that ال limit تبعتها as n tends to infinity
+
+195
+00:23:19,040 --> 00:23:24,440
+بساوي c إذا أنا باخد sequence في المجموعة a
+
+196
+00:23:24,440 --> 00:23:30,410
+وحدودها كلهم أكبر من cو بفرض أن ال limit لل
+
+197
+00:23:30,410 --> 00:23:38,870
+sequence يادي بيساوي العدد c نحتاج أن نظهر عشان
+
+198
+00:23:38,870 --> 00:23:46,250
+نثبت اتنين باقي نثبت أن ال limit نحتاج أن نظهر أن
+
+199
+00:23:46,250 --> 00:23:53,530
+ال limit لل image of the sequence xn as n tends to
+
+200
+00:23:53,530 --> 00:24:01,630
+infinity بساوي Lهيك بنكون أثبتنا أن العبارة 2
+
+201
+00:24:01,630 --> 00:24:10,150
+صحيحة، مصبوط، صح؟ طيب لبرهان ذلك to
+
+202
+00:24:10,150 --> 00:24:11,130
+see this
+
+203
+00:24:16,090 --> 00:24:19,390
+بدأ اثبت ان ال limit لل sequence هذه بساوي عدد L
+
+204
+00:24:19,390 --> 00:24:23,890
+فبستخدم تعريف epsilon capital N لل limit فلازم
+
+205
+00:24:23,890 --> 00:24:31,510
+نبدأ with epsilon أكبر من السفر ب given طيب مش
+
+206
+00:24:31,510 --> 00:24:41,970
+احنا فرضينSince الـ right limit ل F and C موجود أو
+
+207
+00:24:41,970 --> 00:24:46,770
+بالساوي L من تعريف ال right limit there exists
+
+208
+00:24:46,770 --> 00:24:52,230
+delta depends on epsilon positive number بحيث إنه
+
+209
+00:24:52,230 --> 00:25:02,200
+لو كانت ال X تنتمي إلى A و X minus C أكبر من 0أصغر
+
+210
+00:25:02,200 --> 00:25:09,760
+من دلتا هذا معناه بيقدي أنه absolute f of x minus
+
+211
+00:25:09,760 --> 00:25:22,600
+L أصغر من إبسم نسمي ال implication هذي star now
+
+212
+00:25:22,600 --> 00:25:30,880
+for the aboveالدلتا أكبر من السفر لدلتا هذه
+
+213
+00:25:30,880 --> 00:25:34,720
+الاتباع الموجبة لدلتا هذه الاتباع الموجبة لدلتا
+
+214
+00:25:34,720 --> 00:25:36,800
+هذه الاتباع الموجبة لدلتا هذه الاتباع الموجبة
+
+215
+00:25:36,800 --> 00:25:38,440
+لدلتا هذه الاتباع الموجبة لدلتا هذه الاتباع
+
+216
+00:25:38,440 --> 00:25:41,400
+الموجبة لدلتا هذه الاتباع الموجبة لدلتا هذه
+
+217
+00:25:41,400 --> 00:25:41,660
+الاتباع الموجبة لدلتا هذه الاتباع الموجبة لدلتا
+
+218
+00:25:41,660 --> 00:25:42,040
+هذه الاتباع الموجبة لدلتا هذه الاتباع الموجبة
+
+219
+00:25:42,040 --> 00:25:43,160
+لدلتا هذه الاتباع الموجبة لدلتا هذه الاتباع
+
+220
+00:25:43,160 --> 00:25:51,140
+الموجبة لدلتا هذه الاتباع الموجبة لدلتا هذه
+
+221
+00:25:51,140 --> 00:25:57,180
+الاتباع الموجبة لدلتا هذه الاتباع الموجبة لدلت
+
+222
+00:25:57,830 --> 00:26:05,110
+natural number عدد طبيعي بحيث انه لو كان ال N أكبر
+
+223
+00:26:05,110 --> 00:26:12,510
+من أو ساوي capital N فهذا بتضمن ان absolute xn
+
+224
+00:26:12,510 --> 00:26:20,090
+minus c أصغر من delta نسمي ال implication هذه
+
+225
+00:26:20,090 --> 00:26:21,050
+double star
+
+226
+00:26:30,480 --> 00:26:44,680
+hence و بالتالي star and double star imply بيؤدوا
+
+227
+00:26:44,680 --> 00:26:51,860
+إلى ما يلي انه لو كانت ال N أكبر من أو ساوي
+
+228
+00:26:51,860 --> 00:26:56,360
+capital N فمن
+
+229
+00:26:56,360 --> 00:26:57,380
+double star
+
+230
+00:26:59,860 --> 00:27:04,940
+لو كانت n أكبر من أو ساوي capital N فمن double
+
+231
+00:27:04,940 --> 00:27:21,340
+star بطلع absolute xn minus c أصغر من delta هذا
+
+232
+00:27:21,340 --> 00:27:24,800
+بيقدّي أن xn
+
+233
+00:27:26,360 --> 00:27:35,400
+minus C أكبر من سفر أصغر من Delta ليه؟ لأن ال XM
+
+234
+00:27:35,400 --> 00:27:44,420
+موجودة تنتمي لإيه؟ هو أكبر من C، لذلك هذا لأن XM
+
+235
+00:27:44,420 --> 00:27:48,400
+أكبر
+
+236
+00:27:48,400 --> 00:27:58,050
+من Cفبالتالي absolute xn-c أكبر من 0 وبالتالي
+
+237
+00:27:58,050 --> 00:28:06,790
+absolute xn-c absolute عدد موجب بساوي نفسه لأن ال
+
+238
+00:28:06,790 --> 00:28:15,750
+absolute value هنا ل xn-c بساوي xn-c لأن xn أكبر
+
+239
+00:28:15,750 --> 00:28:18,110
+من c وطبعا
+
+240
+00:28:21,830 --> 00:28:32,590
+هذا أكبر من السفر لأن xn لا تساوي c أكبر من c الان
+
+241
+00:28:32,590 --> 00:28:39,310
+من ال star هذا بيقدي by starالـ star بتقول إذا
+
+242
+00:28:39,310 --> 00:28:45,330
+كانت ال X أو هنا في الحالة تبعتنا X in ال X in هذه
+
+243
+00:28:45,330 --> 00:28:49,990
+تنتمي لإيه؟ ال X in هي تنتمي لإيه؟ و بعدين هي عندي
+
+244
+00:28:49,990 --> 00:28:56,470
+X in سالب C أكبر من سفر أصغر من Delta إذا by star
+
+245
+00:28:56,470 --> 00:29:06,950
+بتطلع absolute F of X in minus L أصغر من Y تمام؟
+
+246
+00:29:10,290 --> 00:29:18,790
+الان نلاحظ ان ابسلون was arbitrary ابسلون
+
+247
+00:29:18,790 --> 00:29:28,230
+was arbitrary since
+
+248
+00:29:28,230 --> 00:29:36,890
+ابسلون أكبر من السفر was arbitrary اذا هيك بنكون
+
+249
+00:29:36,890 --> 00:29:43,470
+احنا أثبتناإنه لأي إبسلون أو لكل إبسلون يوجد Delta
+
+250
+00:29:43,470 --> 00:29:50,890
+لأ لكل إبسلون يوجد capital N يعتمد على ال Delta
+
+251
+00:29:50,890 --> 00:29:55,070
+وبالتالي تعتمد على إبسلون لأن ال Delta تعتمد على
+
+252
+00:29:55,070 --> 00:30:01,410
+إبسلون بحيث إنه لكل N أكبر من أو سوى capital N طلع
+
+253
+00:30:01,410 --> 00:30:06,190
+عندي absolute f of xn minus L أصغر من إبسلونإذاً
+
+254
+00:30:06,190 --> 00:30:12,750
+by epsilon capital N definition of limit بيطلع هيك
+
+255
+00:30:12,750 --> 00:30:18,710
+بيكون أثبتنا أنه limit ال sequence f of x n as n
+
+256
+00:30:18,710 --> 00:30:23,710
+tends to infinity بساوي L وهذا اللي بدنا يعني هذا
+
+257
+00:30:23,710 --> 00:30:26,570
+اللي احنا ايه اللي عايزين نثبته
+
+258
+00:30:29,870 --> 00:30:35,490
+إذاً هيك منكون أثبتنا أنه إيه اتنين holds وبالتالي
+
+259
+00:30:35,490 --> 00:30:41,670
+هيك هذا بيكمل برهان واحد implies two okay تمام؟
+
+260
+00:30:41,670 --> 00:30:46,490
+بالمثل ممكن انبرهن اتنين implies one
+
+261
+00:30:55,620 --> 00:31:03,360
+the proof of اتنين implies العبارة
+
+262
+00:31:03,360 --> 00:31:11,200
+التانية implies الأولى is similar is
+
+263
+00:31:11,200 --> 00:31:18,600
+similar to is
+
+264
+00:31:18,600 --> 00:31:22,000
+similar to proof of
+
+265
+00:31:24,130 --> 00:31:34,570
+the sequential criterion for two-sided limit
+
+266
+00:31:34,570 --> 00:31:45,850
+exercises
+
+267
+00:31:45,850 --> 00:31:50,980
+يعني اتمرنوا عليهاانا ارجع لبرهان ال sequential
+
+268
+00:31:50,980 --> 00:31:55,520
+criterion for two-sided limit وشوفوا اقرأوا
+
+269
+00:31:55,520 --> 00:31:59,600
+البرهان و اعملوا التعديلات البسيطة على البرهان لان
+
+270
+00:31:59,600 --> 00:32:03,920
+هنا احنا نتعامل مع right hand limit او limit from
+
+271
+00:32:03,920 --> 00:32:07,500
+the right rather than two-sided limit زي ما عملنا
+
+272
+00:32:07,500 --> 00:32:12,920
+في البرهان تبع واحد implies اتنين okay فحاسيبكم
+
+273
+00:32:12,920 --> 00:32:17,740
+انتوا تكتبوا البرهان تبع اتنين بيقدي لواحدبنفس
+
+274
+00:32:17,740 --> 00:32:21,700
+الطريقة اللى برهنها فى حالة ال two sided limit
+
+275
+00:32:21,700 --> 00:32:30,080
+okay تمام فى اى سؤال طبعا ممكن برضه ايضا يوجد
+
+276
+00:32:30,080 --> 00:32:35,500
+ممكننا نثبت sequential criterion for left hand
+
+277
+00:32:35,500 --> 00:32:42,620
+limit او limit from the left نفس الطريقةOkay فيعني
+
+278
+00:32:42,620 --> 00:32:47,080
+احنا مش هنكتب طبعا نظرية دي هنعتبرها نظرية قائمة و
+
+279
+00:32:47,080 --> 00:32:53,010
+صحيحة و مش بدون برهان okay تمام؟إذن هذه واحدة من
+
+280
+00:32:53,010 --> 00:32:58,650
+النظريات اللي برهناها في section 4.1 و 4.2 و
+
+281
+00:32:58,650 --> 00:33:04,470
+بالمثل كل نظريات اللي برهناهم لـ two sided limit
+
+282
+00:33:04,470 --> 00:33:10,590
+في section 4.1 و 4.2 هنعتبرهم قائمين أو نعتبر
+
+283
+00:33:10,590 --> 00:33:15,330
+نظريات هذه صحيحة لـ left limit و right limit
+
+284
+00:33:22,080 --> 00:33:37,560
+في نظرية أخرى مهمة وهي التعطيل
+
+285
+00:33:37,560 --> 00:33:43,000
+العلاقة بين ال two sided limits و ال one sided
+
+286
+00:33:43,000 --> 00:33:49,100
+limits ف
+
+287
+00:33:51,870 --> 00:34:01,250
+if b function from a to r and let c be a cluster
+
+288
+00:34:01,250 --> 00:34:05,450
+point
+
+289
+00:34:05,450 --> 00:34:08,690
+of
+
+290
+00:34:08,690 --> 00:34:15,310
+المجموعة a تقاطع الفترة المفتوحة from c to
+
+291
+00:34:15,310 --> 00:34:24,070
+infinity and of a تقاطعالـ open interval from
+
+292
+00:34:24,070 --> 00:34:32,250
+negative infinity to c then
+
+293
+00:34:32,250 --> 00:34:42,450
+الـ two-sided limit للـ function f and c بتكون
+
+294
+00:34:42,450 --> 00:34:47,730
+موجودة وبتساوي
+
+295
+00:34:47,730 --> 00:34:54,760
+عدد L if and only ifالـ one-sided limit أو الـ
+
+296
+00:34:54,760 --> 00:35:02,120
+limit from the right the limit at C from the right
+
+297
+00:35:02,120 --> 00:35:12,860
+exist و بساوي L and the limit of F at C from the
+
+298
+00:35:12,860 --> 00:35:19,360
+left exist و بتساوي نفس العدد L وهذه نظرية أخذناها
+
+299
+00:35:19,360 --> 00:35:21,520
+في تفاضل ألف إذا بتذكروا
+
+300
+00:35:24,420 --> 00:35:29,460
+متى ال limit عند نقطة في مجالها او cluster point
+
+301
+00:35:29,460 --> 00:35:34,940
+لمجالها بتكون exist بالساوية عدد اذا كانت ال limit
+
+302
+00:35:34,940 --> 00:35:37,980
+من اليمين موجودة و ال limit من اليسار موجودة و
+
+303
+00:35:37,980 --> 00:35:47,600
+التنتين متساويتين و بساوي نفس العدد هناك
+
+304
+00:35:47,600 --> 00:35:50,500
+بس ماكنش البرهان المطلوب منكم المرة دي احنا
+
+305
+00:35:50,500 --> 00:35:58,420
+مطالبينبالبرهان البرهان يعني كتير سهل ينتج من
+
+306
+00:35:58,420 --> 00:36:06,780
+التعريفات proof ف .. هحاول أبرهنلكم ال F part هذا
+
+307
+00:36:06,780 --> 00:36:16,520
+مسمى ال F part يعني هفرض أنه assume أنه
+
+308
+00:36:16,520 --> 00:36:17,780
+ال one sided limits
+
+309
+00:36:24,700 --> 00:36:29,160
+the limit from the right exist وبساوي L وكذلك
+
+310
+00:36:29,160 --> 00:36:36,000
+limit from the left موجودة
+
+311
+00:36:36,000 --> 00:36:41,840
+وبساوي العدد L وعايز اثبت ان ال limit from the two
+
+312
+00:36:41,840 --> 00:36:48,940
+sides exist اذا هنا هذا الفرض المطلوب
+
+313
+00:37:02,770 --> 00:37:09,030
+أكلم الـ two-sided limit لـ الـ function f at x
+
+314
+00:37:09,030 --> 00:37:14,530
+بساوي c exist و بساوي نفس القيمة أو نفس الأعداد L
+
+315
+00:37:14,530 --> 00:37:27,010
+لبرهان ذلك to see this لبرهان ذلك بنحاول نطبق
+
+316
+00:37:27,010 --> 00:37:33,000
+تعريف epsilon deltaلـ limit of function فبنبدأ
+
+317
+00:37:33,000 --> 00:37:41,140
+بنقول let epsilon أكبر من السفر be given طيب
+
+318
+00:37:41,140 --> 00:37:47,380
+أنا من الفرض أنا فارض تعالى نستفيد من الفرض للوصول
+
+319
+00:37:47,380 --> 00:37:51,720
+إلى المطلوب هذا برهان مباشر البرهان المباشر ده
+
+320
+00:37:51,720 --> 00:37:57,420
+ناخد الفرض بنشتغل عليه بنحط عليه شوية برات و بعدين
+
+321
+00:37:57,420 --> 00:38:05,060
+بنطلع منهالمطلوب فمن الفرض فرضين احنا ان ال limit
+
+322
+00:38:05,060 --> 00:38:14,060
+ل f of x as x tends to c positive لما انه ال limit
+
+323
+00:38:14,060 --> 00:38:20,020
+من اليمين عن c بساوي L y أكبر من سفر given by
+
+324
+00:38:20,020 --> 00:38:25,190
+definitionthere exists delta واحد بالساوي delta
+
+325
+00:38:25,190 --> 00:38:32,830
+واحد تعتمد على epsilon عدد موجب بحيث أنه لو كان x
+
+326
+00:38:32,830 --> 00:38:40,650
+ينتمي إلى a و x minus c أكبر من سفر أصغر من delta
+
+327
+00:38:40,650 --> 00:38:48,350
+واحد فهذا بتضمن أن absolute f of x minus l أصغر من
+
+328
+00:38:48,350 --> 00:38:48,810
+epsilon
+
+329
+00:38:52,780 --> 00:39:00,720
+نسمي الـ implication head star also كذلك بما أن
+
+330
+00:39:00,720 --> 00:39:08,420
+احنا فرضين ان ال limit ل f of x as x tends to c
+
+331
+00:39:08,420 --> 00:39:13,900
+from the left exist وequal نفس العدد L، اذا by
+
+332
+00:39:13,900 --> 00:39:18,980
+definition of left hand limitthere exists delta
+
+333
+00:39:18,980 --> 00:39:21,880
+تانية مش صارت الـ delta هذه تكون نفس الـ delta
+
+334
+00:39:21,880 --> 00:39:27,040
+اللي فوق ماحد بيقدر يجزم ذلك فنسميها delta تانية
+
+335
+00:39:27,040 --> 00:39:32,980
+there exists delta two depends طبعا بالتأكيد تعتمد
+
+336
+00:39:32,980 --> 00:39:38,560
+على إبسلون وعدد موجة بحيث أنه حسب التعريف لكل x
+
+337
+00:39:39,250 --> 00:39:46,270
+تنتمي إلى a و c minus x أكبر من سفر أصغر من delta
+
+338
+00:39:46,270 --> 00:39:54,290
+و 2 طبعا هذا بتضمن أن absolute f of x minus n less
+
+339
+00:39:54,290 --> 00:40:00,710
+than epsilon انسمي ال implication هذه double star
+
+340
+00:40:00,710 --> 00:40:05,390
+خلينا
+
+341
+00:40:05,390 --> 00:40:11,530
+ناخد كالعادة deltaنعرف delta على إنها minimum ال
+
+342
+00:40:11,530 --> 00:40:17,530
+minimum الأصغر بين delta واحد و delta اتنين طبعا
+
+343
+00:40:17,530 --> 00:40:21,890
+هذه بالتأكيد هيطلع الصغيرة بين الاتنين هتكون واحدة
+
+344
+00:40:21,890 --> 00:40:27,770
+منهم وبالتالي تطلع عدد موجب وتعتمد على epsilon إذن
+
+345
+00:40:27,770 --> 00:40:30,930
+هيثبت أن يوجد delta تعتمد على epsilon و ال delta
+
+346
+00:40:30,930 --> 00:40:36,110
+هي عدد موجب الان for this delta تعالى نشوف
+
+347
+00:40:40,450 --> 00:40:49,310
+لو كان x ينتمي ل a و absolute x minus c أكبر من
+
+348
+00:40:49,310 --> 00:40:54,510
+صفر أصغر من دلتا الان
+
+349
+00:40:54,510 --> 00:40:57,850
+بناخد delta بساوي ال minimum ل delta واحد و delta
+
+350
+00:40:57,850 --> 00:41:02,060
+اتنينطبعا بما ان دلتا واحد ودلتا اتنين اعداد موجب
+
+351
+00:41:02,060 --> 00:41:06,080
+اذا دلتا اعداد موجب وكذلك تعتمد على epsilon لان
+
+352
+00:41:06,080 --> 00:41:10,380
+دلتا واحد ودلتا اتنين تعتمد على epsilon الان لو
+
+353
+00:41:10,380 --> 00:41:16,720
+أخدت x تنتمي لمجموعة a و ال x صارت مختلفة عن ال c
+
+354
+00:41:16,720 --> 00:41:23,320
+و المسافة بينها و بين ال c أصغر من دلتا هذا معناه
+
+355
+00:41:23,320 --> 00:41:34,510
+هذا معناه انهال X لا تساوي C وبالتالي
+
+356
+00:41:34,510 --> 00:41:48,230
+ال X ممكن تكون اصغر من C او ال X اكبر من C فهذا
+
+357
+00:41:48,230 --> 00:41:55,630
+بيقدي انه ال .. ال
+
+358
+00:41:55,630 --> 00:42:04,400
+.. ال .. اذا كانت ال Xإذا كانت الـ X أكبر من C لو
+
+359
+00:42:04,400 --> 00:42:08,980
+كانت الـ X أكبر من C فهذا بقدّي أن absolute X
+
+360
+00:42:08,980 --> 00:42:15,200
+minus C بساوي X minus C بصير الـ absolute value
+
+361
+00:42:15,200 --> 00:42:20,560
+هذه عبارة عن X minus C هو أكبر من 0 أصغر من Delta
+
+362
+00:42:20,560 --> 00:42:29,800
+ولو كانتال X أصغر من C فال absolute value هذه
+
+363
+00:42:29,800 --> 00:42:37,360
+بيصير C minus X أكبر من سفر أصغر من Delta في
+
+364
+00:42:37,360 --> 00:42:41,460
+الحالة الأولى ال Delta تبعتي هذه أصغر من أو ساوي
+
+365
+00:42:41,460 --> 00:42:47,120
+Delta واحد صح؟ ال Delta هذه هي ال minimum ل Delta
+
+366
+00:42:47,120 --> 00:42:50,760
+واحد و Delta اتنين وبالتالي أصغر من أو ساوي Delta
+
+367
+00:42:50,760 --> 00:42:58,090
+واحد وبالتالي من ال starإذا كانت x تنتمي إلى a و x
+
+368
+00:42:58,090 --> 00:43:03,990
+minus c أكبر من سفر أصغر من دلتا واحد من ال star
+
+369
+00:43:03,990 --> 00:43:11,770
+بيطلع عندي absolute f of x minus L أصغر من يوإذا
+
+370
+00:43:11,770 --> 00:43:17,510
+كانت ال X أصغر من ال C فبطلع absolute X سالب C
+
+371
+00:43:17,510 --> 00:43:22,870
+بيطلع بيساوي C سالب X أصغر من Delta وطبعا X مستويش
+
+372
+00:43:22,870 --> 00:43:29,190
+C أكبر من 0 وال Delta هذه من تعريفها أصغر من أو
+
+373
+00:43:29,190 --> 00:43:35,300
+يساوي Delta 2باستخدام double star ال implication
+
+374
+00:43:35,300 --> 00:43:41,420
+double star لما يكون ال X تنتمي ل A و C minus X
+
+375
+00:43:41,420 --> 00:43:46,640
+أكبر من 0 أصغر من Delta 2 هذا بيقدر أن absolute F
+
+376
+00:43:46,640 --> 00:43:53,680
+of X minus L أصغر من إبسن إذن في كل الأحوال هذه
+
+377
+00:43:53,680 --> 00:43:58,180
+بتقدر أن absolute F of X minus L أصغر من إبسن
+
+378
+00:43:58,180 --> 00:43:59,400
+تمام؟
+
+379
+00:44:02,170 --> 00:44:06,090
+طب ما هذا هو تعريف epsilon delta لل limit of
+
+380
+00:44:06,090 --> 00:44:12,270
+function صح؟ إذا نيجي بنقول هنا since epsilon أكبر
+
+381
+00:44:12,270 --> 00:44:15,870
+من السفر was arbitrary
+
+382
+00:44:17,410 --> 00:44:22,850
+إذا احنا بنكون أثبتنا لكل إبسلون أكبر من سفر يوجد
+
+383
+00:44:22,850 --> 00:44:27,950
+Delta تعتمد على إبسلون عدد موجب بحيث لكل X تنتمي ل
+
+384
+00:44:27,950 --> 00:44:32,210
+A و Absolute X minus C أكبر من سفر أصغر من Delta
+
+385
+00:44:32,210 --> 00:44:37,810
+طلع عندي Absolute F of X في الحالتين minus L أصغر
+
+386
+00:44:37,810 --> 00:44:41,630
+من إبسلون وبالتالي إذا هذا صحيح لكل إبسلون
+
+387
+00:44:41,630 --> 00:44:45,620
+وبالتالي by إبسلون Delta definition of limitأو
+
+388
+00:44:45,620 --> 00:44:54,600
+function we have أثبتنا أن ال limit ل f of x as x
+
+389
+00:44:54,600 --> 00:45:01,380
+tends to c بساوي العدد L okay تمام، إذا هذا بثبت
+
+390
+00:45:01,380 --> 00:45:04,840
+اللي هو لو كان ال two sided limits موجودين
+
+391
+00:45:04,840 --> 00:45:10,730
+متساويتين، لأ لو كان ال one sided limitsكلا هما
+
+392
+00:45:10,730 --> 00:45:15,530
+موجودة و بساوة قيمة مشتركة L ف ال two sided limit
+
+393
+00:45:15,530 --> 00:45:20,130
+تطلع exist و قيمتها بساوة القيمة المشتركة الان
+
+394
+00:45:20,130 --> 00:45:28,210
+برهان العكس أسهل لذن هكتب هنا ال proof of
+
+395
+00:45:28,210 --> 00:45:36,650
+the canvas is easier أسهل
+
+396
+00:45:38,760 --> 00:45:44,180
+So exercise it يعني
+
+397
+00:45:44,180 --> 00:45:49,780
+اتمرّض عليها لو كانت ال two-sided limit exist فمن
+
+398
+00:45:49,780 --> 00:45:55,840
+السهل أن نثبت أن ال right hand limit exist و ال
+
+399
+00:45:55,840 --> 00:46:00,600
+left hand limit exist و كلهم لهم نفس القيمة okay
+
+400
+00:46:00,600 --> 00:46:05,170
+تمام؟إذا هنوقف هنا و في المحاضرة الجاية ان شاء
+
+401
+00:46:05,170 --> 00:46:09,710
+الله هناخد أمثلة على one-sided limits إما في اتين
+
+402
+00:46:09,710 --> 00:46:13,350
+موجودين و متساوياتين أو اتنين موجودين و مختلفتين
+
+403
+00:46:13,350 --> 00:46:18,190
+أو واحدة موجودة و اتنين مش موجودة و هكذا، هنشوف كل
+
+404
+00:46:18,190 --> 00:46:24,670
+الأنواع و كل ال situations، تمام؟ okay شكرا لكم و
+
+405
+00:46:24,670 --> 00:46:26,550
+نشوفكم ان شاء الله المرة القادمة
+
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/6FDcUXR9Pqo.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/6FDcUXR9Pqo.srt
new file mode 100644
index 0000000000000000000000000000000000000000..01c4b659689b4b94b110f6eaf83a251cee13f645
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/6FDcUXR9Pqo.srt
@@ -0,0 +1,1647 @@
+1
+00:00:20,870 --> 00:00:25,910
+المرة اللي فاتت أو في المحاضرة السابقة عرفنا ال
+
+2
+00:00:25,910 --> 00:00:31,410
+cluster point  وأخذنا أمثلة كيف نجيب ال cluster 
+
+3
+00:00:31,410 --> 00:00:39,710
+points لمجموعة معينة ووجدنا عند المثال الثالث
+
+4
+00:00:59,030 --> 00:01:05,110
+المثال الثالث show that
+
+5
+00:01:05,110 --> 00:01:12,070
+zero is the only cluster
+
+6
+00:01:12,070 --> 00:01:18,290
+point of 
+
+7
+00:01:18,290 --> 00:01:23,570
+the set A 
+
+8
+00:01:26,690 --> 00:01:31,870
+كل واحد على N حيث N that's a number  دكتور هذا
+
+9
+00:01:31,870 --> 00:01:37,210
+مثال ثاني أخذناها ده؟ لأ لأ اللي أخذناه اللي هو ال
+
+10
+00:01:37,210 --> 00:01:48,030
+.. هناخدها هناخدها هناخدها في .. هنا هنا .. هنا
+
+11
+00:01:48,030 --> 00:01:55,250
+هنا اثنين
+
+12
+00:01:59,360 --> 00:02:11,580
+أن Zero is a cluster point  ثلاثة
+
+13
+00:02:11,580 --> 00:02:22,720
+دلتا أكبر من الصفر Be given by Archimedean 
+
+14
+00:02:22,720 --> 00:02:25,880
+property
+
+15
+00:02:30,960 --> 00:02:40,920
+يوجد N تنتمي إلى N بحيث أن واحد على N
+
+16
+00:02:40,920 --> 00:02:54,340
+أصغر من دلتا hence
+
+17
+00:03:01,430 --> 00:03:09,270
+الدلتا نيبر هو صفر لو
+
+18
+00:03:09,270 --> 00:03:25,690
+أخذت xN هو واحد على N فهذا ينتمي إلى
+
+19
+00:03:25,690 --> 00:03:39,580
+المجموعة A وانت ليه لا ال delta number هنا ل ..
+
+20
+00:03:39,580 --> 00:03:44,040
+أو ال x هذا المفروض دلتا ال delta number هو ده
+
+21
+00:03:44,040 --> 00:03:47,860
+اسمه
+
+22
+00:03:47,860 --> 00:03:56,940
+أسوأ؟ إذن
+
+23
+00:03:56,940 --> 00:03:58,460
+هذا لا أي سؤال خالد
+
+24
+00:04:06,200 --> 00:04:10,700
+الدلتا نبر هود للصفر اللي هي الفترة المفتوحة من 
+
+25
+00:04:10,700 --> 00:04:16,100
+سالب دلتا إلى دلتا
+
+26
+00:04:16,100 --> 00:04:21,960
+فهي عندي واحد على N أصغر من دلتا وطبعا أكبر من صفر
+
+27
+00:04:24,160 --> 00:04:27,620
+فواحد على N ينتمي للـDelta neighborhood للصفر
+
+28
+00:04:27,620 --> 00:04:32,400
+وواحد على N ينتمي للمجموعة A وبالتالي يجب أن نكون
+
+29
+00:04:32,400 --> 00:04:38,580
+أثبتنا أنه لأي دلتا أكبر من الصفر أو أي دلتا
+
+30
+00:04:38,580 --> 00:04:44,680
+neighborhood للصفر يتقاطع مع A في نقطة مختلفة عن 
+
+31
+00:04:44,680 --> 00:04:49,000
+الصفر ال
+
+32
+00:04:49,000 --> 00:04:56,810
+X إن جلدها تساوي صفر لاتصار الصفر وبالتالي إذا 
+
+33
+00:04:56,810 --> 00:05:05,470
+هذا يثبت الصفر is a cluster point of
+
+34
+00:05:05,470 --> 00:05:12,310
+الست إذا هذا يثبت ال claim الآن ليه مايكون فيه
+
+35
+00:05:12,310 --> 00:05:14,130
+cluster points أخرى؟
+
+36
+00:05:30,490 --> 00:05:41,150
+إذا كانت X لا تساوي صفر، فهي ليست مجموعة من A 
+
+37
+00:05:41,150 --> 00:05:46,390
+فحاسبكم
+
+38
+00:05:46,390 --> 00:05:47,630
+أنتم تكتبوا البرهان
+
+39
+00:05:50,290 --> 00:06:02,390
+هي صفر وهي واحد وهي نصف وهي ثلث وهي واحد على N وهي
+
+40
+00:06:02,390 --> 00:06:05,390
+واحد على N زائد واحد وهكذا
+
+41
+00:06:16,430 --> 00:06:25,930
+فهنا ثاني two cases case one أن x تنتمي إلى a و
+
+42
+00:06:25,930 --> 00:06:34,690
+الحالة الثانية case two أن x لا تنتمي إلى a ال x
+
+43
+00:06:34,690 --> 00:06:39,530
+دي مش تساوي صفر احنا already اثبتنا أن الصفر
+
+44
+00:06:39,530 --> 00:06:43,850
+cluster point طيب افرض X مش تساوي صفر إذا X ممكن
+
+45
+00:06:43,850 --> 00:06:48,170
+تساوي واحد أو نصف أو ثلث أو واحد على N for some N
+
+46
+00:06:48,170 --> 00:06:53,250
+صح؟ ممكن ماتساويش ولا عنصر ممكن ماتكونش تمثل أي 
+
+47
+00:06:53,250 --> 00:06:58,070
+فلو كانت ال X واحد من العناصر هدول واحد من العناصر
+
+48
+00:06:58,070 --> 00:07:04,630
+الست فبقدر ألاقي أن يوجد بقدر ألاقي دلتا
+
+49
+00:07:04,630 --> 00:07:08,990
+neighborhood للعنصر مثلًا الثلث بقدر ألاقي دلتا
+
+50
+00:07:08,990 --> 00:07:09,490
+neighborhood
+
+51
+00:07:13,860 --> 00:07:19,840
+العنصر اللي بعد هذا هيكون ربع فباخد المسافة الأصغر
+
+52
+00:07:19,840 --> 00:07:26,560
+بين ثلث ربع ثلث ونصف وباخد نصف المسافة دلتا فبيصير
+
+53
+00:07:26,560 --> 00:07:30,920
+عندي هنا دلتا نبر هود للثلث وبتقاطعش مع المجموعة A
+
+54
+00:07:30,920 --> 00:07:38,120
+بالمرة أو في نقطة مختلفة عن الثلث وبالتالي لو كانت 
+
+55
+00:07:38,120 --> 00:07:44,880
+ال X موجودة في A زي الثلث مثلًا فال X ليست cluster
+
+56
+00:07:44,880 --> 00:07:49,860
+point الآن ال X لا تنتمي ل A؟ ال X لا تنتمي ل
+
+57
+00:07:49,860 --> 00:07:55,920
+A؟ معناته هي عنصر هنا زي هذا X أزرق من صفر أو X
+
+58
+00:07:55,920 --> 00:08:01,540
+ممكن تكون جاية بين عنصرين فباخد المسافة بين X و
+
+59
+00:08:01,540 --> 00:08:04,840
+أقرب عنصر لها من اليمين وأقرب عنصر لها من
+
+60
+00:08:04,840 --> 00:08:11,140
+اليسار وباخد نصف المسافة دلتا أو إبسيلون وبكون
+
+61
+00:08:11,140 --> 00:08:17,480
+دلتا نبر هود ل X هذا دلتا نبر هود مش هيتقاطع مع ال A و
+
+62
+00:08:17,480 --> 00:08:20,700
+بالتالي النقطة هذه عمرها ما بتكون cluster point
+
+63
+00:08:20,700 --> 00:08:26,020
+ممكن كمان النقطة هذه ال X مش موجودة فيها ممكن تكون
+
+64
+00:08:26,020 --> 00:08:31,260
+على شمال الصفر أو على يمين الواحد فلو كانت على يمين
+
+65
+00:08:31,260 --> 00:08:35,560
+الواحد خد نصف المسافة هي دلتا إذا هي دلتا نبر
+
+66
+00:08:35,560 --> 00:08:39,420
+هود ل X مابتقاطعش معاه بالمرة بالتالي X مابت
+
+67
+00:08:39,420 --> 00:08:44,440
+cluster هنا لو كانت X أصغر من صفر فخد نصف المسافة
+
+68
+00:08:44,440 --> 00:08:48,960
+بين X و 0 على إنها دلتا وبالتالي كون delta
+
+69
+00:08:48,960 --> 00:08:52,460
+neighborhood ل X هذا delta neighborhood بتقاطعش مع
+
+70
+00:08:52,460 --> 00:08:56,240
+A بالمرة وبالتالي X ليست cluster point إذا في كل
+
+71
+00:08:56,240 --> 00:09:01,860
+الأحوال X ليست cluster point سواء كانت في A أو مش
+
+72
+00:09:01,860 --> 00:09:05,420
+موجودة إذا كانت X مختلفة عن الصفر فليست cluster
+
+73
+00:09:05,420 --> 00:09:14,930
+point okay إذا zero is the only النقطة الوحيدة مافيش
+
+74
+00:09:14,930 --> 00:09:18,990
+نقطة غيرها بتكون cluster point بالمثل ممكن نثبت 
+
+75
+00:09:18,990 --> 00:09:29,650
+مثال آخر f
+
+76
+00:09:29,650 --> 00:09:35,710
+i بساوي ال
+
+77
+00:09:35,710 --> 00:09:39,830
+unit technological interval and
+
+78
+00:09:51,210 --> 00:10:02,710
+IQ بساوي I تقاطع ال rational numbers then 
+
+79
+00:10:02,710 --> 00:10:13,170
+every x تنتمي ل I is a cluster point a cluster 
+
+80
+00:10:13,170 --> 00:10:15,250
+point of IQ
+
+81
+00:10:18,800 --> 00:10:26,900
+إذا I هي الفترة المغلقة من صفر لواحد IQ هي كل
+
+82
+00:10:26,900 --> 00:10:31,800
+الأعداد النسبية الموجودة في الفترة المغلقة من صفر
+
+83
+00:10:31,800 --> 00:10:38,340
+لواحد ممكن إثبات إن كل X في الفترة المغلقة I هي 
+
+84
+00:10:38,340 --> 00:10:45,940
+cluster point للمجموعة IQ وذلك باستخدام ال density
+
+85
+00:10:45,940 --> 00:10:52,060
+theorem proof use
+
+86
+00:10:52,060 --> 00:11:06,500
+the density theorem فحاسبكم
+
+87
+00:11:06,500 --> 00:11:15,920
+انتوا تكتبوا البرهان خدي أي x في I واخذتي أن أي
+
+88
+00:11:17,780 --> 00:11:22,140
+أعملكم برهان هكذا بدون أن أكتب أي شيء هذه الفترة I
+
+89
+00:11:22,140 --> 00:11:29,720
+من صفر إلى واحد هذه الفترة المغلقة من ملفة أن كل X
+
+90
+00:11:29,720 --> 00:11:35,800
+كل X فيها هو cluster point لمجموعة الأعداد النسبية
+
+91
+00:11:35,800 --> 00:11:42,520
+في I ففي عندي ثلاثة حالات MX هتكون أكبر من صفر أصغر
+
+92
+00:11:42,520 --> 00:11:48,060
+من واحد يعني نقطة داخلية ليست نقطة طرفها طبعًا هي
+
+93
+00:11:48,060 --> 00:11:52,460
+لو أخذت أي دلتا عدد موجب وكونت دلتا
+
+94
+00:11:52,460 --> 00:11:57,380
+neighborhood لل X فال delta neighborhood هذا 
+
+95
+00:11:57,380 --> 00:12:05,920
+هيتقاطع مع المجموعة A IQ حسب نظرية الكثافة أي فترة
+
+96
+00:12:05,920 --> 00:12:11,520
+مفتوحة زي هذه تحتوي rational number صح؟ وبالتالي أي
+
+97
+00:12:11,520 --> 00:12:16,600
+دلتا neighborhood ل ال X هيتقاطع مع المجموعة IQ
+
+98
+00:12:16,600 --> 00:12:24,160
+في نقطة R مختلفة عن ال X حسب نظرية الكثافة
+
+99
+00:12:24,160 --> 00:12:28,040
+وبالتالي حسب التعريف إذا ال X هذه اللي هي نقطة
+
+100
+00:12:28,040 --> 00:12:33,700
+داخلية is a cluster point لمن؟
+
+101
+00:12:33,700 --> 00:12:40,260
+للمجموعة IQ لو كانت ال .. ال .. ال x هي نقطة الطرف
+
+102
+00:12:40,260 --> 00:12:46,960
+الحالة الثانية لما x تكون هي صفر لما x تكون بساوي
+
+103
+00:12:46,960 --> 00:12:52,160
+صفر وخدي أي دلتا neighborhood لأن هاي سالب دلتا
+
+104
+00:12:52,160 --> 00:12:56,560
+موجب دلتا فالفترة
+
+105
+00:12:56,560 --> 00:13:01,200
+هذه تتقطع يعني
+
+106
+00:13:01,200 --> 00:13:07,230
+هاي دلتا هادي دلتا و هادي نقطة صفر الآن الفترة 
+
+107
+00:13:07,230 --> 00:13:12,870
+هذه بقدر ألاقي فيها rational number حسب مباريك 
+
+108
+00:13:12,870 --> 00:13:16,970
+الكثافة موجود بين صفر و دلتا و ال rational number
+
+109
+00:13:16,970 --> 00:13:23,550
+هذا موجود في ال unit closed interval وبالتالي كل 
+
+110
+00:13:23,550 --> 00:13:28,450
+دلتا neighborhood للصفر يتقاطع 
+
+111
+00:13:28,450 --> 00:13:33,670
+مع المجموع IQ في نقطة R مختلفة عن الصفر وبالتالي
+
+112
+00:13:33,670 --> 00:13:37,910
+الصفر هو cluster point بالمثل ممكن تثبتوا بالواحد
+
+113
+00:13:37,910 --> 00:13:41,970
+cluster point لأن أي دلتا neighborhood للواحد
+
+114
+00:13:43,720 --> 00:13:48,560
+هيحتوي حسب نظرية الكثافة لو هذا كان واحد ثالث دلتا 
+
+115
+00:13:48,560 --> 00:13:54,510
+في النقطة هذه وهذه واحد فبنقدر نلاقي Rrational 
+
+116
+00:13:54,510 --> 00:13:58,650
+number حسب مجرد كدفة بين واحد سالب دلتا وواحد
+
+117
+00:13:58,650 --> 00:14:03,710
+وبالتالي ال دلتا neighborhood هذا مركزه واحد و
+
+118
+00:14:03,710 --> 00:14:08,490
+نصف قطره دلتا هيتقاطع مع ال IQ في نقطة مختلفة عن
+
+119
+00:14:08,490 --> 00:14:13,250
+الواحد وبالتالي واحد cluster point الآن لأن هسيبكم
+
+120
+00:14:13,250 --> 00:14:15,650
+تكتبوا البرهان بالتفصيل
+
+121
+00:14:19,850 --> 00:14:25,250
+Okay إذاً هيك يعني عندنا عدة أمثلة على ال cluster 
+
+122
+00:14:25,250 --> 00:14:32,490
+points نرجع لمفهوم ال limit ل ال function اللي هو
+
+123
+00:14:32,490 --> 00:14:47,650
+كان عنوان ال section تبعنا إذا
+
+124
+00:14:47,650 --> 00:14:48,750
+هنا definition
+
+125
+00:14:55,450 --> 00:15:07,150
+دع الـ f يكون دالة من a إلى r دالة 
+
+126
+00:15:07,150 --> 00:15:19,710
+في أين a مجموعة جزئية من r و c مجموعة جزئية من الـ
+
+127
+00:15:19,710 --> 00:15:22,090
+set A
+
+128
+00:15:26,660 --> 00:15:35,260
+العدد number L هو ليمت
+
+129
+00:15:35,260 --> 00:15:39,440
+للـ دالة 
+
+130
+00:15:39,440 --> 00:15:44,440
+f at 
+
+131
+00:15:44,440 --> 00:15:59,020
+x بس you see إذا تحقق الشرط التالي for any epsilon 
+
+132
+00:15:59,020 --> 00:16:05,340
+أكبر من صفر يوجد دلتا تعتمد على إبسيلون عدد موجب
+
+133
+00:16:05,340 --> 00:16:14,690
+بحيث أنه لكل x تنتمي إلى a والمسافة بين .. وال X
+
+134
+00:16:14,690 --> 00:16:23,090
+هذا يختلف عن ال C والمسافة بينها وبين ال C أصغر
+
+135
+00:16:23,090 --> 00:16:30,030
+من دلتا لازم هذا يضمن أن absolute F of X minus L
+
+136
+00:16:30,030 --> 00:16:41,010
+أصغر من دلتا إذن هذا شرط .. هذا شرط أو بسميه أنا
+
+137
+00:16:41,010 --> 00:16:49,470
+بسميه إبسيلون دلتا definition إبسيلون دلتا
+
+138
+00:16:49,470 --> 00:16:54,310
+definition of limit للـ
+
+139
+00:16:54,310 --> 00:16:58,550
+limit of a function الـ Limit لـ function f of x 
+
+140
+00:16:58,550 --> 00:17:03,590
+بالساوي أو L هي عبارة عن Limit لـ function f of x 
+
+141
+00:17:03,590 --> 00:17:09,970
+and x = C إذا لو أعطتوني أي إبسلون عدد موجب
+
+142
+00:17:09,970 --> 00:17:15,570
+لازم أنا أرد عليها Delta عدد موجب آخر يعتمد على
+
+143
+00:17:15,570 --> 00:17:20,750
+إبسلون بحيث أنه لكل X في المجموعة A اللي هو ال
+
+144
+00:17:20,750 --> 00:17:27,300
+domain  تبع ال function و X هذه مختلفة لا تساوي C
+
+145
+00:17:27,300 --> 00:17:33,360
+يعني المتباينة هذه معناها X لا تساوي C إذاً لكل x في
+
+146
+00:17:33,360 --> 00:17:38,200
+A مختلفة عن الـc والمسافة بينها وبين الـc أصغر من
+
+147
+00:17:38,200 --> 00:17:42,580
+دلتا لازم تطلع المسافة بين f of x و L أصغر من
+
+148
+00:17:42,580 --> 00:17:47,220
+إبسلون هذا الكلام اتحقق لكل إبسلون أكبر من الصفر
+
+149
+00:17:47,220 --> 00:17:51,640
+فبنقول إن العدد L is a limit of the function f عند
+
+150
+00:17:51,640 --> 00:17:52,360
+النقطة c
+
+151
+00:17:59,170 --> 00:18:07,710
+من هذا التعريف بينتج على طول اه طيب in this case
+
+152
+00:18:07,710 --> 00:18:13,930
+in this case we 
+
+153
+00:18:13,930 --> 00:18:26,600
+say انه if converges if converges to the number L
+
+154
+00:18:26,600 --> 00:18:39,520
+at X = C and we write ونكتب الحالة هذه limit ل
+
+155
+00:18:39,520 --> 00:18:46,940
+F of X لما X تقول إلى C = L أو ممكن نكتب limit
+
+156
+00:18:46,940 --> 00:18:54,260
+F as X tends to C = L أو ممكن نكتب
+
+157
+00:19:01,220 --> 00:19:11,260
+أو ممكن نكتب f of x tends to L as x tends to c كل
+
+158
+00:19:11,260 --> 00:19:16,360
+هدول معناهم أن العدد L limit لل function f and x
+
+159
+00:19:16,360 --> 00:19:17,360
+= c
+
+160
+00:19:22,850 --> 00:19:30,090
+ف limit f of x as x tends to c does not exist،
+
+161
+00:19:30,090 --> 00:19:37,430
+يعني مافيش عدد L زي هذا، we say ان ال function f
+
+162
+00:19:37,430 --> 00:19:45,590
+diverges، diverges at x = c
+
+163
+00:19:50,120 --> 00:19:55,320
+الآن نستطيع ان نثبت ان ال limit لل function هذه
+
+164
+00:19:55,320 --> 00:20:00,040
+النقطة لو كانت موجودة لو كان ال function لها limit
+
+165
+00:20:00,040 --> 00:20:07,120
+فlimit هذه لازم تكون unique ال
+
+166
+00:20:07,120 --> 00:20:19,320
+function if from A to R can have only
+
+167
+00:20:39,940 --> 00:20:44,760
+والبرهان شبه البرهان الـ uniqueness of the limit
+
+168
+00:20:44,760 --> 00:20:50,700
+of a sequence we use epsilon over two argument
+
+169
+00:20:51,860 --> 00:20:54,780
+استنتاج epsilon على اتنين او برهان epsilon على
+
+170
+00:20:54,780 --> 00:21:01,020
+اتنين هنشوف مع بعض هاي برهان تروح left epsilon
+
+171
+00:21:01,020 --> 00:21:04,300
+أكبر
+
+172
+00:21:04,300 --> 00:21:11,720
+من السفر ب given since
+
+173
+00:21:11,720 --> 00:21:19,120
+طب
+
+174
+00:21:19,120 --> 00:21:25,040
+خليني الأول أبرهان النظرية هذه بدي أفرض أنه فيه
+
+175
+00:21:25,040 --> 00:21:34,640
+two limits assume أن ال limit ل f of x as x tends
+
+176
+00:21:34,640 --> 00:21:44,340
+to c = عدد الواحد and limit أيضا ل f of x as x
+
+177
+00:21:44,340 --> 00:21:50,340
+tends to c = عدد تاني الاتنين وعشان أثبت
+
+178
+00:21:50,340 --> 00:21:57,860
+النظرية لازم أثبت الإدعاء التالي إن ال1 = ال4
+
+179
+00:21:57,860 --> 00:22:07,720
+فلبرهان ذلك let epsilon أكبر من السفر be given
+
+180
+00:22:07,720 --> 00:22:19,500
+since مما أننا فرضين أن ال limit لأف as x tends to
+
+181
+00:22:19,500 --> 00:22:28,420
+c = الواحد then by definition by epsilon
+
+182
+00:22:28,420 --> 00:22:33,180
+delta definition of limit there exists delta one
+
+183
+00:22:33,180 --> 00:22:39,830
+depends on epsilon positive number بحيث أنه لو كان
+
+184
+00:22:39,830 --> 00:22:46,150
+x ينتمي إلى a و |x - c| أصغر من delta
+
+185
+00:22:46,150 --> 00:22:54,850
+one أكبر من سفر فهذا بيقدي أن |f of x
+
+186
+00:22:54,850 --> 00:23:02,530
+- l one| أصغر من epsilon على 2 عشان الاستنتاج
+
+187
+00:23:02,530 --> 00:23:05,510
+هذا واحد
+
+188
+00:23:08,770 --> 00:23:13,810
+Also ايضا احنا
+
+189
+00:23:13,810 --> 00:23:20,610
+فرضين من ال limit لل function f of x as x tends to
+
+190
+00:23:20,610 --> 00:23:27,990
+c = عدد تاني ال اتنين then
+
+191
+00:23:27,990 --> 00:23:35,650
+for the same for same epsilon أكبر من ستة نفس ال
+
+192
+00:23:35,650 --> 00:23:43,140
+epsilon يعنيلنفس الـ Epsilon بما أن limit ل F of X
+
+193
+00:23:43,140 --> 00:23:48,940
+and X = C = L2 نجد Delta 2 تعتمد على
+
+194
+00:23:48,940 --> 00:23:53,940
+Epsilon على 2 وبالتالي على Epsilon عدد موجد بحيث
+
+195
+00:23:53,940 --> 00:24:00,300
+أنه لو كان X ينتمي إلى A و |X - C| أصغر
+
+196
+00:24:00,300 --> 00:24:07,350
+من Delta 2 أكبر من 0 فهذا أكيد بيقدّي أنه |
+
+197
+00:24:07,350 --> 00:24:14,510
+f of x - L2| أصغر من epsilon على 2 ال sum ال
+
+198
+00:24:14,510 --> 00:24:22,030
+implication هي D2 الآن بناخد .. بنعرف delta ل L
+
+199
+00:24:22,030 --> 00:24:31,230
+Delta = ال minimum ل delta واحد و delta اتنين
+
+200
+00:24:32,560 --> 00:24:37,340
+طبعا هذا بيطلع عدد
+
+201
+00:24:37,340 --> 00:24:43,200
+موجب لإن دلتا واحد ودلتا اتنين عدد موجب وكذلك دلتا
+
+202
+00:24:43,200 --> 00:24:46,200
+دي تعتمد على ابسلون لإن دلتا واحد ودلتا اتنين
+
+203
+00:24:46,200 --> 00:24:52,720
+يعتمدوا على ابسلون then
+
+204
+00:24:52,720 --> 00:24:55,860
+by
+
+205
+00:24:55,860 --> 00:25:07,520
+واحد and اتنين نحصل على التالي، لو كان x ينتقل إلى
+
+206
+00:25:07,520 --> 00:25:14,980
+a و |x - c| أصغر من delta أكبر من سفر
+
+207
+00:25:14,980 --> 00:25:26,590
+فهذا هيقدر أن |L1 - L2| = |
+
+208
+00:25:26,590 --> 00:25:39,610
+L1 - F of X + F of X - L2| إذا انطلعت
+
+209
+00:25:39,610 --> 00:25:46,590
+أنا F of X ورجعتها فكأني ما جيرتش حاجة وهذا بيطلع
+
+210
+00:25:46,590 --> 00:25:50,460
+باستخدام ال triangle inequality بالترائنجل الـ
+
+211
+00:25:50,460 --> 00:25:54,900
+equality لـ | مجموعة حاجتين أصغر من
+
+212
+00:25:54,900 --> 00:26:00,920
+لو يساوي |L1 - F of X| + |F
+
+213
+00:26:00,920 --> 00:26:07,980
+of X - L2| الآن
+
+214
+00:26:07,980 --> 00:26:13,340
+باستخدام ال implication واحد، اللحظة أن الـ delta
+
+215
+00:26:13,340 --> 00:26:17,960
+هي ال minimum لـ delta واحد و delta اتنين وبالتالي
+
+216
+00:26:17,960 --> 00:26:24,340
+الـ delta هذه أصغر من delta واحد فحسب ال
+
+217
+00:26:24,340 --> 00:26:28,600
+implication واحد لما يكون x ينتمي ل a و |x 
+
+218
+00:26:28,600 --> 00:26:33,800
+- c| أصغر من delta واحد فانا بقدم ال |
+
+219
+00:26:33,800 --> 00:26:40,460
+value هذه أصغر من epsilon على 2 كذلك باستخدام ال
+
+220
+00:26:40,460 --> 00:26:45,060
+implication 2 أنا عندي الـ delta هذه هي الـ minimum
+
+221
+00:26:45,060 --> 00:26:51,760
+لـ delta 1 و delta 2 وبالتالي أصغر من delta 2 فبال
+
+222
+00:26:51,760 --> 00:26:55,680
+implication 2 بتقول لو كان x ينتمي ل a و |
+
+223
+00:26:55,680 --> 00:27:00,320
+x - c| أصغر من delta 2 فال | ل f
+
+224
+00:27:00,320 --> 00:27:07,500
+of x - l2| < epsilon على 2 هذا بيساوي
+
+225
+00:27:07,500 --> 00:27:16,080
+epsilon إذا أنا طلع عندي أثبتت أن |L1 -
+
+226
+00:27:16,080 --> 00:27:22,540
+L2| < إبسلون طبعا أكيد أكبر من أو يساوي سفر و
+
+227
+00:27:22,540 --> 00:27:28,600
+الآن هذا صحيح لكل أبسلون أكبر من السفر لأن one
+
+228
+00:27:28,600 --> 00:27:34,500
+hand هنا ال epsilon was arbitrary given الإبسلون
+
+229
+00:27:34,500 --> 00:27:38,660
+was arbitrarily يعني نقول since this holds for
+
+230
+00:27:38,660 --> 00:27:43,160
+every إبسلون في لمّة أخدناها في بداية ال course
+
+231
+00:27:43,160 --> 00:27:48,820
+بتقول لو في عندي عدد حفيفي a أكبر من أو يساوي سفر و
+
+232
+00:27:48,820 --> 00:27:53,940
+أصغر من إبسلون فإن إبسلون أكبر من السفر فهذا بيقدي
+
+233
+00:27:53,940 --> 00:28:00,160
+أن a = سفر أخد ايه هنا الـ | ل L1
+
+234
+00:28:00,160 --> 00:28:09,140
+- L2| فحسب النمة هذه هيطلع عندى هذا بقدر اللي
+
+235
+00:28:09,140 --> 00:28:15,600
+قدر انه |L1 - L2| = سفر وبالتالي
+
+236
+00:28:15,600 --> 00:28:24,600
+بيطلع عندى L1 = L2 وهو المطلوب إذا أنا فرقت إن
+
+237
+00:28:24,600 --> 00:28:28,860
+الـ function إلها two limits عن نقطة فطلع الـ two
+
+238
+00:28:28,860 --> 00:28:32,680
+limits متساوياتين وبالتالي لو كانت الـ function
+
+239
+00:28:32,680 --> 00:28:37,240
+إلها limit عن نقطة فال limit تطلع وحيدة unique،
+
+240
+00:28:37,240 --> 00:28:43,200
+بقى ده؟ في أي سؤال أو أي سؤال على البرهانة؟
+
+241
+00:29:02,290 --> 00:29:15,590
+ناخد ملاحظة هنا الـ
+
+242
+00:29:15,590 --> 00:29:27,330
+epsilon delta definition of limit of a function f
+
+243
+00:29:27,330 --> 00:29:29,270
+from a to r
+
+244
+00:29:32,670 --> 00:29:40,250
+the inequality المتباينة
+
+245
+00:29:40,250 --> 00:29:48,030
+اللي هي |x - c| > 0 <
+
+246
+00:29:48,030 --> 00:29:58,470
+دولتان means حاجتين الجزء هذا معناه أن |x
+
+247
+00:29:58,470 --> 00:30:09,330
+- c| لا يساوى سفروبالتالي X لا يساوي C إذاً هذا
+
+248
+00:30:09,330 --> 00:30:17,870
+يعني أن X لا تساوي C المتباينة
+
+249
+00:30:17,870 --> 00:30:23,410
+التانية اللي هي |X - C| أصغر من Delta
+
+250
+00:30:23,410 --> 00:30:31,170
+هذه بتكافئ أن X - C < Delta > 0
+
+251
+00:30:31,170 --> 00:30:39,850
+Delta صح؟ وهذه بتكافئ أن X > C - Delta
+
+252
+00:30:39,850 --> 00:30:47,870
+< C + Delta وهذا معناه أن X تنتمي لـ
+
+253
+00:30:47,870 --> 00:30:56,450
+Delta Neverhood لـ C اللي هو الفترة اللي اتروها من
+
+254
+00:30:56,450 --> 00:30:59,410
+C - Delta ل C + Delta
+
+255
+00:31:06,690 --> 00:31:12,550
+إذا المتباينة دي معناها x لا تساوي c and x تنتمي
+
+256
+00:31:12,550 --> 00:31:18,190
+لل delta اللي برجود لل c اللي هو الفترة المفتوحة
+
+257
+00:31:18,190 --> 00:31:25,170
+من c - delta إلى c + delta كذلك
+
+258
+00:31:25,170 --> 00:31:29,570
+المتباينة also
+
+259
+00:31:33,070 --> 00:31:37,490
+الإي نكواليتي المتباينة
+
+260
+00:31:37,490 --> 00:31:43,450
+اللي هي |f of x - L| أصغر من إبسلون
+
+261
+00:31:43,450 --> 00:31:46,490
+means
+
+262
+00:31:46,490 --> 00:31:52,830
+لو جينا الحل المتباينة هذه في f of x
+
+263
+00:32:05,840 --> 00:32:11,920
+فهي عندي f of x - L < إبسلون >
+
+264
+00:32:11,920 --> 00:32:17,480
+0 حل المتباينة هذه في f of x اجمع L على
+
+265
+00:32:17,480 --> 00:32:24,880
+كل أطراف فبطلع f of x < L + إبسلون >
+
+266
+00:32:24,880 --> 00:32:33,940
+L - إبسلون فهذا معناه أن f of x belongs to
+
+267
+00:32:33,940 --> 00:32:38,520
+the epsilon neighborhood للعدد L اللي هو الفترة
+
+268
+00:32:38,520 --> 00:32:44,040
+المفتوحة from L - إبسلون إلى L +
+
+269
+00:32:44,040 --> 00:32:57,040
+إبسلون مظبوط؟ صحيح؟ وبالتالي إن هذا بيقود إلى
+
+270
+00:32:57,040 --> 00:32:59,780
+المتيجة التالية
+
+271
+00:33:06,460 --> 00:33:20,660
+دع الـ F تعمل من A إلى R وC مقاومة مقاومة من A
+
+272
+00:33:20,660 --> 00:33:32,220
+ثم التقالير ممتازة التقالير ممتازة
+
+273
+00:33:36,480 --> 00:33:43,660
+Limit f of x as x tends to c = عدد delta اللي
+
+274
+00:33:43,660 --> 00:33:51,460
+هو تعريف epsilon delta هذا معناه أنه لكل epsilon
+
+275
+00:33:51,460 --> 00:33:57,640
+أكبر من السفر يوجد delta تعتمد على epsilon عدد
+
+276
+00:33:57,640 --> 00:33:58,180
+موجد
+
+277
+00:34:03,020 --> 00:34:10,300
+Such that لو كان x ينتمي ل a و |x - c|
+
+278
+00:34:10,300 --> 00:34:16,340
+أصغر من delta أكبر من ستر فهذا لازم يتضمن أن
+
+279
+00:34:16,340 --> 00:34:23,580
+|f of x - L| أصغر من أصغر يعني معنى أخر
+
+280
+00:34:23,580 --> 00:34:29,200
+L هي limit لل function f of x هو بتالي تعريف 
+
+281
+00:34:29,200 --> 00:34:33,300
+epsilon delta  المتحفان هذا تعريف epsilon دلتا
+
+282
+00:34:33,300 --> 00:34:42,000
+بكاذب ال-neverhood definition ال
+
+283
+00:34:42,000 --> 00:34:54,940
+neverhood definition of limit وهو
+
+284
+00:34:54,940 --> 00:34:57,720
+أن for every
+
+285
+00:35:02,320 --> 00:35:06,700
+for every epsilon
+
+286
+00:35:06,700 --> 00:35:12,480
+neighborhood V
+
+287
+00:35:12,480 --> 00:35:22,920
+epsilon of L there exists delta neighborhood V
+
+288
+00:35:22,920 --> 00:35:30,280
+delta of C بحيث
+
+289
+00:35:32,120 --> 00:35:46,700
+إن لو كانت X تنتمي إلى A ومختلفة عن الـC وموجودة
+
+290
+00:35:46,700 --> 00:35:53,200
+أيضًا في الـDelta neighborhood لـC فلازم هذا يقدر إن
+
+291
+00:35:53,200 --> 00:36:01,600
+صورة X لازم تنتمي للـEpsilon neighborhood لـL
+
+292
+00:36:06,140 --> 00:36:13,440
+و هذا بالظبط عملنا آخر remark، prove it
+
+293
+00:36:13,440 --> 00:36:19,240
+follows from
+
+294
+00:36:19,240 --> 00:36:32,800
+above remark write
+
+295
+00:36:32,800 --> 00:36:33,380
+it down
+
+296
+00:36:40,630 --> 00:36:44,690
+حاولوا تكتبوا البرهان، البرهان فسرناها هنا
+
+297
+00:36:44,690 --> 00:37:00,590
+واضحناها من ال-remark خلينا نشوف خلينا
+
+298
+00:37:00,590 --> 00:37:10,530
+نرسم رسمها في المحور X نحو الـ y وهي ال-origin وخفض
+
+299
+00:37:10,530 --> 00:37:16,270
+أنه يوجد function مثل هذه فهذا بالحقيقة function y
+
+300
+00:37:16,270 --> 00:37:23,070
+بسهولة f of x وهي c نقطة في المجال تبع الدالة أو
+
+301
+00:37:23,070 --> 00:37:34,330
+حتى لو ما كانت موجودة c is the cluster point وهي
+
+302
+00:37:34,330 --> 00:37:35,870
+هذا عدد حقيقي
+
+303
+00:37:38,510 --> 00:37:44,570
+فده عدد حقيقي فمعنى
+
+304
+00:37:44,570 --> 00:37:50,770
+أن limit لل-F and X بالساوية C بالساوية L معناه
+
+305
+00:37:50,770 --> 00:37:57,210
+لأي أبسلون أكبر من الصفر أي لأي أبسلون أكبر من
+
+306
+00:37:57,210 --> 00:38:24,500
+الصفر ممكن أنا أقول epsilon neighborhood لأي
+
+307
+00:38:24,500 --> 00:38:33,180
+epsilon أكبر من 0 يعني بقدر أكون إبسلون نبرهود
+
+308
+00:38:33,180 --> 00:38:37,660
+بإبسلون
+
+309
+00:38:37,660 --> 00:38:44,180
+لإل فلو كان هذا given given إبسلون يعني given
+
+310
+00:38:44,180 --> 00:38:52,920
+إبسلون neighborhood لإل بقدر أجيل
+
+311
+00:38:52,920 --> 00:38:56,200
+أرد عليه
+
+312
+00:39:01,810 --> 00:39:07,770
+الدلتا Delta neighborhood هذا عبارة عن Delta 
+
+313
+00:39:07,770 --> 00:39:15,250
+neighborhood لـ C إذا أنا أخدت أعطتوني إبسلون بقدر
+
+314
+00:39:15,250 --> 00:39:20,570
+أكون إبسلون neighborhood لـ L فبقدر أرد عليه ال
+
+315
+00:39:20,570 --> 00:39:24,110
+Delta neighborhood لـ C في الفترة المفتوحة هذه
+
+316
+00:39:25,810 --> 00:39:31,550
+بالتالي، إذا كان هناك مجتمع إبسلن لـL، فهناك مجتمع
+
+317
+00:39:31,550 --> 00:39:38,230
+Delta لـC أو هناك مجتمع Delta عدد موجب بحيث إن لو
+
+318
+00:39:38,230 --> 00:39:42,930
+كانت الـX موجودة في A، والمتباينة هذه تتحقق، طب X
+
+319
+00:39:42,930 --> 00:39:47,550
+موجودة في A، والمتباينة هذه تتحقق، معناته X موجودة
+
+320
+00:39:47,550 --> 00:39:51,990
+في A ومختلفة عن الـC، و X موجودة في الـDelta
+
+321
+00:39:51,990 --> 00:39:57,400
+neighborhood هذه معناته X تنتمي لـA ومختلفة عن الـC
+
+322
+00:39:57,400 --> 00:40:02,480
+و X موجودة في الـDelta neighborhood هذا الشرط هذا بيقدي
+
+323
+00:40:02,480 --> 00:40:08,760
+أن المتباينة هذه تتحقق المتباينة هذه تتحقق معناه أن
+
+324
+00:40:08,760 --> 00:40:14,840
+الـ F of X صورة X تنتمي للـEpsilon neighborhood لـL فهو واضح أن 
+
+325
+00:40:14,840 --> 00:40:20,020
+هذا التعريف بيقدي للتعريف هذا باستخدام تفسير ال
+
+326
+00:40:20,020 --> 00:40:31,340
+remark والعكس طبعًا صحيح .. صحيح okay تمام؟ إذا هذا
+
+327
+00:40:31,340 --> 00:40:36,270
+بنسميه الـ .. هذا التعريف بنسميه الـ neighborhood
+
+328
+00:40:36,270 --> 00:40:40,630
+definition للـ limit of a function والتعريف دا أو
+
+329
+00:40:40,630 --> 00:40:45,810
+هذا بنسميه الـ epsilon delta definition
+
+330
+00:40:45,810 --> 00:40:52,930
+of the limit of a function تمام؟ وهذا طبعًا زي ال
+
+331
+00:40:52,930 --> 00:40:57,490
+epsilon capital N definition للlimit of a sequence
+
+332
+00:40:57,490 --> 00:41:00,870
+وبعد هي فكرين عرفنا ال-neighborhood definition
+
+333
+00:41:00,870 --> 00:41:05,640
+للlimit of a sequence هذا يعني يكافئ الكلام اللي
+
+334
+00:41:05,640 --> 00:41:09,720
+هنا إذا الآن في عندي تعريفين لل-limit of a
+
+335
+00:41:09,720 --> 00:41:14,060
+function at a point أو at a cluster point الدارس
+
+336
+00:41:14,060 --> 00:41:17,080
+التعريف اللي هنستخدمه أكثر هو epsilon delta
+
+337
+00:41:17,080 --> 00:41:23,580
+definition of the limit أكثر من ال-neighborhood
+
+338
+00:41:23,580 --> 00:41:26,920
+definition لكن أنا ما منعش أن أنا في أوقات معينة
+
+339
+00:41:26,920 --> 00:41:30,560
+أستخدم ال-neighborhood definition طيب نأخذ بعض
+
+340
+00:41:30,560 --> 00:41:38,200
+الأمثلة على كيفية إثبات إن الـ limit لـ certain
+
+341
+00:41:38,200 --> 00:41:43,980
+function is a certain number by
+
+342
+00:41:43,980 --> 00:41:49,020
+using epsilon delta definition نشوف مع بعض، وهذا
+
+343
+00:41:49,020 --> 00:41:54,100
+يعني بوازي الكلام اللي عملناه بالنسبة لل-limits of
+
+344
+00:41:54,100 --> 00:42:00,240
+sequences فإذا هنا في الأمثلة في كل الأمثلة التالية
+
+345
+00:42:00,240 --> 00:42:04,500
+عايزين نستخدم ال-definition of أو epsilon delta
+
+346
+00:42:04,500 --> 00:42:07,520
+definition أو ال-neighborhood definition لل-limit
+
+347
+00:42:07,520 --> 00:42:10,720
+of a function to prove إنه limit عن نقطة محددة
+
+348
+00:42:10,720 --> 00:42:18,760
+بساوي عدد محدد فمثلًا إن نثبت إنه limit لدالة ثابت
+
+349
+00:42:18,760 --> 00:42:22,240
+B لما X تقول لها C بساوي B
+
+350
+00:42:25,710 --> 00:42:30,410
+فلت هنا عندي فان الـ function تبعتي f of x بالساوية
+
+351
+00:42:30,410 --> 00:42:38,130
+ثابت B لكل x تنتمي إلى R فان الـ function تبعتي
+
+352
+00:42:38,130 --> 00:42:43,130
+ده اللي ثابتة وبالتالي إذا هنا لثبات إن ال-limit
+
+353
+00:42:43,130 --> 00:42:45,290
+تبعتها بالساوية B
+
+354
+00:42:48,460 --> 00:42:50,340
+أكبر من الصفر أكبر من الصفر أكبر من الصفر أكبر من
+
+355
+00:42:50,340 --> 00:42:50,500
+أكبر من الصفر أكبر من الصفر أكبر من الصفر أكبر من
+
+356
+00:42:50,500 --> 00:42:51,200
+الصفر أكبر من الصفر أكبر من الصفر أكبر من الصفر
+
+357
+00:42:51,200 --> 00:42:52,980
+أكبر من الصفر أكبر من الصفر أكبر من الصفر أكبر من
+
+358
+00:42:52,980 --> 00:42:53,500
+الصفر أكبر من الصفر أكبر من الصفر أكبر من الصفر
+
+359
+00:42:53,500 --> 00:42:54,560
+أكبر من الصفر أكبر من الصفر أكبر من الصفر أكبر من
+
+360
+00:42:54,560 --> 00:42:55,760
+الصفر أكبر من الصفر أكبر من الصفر أكبر من الصفر
+
+361
+00:42:55,760 --> 00:43:05,680
+أكبر من الصفر أكبر من الصفر أكبر من الصفر أكبر
+
+362
+00:43:05,680 --> 00:43:06,180
+من الصفر
+
+363
+00:43:18,680 --> 00:43:23,060
+تعال نشوف ال-implication ال-Delta هذه works ولا
+
+364
+00:43:23,060 --> 00:43:27,560
+لأ ف أنا عندي إن لو كانت الـ X تنتمي لـ A طبعًا الـ A
+
+365
+00:43:27,560 --> 00:43:31,700
+مجال الدالة هنا هو كل الأعداد الحقيقية و absolute
+
+366
+00:43:31,700 --> 00:43:38,610
+X minus C أكبر من 0 أصغر من Delta هل هذا بيقدر
+
+367
+00:43:38,610 --> 00:43:43,910
+لأبسليوت f of x minus b أصغر من إبسلون بالتأكيد
+
+368
+00:43:43,910 --> 00:43:48,910
+أنا عندي f of x بالساوية B سالب ال-limit اللي هي
+
+369
+00:43:48,910 --> 00:43:55,090
+B فهذا بيطلع أبسليوت الصفر بيطلع صفر والصفر هذا
+
+370
+00:43:55,090 --> 00:44:02,040
+أصغر من أي إبسلون موجبة إذا حصلت تعريف Epsilon Delta
+
+371
+00:44:02,040 --> 00:44:06,560
+يعني أثبتت أنه لأي Epsilon أي Delta موجبة works
+
+372
+00:44:06,560 --> 00:44:10,580
+تعمل تعطيل ال-implication وبالتالي by definition
+
+373
+00:44:10,580 --> 00:44:20,140
+limit F of X as X tends to C بساوي D طيب
+
+374
+00:44:20,140 --> 00:44:27,120
+نأخذ كمان مثال لو أخذت ال-identity function
+
+375
+00:44:34,810 --> 00:44:40,290
+بنثبت إن limit ده identity function لما x تقول إلى
+
+376
+00:44:40,290 --> 00:44:47,110
+أي عدد حقيقي c بساوي c نستخدم
+
+377
+00:44:47,110 --> 00:44:52,310
+تعريف epsilon delta let epsilon أكبر من الصفر be
+
+378
+00:44:52,310 --> 00:44:59,370
+given المرة هذه بدي أرد على ال-epsilon هذه ال
+
+379
+00:44:59,370 --> 00:45:05,230
+Delta تعتمد عليها هأختار ال-Delta بساوي epsilon
+
+380
+00:45:05,230 --> 00:45:10,430
+يختار Delta بساوي epsilon يختار Delta بساوي epsilon
+
+381
+00:45:10,430 --> 00:45:10,530
+يختار Delta بساوي epsilon يختار Delta بساوي epsilon
+
+382
+00:45:10,530 --> 00:45:12,470
+يختار Delta بساوي epsilon يختار Delta بساوي epsilon
+
+383
+00:45:12,470 --> 00:45:17,090
+يختار Delta بساوي epsilon يختار Delta بساوي epsilon
+
+384
+00:45:17,090 --> 00:45:19,890
+يختار Delta بساوي epsilon يختار Delta بساوي epsilon
+
+385
+00:45:19,890 --> 00:45:23,450
+يختار Delta بساوي epsilon يختار Delta بساوي epsilon
+
+386
+00:45:23,450 --> 00:45:29,320
+يختار Delta بساوي epsilon يختار Delta هي عبارة عن ال
+
+387
+00:45:29,320 --> 00:45:33,500
+identity function لكل X المجال تبعها كل الأعداد
+
+388
+00:45:33,500 --> 00:45:40,060
+الحقيقية فلو كانت X تنتمي لـ A اللي هي R و Absolute
+
+389
+00:45:40,060 --> 00:45:46,100
+X minus C أكبر من Zero أصغر من Delta فخلينا نشوف
+
+390
+00:45:46,100 --> 00:45:52,880
+هل بيطلع Absolute F of X minus ال-L اللي هو C أصغر
+
+391
+00:45:52,880 --> 00:45:56,240
+من Epsilon هنشوف
+
+392
+00:45:57,800 --> 00:46:04,860
+طيب نعوض عن F of X بالساوية X minus C طب أنا عند ال
+
+393
+00:46:04,860 --> 00:46:10,320
+X هذه موجودة في R و المسافة بينها ومختلفة عن الـ C
+
+394
+00:46:10,320 --> 00:46:13,880
+والمسافة بينهم أصغر من Delta اللي أنا باخدها
+
+395
+00:46:13,880 --> 00:46:19,420
+بالساوية epsilon إذا ال-absolute X minus C من هنا أصغر من
+
+396
+00:46:19,420 --> 00:46:27,090
+Delta اللي هي epsilon وبالتالي درهين أثبتت أنه لأي إبسلون
+
+397
+00:46:27,090 --> 00:46:32,990
+يوجد Delta اللي هي إبسلون نفسها بحيث لكل x في a إذا
+
+398
+00:46:32,990 --> 00:46:36,570
+كان absolute x minus c أكبر من صفر أصغر من Delta
+
+399
+00:46:36,570 --> 00:46:40,490
+هذا بيقدر أن absolute f of x minus l أصغر من
+
+400
+00:46:40,490 --> 00:46:47,650
+إبسلون since إبسلون أكبر من الصفر was arbitrary
+
+401
+00:46:52,350 --> 00:47:00,830
+we get from the definition هذا الكلام صحيح لكل
+
+402
+00:47:00,830 --> 00:47:06,690
+epsilon وبالتالي by definition بيطلع عندي limit ال
+
+403
+00:47:06,690 --> 00:47:10,430
+function f of x اللي هي ال-identity function لما x
+
+404
+00:47:10,430 --> 00:47:19,750
+تقوى لـ c بساوي c وهو المطلوب تمام okay إذا المرة
+
+405
+00:47:19,750 --> 00:47:27,480
+الجاية هنثبت إن limited ده لتربعية لما x أو لـ c
+
+406
+00:47:27,480 --> 00:47:33,280
+بساوي c تربيع وهذا موجود طبعًا في الكتاب وفي كمان
+
+407
+00:47:33,280 --> 00:47:37,780
+أمثلة أخرى فأرجو أنكم تقرأوا الأمثلة هذه من الكتاب
+
+408
+00:47:37,780 --> 00:47:44,350
+وتحضروها للمحاضرة الجاية وتشوفوا كيف تم استخدام
+
+409
+00:47:44,350 --> 00:47:49,410
+تعريف epsilon delta في إثبات إن ال-limit لدالة زهر
+
+410
+00:47:49,410 --> 00:47:53,530
+الدالة التربعية بساوي C تربيع عند أي نقطة C okay
+
+411
+00:47:53,530 --> 00:47:58,270
+تمام؟ في أي سؤال أو إيضاح؟ إذا نكتفي بهذا القدر
+
+412
+00:47:58,270 --> 00:48:02,410
+وإن شاء الله اللي أنا أكملّه في المحاضرة القادمة
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/6FDcUXR9Pqo_raw.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/6FDcUXR9Pqo_raw.srt
new file mode 100644
index 0000000000000000000000000000000000000000..83ec7812b0b57f60e4e6f41899c197f715af2358
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/6FDcUXR9Pqo_raw.srt
@@ -0,0 +1,1652 @@
+1
+00:00:20,870 --> 00:00:25,910
+المرة اللى فاتت او في المحاضرة للصادفة عرفنا ال
+
+2
+00:00:25,910 --> 00:00:31,410
+cluster point و أخدنا أمثلة كيف نجيب ال cluster
+
+3
+00:00:31,410 --> 00:00:39,710
+points لمجموعة معينة و وجفنا عند المثال التالت
+
+4
+00:00:59,030 --> 00:01:05,110
+المثال التالت show that
+
+5
+00:01:05,110 --> 00:01:12,070
+zero is the only cluster
+
+6
+00:01:12,070 --> 00:01:18,290
+point of
+
+7
+00:01:18,290 --> 00:01:23,570
+the set A7
+
+8
+00:01:26,690 --> 00:01:31,870
+كل واحد على N حيث و N that's a number دكتور هذا
+
+9
+00:01:31,870 --> 00:01:37,210
+مثال تاني أخدناها ده؟ لأ لأ اللي أخدناه اللي هو ال
+
+10
+00:01:37,210 --> 00:01:48,030
+.. هناخدها هناخدها هناخدها ف .. هنا هنا .. هنا
+
+11
+00:01:48,030 --> 00:01:55,250
+هنا اتنين
+
+12
+00:01:59,360 --> 00:02:11,580
+إن Zero is a cluster point تلت
+
+13
+00:02:11,580 --> 00:02:22,720
+Delta أكبر من السفل Be given by Archimedean
+
+14
+00:02:22,720 --> 00:02:25,880
+property
+
+15
+00:02:30,960 --> 00:02:40,920
+يوجد capital N ينتمي إلى N بحيث انه واحد على N
+
+16
+00:02:40,920 --> 00:02:54,340
+أصغر من نفسه hence
+
+17
+00:03:01,430 --> 00:03:09,270
+الدلتا نيبر هو zero لو
+
+18
+00:03:09,270 --> 00:03:25,690
+أخدت xN هو واحد على ن فهذا ينتمي إلى
+
+19
+00:03:25,690 --> 00:03:39,580
+المجموعة Aوانت ليه لا ال delta number هون ل ..
+
+20
+00:03:39,580 --> 00:03:44,040
+او ال x هذا المفروض delta ال delta number هو ده
+
+21
+00:03:44,040 --> 00:03:47,860
+اسمه
+
+22
+00:03:47,860 --> 00:03:56,940
+أسوأ؟ إذن
+
+23
+00:03:56,940 --> 00:03:58,460
+هذا لا أي سؤال خالد
+
+24
+00:04:06,200 --> 00:04:10,700
+الدلتا نبقى رهود للسفر اللي هي الفترة المفتوحة من
+
+25
+00:04:10,700 --> 00:04:16,100
+سالب دلتا إلى دلتا
+
+26
+00:04:16,100 --> 00:04:21,960
+فهي عندي واحد على N أصغر من دلتا وطبعا أكبر من سفر
+
+27
+00:04:24,160 --> 00:04:27,620
+فواحد على أن ينتمي للـDelta neighborhood للسفر
+
+28
+00:04:27,620 --> 00:04:32,400
+وواحد على أن ينتمي للمجموعة A وبالتالي يجب أن نكون
+
+29
+00:04:32,400 --> 00:04:38,580
+أثبتنا أنه لأي Delta أكبر من السفر أو أي Delta
+
+30
+00:04:38,580 --> 00:04:44,680
+neighborhood للسفر يتقاطع مع A في نقطة مختلفة عن
+
+31
+00:04:44,680 --> 00:04:49,000
+السفر ال
+
+32
+00:04:49,000 --> 00:04:56,810
+X انها جلدها تساوي سفرلاتسار السفر وبالتالي إذا
+
+33
+00:04:56,810 --> 00:05:05,470
+هذا بثبت السفر is a cluster point of
+
+34
+00:05:05,470 --> 00:05:12,310
+الست إذا هذا بثبت ال claim الآن ليه مايكون فيه
+
+35
+00:05:12,310 --> 00:05:14,130
+cluster points أخرى؟
+
+36
+00:05:30,490 --> 00:05:41,150
+إذا كانت X لا تساوي سفر، فهي ليست مجموعة من A
+
+37
+00:05:41,150 --> 00:05:46,390
+فحاسيبكم
+
+38
+00:05:46,390 --> 00:05:47,630
+أنتم تكتبوا البرهان
+
+39
+00:05:50,290 --> 00:06:02,390
+هي سفر وهي واحد وهي نص وهي تلت وهي واحد على N وهي
+
+40
+00:06:02,390 --> 00:06:05,390
+واحد على N زائد واحد وهكذا
+
+41
+00:06:16,430 --> 00:06:25,930
+فهنا تاندي two cases case one ان x تنتمي الى a و
+
+42
+00:06:25,930 --> 00:06:34,690
+الحلقة التانية case two ان x لا تنتمي الى a ال x
+
+43
+00:06:34,690 --> 00:06:39,530
+دي مستويش السفر احنا already اثبتنا ان السفر
+
+44
+00:06:39,530 --> 00:06:43,850
+cluster pointطيب افرض X مستويش سفر إذا X ممكن
+
+45
+00:06:43,850 --> 00:06:48,170
+تساوي واحد أو نص أو تلت أو واحد على N for some N
+
+46
+00:06:48,170 --> 00:06:53,250
+صح؟ ممكن ماتساويش ولا عنصر ممكن ماتكونش تم تمل أيه
+
+47
+00:06:53,250 --> 00:06:58,070
+فلو كانت ال X واحد من العناصر هدول واحد من العناصر
+
+48
+00:06:58,070 --> 00:07:04,630
+الست فبقدر ألاقي أن يوجد بقدر ألاقي delta
+
+49
+00:07:04,630 --> 00:07:08,990
+neighborhood للعنصر مثلا التلت بقدر ألاقي delta
+
+50
+00:07:08,990 --> 00:07:09,490
+neighborhood
+
+51
+00:07:13,860 --> 00:07:19,840
+الأنصر اللي بعد هذا هيكون ربع فباخد المسافة الأصغر
+
+52
+00:07:19,840 --> 00:07:26,560
+بين تلت ربع تلت و نص و باخد نص المسافة ديلتا فبصير
+
+53
+00:07:26,560 --> 00:07:30,920
+عند هنا ديلتا نبرهود للتلت و بتقاطعش مع المجموعة A
+
+54
+00:07:30,920 --> 00:07:38,120
+بالمرة أو في نقطة مختلفة عن التلتوبالتالي لو كانت
+
+55
+00:07:38,120 --> 00:07:44,880
+ال X موجودة في A زي التلت مثلا فال X ليست cluster
+
+56
+00:07:44,880 --> 00:07:49,860
+point الآن ال X لا تنتمي ل أيه؟ ال X لا تنتمي ل
+
+57
+00:07:49,860 --> 00:07:55,920
+أيه؟ معناته هي عنصر هنا زي هذا X أزرق من سفر أو X
+
+58
+00:07:55,920 --> 00:08:01,540
+ممكن تكون جاية بين عنصرين فباخد المسافة بين X و
+
+59
+00:08:01,540 --> 00:08:04,840
+أقرب عنصر إلها من اليمين و أقرب عنصر إلها من
+
+60
+00:08:04,840 --> 00:08:11,140
+اليسارو باخد نص المسافة ديلتا او ابسلان و بكوّن
+
+61
+00:08:11,140 --> 00:08:17,480
+دلتا نبرود ل X هذا دلتا نبرود مش هيتقاطع مع ال 6 و
+
+62
+00:08:17,480 --> 00:08:20,700
+بالتالي النقطة هذه عمرها ما بتكون cluster point
+
+63
+00:08:20,700 --> 00:08:26,020
+ممكن كمان النقطة هذه ال X مش موجودة فيها ممكن تكون
+
+64
+00:08:26,020 --> 00:08:31,260
+على شمال السفر او على يمين الواحدفلو كانت على يمين
+
+65
+00:08:31,260 --> 00:08:35,560
+الواحد خد نص المسافة هي دلتا إذا هي دلتا نبقى
+
+66
+00:08:35,560 --> 00:08:39,420
+روحود ل X مابتخطعش معاه بالمرة بالتالي X مابت
+
+67
+00:08:39,420 --> 00:08:44,440
+cluster هنا لو كانت X أصغر من سفر فخد نص المسافة
+
+68
+00:08:44,440 --> 00:08:48,960
+بين X و 0 على إنها دلتاوبالتالي كونة delta
+
+69
+00:08:48,960 --> 00:08:52,460
+neighborhood ل X هذا delta neighborhood بتقاطعش مع
+
+70
+00:08:52,460 --> 00:08:56,240
+A بالمرة وبالتالي X ليست cluster point إذا في كل
+
+71
+00:08:56,240 --> 00:09:01,860
+الأحوال X ليست cluster point سواء كانت في A أو مش
+
+72
+00:09:01,860 --> 00:09:05,420
+موجودة إذا كانت X مختلفة عن الصفر فليست cluster
+
+73
+00:09:05,420 --> 00:09:14,930
+point okay إذا zero is the only نقطة الوحيدةمافيش
+
+74
+00:09:14,930 --> 00:09:18,990
+نقطة غيرها بتكون cluster point بالمثل ممكن نثبت
+
+75
+00:09:18,990 --> 00:09:29,650
+مثال أخر f
+
+76
+00:09:29,650 --> 00:09:35,710
+i بساوي ال
+
+77
+00:09:35,710 --> 00:09:39,830
+unit technological interval and
+
+78
+00:09:51,210 --> 00:10:02,710
+IQ بساوي I تقاطع ال rational numbers then
+
+79
+00:10:02,710 --> 00:10:13,170
+every x تنهي ل I is a cluster point a cluster
+
+80
+00:10:13,170 --> 00:10:15,250
+point of IQ
+
+81
+00:10:18,800 --> 00:10:26,900
+إذا I هي الفترة المغلقة من صفر لواحد IQ هي كل
+
+82
+00:10:26,900 --> 00:10:31,800
+الأعداد المسبية الموجودة في الفترة المغلقة من صفر
+
+83
+00:10:31,800 --> 00:10:38,340
+لواحد ممكن إثبات إن كل X في الفترة المغلقة I هي
+
+84
+00:10:38,340 --> 00:10:45,940
+cluster point للمجموعة IQ وذلك باستخدام ال density
+
+85
+00:10:45,940 --> 00:10:52,060
+theoremproof use
+
+86
+00:10:52,060 --> 00:11:06,500
+the density theorem فحاسيبكم
+
+87
+00:11:06,500 --> 00:11:15,920
+انتوا تكتبوا البرهان خدي أي x في I واخدتي انه أي
+
+88
+00:11:17,780 --> 00:11:22,140
+أعملكم برهان هكذا بدون أن أكتب أي شيء هذه الفترة I
+
+89
+00:11:22,140 --> 00:11:29,720
+من صفر إلى واحد هذه الفترة المغلقة من ملفة أن كل X
+
+90
+00:11:29,720 --> 00:11:35,800
+كل X فيها هو cluster point لمجموعة الأعداد المسمية
+
+91
+00:11:35,800 --> 00:11:42,520
+في I ففي عندي تلت حالات MX هتكون أكبر من صفر أصغر
+
+92
+00:11:42,520 --> 00:11:48,060
+من واحد يعني نقطة داخليها ليست نقطة طرفو طبعا هي
+
+93
+00:11:48,060 --> 00:11:52,460
+لو أخدت أي delta عدد موجب و كوّنت delta
+
+94
+00:11:52,460 --> 00:11:57,380
+neighborhood لل X فال delta neighborhood هذا
+
+95
+00:11:57,380 --> 00:12:05,920
+هيتقاطع مع المجموعة A IQ حسب نظرية الكثافة أي فترة
+
+96
+00:12:05,920 --> 00:12:11,520
+مفتوحة زي هذه تحتوي rational number صح؟وبالتالي اي
+
+97
+00:12:11,520 --> 00:12:16,600
+delta neighborhood ل ال X هيتقاطع مع المجموعة IQ
+
+98
+00:12:16,600 --> 00:12:24,160
+في نقطة R مختلفة عن ال X حسب نظرية الكثارة
+
+99
+00:12:24,160 --> 00:12:28,040
+وبالتالي حسب التعريف اذا ال X هذه اللي هي نقطة
+
+100
+00:12:28,040 --> 00:12:33,700
+داخلية is a cluster point لمن؟
+
+101
+00:12:33,700 --> 00:12:40,260
+للمجموعة IQلو كانت ال .. ال .. ال x هي نقطة الطرف
+
+102
+00:12:40,260 --> 00:12:46,960
+الحلقة التانية لما x تكون هي سفر لما x تكون بساوي
+
+103
+00:12:46,960 --> 00:12:52,160
+سفر وخدي أي delta neighborhood لأن هاي سالب delta
+
+104
+00:12:52,160 --> 00:12:56,560
+موجب delta فالفترة
+
+105
+00:12:56,560 --> 00:13:01,200
+هذه تتقطع يعني
+
+106
+00:13:01,200 --> 00:13:07,230
+هاي delta هادي delta و هادي نقطة سفرالان الفترة
+
+107
+00:13:07,230 --> 00:13:12,870
+هذه بقدر الاقي فيها rational number حسب مباريك
+
+108
+00:13:12,870 --> 00:13:16,970
+الكفافة موجود بين سفر و دلتا و ال rational number
+
+109
+00:13:16,970 --> 00:13:23,550
+هذا موجود في ال unit closed intervalوبالتالي كل
+
+110
+00:13:23,550 --> 00:13:28,450
+delta neighborhood للصفر يتقاطع
+
+111
+00:13:28,450 --> 00:13:33,670
+مع المجموع IQ في نقطة R مختلفة عن الصفر وبالتالي
+
+112
+00:13:33,670 --> 00:13:37,910
+الصفر هو cluster point بالمثل ممكن تباتل بالواحد
+
+113
+00:13:37,910 --> 00:13:41,970
+cluster point لأن اي delta neighborhood للواحد
+
+114
+00:13:43,720 --> 00:13:48,560
+هيحتوي حسب نظرية الكثافة لو هذا كان واحد ثالث دلتا
+
+115
+00:13:48,560 --> 00:13:54,510
+في النقطة هذه وهذه واحد فبنقدر نلاقي Rrational
+
+116
+00:13:54,510 --> 00:13:58,650
+number حسب مجرد كدفة بين واحد سالب دلتا وواحد
+
+117
+00:13:58,650 --> 00:14:03,710
+وبالتالي ال delta neighborhood هذا المركزه واحد و
+
+118
+00:14:03,710 --> 00:14:08,490
+نصف خطره دلتا هيتقاطع مع ال IQ في نقطة مختلفة عن
+
+119
+00:14:08,490 --> 00:14:13,250
+الواحد وبالتالي واحد cluster point الان لان هسيبكم
+
+120
+00:14:13,250 --> 00:14:15,650
+تكتبوا البرهان بالتفصيل
+
+121
+00:14:19,850 --> 00:14:25,250
+Okay إذاً هيك يعني عندنا عدة أمثلة على ال cluster
+
+122
+00:14:25,250 --> 00:14:32,490
+points نرجع لمفهوم ال limit ل ال function اللي هو
+
+123
+00:14:32,490 --> 00:14:47,650
+كان عنوان ال section تبعنا اذا
+
+124
+00:14:47,650 --> 00:14:48,750
+هنا definition
+
+125
+00:14:55,450 --> 00:15:07,150
+دع الـ f يكون مفعولًا من a إلى r مفعولًا
+
+126
+00:15:07,150 --> 00:15:19,710
+في أين a مجزرة من r و c مجزرة من الـ
+
+127
+00:15:19,710 --> 00:15:22,090
+set A
+
+128
+00:15:26,660 --> 00:15:35,260
+المعنى number L هو مقال
+
+129
+00:15:35,260 --> 00:15:39,440
+للمعنى
+
+130
+00:15:39,440 --> 00:15:44,440
+f at
+
+131
+00:15:44,440 --> 00:15:59,020
+xبس you see إذا تحقق الشرط التالي for any epsilon
+
+132
+00:15:59,020 --> 00:16:05,340
+أكبر من سفر يوجد delta تعتمد على epsilon عدد موجد
+
+133
+00:16:05,340 --> 00:16:14,690
+بحيث أنه لكل x ينتمي إلى aو المسافة بين .. و ال X
+
+134
+00:16:14,690 --> 00:16:23,090
+هذا يختلف عن ال C و المسافة بينها و بين ال C أصغر
+
+135
+00:16:23,090 --> 00:16:30,030
+من Delta لازم هذا يضمن أن absolute F of X minus L
+
+136
+00:16:30,030 --> 00:16:41,010
+أصغر من Delta إذن هذا شرط .. هذا شرط أو بسميهأنا
+
+137
+00:16:41,010 --> 00:16:49,470
+بسميه ابسلون دلتا definition ابسلون دلتا
+
+138
+00:16:49,470 --> 00:16:54,310
+definition of limit لل
+
+139
+00:16:54,310 --> 00:16:58,550
+limit of a functionالـ Limit لـ function f of x
+
+140
+00:16:58,550 --> 00:17:03,590
+بالساوي أو L هي عبارة عن Limit لـ function f of x
+
+141
+00:17:03,590 --> 00:17:09,970
+and x بالساوي C إذا لو أعطتوني أي إبسلون عدد موجب
+
+142
+00:17:09,970 --> 00:17:15,570
+لازم أنا أرد عليها Delta عدد موجب آخر يعتمد على
+
+143
+00:17:15,570 --> 00:17:20,750
+إبسلون بحيث أنه لكل X في المجموعة A اللي هو ال
+
+144
+00:17:20,750 --> 00:17:27,300
+domain تبع ال functionو X هذه مختلفة لا تساوي C
+
+145
+00:17:27,300 --> 00:17:33,360
+يعني المتباين هذه معناها X لا تساوي Cإذاً لكل x في
+
+146
+00:17:33,360 --> 00:17:38,200
+a مختلفة عن الـc والمسافة بينها وبين الـc أصغر من
+
+147
+00:17:38,200 --> 00:17:42,580
+دلتا لازم تطلع المسافة بين f of x و L أصغر من
+
+148
+00:17:42,580 --> 00:17:47,220
+إبسلون هذا الكلام اتحقق لكل إبسلون أكبر من الصفر
+
+149
+00:17:47,220 --> 00:17:51,640
+فبنقول إن العدد L is a limit of the function f عند
+
+150
+00:17:51,640 --> 00:17:52,360
+النقطة c
+
+151
+00:17:59,170 --> 00:18:07,710
+من هذا التعريف بينتج على طول اه طيب in this case
+
+152
+00:18:07,710 --> 00:18:13,930
+in this case we
+
+153
+00:18:13,930 --> 00:18:26,600
+say انه if converges if convergesto the number L
+
+154
+00:18:26,600 --> 00:18:39,520
+at X بساوي C and we write ونكتب الحالة هذه limit ل
+
+155
+00:18:39,520 --> 00:18:46,940
+F of X لما X تقول إلى C بساوي L أو ممكن نكتب limit
+
+156
+00:18:46,940 --> 00:18:54,260
+F as X tends to C بساوي L أو ممكن نكتب
+
+157
+00:19:01,220 --> 00:19:11,260
+أو ممكن نكتب f of x tends to L as x tends to c كل
+
+158
+00:19:11,260 --> 00:19:16,360
+هدول معناهم أن العدد L limit لل function f and x
+
+159
+00:19:16,360 --> 00:19:17,360
+سوى c
+
+160
+00:19:22,850 --> 00:19:30,090
+ف limit f of x as x tends to c does not exist،
+
+161
+00:19:30,090 --> 00:19:37,430
+يعني مافيش عدد L زي هذا، we say ان ال function f
+
+162
+00:19:37,430 --> 00:19:45,590
+diverges، diverges at x less than c
+
+163
+00:19:50,120 --> 00:19:55,320
+الان نستطيع ان نثبت ان ال limit لل function هذه
+
+164
+00:19:55,320 --> 00:20:00,040
+النقطة لو كانت موجودة لو كان ال function لها limit
+
+165
+00:20:00,040 --> 00:20:07,120
+فlimit هذه لازم تكون unique ال
+
+166
+00:20:07,120 --> 00:20:19,320
+function if from A to R can have only
+
+167
+00:20:39,940 --> 00:20:44,760
+والبرهان شبه البرهان الـ uniqueness of the limit
+
+168
+00:20:44,760 --> 00:20:50,700
+of a sequence we use epsilon over two argument
+
+169
+00:20:51,860 --> 00:20:54,780
+استنتاج epsilon على اتنين او برهان epsilon على
+
+170
+00:20:54,780 --> 00:21:01,020
+اتنين هنشوف مع بعض هاي برهان تروح left epsilon
+
+171
+00:21:01,020 --> 00:21:04,300
+اكبر
+
+172
+00:21:04,300 --> 00:21:11,720
+من السفر ب given since
+
+173
+00:21:11,720 --> 00:21:19,120
+طب
+
+174
+00:21:19,120 --> 00:21:25,040
+خليني الأوللبرهانة النظرية هذه بدي أفرض أنه فيه
+
+175
+00:21:25,040 --> 00:21:34,640
+two limits assume أن ال limit ل f of x as x tends
+
+176
+00:21:34,640 --> 00:21:44,340
+to c بساوي عدد الواحد and limit أيضا ل f of x as x
+
+177
+00:21:44,340 --> 00:21:50,340
+tends to c بساوي عدد تاني الاتنينوعشان أثبت
+
+178
+00:21:50,340 --> 00:21:57,860
+النظرية لازم أثبت الإدعاء التالي إن ال1 بساوي ال4
+
+179
+00:21:57,860 --> 00:22:07,720
+فلبرهان ذلك let epsilon أكبر من السفر be given
+
+180
+00:22:07,720 --> 00:22:19,500
+since مما أننا فرضين أن ال limitلأف as x tends to
+
+181
+00:22:19,500 --> 00:22:28,420
+c بالساوي الواحد then by definition by epsilon
+
+182
+00:22:28,420 --> 00:22:33,180
+delta definition of limit there exists delta one
+
+183
+00:22:33,180 --> 00:22:39,830
+depends on epsilon positive numberبحيث أنه لو كان
+
+184
+00:22:39,830 --> 00:22:46,150
+x ينتمي إلى a و absolute x minus c أصغر من delta
+
+185
+00:22:46,150 --> 00:22:54,850
+one أكبر من سفر فهذا بيقدي أن absolute f of x
+
+186
+00:22:54,850 --> 00:23:02,530
+minus l one أصغر من epsilon عتنين عشان الاستنتاج
+
+187
+00:23:02,530 --> 00:23:05,510
+هذا واحد
+
+188
+00:23:08,770 --> 00:23:13,810
+Also ايضا احنا
+
+189
+00:23:13,810 --> 00:23:20,610
+فرضين من ال limit لل function f of x as x tends to
+
+190
+00:23:20,610 --> 00:23:27,990
+c بساوي عدد تاني ال اتنين then
+
+191
+00:23:27,990 --> 00:23:35,650
+for the same for same epsilon اكبر من ستة نفس ال
+
+192
+00:23:35,650 --> 00:23:43,140
+epsilon يعنيلنفس الـ Epsilon بما أن limit ل F of X
+
+193
+00:23:43,140 --> 00:23:48,940
+and X بساوية C بساوية L2 نجد Delta 2 تعتمد على
+
+194
+00:23:48,940 --> 00:23:53,940
+Epsilon على 2 وبالتالي على Epsilon عدد موجد بحيث
+
+195
+00:23:53,940 --> 00:24:00,300
+أنه لو كان X ينتمي إلى A و Absolute X minus C أصغر
+
+196
+00:24:00,300 --> 00:24:07,350
+من Delta 2 أكبر من 0فهذا أكيد بيقدّي أنه absolute
+
+197
+00:24:07,350 --> 00:24:14,510
+f of x minus L2 أصغر من epsilon على 2 ال sum ال
+
+198
+00:24:14,510 --> 00:24:22,030
+implication هي D2 الآن بناخد .. بنعرف delta ل L
+
+199
+00:24:22,030 --> 00:24:31,230
+Delta بتساوي ال minimum ل delta واحد و delta اتنين
+
+200
+00:24:32,560 --> 00:24:37,340
+طبعا هذا بيطلع عدد
+
+201
+00:24:37,340 --> 00:24:43,200
+موجب لإن دلتا واحد ودلتا اتنين عدد موجب وكذلك دلتا
+
+202
+00:24:43,200 --> 00:24:46,200
+دي تعتمد على ابسلون لإن دلتا واحد ودلتا اتنين
+
+203
+00:24:46,200 --> 00:24:52,720
+يعتمدوا على ابسلون then
+
+204
+00:24:52,720 --> 00:24:55,860
+by
+
+205
+00:24:55,860 --> 00:25:07,520
+واحد and اتنيننحصل على التالي، لو كان x ينتقل إلى
+
+206
+00:25:07,520 --> 00:25:14,980
+a و absolute x minus c أصغر من delta أكبر من سفر
+
+207
+00:25:14,980 --> 00:25:26,590
+فهذا هيقدر أن absolute L1 minus L2بساوي absolute
+
+208
+00:25:26,590 --> 00:25:39,610
+L1 minus F of X زائد F of X minus L2 إذا انطلعت
+
+209
+00:25:39,610 --> 00:25:46,590
+أنا F of X ورجعتها فكأني ما جيرتش حاجة وهذا بيطلع
+
+210
+00:25:46,590 --> 00:25:50,460
+باستخدام ال triangle inequalityبالترائنجل الـ
+
+211
+00:25:50,460 --> 00:25:54,900
+equality لـ absolute value لمجموعة حاجتين أصغر من
+
+212
+00:25:54,900 --> 00:26:00,920
+لو يساوي absolute L1 minus F of X ذات absolute F
+
+213
+00:26:00,920 --> 00:26:07,980
+of X minus L2 الآن
+
+214
+00:26:07,980 --> 00:26:13,340
+باستخدام ال implication واحد، اللحظة أن الـ delta
+
+215
+00:26:13,340 --> 00:26:17,960
+هي ال minimum لـ delta واحد و delta اتنينوبالتالي
+
+216
+00:26:17,960 --> 00:26:24,340
+الـ delta هذه أصغر من delta واحد فحسب ال
+
+217
+00:26:24,340 --> 00:26:28,600
+implication واحد لما يكون x ينتمي ل a و absolute x
+
+218
+00:26:28,600 --> 00:26:33,800
+minus c أصغر من delta واحد فانا بقدم ال absolute
+
+219
+00:26:33,800 --> 00:26:40,460
+value هذه أصغر من y على 2 كذلك باستخدام ال
+
+220
+00:26:40,460 --> 00:26:45,060
+implication 2أنا عندي الـ delta هذه هي الـ minimum
+
+221
+00:26:45,060 --> 00:26:51,760
+لـ delta 1 و delta 2 وبالتالي أصغر من delta 2 فبال
+
+222
+00:26:51,760 --> 00:26:55,680
+implication 2 بتقول لو كان x ينتمي ل a و absolute
+
+223
+00:26:55,680 --> 00:27:00,320
+x minus c أصغر من delta 2 فال absolute value ل f
+
+224
+00:27:00,320 --> 00:27:07,500
+of x minus l2 less than epsilon over 2 هذا بيساوي
+
+225
+00:27:07,500 --> 00:27:16,080
+epsilonإذا أنا طلع عندي أثبتت أن absolute L1 minus
+
+226
+00:27:16,080 --> 00:27:22,540
+L2 أكبر من أبسلون طبعا أكيد أكبر من أو ساوى سفر و
+
+227
+00:27:22,540 --> 00:27:28,600
+الأن هذا صحيح لكل أبسلون أكبر من السفر لأن one
+
+228
+00:27:28,600 --> 00:27:34,500
+hand هنا ال epsilon was arbitrary givenالإبسلون
+
+229
+00:27:34,500 --> 00:27:38,660
+was arbitrarily يعني نقول since this holds for
+
+230
+00:27:38,660 --> 00:27:43,160
+every إبسلون في لمّة أخدناها في بداية ال course
+
+231
+00:27:43,160 --> 00:27:48,820
+بتقول لو في عندي عدد حفيفي a أكبر من أو ساوي سفر و
+
+232
+00:27:48,820 --> 00:27:53,940
+أصغر من إبسلون فإن إبسلون أكبر من السفر فهذا بيقدي
+
+233
+00:27:53,940 --> 00:28:00,160
+أن a بيساوي سفرأخد ايه هنا الـ absolute value ل L1
+
+234
+00:28:00,160 --> 00:28:09,140
+minus L2 فحسب النمة هذه هيطلع عندى هذا بقدر اللي
+
+235
+00:28:09,140 --> 00:28:15,600
+قدر انه absolute L1 minus L2 بالساوية سفر وبالتالي
+
+236
+00:28:15,600 --> 00:28:24,600
+بطلع عندى L1 بساوية L2 وهو المطلوبإذا أنا فرقت إن
+
+237
+00:28:24,600 --> 00:28:28,860
+الـ function إلها two limits عن نقطة فطلع الـ two
+
+238
+00:28:28,860 --> 00:28:32,680
+limits متساوياتين وبالتالي لو كانت الـ function
+
+239
+00:28:32,680 --> 00:28:37,240
+إلها limit عن نقطة فال limit تطلع وحيدة unique،
+
+240
+00:28:37,240 --> 00:28:43,200
+بقى ده؟ في أي سؤال أو أي سؤال على البرهانة؟
+
+241
+00:29:02,290 --> 00:29:15,590
+ناخد ملاحظة هنا الـ
+
+242
+00:29:15,590 --> 00:29:27,330
+epsilon delta definition of limit of a function f
+
+243
+00:29:27,330 --> 00:29:29,270
+from a to r
+
+244
+00:29:32,670 --> 00:29:40,250
+the inequality المتباينة
+
+245
+00:29:40,250 --> 00:29:48,030
+اللي هي absolute x minus c أكبر من سفر أصغر من
+
+246
+00:29:48,030 --> 00:29:58,470
+دولتان means حاجتين الجزء هذا معناه أن absolute x
+
+247
+00:29:58,470 --> 00:30:09,330
+minus c لا يساوى سفروبالتالي X لا يساوي C إذاً هذا
+
+248
+00:30:09,330 --> 00:30:17,870
+يعني أن X لا تساوي C المتباينة
+
+249
+00:30:17,870 --> 00:30:23,410
+التانية اللي هي absolute X minus C أصغر من Delta
+
+250
+00:30:23,410 --> 00:30:31,170
+هذه بتكافئ أن X minus C أصغر من Delta أكبر من ثالث
+
+251
+00:30:31,170 --> 00:30:39,850
+Delta صح؟وهذه بتكافئ أن X أكبر من C Negative Delta
+
+252
+00:30:39,850 --> 00:30:47,870
+أصغر من C زائد Delta وهذا معناه أن X تنتمي لـ
+
+253
+00:30:47,870 --> 00:30:56,450
+Delta Neverhood لـ C اللي هو الفترة اللي اتروها من
+
+254
+00:30:56,450 --> 00:30:59,410
+C Minus Delta ل C Plus Delta
+
+255
+00:31:06,690 --> 00:31:12,550
+إذا المتباينة دي معناها x لا تساوي c and x تنتمي
+
+256
+00:31:12,550 --> 00:31:18,190
+لل delta اللي برجود لل c اللي هو الفترة المفتوحة
+
+257
+00:31:18,190 --> 00:31:25,170
+من c سالم negative delta إلى c plus delta كذلك
+
+258
+00:31:25,170 --> 00:31:29,570
+المتباينة also
+
+259
+00:31:33,070 --> 00:31:37,490
+الإي نكواليتي المتباينة
+
+260
+00:31:37,490 --> 00:31:43,450
+اللي هي absolute f of x minus L أصغر من إبسلون
+
+261
+00:31:43,450 --> 00:31:46,490
+means
+
+262
+00:31:46,490 --> 00:31:52,830
+لو جينا الحل المتباينة هذه في f of x
+
+263
+00:32:05,840 --> 00:32:11,920
+فهي عندي f of x minus L أصغر من إبسلون أكبر من
+
+264
+00:32:11,920 --> 00:32:17,480
+سالم إبسلون حل المتباينة هذه في f of x اجمع L على
+
+265
+00:32:17,480 --> 00:32:24,880
+كل أطراف فبطلع f of x أصغر من L زاد إبسلون أكبر من
+
+266
+00:32:24,880 --> 00:32:33,940
+L ميجا تل إبسلون فهذا معناه أن f of xbelongs to
+
+267
+00:32:33,940 --> 00:32:38,520
+the epsilon neighborhood للعدد L اللي هو الفترة
+
+268
+00:32:38,520 --> 00:32:44,040
+المفتوحة from L negative epsilon إلى L plus
+
+269
+00:32:44,040 --> 00:32:57,040
+epsilon مظبوط؟ صحيح؟ وبالتالي إن هذا بيقود إلى
+
+270
+00:32:57,040 --> 00:32:59,780
+المتيجة التالية
+
+271
+00:33:06,460 --> 00:33:20,660
+دع الـ F تعمل من A إلى R وC مقاومة مقاومة من A
+
+272
+00:33:20,660 --> 00:33:32,220
+ثم التقالير ممتازة التقالير ممتازة
+
+273
+00:33:36,480 --> 00:33:43,660
+Limit f of x as x tends to c بساوية عدد delta اللي
+
+274
+00:33:43,660 --> 00:33:51,460
+هو تعريف epsilon delta هذا معناه أنه لكل epsilon
+
+275
+00:33:51,460 --> 00:33:57,640
+أكبر من السفر يوجد delta تعتمد على epsilon عدد
+
+276
+00:33:57,640 --> 00:33:58,180
+موجد
+
+277
+00:34:03,020 --> 00:34:10,300
+Such that لو كان x ينتمي ل a و absolute x minus c
+
+278
+00:34:10,300 --> 00:34:16,340
+أصغر من delta أكبر من ستر فهذا لازم يتضمن أن
+
+279
+00:34:16,340 --> 00:34:23,580
+absolute f of x minus L أصغر من أصغر يعني معنى أخر
+
+280
+00:34:23,580 --> 00:34:29,200
+L هي limit لل function f of x هو بتالي تعريف
+
+281
+00:34:29,200 --> 00:34:33,300
+epsilon delta متحقفالان هذا تعريف epsilon دلتا
+
+282
+00:34:33,300 --> 00:34:42,000
+بكاذب ال neverhood definition ال
+
+283
+00:34:42,000 --> 00:34:54,940
+neverhood definition of limit وهو
+
+284
+00:34:54,940 --> 00:34:57,720
+ان for every
+
+285
+00:35:02,320 --> 00:35:06,700
+for every epsilon
+
+286
+00:35:06,700 --> 00:35:12,480
+neighborhood V
+
+287
+00:35:12,480 --> 00:35:22,920
+epsilon of L there exists delta neighborhood V
+
+288
+00:35:22,920 --> 00:35:30,280
+delta of C بحيث
+
+289
+00:35:32,120 --> 00:35:46,700
+إن لو كانت X تنتمي إلى A ومختلفة عن الـC وموجودة
+
+290
+00:35:46,700 --> 00:35:53,200
+أيضًا في الـDelta نبرهود لـC فلازم هذا يقدر إن
+
+291
+00:35:53,200 --> 00:36:01,600
+صورة X لازم تنتمي للـY نبرهود لـA
+
+292
+00:36:06,140 --> 00:36:13,440
+و هذا بالظبط عملنا اخر remark، prove it
+
+293
+00:36:13,440 --> 00:36:19,240
+follows from
+
+294
+00:36:19,240 --> 00:36:32,800
+above remark write
+
+295
+00:36:32,800 --> 00:36:33,380
+it down
+
+296
+00:36:40,630 --> 00:36:44,690
+حاولوا تكتبوا البرهان، البرهان فسرناها هنا
+
+297
+00:36:44,690 --> 00:37:00,590
+واضحناها من ال remark خلينا نشوف خلينا
+
+298
+00:37:00,590 --> 00:37:10,530
+نرسم رسمها في المحور Xنحو الـ y وهي ال origin وخفض
+
+299
+00:37:10,530 --> 00:37:16,270
+انه يوجد function مثل هذه فهذا بالحقيقة function y
+
+300
+00:37:16,270 --> 00:37:23,070
+بسهولة f of x وهي c نقطة في المجال تبع الدالة او
+
+301
+00:37:23,070 --> 00:37:34,330
+حتى لو ماكنتش موجودة c is the cluster point وهي
+
+302
+00:37:34,330 --> 00:37:35,870
+هذا عدد حقيقي
+
+303
+00:37:38,510 --> 00:37:44,570
+فده عدد حقيقي فمعنى
+
+304
+00:37:44,570 --> 00:37:50,770
+ان limit لل F and X بالساوية C بالساوية L معناه
+
+305
+00:37:50,770 --> 00:37:57,210
+لأي أبسلون أكبر من السفر أي لأي أبسلون أكبر من
+
+306
+00:37:57,210 --> 00:38:24,500
+السفر ممكن أناأكول epsilon neighborhood لأي
+
+307
+00:38:24,500 --> 00:38:33,180
+epsilon أكبر من 0 يعني بقدر أكوّنإبسلون نبرهود
+
+308
+00:38:33,180 --> 00:38:37,660
+بإبسلون
+
+309
+00:38:37,660 --> 00:38:44,180
+لإل فلو كان هذا given given إبسلون يعني given
+
+310
+00:38:44,180 --> 00:38:52,920
+إبسلون نبرهود لإل بقدر أجيل
+
+311
+00:38:52,920 --> 00:38:56,200
+أرد عليه
+
+312
+00:39:01,810 --> 00:39:07,770
+الدلتا دلتا neighborhood هذا عبارة عن delta
+
+313
+00:39:07,770 --> 00:39:15,250
+neighborhood ل C إذا انا أخدت اعطتوني إبسلون بقدر
+
+314
+00:39:15,250 --> 00:39:20,570
+أكون إبسلون neighborhood ل L فبقدر أرد عليه ال
+
+315
+00:39:20,570 --> 00:39:24,110
+delta neighborhood ل C في الفترة المفتوحة هذه
+
+316
+00:39:25,810 --> 00:39:31,550
+بالتالي، إذا كان هناك مجتمع إبسلن لـL، فهناك مجتمع
+
+317
+00:39:31,550 --> 00:39:38,230
+Delta لـC أو هناك مجتمع Delta عدد موجب بحيث إن لو
+
+318
+00:39:38,230 --> 00:39:42,930
+كانت الـX موجودة في A، و المتباين هذا تتحقق، طب X
+
+319
+00:39:42,930 --> 00:39:47,550
+موجودة في A، و المتباين هذا تتحقق، معناته X موجودة
+
+320
+00:39:47,550 --> 00:39:51,990
+في A و مختلفة عن الـC، و X موجودة في الـDelta
+
+321
+00:39:51,990 --> 00:39:57,400
+neighborhoodهذه معناته X تنتمي لـA ومختلفة عن الـC
+
+322
+00:39:57,400 --> 00:40:02,480
+و X موجودة في الـDelta نبرهود هذا الشرط هذا بيقدّي
+
+323
+00:40:02,480 --> 00:40:08,760
+أن المتباين هذا تتحقق المتباين هذا تتحقق معناه أن
+
+324
+00:40:08,760 --> 00:40:14,840
+ال F of X صورة X تنتمي للـY نبرهود لـB فهو واضح أن
+
+325
+00:40:14,840 --> 00:40:20,020
+هذا التعريف بيقدّي للتعريف هذا باستخدام تفسير ال
+
+326
+00:40:20,020 --> 00:40:31,340
+remarkو العكس طبعا صحيح .. صحيح okay تمام؟ اذا هذا
+
+327
+00:40:31,340 --> 00:40:36,270
+بنسميه ال .. هذا التعريفبنسمي الـ neighborhood
+
+328
+00:40:36,270 --> 00:40:40,630
+definition للـ limit of a function والتعريف دا أو
+
+329
+00:40:40,630 --> 00:40:45,810
+هذا بنسمي الـ epsilon delta definition
+
+330
+00:40:45,810 --> 00:40:52,930
+of the limit of a function تمام؟ وهذا طبعا زي ال
+
+331
+00:40:52,930 --> 00:40:57,490
+epsilon capital N definition للlimit of a sequence
+
+332
+00:40:57,490 --> 00:41:00,870
+وبعد هي فكرين أعرفنا ال neighborhood definition
+
+333
+00:41:00,870 --> 00:41:05,640
+للlimit of a sequenceهذا يعني يكافئ الكلام اللي
+
+334
+00:41:05,640 --> 00:41:09,720
+هنا إذا الأن في عندي تعريفين لل limit of a
+
+335
+00:41:09,720 --> 00:41:14,060
+function at a point أو at a cluster point الدارش
+
+336
+00:41:14,060 --> 00:41:17,080
+التعريف اللي هنستخدمه أكتر هو epsilon delta
+
+337
+00:41:17,080 --> 00:41:23,580
+definition of the limit أكتر من ال neighborhood
+
+338
+00:41:23,580 --> 00:41:26,920
+definition لكن أنا ممنعش أن أنا في أوقات معينة
+
+339
+00:41:26,920 --> 00:41:30,560
+أستخدم ال neighborhood definition طيب ناخد بعض
+
+340
+00:41:30,560 --> 00:41:38,200
+الأمثلةعلى كيفية إثبات إن الـ limit لـ certain
+
+341
+00:41:38,200 --> 00:41:43,980
+function is a certain number by
+
+342
+00:41:43,980 --> 00:41:49,020
+using epsilon delta definition نشوف مع بعض، وهذا
+
+343
+00:41:49,020 --> 00:41:54,100
+يعني بوازي الكلام اللي عملناه بالنسبة لل limits of
+
+344
+00:41:54,100 --> 00:42:00,240
+sequencesفإذا هنا في الامثلة في كل الامثلة التالية
+
+345
+00:42:00,240 --> 00:42:04,500
+عايزين نستخدم ال definition of أو epsilon delta
+
+346
+00:42:04,500 --> 00:42:07,520
+definition أو ال neighborhood definition لل limit
+
+347
+00:42:07,520 --> 00:42:10,720
+of a function to prove إنه limit عن نقطة محددة
+
+348
+00:42:10,720 --> 00:42:18,760
+بساوي عدد محدد فمثلا إن نثبت إنه limit لدالة ثابت
+
+349
+00:42:18,760 --> 00:42:22,240
+B لما X تقولها C بساوي B
+
+350
+00:42:25,710 --> 00:42:30,410
+فلت هنا عندي فان الـ function تبعتي f of x بالساوي
+
+351
+00:42:30,410 --> 00:42:38,130
+ثابت بي لكل x تنتمي إلى R فان الـ function تبعتي
+
+352
+00:42:38,130 --> 00:42:43,130
+ده اللي ثابتة وبالتالي اذا هنا لثبات ان ال limit
+
+353
+00:42:43,130 --> 00:42:45,290
+تبعتها بالساوي بي
+
+354
+00:42:48,460 --> 00:42:50,340
+أكبر من السفر أكبر من السفر أكبر من السفر أكبر من
+
+355
+00:42:50,340 --> 00:42:50,340
+السفر أكبر من السفر أكبر من السفر أكبر من السفر
+
+356
+00:42:50,340 --> 00:42:50,500
+أكبر من السفر أكبر من السفر أكبر من السفر أكبر من
+
+357
+00:42:50,500 --> 00:42:51,200
+السفر أكبر من السفر أكبر من السفر أكبر من السفر
+
+358
+00:42:51,200 --> 00:42:52,980
+أكبر من السفر أكبر من السفر أكبر من السفر أكبر من
+
+359
+00:42:52,980 --> 00:42:53,500
+السفر أكبر من السفر أكبر من السفر أكبر من السفر
+
+360
+00:42:53,500 --> 00:42:54,560
+أكبر من السفر أكبر من السفر أكبر من السفر أكبر من
+
+361
+00:42:54,560 --> 00:42:55,760
+السفر أكبر من السفر أكبر من السفر أكبر من السفر
+
+362
+00:42:55,760 --> 00:43:05,680
+أكبر من السفر أكبر من السفر أكبر من السفر أكبر
+
+363
+00:43:05,680 --> 00:43:06,180
+من السفر
+
+364
+00:43:18,680 --> 00:43:23,060
+تعالى نشوف ال implication ال delta هذه works ولا
+
+365
+00:43:23,060 --> 00:43:27,560
+لأ فانا عندي ان لو كانت ال X تنتمي ل A طبعا ال A
+
+366
+00:43:27,560 --> 00:43:31,700
+مجال الدالة هنا هو كل العدالة الحقيقية و absolute
+
+367
+00:43:31,700 --> 00:43:38,610
+X minus C أكبر من 0 أصغر من Deltaهل هذا بيقدر
+
+368
+00:43:38,610 --> 00:43:43,910
+لأبسليوت f of x minus b أصغر من إبسلون بالتأكيد
+
+369
+00:43:43,910 --> 00:43:48,910
+أنا عندي f of x بالساوي بيه سالب ال limit اللي هي
+
+370
+00:43:48,910 --> 00:43:55,090
+بيه فهذا بيطلع أبسليوت السفر بيطلع سفر والسفر هذا
+
+371
+00:43:55,090 --> 00:44:02,040
+أصغر من أي إبسلون موجةإذا حصلت تعريف Epsilon Delta
+
+372
+00:44:02,040 --> 00:44:06,560
+يعني أثبتت أنه لأي Epsilon أي Delta موجبة works
+
+373
+00:44:06,560 --> 00:44:10,580
+تعمل تعطيل ال implication وبالتالي by definition
+
+374
+00:44:10,580 --> 00:44:20,140
+limit F of X as X tends to C بساوي D طيب
+
+375
+00:44:20,140 --> 00:44:27,120
+ناخد كمان مثال لو أخدت ال identity function
+
+376
+00:44:34,810 --> 00:44:40,290
+بنثبت ان limit ده identity function لما x تقول الى
+
+377
+00:44:40,290 --> 00:44:47,110
+اي عدد حقيقى c بساوي c نستخدم
+
+378
+00:44:47,110 --> 00:44:52,310
+تعريف epsilon delta let epsilon أكبر من السفر be
+
+379
+00:44:52,310 --> 00:44:59,370
+given المرة هذه بدي ارد على ال epsilon هذه ال
+
+380
+00:44:59,370 --> 00:45:05,230
+delta تعتمد عليها هختار ال deltaبساول ابسلون
+
+381
+00:45:05,230 --> 00:45:10,430
+بيختار دلتا بساول ابسلون بيختار دلتا بساول ابسلون
+
+382
+00:45:10,430 --> 00:45:10,530
+بيختار دلتا بساول ابسلون بيختار دلتا بساول ابسلون
+
+383
+00:45:10,530 --> 00:45:12,470
+بيختار دلتا بساول ابسلون بيختار دلتا بساول ابسلون
+
+384
+00:45:12,470 --> 00:45:17,090
+بيختار دلتا بساول ابسلون بيختار دلتا بساول ابسلون
+
+385
+00:45:17,090 --> 00:45:19,890
+بيختار دلتا بساول ابسلون بيختار دلتا بساول ابسلون
+
+386
+00:45:19,890 --> 00:45:23,450
+بيختار دلتا بساول ابسلون بيختار دلتا بساول ابسلون
+
+387
+00:45:23,450 --> 00:45:29,320
+بيختار دلتا بساول ابسلون بيختار دلتهي عبارة عن ال
+
+388
+00:45:29,320 --> 00:45:33,500
+identity function لكل X المجال تبعها كل الأعداد
+
+389
+00:45:33,500 --> 00:45:40,060
+الحقيقية فلو كانت X تنتمي ل A اللي هي R و Absolute
+
+390
+00:45:40,060 --> 00:45:46,100
+X minus C أكبر من Zero أصغر من Delta فخلينا نشوف
+
+391
+00:45:46,100 --> 00:45:52,880
+هل بيطلع Absolute F of X minus ال L اللي هو C أصغر
+
+392
+00:45:52,880 --> 00:45:56,240
+من Epsilon هنشوف
+
+393
+00:45:57,800 --> 00:46:04,860
+طيب نعوض عن F of X بالساوي X minus C طب أنا عند ال
+
+394
+00:46:04,860 --> 00:46:10,320
+X هذه موجودة في R و المسافة بينها و مختلفة عن ال C
+
+395
+00:46:10,320 --> 00:46:13,880
+و المسافة بينهم أصغر من Delta اللي أنا باخدها
+
+396
+00:46:13,880 --> 00:46:19,420
+بالساوي Y إذا ال absolute X minus C من هنا أصغر من
+
+397
+00:46:19,420 --> 00:46:27,090
+Delta اللي هي Yوبالتالي درهين أثبتت أنه لأي إبسلون
+
+398
+00:46:27,090 --> 00:46:32,990
+يوجد دلتا اللي هي إبسلون نفسها بحيث لكل x في a إذا
+
+399
+00:46:32,990 --> 00:46:36,570
+كان absolute x minus c أكبر من صفر أصغر من دلتا
+
+400
+00:46:36,570 --> 00:46:40,490
+هذا بيقدر أن absolute f of x minus l أصغر من
+
+401
+00:46:40,490 --> 00:46:47,650
+إبسلون since إبسلون أكبر من الصفر was arbitrary
+
+402
+00:46:52,350 --> 00:47:00,830
+we get from the definition هذا الكلام صحيح لكل
+
+403
+00:47:00,830 --> 00:47:06,690
+epsilon وبالتالي by definition بطلع عندي limit ال
+
+404
+00:47:06,690 --> 00:47:10,430
+function f of x اللي هي ال identity function لما x
+
+405
+00:47:10,430 --> 00:47:19,750
+تقوى ل c بساوي c وهو المطلوب تمام okay إذا المرة
+
+406
+00:47:19,750 --> 00:47:27,480
+الجايةهنثبت ان limited ده لتربعية لما x او ل c
+
+407
+00:47:27,480 --> 00:47:33,280
+بساوي c تربية وهذا موجود طبعا في الكتاب وفي كمان
+
+408
+00:47:33,280 --> 00:47:37,780
+أمثلة أخرى فارجو أنكم تقرؤوا الأمثلة هذه من الكتاب
+
+409
+00:47:37,780 --> 00:47:44,350
+و تحضروها للمحاضرة الجايةوتشوفوا كيف تم استخدام
+
+410
+00:47:44,350 --> 00:47:49,410
+تعريف epsilon delta في اثبات ان ال limit لدالة زهر
+
+411
+00:47:49,410 --> 00:47:53,530
+الدالة التربعية بساوي C تربية عند اي نقطة C okay
+
+412
+00:47:53,530 --> 00:47:58,270
+تمام؟ في اي سؤال او افسار؟ اذا نكتفي بهذا القدر
+
+413
+00:47:58,270 --> 00:48:02,410
+وان شاء الله اللي انا تكمله في المحاضرة القادمة
+
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/7KfEZYA9kIA_postprocess.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/7KfEZYA9kIA_postprocess.srt
new file mode 100644
index 0000000000000000000000000000000000000000..8579b3b6d2387280d319113a30c60ff472a8377d
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/7KfEZYA9kIA_postprocess.srt
@@ -0,0 +1,1592 @@
+1
+00:00:21,840 --> 00:00:28,120
+المحاضرة اللى فاتت بدينا في عرض بعض ال
+
+2
+00:00:28,120 --> 00:00:32,760
+applications of the soprano property و بعتقد ان
+
+3
+00:00:32,760 --> 00:00:37,680
+احنا اخدنا اول مثال اللى هو المثال هذا مظبوط
+
+4
+00:00:37,680 --> 00:00:40,980
+فقولنا
+
+5
+00:00:40,980 --> 00:00:46,160
+انه المثال هذا لو اخدت اى bounded set bounded
+
+6
+00:00:46,160 --> 00:00:56,150
+above وعرفت المجموع a زاد sبالطريقة هذه فأثبتنا
+
+7
+00:00:56,150 --> 00:01:00,770
+وممكن بسهولة أثبات أن ال supremum للمجموعة الجديدة
+
+8
+00:01:00,770 --> 00:01:09,870
+A plus S بتساوي A plus ال supremum ل S وشوفنا
+
+9
+00:01:09,870 --> 00:01:15,630
+البرهان بالتفصيل المرة اللي فاتت وكان هنا البرهان
+
+10
+00:01:15,630 --> 00:01:18,390
+بعتمد على أن ال 6 اللي bounded above
+
+11
+00:01:32,250 --> 00:01:36,230
+الست اس هى dependent above لإن السوبرمان تباعها
+
+12
+00:01:36,230 --> 00:01:42,940
+exist by the superman propertyوشوفنا بعد هيك أنه
+
+13
+00:01:42,940 --> 00:01:49,660
+ال .. العدد a زائد u بيطلع upper bound للست هذه و
+
+14
+00:01:49,660 --> 00:01:53,500
+بعدين أثبتنا أن هذا العدد هو أصغر upper bound أو
+
+15
+00:01:53,500 --> 00:01:59,320
+supremum للست هذه وبالتالي هيك بنكون أثبتنا أن
+
+16
+00:01:59,320 --> 00:02:03,760
+supremum للست هذه موجود و بيساوي العدد a زائد u
+
+17
+00:02:03,760 --> 00:02:09,420
+اللي هو a زائد supremum S المثال التاني
+
+18
+00:02:16,520 --> 00:02:20,320
+لو أخدت two functions المجال ال domain تبعهم
+
+19
+00:02:20,320 --> 00:02:25,300
+مجموعة D subset من R وكتبت
+
+20
+00:02:25,300 --> 00:02:29,280
+F of D علي أنها مجموعة كل العناصر F of X حيث و X
+
+21
+00:02:29,280 --> 00:02:34,400
+متملة D فال set F of D هذه هي ال range تبع ال
+
+22
+00:02:34,400 --> 00:02:39,120
+function F صح؟ هي المدى تبع ال function F و كذلك
+
+23
+00:02:39,120 --> 00:02:46,000
+ال set G of D هي ال range تبع ال function G
+
+24
+00:02:48,510 --> 00:02:53,250
+فلو فرضنا ان ال set f of d و ال set g of d bounded
+
+25
+00:02:53,250 --> 00:03:01,530
+set R فطبعا حسب ال supremum property المجموعات دول
+
+26
+00:03:01,530 --> 00:03:06,430
+كل واحدة لها supremum كذلك حسب ال infimum property
+
+27
+00:03:07,290 --> 00:03:11,050
+المجمعتين هذول كل واحدة فيهم إلها infimum، الـ
+
+28
+00:03:11,050 --> 00:03:15,350
+infimum تبعهم exist إذا نفرض إن المجمعتين هذول
+
+29
+00:03:15,350 --> 00:03:18,570
+bounded عشان إيه نضمن وجود ال suprem والinfimum
+
+30
+00:03:18,570 --> 00:03:26,450
+لكل واحدة منهم الآن في عندي بدي أبرن حاجة تانية لو
+
+31
+00:03:26,450 --> 00:03:31,930
+كان الفرض f of x أصغر من أو ساوي g of x بتحقق لكل
+
+32
+00:03:31,930 --> 00:03:38,040
+x ديفبطلع ال supremum للمجموعة F of D بطلع أصغر من
+
+33
+00:03:38,040 --> 00:03:44,660
+لو ساوي ال supremum للمجموعة G of D وبرهان هذا
+
+34
+00:03:44,660 --> 00:03:54,220
+البرهان يعني سهل أنا كاتب انه easy exercise لكن
+
+35
+00:03:54,220 --> 00:04:02,780
+ممكن تبرهنه ممكن تبرهنه كل سهولة فهي نكتب ال proof
+
+36
+00:04:06,320 --> 00:04:14,320
+of part one للجزء الأول فخلّينا
+
+37
+00:04:14,320 --> 00:04:19,400
+نثبت fix x
+
+38
+00:04:19,400 --> 00:04:29,400
+ينتمي إلى d ناخد عنصر x ينتمي إلى d عشوائي by
+
+39
+00:04:29,400 --> 00:04:31,240
+hypothesis من الفرض
+
+40
+00:04:33,710 --> 00:04:40,970
+من الفرض أنا عندي f of x أصغر من أو يساوي g of x
+
+41
+00:04:40,970 --> 00:04:52,470
+لل x هذه و لأي x دي صح هذا من الفرض و g of x g of
+
+42
+00:04:52,470 --> 00:05:00,550
+x أصغر من أو يساوي ال supremum لل set g of d
+
+43
+00:05:04,610 --> 00:05:14,410
+طبعا هذا زي ما قلنا exist by supremum property
+
+44
+00:05:14,410 --> 00:05:20,970
+باستخدام خاصية التمام ال
+
+45
+00:05:20,970 --> 00:05:26,910
+supremum .. هذا .. هذا عنصر في set هذا g of x عنصر
+
+46
+00:05:26,910 --> 00:05:32,550
+في set g of d صح؟وهذا upper bound ال suprem ل g of
+
+47
+00:05:32,550 --> 00:05:38,690
+d و هذا عنصر في ال 6 g of d فهذا أكيد أزم أو ساوي
+
+48
+00:05:38,690 --> 00:05:43,610
+ال upper bound لل 6 اللي بينتمي إليها فهذا صحيح
+
+49
+00:05:43,610 --> 00:05:56,610
+الآن هذا صحيح لكل x since x belonged to D was
+
+50
+00:05:56,610 --> 00:05:57,610
+arbitrarily
+
+51
+00:06:03,450 --> 00:06:10,110
+arbitrary ان ان بطلع اندي F of X أصغر من أو ساوي
+
+52
+00:06:10,110 --> 00:06:20,490
+ال supremum ل G of D وهذا صحيح لكل X في D هذا
+
+53
+00:06:20,490 --> 00:06:29,900
+معناه انه العدد هذاهذا العدد أكبر من أو ساوي كل
+
+54
+00:06:29,900 --> 00:06:36,960
+عناصر ال set F of D صح؟ هاي هذا معناه أن ال
+
+55
+00:06:36,960 --> 00:06:47,600
+supremum ل set G of D is an upper bound an upper
+
+56
+00:06:47,600 --> 00:06:50,860
+bound
+
+57
+00:06:50,860 --> 00:06:53,780
+لمين؟
+
+58
+00:06:54,920 --> 00:07:01,100
+of set f of d بصح؟
+
+59
+00:07:01,100 --> 00:07:07,040
+لأن هيكل أنصر f of x في f of d أصغر من أول ساول
+
+60
+00:07:07,040 --> 00:07:18,980
+عدد هذا، صح؟ طيب since ال supremum لset f of d
+
+61
+00:07:18,980 --> 00:07:25,890
+existin r طبعا برضه by supremum property لأن احنا
+
+62
+00:07:25,890 --> 00:07:31,890
+فرضين ان ال set هذه bounded صح فال supremum تبعها
+
+63
+00:07:31,890 --> 00:07:37,110
+موجود الآن ال set هذه ال supremum تبعها موجود
+
+64
+00:07:37,110 --> 00:07:42,750
+والعدد هذا هذا العدد عبارة عن upper bound ل set
+
+65
+00:07:42,750 --> 00:07:46,850
+إذا ما العلاقة بين ال upper bound هذا لل set وال
+
+66
+00:07:46,850 --> 00:07:53,650
+supremum لل setفي واحد اكبر من او ساوى التاني لان
+
+67
+00:07:53,650 --> 00:07:59,770
+بما ان هذا الكلام صحيح نحن نحن نحن نحن نحن نحن نحن
+
+68
+00:07:59,770 --> 00:08:01,050
+نحن نحن نحن نحن نحن نحن نحن نحن نحن نحن نحن نحن
+
+69
+00:08:01,050 --> 00:08:01,350
+نحن نحن نحن نحن نحن نحن نحن نحن نحن نحن نحن نحن
+
+70
+00:08:01,350 --> 00:08:04,050
+نحن نحن نحن نحن نحن نحن نحن نحن نحن نحن نحن نحن
+
+71
+00:08:04,050 --> 00:08:05,880
+نحن نحن نحن نحن نحن نحن نحن نحن نحن نحن نحنهذا
+
+72
+00:08:05,880 --> 00:08:10,820
+أصغر upper bound للست f of d وهذا upper bound للست
+
+73
+00:08:10,820 --> 00:08:15,440
+f of d إذا ال supremum بيطلع أصغر من أو ساوي ال
+
+74
+00:08:15,440 --> 00:08:22,480
+upper bound اللي هو supremum g of d وهو المطلوب
+
+75
+00:08:22,480 --> 00:08:29,800
+وهذا بثبت الجزء الأول okay تمام إذا الجزء الأول مش
+
+76
+00:08:29,800 --> 00:08:33,680
+صعب وهنا أثبتنا واضح
+
+77
+00:08:37,050 --> 00:08:42,310
+برهان الجزء التاني برضه شبيه فيه الجزء التاني، إيش
+
+78
+00:08:42,310 --> 00:08:47,510
+بيقول ليه؟ الفرض، لاحظوا الفرق بين الفرض تبع الجزء
+
+79
+00:08:47,510 --> 00:08:54,910
+التاني والجزء الأول الفرض
+
+80
+00:08:54,910 --> 00:09:00,210
+هنا إن f of x أصغر من أو ساوي g of y لكل x و y في
+
+81
+00:09:00,210 --> 00:09:00,450
+D
+
+82
+00:09:04,010 --> 00:09:09,170
+هذا أشمل وهذا أعمل من هذا وهذا أقوى لتاعتي لاحظوا
+
+83
+00:09:09,170 --> 00:09:14,690
+إنه لو هذا صح فهذا بيطلع صح اللي فور لكن الأكس مش
+
+84
+00:09:14,690 --> 00:09:18,430
+صحيح طيب
+
+85
+00:09:18,430 --> 00:09:22,130
+إذا .. إذا هذا الكلام صحيح فهذا بيقدي إن ال
+
+86
+00:09:22,130 --> 00:09:26,410
+supremum ل F of D بيطلع أصغر من أو ساوى الinfimum
+
+87
+00:09:26,410 --> 00:09:31,110
+لست G of D نشوف
+
+88
+00:09:31,110 --> 00:09:32,710
+ال .. نبرهن الكلام هذا
+
+89
+00:09:50,270 --> 00:10:02,090
+البرهان الجزء التاني البرهان
+
+90
+00:10:02,090 --> 00:10:05,030
+الجزء التاني هذا conditional statement هاي الفرض
+
+91
+00:10:05,030 --> 00:10:11,370
+وهي النتيجة ال conclusion فبنفرض ان الفرض هذا صحيح
+
+92
+00:10:11,370 --> 00:10:23,770
+و بنثبت يثبت يثبت أنصر Y في Dمن الفرض بيطلع عندي f
+
+93
+00:10:23,770 --> 00:10:29,530
+of x أصغر من أو ساوي g of y وهذا صحيح لكل x في دي
+
+94
+00:10:29,530 --> 00:10:38,280
+و ال y ثابت يعني هذا من الفرض صحيح لكل x في ديطيب،
+
+95
+00:10:38,280 --> 00:10:45,040
+الان هذا معناه ان العدد هذا g of y هي في y أنصة
+
+96
+00:10:45,040 --> 00:10:49,600
+ثابت هي أكبر .. هذا العدد أكبر من او ساوي كل ال F
+
+97
+00:10:49,600 --> 00:10:54,100
+of X لكل X دي معناه هذا upper bound لل set F of D
+
+98
+00:10:54,100 --> 00:10:59,020
+الان g of y عبارة عن upper bound لل set F of D من
+
+99
+00:10:59,020 --> 00:11:01,860
+هنا، مظبوط؟ تمام؟
+
+100
+00:11:04,040 --> 00:11:07,840
+وبالتالي ال least upper bound ل F of D بيطلع أصلا
+
+101
+00:11:07,840 --> 00:11:12,080
+أو ساوي ال upper bound ل F of D اللي هو G of Y لأن
+
+102
+00:11:12,080 --> 00:11:13,620
+هذه المتباينة صحيحة
+
+103
+00:11:18,050 --> 00:11:22,770
+اخترناها was arbitrary fixed احنا اخترناها عشوائي
+
+104
+00:11:22,770 --> 00:11:27,470
+arbitrary وثبتناها ان الكلام المتبين هذه الآن صحيح
+
+105
+00:11:27,470 --> 00:11:33,110
+لكل y ان المتبين هذه صحيحة true for every y في D
+
+106
+00:11:33,110 --> 00:11:39,510
+هذا معناه من المتبين هذه ال percentage انه العدد
+
+107
+00:11:39,510 --> 00:11:45,350
+ال supremum ل F of Dهذا عبارة عن lower bound
+
+108
+00:11:45,350 --> 00:11:51,030
+لمجموعة العناصر g of y حيث y ينتمي ل d يعني العدد
+
+109
+00:11:51,030 --> 00:11:58,210
+هذا عبارة عن lower bound للست g of d عظيم صح؟ طيب
+
+110
+00:11:58,210 --> 00:12:04,230
+ال inform ل g of d existو هذا العدد lower bound
+
+111
+00:12:04,230 --> 00:12:08,950
+للست هذه و ال infimum هذا عبارة عن ال greatest
+
+112
+00:12:08,950 --> 00:12:12,970
+lower bound ل G و D إذا ال greatest lower bound
+
+113
+00:12:12,970 --> 00:12:18,810
+دايما بيكون أكبر من أو يساوي أي lower bound إذا ال
+
+114
+00:12:18,810 --> 00:12:23,090
+lower bound هذا أصغر من أو يساوي ال greatest lower
+
+115
+00:12:23,090 --> 00:12:28,610
+bound ل G و D و هذا اللي هو هذا النتيجة اللي احنا
+
+116
+00:12:28,610 --> 00:12:34,800
+عايزين نصل لها okay تمامواضح؟ إذن هذا برهاني جزء
+
+117
+00:12:34,800 --> 00:12:48,220
+التاني الآن في ملاحظة الملاحظة هذه بتقول إنه يعني
+
+118
+00:12:48,220 --> 00:12:56,120
+ممكن طالبة طلعت تسأل أو تستفسر أو تتساءل طب ما هذا
+
+119
+00:12:56,120 --> 00:13:01,400
+الشرط تبين زي هذا مافيش فرق بينهمفاحنا بنقول لأ
+
+120
+00:13:01,400 --> 00:13:05,480
+هذا الشرط التحت أقوى من اللي فوق اللي تحت لو كان
+
+121
+00:13:05,480 --> 00:13:09,160
+التحت صحيح بيقدي للي فوق لكن لو كان اللي فوق صحيح
+
+122
+00:13:09,160 --> 00:13:14,300
+هذا ما بيقدي للي تحت هذا الشرط أقوى من اللي فوق
+
+123
+00:13:14,300 --> 00:13:20,240
+فممكن واحدة فيكم تسأل تقول طب لو احنا أخدنا الفرض
+
+124
+00:13:20,240 --> 00:13:24,840
+هذا لو فرضنا أن هذا الكلام صح هل ممكن نحصل على
+
+125
+00:13:24,840 --> 00:13:30,580
+نتيجة اللي تحتفالإجابة لأ، الإجابة لأ، هذا مش
+
+126
+00:13:30,580 --> 00:13:36,900
+ممكن، إذا ال .. لو شيلنا الفرض هذا و بدلناه بالفرض
+
+127
+00:13:36,900 --> 00:13:41,820
+اللي فوق فالنتيجة هذه لا يمكن نحصل عليها، مش شرط
+
+128
+00:13:41,820 --> 00:13:53,110
+تكونصحيحة او مثال يوضح انه لا يمكن استبدال الفرض
+
+129
+00:13:53,110 --> 00:13:58,610
+تبع الجزء التاني بالفرض تبع الجزء الأول ونحصل نحصل
+
+130
+00:13:58,610 --> 00:14:00,630
+على نتيجة الجزء التاني
+
+131
+00:14:12,790 --> 00:14:16,530
+فناخد على سبيل المثال او counterexample بيسميه في
+
+132
+00:14:16,530 --> 00:14:22,910
+رياضيات لو أخدت f of x بالساوي x تربية دالة تربية
+
+133
+00:14:22,910 --> 00:14:26,830
+و g of x ال identity function و أخدت ال domain
+
+134
+00:14:26,830 --> 00:14:30,950
+المشترك لf و g ال closed unit interval
+
+135
+00:14:34,300 --> 00:14:40,040
+فطبعا بنلاحظ أن f of x اللي هي x تربية لكل x في ال
+
+136
+00:14:40,040 --> 00:14:45,220
+closed unit interval x تربية أصغر من أو ساوى x،
+
+137
+00:14:45,220 --> 00:14:51,180
+مظبوط؟و X بيساوي G of X فهي في عندي ال two
+
+138
+00:14:51,180 --> 00:14:54,460
+functions هدول بالمناسبة ال two functions هدول
+
+139
+00:14:54,460 --> 00:14:59,220
+كلاهم كلاهم bounded bounded below by zero bounded
+
+140
+00:14:59,220 --> 00:15:08,940
+above by ال range تبعهم ال range تبعهم F of D و G
+
+141
+00:15:08,940 --> 00:15:14,900
+of D ك sets كمجموعات بطلوا subset من المجموعة من
+
+142
+00:15:14,900 --> 00:15:20,520
+السفر لواحدوبالتالي كلا هما bounded above by واحد
+
+143
+00:15:20,520 --> 00:15:27,000
+و bounded below by سفره إذن
+
+144
+00:15:27,000 --> 00:15:32,420
+هذه المجموعات هي bounded وهي عند ال function f of
+
+145
+00:15:32,420 --> 00:15:36,860
+x أصغر من أو ساوي g of x لكل x دي هذا الفرض تبع
+
+146
+00:15:36,860 --> 00:15:41,600
+الجزء واحد اللي شوفناه قبل شوية لكن النتيجة تبع
+
+147
+00:15:41,600 --> 00:15:45,560
+الجزء التالي لا تتحقق تعالى نشوف هي ال supremum ل
+
+148
+00:15:45,560 --> 00:15:52,190
+f of dهي مجموعة f of d الواحد
+
+149
+00:15:52,190 --> 00:15:56,430
+أكبر
+
+150
+00:15:56,430 --> 00:16:00,090
+من السفر السفر
+
+151
+00:16:00,090 --> 00:16:05,650
+برضه عبارة عن greatest lower bound أو الانفم من
+
+152
+00:16:05,650 --> 00:16:09,950
+المجموعة هذه واضح ان السفر lower bound للسفر هذه
+
+153
+00:16:09,950 --> 00:16:15,580
+وهو greatest lower boundإذاً هي عند الـ supremum
+
+154
+00:16:15,580 --> 00:16:20,220
+لـ F of D أكبر من الـ infimum لـ G of D وهذا نفي
+
+155
+00:16:20,220 --> 00:16:23,700
+نتيجة نتيجة نتيجة نتيجة نتيجة نتيجة نتيجة نتيجة
+
+156
+00:16:23,700 --> 00:16:24,240
+نتيجة نتيجة نتيجة نتيجة نتيجة نتيجة نتيجة نتيجة
+
+157
+00:16:24,240 --> 00:16:26,120
+نتيجة نتيجة نتيجة نتيجة نتيجة نتيجة نتيجة نتيجة
+
+158
+00:16:26,120 --> 00:16:26,240
+نتيجة نتيجة نتيجة نتيجة نتيجة نتيجة نتيجة نتيجة
+
+159
+00:16:26,240 --> 00:16:26,440
+نتيجة نتيجة نتيجة نتيجة نتيجة نتيجة نتيجة نتيجة
+
+160
+00:16:26,440 --> 00:16:32,560
+نتيجة نتيجة نتيجة نتيجة نتيجة نتيجة نتيجة نتيجة
+
+161
+00:16:49,380 --> 00:16:56,900
+كنتيجة على الـ completeness property في عندي نتيجة
+
+162
+00:16:56,900 --> 00:17:05,420
+كتير مهمة و هنستخدمها كتير معناها اللي هو ال
+
+163
+00:17:05,420 --> 00:17:10,120
+material اللي هناخدها لاحقا اللي هو ال Archimedean
+
+164
+00:17:10,120 --> 00:17:16,220
+property او خاصية Archimedes ايه الخاصية هذه بتقول
+
+165
+00:17:17,950 --> 00:17:23,890
+لأي عدد حقيقي x في عدد طبيعي أكبر منه، أعطيني أي
+
+166
+00:17:23,890 --> 00:17:29,650
+عدد حقيقي x سواء كان سفر أو موجب أو سالب، بقدر
+
+167
+00:17:29,650 --> 00:17:36,970
+أعطيكي عدد طبيعي أكبر منه أو بقدر أوجدلك عدد طبيعي
+
+168
+00:17:36,970 --> 00:17:42,760
+يكون أكبر منهالبرهان تبع النظرية هذه بيعتمد على
+
+169
+00:17:42,760 --> 00:17:47,040
+الـ completeness property فلبرهان ذلك نبدأ بالـ
+
+170
+00:17:47,040 --> 00:17:54,320
+Fix X في R ونثبتها ونعمل برهان بالتناقل نحن عايزين
+
+171
+00:17:54,320 --> 00:17:58,840
+نثبت أنه للـ Fix X اللي احنا ثبتناها يوجد
+
+172
+00:18:01,850 --> 00:18:07,810
+عايزين نثبت العبارة أن العبارة هذه تكون صحيحة يوجد
+
+173
+00:18:07,810 --> 00:18:12,430
+عدد طبيعي أكبر من X فبدا أعمل برهان بالتناقض بدا
+
+174
+00:18:12,430 --> 00:18:17,610
+أفرض أن نفي العبارة هذه هو الصح إذا ن assume ال
+
+175
+00:18:17,610 --> 00:18:21,030
+contrary أن نفي العبارة هذه الصح طب نفي العبارة
+
+176
+00:18:21,030 --> 00:18:27,750
+هذه الصح there exist ما بصير لكل N في Nعكس
+
+177
+00:18:27,750 --> 00:18:32,730
+المتباينة هذه اللي هو n أصغر من أو ساوي x إذن هنا
+
+178
+00:18:32,730 --> 00:18:37,550
+ال contrary أو النفي، نفي النتيجة هذه معناها أن كل
+
+179
+00:18:37,550 --> 00:18:44,610
+الأعداد الطبيعية أصغر من أو ساوي x هذا معناه أن ال
+
+180
+00:18:44,610 --> 00:18:51,230
+x هذا upper bound لست nوبالتالي الـ sit-in إلها
+
+181
+00:18:51,230 --> 00:18:54,850
+upper bound أو bounded above، إذا by the supremum
+
+182
+00:18:54,850 --> 00:19:00,590
+أو completeness of property، الـ sit-in بطلع يوجد
+
+183
+00:19:00,590 --> 00:19:04,970
+إلها supremum، الـ supremum تبعها exist and are،
+
+184
+00:19:04,970 --> 00:19:12,410
+سميه، فلينا نسميه you، فلينا نسميه you، تمام؟ في
+
+185
+00:19:12,410 --> 00:19:19,340
+لمة واحد اتناشلما واحدة اثناء عاش كده بتقول لو كان
+
+186
+00:19:19,340 --> 00:19:28,300
+U أو U بساوي ال superman لست S if and only if لكل
+
+187
+00:19:28,300 --> 00:19:35,920
+epsilon أكبر من الصفر نقدر نلاقي S epsilon في الست
+
+188
+00:19:35,920 --> 00:19:42,460
+S بحيث انه U سالب epsilon أصغر من S epsilon
+
+189
+00:19:45,010 --> 00:19:50,110
+طب اقل انا عندى فيه U بساوية Supremum ل N S بساوية
+
+190
+00:19:50,110 --> 00:19:55,450
+6 N كل الاعداد الطبيعية هى عندى Supremum ل N اللى
+
+191
+00:19:55,450 --> 00:20:01,890
+هو U exist اذا حسب لمّة واحد اتناش لو اخدت ابسلون
+
+192
+00:20:01,890 --> 00:20:06,670
+لو اخدت ابسلون بالساوية واحد هذا عدد موجب اذا لهذا
+
+193
+00:20:06,670 --> 00:20:11,690
+الابسلون بقدر الاقي عدد S epsilon هسمي M هنا بدل S
+
+194
+00:20:11,690 --> 00:20:16,930
+epsilon فى اللمةعدد طبيعي بحيث أنه لما أخد U minus
+
+195
+00:20:16,930 --> 00:20:20,670
+epsilon اللي هو الواحد هذا بيطلع أصغر من S epsilon
+
+196
+00:20:20,670 --> 00:20:25,050
+اللي هو M إذاً هذا نحصل عليه من لمّة واحدة واثنين
+
+197
+00:20:25,050 --> 00:20:30,870
+عشر طيب المتباين هذه ودي واحد نجري واحد علي مين
+
+198
+00:20:30,870 --> 00:20:35,010
+فبيطلع U أصغر من M زياد واحد طيب ال M عدد طبيعي
+
+199
+00:20:35,010 --> 00:20:40,130
+إذاً M زياد واحد عدد طبيعي صح؟ إذاً هذا M زياد
+
+200
+00:20:40,130 --> 00:20:47,360
+واحد عدد طبيعي وأكبر من Uو U قلنا ال U هو ال
+
+201
+00:20:47,360 --> 00:20:50,520
+supremum ل N يعني upper bound بيطلع upper bound ل
+
+202
+00:20:50,520 --> 00:20:55,860
+N فكيف U upper bound ل set N للعداد الطبيعية و في
+
+203
+00:20:55,860 --> 00:20:59,620
+عنصر في العداد الطبيعية أكبر منه لأن هذا بيديني
+
+204
+00:20:59,620 --> 00:21:06,380
+تناقض لكون U هو upper bound ل set للعداد الطبيعية
+
+205
+00:21:06,380 --> 00:21:13,060
+إذا وصلنا إلى تناقض وبالتالي هذا بكمل البرهانةإذا
+
+206
+00:21:13,060 --> 00:21:16,980
+الفرض تبعنا التناقض هذا تقول إن ال assumption
+
+207
+00:21:16,980 --> 00:21:24,720
+تبعنا هذا إن الكلام هذا صح كان خطر إذا الصح نفيه
+
+208
+00:21:24,720 --> 00:21:29,480
+اللي هو المطلوب okay تمام إذا هذه ال Archimedean
+
+209
+00:21:29,480 --> 00:21:35,460
+property هذه ال Archimedean property الآن ال
+
+210
+00:21:35,460 --> 00:21:39,580
+Archimedean property هذه أو خاصية Archimedes إلها
+
+211
+00:21:39,580 --> 00:21:45,520
+صور أخرى متعددةو هذه الصور هي موجودة في كوريلري
+
+212
+00:21:45,520 --> 00:21:50,700
+واحد ستة عشر اذا
+
+213
+00:21:50,700 --> 00:21:58,060
+النتيجة هذه في ان صور اخرى ل ال Archimedean
+
+214
+00:21:58,060 --> 00:22:06,500
+property ف
+
+215
+00:22:07,840 --> 00:22:11,520
+Alternative forms يعني صور أخرى لـArchimedean
+
+216
+00:22:11,520 --> 00:22:16,520
+property، let YUZ be positive real numbers، إذن
+
+217
+00:22:16,520 --> 00:22:19,760
+YUZ تنتمي لمجموعة الأعداد الحقيقية المهمة
+
+218
+00:22:22,550 --> 00:22:28,990
+أول نتيجة يوجد n عدد طبيعي بحيث ان الـ z أصغر من n
+
+219
+00:22:28,990 --> 00:22:35,410
+مضروب في y إذا لو فيندي عددين حقيقين موجبين z وy
+
+220
+00:22:35,410 --> 00:22:39,790
+بقدر ألاقي عدد طبيعي بحيث ان ال z أصغر من n مضروب
+
+221
+00:22:39,790 --> 00:22:49,740
+في y كذلك لأي عدد حقيقي موجب yبقدر ألاقي عدد طبيعي
+
+222
+00:22:49,740 --> 00:22:54,740
+مقلوبه أصغر من العدد الموجب Y طبعا مقلوب العدد
+
+223
+00:22:54,740 --> 00:22:59,220
+الطبيعي دائما موجب كذلك
+
+224
+00:22:59,220 --> 00:23:04,820
+لأي عدد حقيقي موجب Z بقدر ألاقي عدد طبيعي بحيث أن
+
+225
+00:23:04,820 --> 00:23:09,920
+العدد الموجب Z أكبر من أو ساوي N سالب واحد وأصغر
+
+226
+00:23:09,920 --> 00:23:16,770
+من N إذن التلات خواص هدولةكل واحدة منهم بنسميها
+
+227
+00:23:16,770 --> 00:23:20,730
+Archimedean property أو صورة أخرى من ال
+
+228
+00:23:20,730 --> 00:23:25,590
+Archimedean property الجزء
+
+229
+00:23:25,590 --> 00:23:30,250
+الأخير هذا هو عبارة عن مثال وليس ال Archimedean
+
+230
+00:23:30,250 --> 00:23:37,810
+يعني هذا استثناء يعني مجرد set بالساوي ال sequence
+
+231
+00:23:37,810 --> 00:23:44,140
+واحد على nمتتالية العداد الحقيقية 1 على N حيث N
+
+232
+00:23:44,140 --> 00:23:49,540
+عدد طبيعي فال set هذه هنثبت أن ال infimum إلها هو
+
+233
+00:23:49,540 --> 00:23:59,860
+السفر طيب إذا نشوف و نثبت العزاء الأولى الجزء
+
+234
+00:23:59,860 --> 00:24:00,780
+الأول
+
+235
+00:24:06,710 --> 00:24:15,270
+الجزء A لإثبات الجزء A خلّينا نعرف X بساوي Z على Y
+
+236
+00:24:15,270 --> 00:24:19,930
+طبعا Z وY أعداد حقيقية موجبة إذن خارج قسمتهم أعداد
+
+237
+00:24:19,930 --> 00:24:26,090
+موجب إذن هذا عبارة عن عدد حقيقي موجب يعني ال X هذا
+
+238
+00:24:26,090 --> 00:24:33,170
+عبارة عن real number وموجب فحسب ال Archimedean
+
+239
+00:24:33,170 --> 00:24:42,860
+propertyلأي x عدد حقيقي يوجد عدد طبيعي أكبر من الـ
+
+240
+00:24:42,860 --> 00:24:48,000
+x إذا الـ x اللي أنا أخده z على y بقدر ألاقي عدد
+
+241
+00:24:48,000 --> 00:24:53,440
+طبيعي n أكبر منه يعني z على y أصغر من n لو ضربت
+
+242
+00:24:53,440 --> 00:25:01,550
+المتباينة هذه في yy عدد موجب فهيصير عندي z أصغر من
+
+243
+00:25:01,550 --> 00:25:08,110
+n في y وهذه هي النتيجة تبع الجزء الأول okay إذا
+
+244
+00:25:08,110 --> 00:25:13,270
+هيك يكون أثبتنا الجزء الأول واضح لإثبات الجزء
+
+245
+00:25:13,270 --> 00:25:19,410
+التاني لو أخدنا في الجزء الأول لو أخدت z بساوي
+
+246
+00:25:19,410 --> 00:25:30,500
+واحد فهيصير عندي واحد1 أصغر من n في y ال z هذا عدد
+
+247
+00:25:30,500 --> 00:25:35,780
+موجب فلو أخد ال z بالساعة واحد هذا عدد موجب فحسب
+
+248
+00:25:35,780 --> 00:25:41,420
+النتيجة a بيطلع عند z أصغر من n يوجد عدد طبيعي n
+
+249
+00:25:41,420 --> 00:25:48,080
+بحيث ان z أصغر من ny يعني 1 أصغر من ny الآن نضرب
+
+250
+00:25:48,080 --> 00:25:53,910
+في 1 على n 1 على n عدد موجب لو ضربنا الطرفينبالعدد
+
+251
+00:25:53,910 --> 00:25:57,850
+الموجة بواحد علينا بيطلع واحد علينا أصغر من Y وهذا
+
+252
+00:25:57,850 --> 00:26:01,330
+اللي احنا عايزينه تمام ان هذا برهان الجزء التاني
+
+253
+00:26:01,330 --> 00:26:14,310
+لبرهان الجزء التالت الجزء
+
+254
+00:26:14,310 --> 00:26:14,730
+C
+
+255
+00:26:18,400 --> 00:26:23,700
+بنثبت أنه لأي عدد حقيقي موجب Z فيه عدد طبيعي بحيث
+
+256
+00:26:23,700 --> 00:26:30,940
+أن Z محصورة بين N سارب واحد و M تمام نعرف الست EZ
+
+257
+00:26:30,940 --> 00:26:36,380
+على إنها كل الأعداد الطبيعية M اللي بتكون أكبر من
+
+258
+00:26:36,380 --> 00:26:46,880
+Z الآن هذه المجموعة غير خالية لأنه
+
+259
+00:26:51,070 --> 00:26:57,610
+لأن الـ z هذا عدد موجب وبالتالي في الآخر هو عدد
+
+260
+00:26:57,610 --> 00:27:01,950
+حقيقي فby Archimedean property
+
+261
+00:27:10,880 --> 00:27:17,220
+اللي هي 115 رقمها نظرية 115 بتقول أي عدد حقيقي z
+
+262
+00:27:17,220 --> 00:27:26,880
+يوجد عدد .. يوجد عدد طبيعي يوجد m في n بحيث ان z
+
+263
+00:27:26,880 --> 00:27:32,820
+أصغر من n اذا
+
+264
+00:27:32,820 --> 00:27:42,120
+المجموعة هذهعلى الأقل فيها وانصر واحد اللي هو الـ
+
+265
+00:27:42,120 --> 00:27:49,100
+M هذا او خليني اسميه MZ تمام
+
+266
+00:27:49,100 --> 00:27:58,000
+الـ Archimedean property تضمن انه للعدد Z هذا اللي
+
+267
+00:27:58,000 --> 00:28:05,100
+هو يعني احنا فرضين ان العدد موجبالـ set هذه بقدر
+
+268
+00:28:05,100 --> 00:28:10,460
+ألاقي عدد طبيعي MZ أكبر من Z وبالتالي المجموعة هذه
+
+269
+00:28:10,460 --> 00:28:15,580
+تحتوي تحتوي على العنصر هذا على الأقل لأن هذه
+
+270
+00:28:15,580 --> 00:28:22,720
+مجموعة جار خالية واضحة النقطة هذه؟ الآن في خاصية
+
+271
+00:28:22,720 --> 00:28:29,920
+الترتيب أو بنسميها ال will ordering propertyوهذه
+
+272
+00:28:29,920 --> 00:28:34,400
+في الحقيقة بتدرسها في نهاية في أخر chapter في
+
+273
+00:28:34,400 --> 00:28:40,640
+مبادئ رياضيات ال will ordering property بتقول ان
+
+274
+00:28:40,640 --> 00:28:46,240
+every non-empty subset of N has a least element
+
+275
+00:28:46,240 --> 00:28:51,020
+يعني أي مجموعة غير خالية من مجموعة الأعداد
+
+276
+00:28:51,020 --> 00:28:55,880
+الطبيعية لازم اللي جي لها least element لازم يكون
+
+277
+00:28:55,880 --> 00:29:00,520
+لها أصغر عنصريعني خدي انت على الجربة حتى خدي اي
+
+278
+00:29:00,520 --> 00:29:04,060
+مجموعة جزئية من العدالة الطبيعية هتجد ان فيها أنصر
+
+279
+00:29:04,060 --> 00:29:08,620
+فيها هو أصغر أنصر فهذا طبعا حسب ال will ordering
+
+280
+00:29:08,620 --> 00:29:12,880
+property يعني درس المبادئ و انا شخصيا لما بدرس
+
+281
+00:29:12,880 --> 00:29:16,400
+مبادئ بحاول يعني امر عليها او اعطيها حتى لو يعني
+
+282
+00:29:16,400 --> 00:29:21,620
+بصورة مختصرة بقرابش الناس التانية لما بدرسوا
+
+283
+00:29:21,620 --> 00:29:25,340
+المبادئ بعتقد ممكن موصلوش اليها لكن مش مشكلة هاي
+
+284
+00:29:25,340 --> 00:29:26,400
+نحن بنحكيلكم عنها
+
+285
+00:29:29,700 --> 00:29:35,480
+إذا هي عندي هذه عبارة عن subset من مجموعة الأعداد
+
+286
+00:29:35,480 --> 00:29:40,060
+الطبيعية و non-empty إذا لازم يكون في إلها least
+
+287
+00:29:40,060 --> 00:29:45,640
+element إذا بقدر ألاقي NZ في مجموعة الأعداد
+
+288
+00:29:45,640 --> 00:29:49,300
+الطبيعية و هذا ال NZ هو least element لل set هذه
+
+289
+00:29:49,300 --> 00:29:56,530
+الغير خالية okay تمامإذا هنا يوجد أنصر nz عدد
+
+290
+00:29:56,530 --> 00:30:02,390
+طبيعي وهذا العدد الطبيعي هو ال least element ل
+
+291
+00:30:02,390 --> 00:30:09,530
+easy طيب
+
+292
+00:30:09,530 --> 00:30:17,350
+الآن هذا أصغر أنصر في الست هذه يعني معناه nz لو
+
+293
+00:30:17,350 --> 00:30:25,080
+طرحت من nzطرحت منها واحد فطبعا هذا أصغر من NZ هذا
+
+294
+00:30:25,080 --> 00:30:34,920
+أصغر من NZ صح؟ مظبوط؟ و هذا أصغر أنصر لل set easy
+
+295
+00:30:34,920 --> 00:30:41,700
+هذا أصغر أنصر و هذا أصغر منه إذا هذا الأنصر مش
+
+296
+00:30:41,700 --> 00:30:49,690
+ممكن يكون موجود بال set easy صح؟لأن هذا أصغر من
+
+297
+00:30:49,690 --> 00:30:53,370
+أصغر
+
+298
+00:30:53,370 --> 00:30:59,410
+عنصر في ال set طيب،
+
+299
+00:30:59,410 --> 00:31:04,290
+معناه أن هذا نز سالب واحد ما هوش في ez
+
+300
+00:31:09,210 --> 00:31:13,650
+مدين هذا الأنصار مش موجود في set easy هذا هي
+
+301
+00:31:13,650 --> 00:31:21,730
+معناته بيحققش الصفة المميزة لل set easy متى
+
+302
+00:31:21,730 --> 00:31:27,210
+الانصار بيكون موجود هنا إذا بيحقق الصفة هذه أو
+
+303
+00:31:27,210 --> 00:31:30,390
+المتبينة هذه طب إذا كان الأنصار لا ينتمي لل set
+
+304
+00:31:30,390 --> 00:31:36,240
+معناته بيحققشالمتبينة دي بحقق ما فيها إذا هي بحقق
+
+305
+00:31:36,240 --> 00:31:43,740
+ما فيها هاي NZ-1 بدل ما يكون أكبر بصير أصغر من أو
+
+306
+00:31:43,740 --> 00:31:47,900
+يساوي ال Z إذا كون العنصر هذا مش موجود في EZ
+
+307
+00:31:47,900 --> 00:31:56,560
+معناته بيطلع أصغر من أو يساوي ال Z وال Z هو أصغر
+
+308
+00:31:56,560 --> 00:31:59,440
+عنصر لل set EZ
+
+309
+00:32:06,800 --> 00:32:16,820
+ف ال z أصغر من n احنا قلنا انه ال ..
+
+310
+00:32:16,820 --> 00:32:18,760
+او أصغر من ال nz
+
+311
+00:32:44,130 --> 00:32:50,890
+الان زي هذا انصر يعني
+
+312
+00:32:50,890 --> 00:32:57,270
+هذا بينتمي الى الست easy لأنه أصغر عنصر فيها
+
+313
+00:32:57,270 --> 00:33:06,070
+فينتمي إليها فان زي ينتمي ل easy معناته الان زي
+
+314
+00:33:06,070 --> 00:33:11,050
+هذا أكبر من ال z الان زي أكبر من ال z ومن هنا ان
+
+315
+00:33:11,050 --> 00:33:17,910
+زي سلب واحد مش موجود في easyفهو أصغر من أو يساوي
+
+316
+00:33:17,910 --> 00:33:24,290
+ال Z وبالتالي هيك بنكون أثبتنا المتباينة هذه اللي
+
+317
+00:33:24,290 --> 00:33:29,090
+هو اللي احنا عايزينه في الجزء C لأن هيك بنكون
+
+318
+00:33:29,090 --> 00:33:34,420
+كملنا برهان الجزء Cالأقل بالنسبة للجزء الأخير هذا
+
+319
+00:33:34,420 --> 00:33:42,460
+يعني عبارة عن ليس مش alternative form لل
+
+320
+00:33:42,460 --> 00:33:46,180
+Archimedean property ليس صورة أخرى لخاصية
+
+321
+00:33:46,180 --> 00:33:51,500
+Archimedean بس مجرد مثال، مجرد مثال أندي ست و الست
+
+322
+00:33:51,500 --> 00:33:56,290
+هذي boundedbounded above by one bounded below by
+
+323
+00:33:56,290 --> 00:34:02,570
+zero لبرهان
+
+324
+00:34:02,570 --> 00:34:12,350
+ذلك البرهان سهل نشوف
+
+325
+00:34:12,350 --> 00:34:12,950
+البرهان
+
+326
+00:34:29,410 --> 00:34:34,370
+كمان مرة ال set هذه هي عبارة عن .. نكتبها إيش هي
+
+327
+00:34:34,370 --> 00:34:37,710
+ال
+
+328
+00:34:37,710 --> 00:34:44,490
+set is عبارة عن ال set of all واحد على n حيث n is
+
+329
+00:34:44,490 --> 00:34:45,650
+natural number
+
+330
+00:34:51,720 --> 00:34:59,580
+واضح أن السفر أصغر من أو ساوي واحد على N لكل N
+
+331
+00:34:59,580 --> 00:35:11,180
+ينتمي إلى N صح؟ وبالتالي إذا zero is lower lower
+
+332
+00:35:11,180 --> 00:35:22,090
+bound لمين of set S وبالتالي ال infimumإذا it has
+
+333
+00:35:22,090 --> 00:35:25,890
+an infimum by the infimum property الـ infimum
+
+334
+00:35:25,890 --> 00:35:30,630
+property بتقول كل set bounded below بيكون ال في
+
+335
+00:35:30,630 --> 00:35:37,070
+إلها infimum say w بيساوي infimum s إذا هنا say
+
+336
+00:35:37,070 --> 00:35:41,290
+دعنا نسمي ال infimum هذا اللي إحنا ضمنين وجوده
+
+337
+00:35:41,290 --> 00:35:48,760
+باستخدام ال infimum property دعنا نسميه wتمام؟ إذا
+
+338
+00:35:48,760 --> 00:35:55,540
+ال .. ال w هذا هو أكبر .. هو أكبر lower bound لست
+
+339
+00:35:55,540 --> 00:36:02,640
+S و السفر lower bound إذا أكيد ال w أكبر من أوسعه
+
+340
+00:36:02,640 --> 00:36:09,100
+و سفر صح؟ السفر قلنا هايه lower bound لست و ال w
+
+341
+00:36:09,100 --> 00:36:11,960
+هو ال infimum اللي هو أكبر lower bound إذا ال w
+
+342
+00:36:11,960 --> 00:36:16,830
+أكبر من او .. أكبر من أوسعه و سفرطب احنا عايزين
+
+343
+00:36:16,830 --> 00:36:22,630
+نثبت .. احنا عايزين في النهاية نثبت ان الـ W هذا
+
+344
+00:36:22,630 --> 00:36:27,490
+اللي هو الـ infimum بساوء سفر هذا اللي عايزين
+
+345
+00:36:27,490 --> 00:36:33,570
+نثبته انا عندي W أكبر من أو ساوء سفر لكن انا بدي
+
+346
+00:36:33,570 --> 00:36:39,750
+أثبت ان الـ W بساوء سفر، تمام؟
+
+347
+00:36:39,750 --> 00:36:41,510
+فلإثبات ذلك
+
+348
+00:36:47,400 --> 00:36:54,780
+خلّينا ناخد أي إبسلون أكبر من السفر فحسب
+
+349
+00:36:54,780 --> 00:36:59,600
+الـ Archimedean property اللي هو الجزء بي المكافلة
+
+350
+00:36:59,600 --> 00:37:04,640
+Archimedean property لأي عدد موجة بإبسلون بقدر
+
+351
+00:37:04,640 --> 00:37:08,880
+ألاقي عدد طبيعي مقلوب وأصغر من إبسلون، صح؟ هذا
+
+352
+00:37:08,880 --> 00:37:12,000
+الجزء بي من النتيجة
+
+353
+00:37:14,540 --> 00:37:18,960
+إن أنا في عيندي هي 1 على n أصغر من epsilon يوجد
+
+354
+00:37:18,960 --> 00:37:24,760
+انها ده الطبيعي بحيث 1 على n أصغر من epsilon و 1
+
+355
+00:37:24,760 --> 00:37:30,700
+على n هده عنصر ال 1 على n هده عبارة عن عنصر في ال
+
+356
+00:37:30,700 --> 00:37:37,180
+6S و ال W هده lower bound إلها ال W هده هو ال
+
+357
+00:37:37,180 --> 00:37:44,890
+minimum لل 6Sو 1 على N عنصر في S إذا ال W بطلع
+
+358
+00:37:44,890 --> 00:37:48,490
+أصغر من أو ساوي أي عنصر في ال set لأنه lower bound
+
+359
+00:37:48,490 --> 00:37:53,830
+صح؟ و قبل شوية قلنا إن ال W هي U بس نتجنا إن ال W
+
+360
+00:37:53,830 --> 00:37:57,990
+اللي هو ال infimum أكبر من أو ساوي السفر اللي هو
+
+361
+00:37:57,990 --> 00:38:02,190
+lower bound وهذا أكبر lower bound الآن هذه ال
+
+362
+00:38:02,190 --> 00:38:06,850
+epsilon عشوائية إن الكلام هذا صحيح لكل epsilon
+
+363
+00:38:06,850 --> 00:38:13,170
+أكبر من السفرإذا في عندي نظرية واحد تمانية بتقول
+
+364
+00:38:13,170 --> 00:38:19,630
+ليه؟ كانت بتقول إن لو كان ال a عدد غير سالب و أصغر
+
+365
+00:38:19,630 --> 00:38:24,810
+من epsilon لكل epsilon أكبر من السفر فهذا بيقدي إن
+
+366
+00:38:24,810 --> 00:38:33,630
+a بساوي سفر، صح؟ هذه نظرية واحد تمانية، صح؟هي الـ
+
+367
+00:38:33,630 --> 00:38:39,230
+W التي هي الـ A أكبر من أو يساوي سفر وأصغر من
+
+368
+00:38:39,230 --> 00:38:44,590
+إبسلون لكل إبسلون عدد موجة فحسب النظرية هذه بيطلع
+
+369
+00:38:44,590 --> 00:38:50,590
+W بساوي سفر وهذا اللي احنا عايزين نثبته، تمام؟ إذن
+
+370
+00:38:50,590 --> 00:38:56,050
+هذا بيثبت أن الـ infimum للست دي أو لل sequence
+
+371
+00:38:56,050 --> 00:39:03,650
+واحد على N هو السفر، تمام؟وهنا استخدمنا في البرهان
+
+372
+00:39:03,650 --> 00:39:09,010
+الـ Archimedean property الصورة بيه من ال
+
+373
+00:39:09,010 --> 00:39:24,610
+Archimedean property في
+
+374
+00:39:24,610 --> 00:39:27,390
+النظرية هذه احنا أثبتنا قبل هيك
+
+375
+00:39:32,670 --> 00:39:41,530
+احنا أثبتنا سابقا في
+
+376
+00:39:41,530 --> 00:39:51,490
+السابق أثبتنا أنه في كان نظرية أو مثال بتقول أن
+
+377
+00:39:51,490 --> 00:39:55,550
+جذر 2 is not a rational number
+
+378
+00:39:58,290 --> 00:40:04,470
+أو العدد جدر اتنين is irrational نعم مظبوط فطبعا
+
+379
+00:40:04,470 --> 00:40:08,730
+في البرهان هذا اعتمدنا في البرهان على انه جدر
+
+380
+00:40:08,730 --> 00:40:12,850
+اتنين هذا عدد حقيقي يعني exist هو احد العداد
+
+381
+00:40:12,850 --> 00:40:20,950
+الحقيقية وفرضنا عملنا برهان غير مباشر فرضنا انه
+
+382
+00:40:20,950 --> 00:40:26,450
+جدر اتنين ينتمي ل Q او عدد نسبي ووصلنا الى تنظيم
+
+383
+00:40:26,450 --> 00:40:32,380
+تماماليوم بنرجع للوراء شوية و بنقول احنا هنا في
+
+384
+00:40:32,380 --> 00:40:36,220
+النظرية هذه في البرهان او في النظرية هذه افترضنا
+
+385
+00:40:36,220 --> 00:40:42,140
+جدلا او افترضنا مسبقا ان جدر اتنين هذا عدد حقيقي
+
+386
+00:40:42,140 --> 00:40:47,600
+اليوم هنرجع و نثبت ان existence of جدر اتنين يعني
+
+387
+00:40:47,600 --> 00:40:51,720
+جدر اتنين هذا بنثبت ان هو فعلا عدد حقيقي مش عدد
+
+388
+00:40:51,720 --> 00:40:53,040
+اخر مش عدد تقيم
+
+389
+00:40:55,660 --> 00:41:02,360
+فهذا يعني البرهان او
+
+390
+00:41:02,360 --> 00:41:05,560
+نظريها دي بالظبط بتقول انه جذر اتنين وعدد حقيقي
+
+391
+00:41:05,560 --> 00:41:14,760
+يعني يوجد عدد حقيقي موجب x ومربعه هو اتنين okay
+
+392
+00:41:16,030 --> 00:41:20,890
+فبرهان النظرية هذه يعني ممكن شوية طويل لكن موجود
+
+393
+00:41:20,890 --> 00:41:29,250
+عندكم بالتفصيل ويعني انجزة إلى أعزاء ويعني مش صعب
+
+394
+00:41:29,250 --> 00:41:35,490
+أنكم يعني تقرؤوا بمجتمعهم و تفهموه فأرجو أنكم
+
+395
+00:41:35,490 --> 00:41:39,990
+تقرؤوا البرهان و تحاولوا تفهموه و ممكن يعني المرة
+
+396
+00:41:39,990 --> 00:41:45,510
+الجاية ان شاء اللهنسأل نحاول نمر عليه او نحاول
+
+397
+00:41:45,510 --> 00:41:52,090
+نبره نقصر عليه، طبعا؟ اذا نكتفي بهذا القدر و نكمل
+
+398
+00:41:52,090 --> 00:41:53,230
+ان شاء الله المرة الجاية
+
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/7KfEZYA9kIA_raw.json b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/7KfEZYA9kIA_raw.json
new file mode 100644
index 0000000000000000000000000000000000000000..96c832614a44954caec9a09783d2c2579ad392fe
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/7KfEZYA9kIA_raw.json
@@ -0,0 +1 @@
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0.859619140625}, {"start": 70.37, "end": 71.17, "word": " بالتفصيل", "probability": 0.97529296875}, {"start": 71.17, "end": 71.55, "word": " المرة", "probability": 0.893798828125}, {"start": 71.55, "end": 71.73, "word": " اللي", "probability": 0.859130859375}, {"start": 71.73, "end": 72.35, "word": " فاتت", "probability": 0.9876302083333334}, {"start": 72.35, "end": 73.71, "word": " وكان", "probability": 0.91357421875}, {"start": 73.71, "end": 74.05, "word": " هنا", "probability": 0.99267578125}, {"start": 74.05, "end": 75.63, "word": " البرهان", "probability": 0.95751953125}, {"start": 75.63, "end": 76.25, "word": " بعتمد", "probability": 0.923583984375}, {"start": 76.25, "end": 76.43, "word": " على", "probability": 0.83740234375}, {"start": 76.43, "end": 76.67, "word": " أن", "probability": 0.5478515625}, {"start": 76.67, "end": 77.01, "word": " ال", "probability": 0.91162109375}, {"start": 77.01, "end": 77.29, "word": " 6", "probability": 0.3505859375}, {"start": 77.29, "end": 77.55, "word": " اللي", "probability": 0.6270751953125}, {"start": 77.55, "end": 77.83, "word": " bounded", "probability": 0.90625}, {"start": 77.83, "end": 78.39, "word": " above", "probability": 0.9609375}], "temperature": 1.0}, {"id": 3, "seek": 9935, "start": 92.25, "end": 99.35, "text": "الست اس هى dependent above لإن السوبرمان تباعها exist by the superman property", "tokens": [6027, 14851, 24525, 8032, 7578, 12334, 3673, 5296, 28814, 1863, 21136, 2407, 26890, 2304, 7649, 6055, 3555, 45761, 11296, 2514, 538, 264, 1687, 1601, 4707], "avg_logprob": -0.7451922847674444, "compression_ratio": 1.0505050505050506, "no_speech_prob": 0.0, "words": [{"start": 92.25, "end": 93.29, "word": "الست", "probability": 0.57568359375}, {"start": 93.29, "end": 93.97, "word": " اس", "probability": 0.46875}, {"start": 93.97, "end": 94.17, "word": " هى", "probability": 0.2466583251953125}, {"start": 94.17, "end": 94.47, "word": " dependent", "probability": 0.11651611328125}, {"start": 94.47, "end": 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{"start": 1257.46, "end": 1257.88, "word": " أكبر", "probability": 0.8860677083333334}, {"start": 1257.88, "end": 1258.42, "word": " منه", "probability": 0.98486328125}, {"start": 1258.42, "end": 1258.84, "word": " لأن", "probability": 0.6097412109375}, {"start": 1258.84, "end": 1259.06, "word": " هذا", "probability": 0.84326171875}, {"start": 1259.06, "end": 1259.62, "word": " بيديني", "probability": 0.82513427734375}, {"start": 1259.62, "end": 1260.96, "word": " تناقض", "probability": 0.966552734375}, {"start": 1260.96, "end": 1261.82, "word": " لكون", "probability": 0.988037109375}, {"start": 1261.82, "end": 1262.32, "word": " U", "probability": 0.98681640625}, {"start": 1262.32, "end": 1262.7, "word": " هو", "probability": 0.98779296875}, {"start": 1262.7, "end": 1263.08, "word": " upper", "probability": 0.89208984375}, {"start": 1263.08, "end": 1263.54, "word": " bound", "probability": 0.90625}, {"start": 1263.54, "end": 1264.08, "word": " ل", "probability": 0.93505859375}, {"start": 1264.08, "end": 1264.42, "word": " set", "probability": 0.75732421875}, {"start": 1264.42, "end": 1265.6, "word": " للعداد", "probability": 0.8727213541666666}, {"start": 1265.6, "end": 1266.38, "word": " الطبيعية", "probability": 0.9888916015625}, {"start": 1266.38, "end": 1267.06, "word": " إذا", "probability": 0.58447265625}, {"start": 1267.06, "end": 1267.56, "word": " وصلنا", "probability": 0.8719075520833334}, {"start": 1267.56, "end": 1267.74, "word": " إلى", "probability": 0.82763671875}, {"start": 1267.74, "end": 1268.36, "word": " تناقض", "probability": 0.98291015625}, {"start": 1268.36, "end": 1268.9, "word": " وبالتالي", "probability": 0.8701171875}, {"start": 1268.9, "end": 1269.54, "word": " هذا", "probability": 0.9169921875}, {"start": 1269.54, "end": 1270.14, "word": " بكمل", "probability": 0.8221028645833334}, {"start": 1270.14, "end": 1270.88, "word": " البرهانة", "probability": 0.81142578125}], "temperature": 1.0}, {"id": 48, "seek": 130116, "start": 1272.76, "end": 1301.16, "text": "إذا الفرض تبعنا التناقض هذا تقول إن ال assumption تبعنا هذا إن الكلام هذا صح كان خطر إذا الصح نفيه اللي هو المطلوب okay تمام إذا هذه ال Archimedean property هذه ال Archimedean property الآن ال Archimedean property هذه أو خاصية Archimedes إلها صور أخرى متعددة", "tokens": [28814, 15730, 27188, 43042, 6055, 3555, 3615, 8315, 16712, 8315, 4587, 11242, 23758, 6055, 39648, 36145, 2423, 15302, 6055, 3555, 3615, 8315, 23758, 36145, 2423, 28820, 10943, 23758, 20328, 5016, 25961, 16490, 9566, 2288, 11933, 15730, 31767, 5016, 8717, 41185, 3224, 13672, 1829, 31439, 9673, 9566, 1211, 37746, 1392, 46811, 10943, 11933, 15730, 29538, 2423, 10984, 332, 4858, 282, 4707, 29538, 2423, 10984, 332, 4858, 282, 4707, 6024, 48506, 2423, 10984, 332, 4858, 282, 4707, 29538, 34051, 16490, 33546, 10632, 10984, 332, 48490, 11933, 1211, 11296, 20328, 13063, 5551, 34740, 7578, 44650, 22488, 41891], "avg_logprob": -0.247532908540023, "compression_ratio": 1.875, "no_speech_prob": 0.0, 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"probability": 0.619140625}, {"start": 1295.46, "end": 1296.12, "word": " Archimedean", "probability": 0.962890625}, {"start": 1296.12, "end": 1296.74, "word": " property", "probability": 0.96435546875}, {"start": 1296.74, "end": 1297.14, "word": " هذه", "probability": 0.91015625}, {"start": 1297.14, "end": 1297.32, "word": " أو", "probability": 0.8720703125}, {"start": 1297.32, "end": 1297.74, "word": " خاصية", "probability": 0.9724934895833334}, {"start": 1297.74, "end": 1298.56, "word": " Archimedes", "probability": 0.9729817708333334}, {"start": 1298.56, "end": 1299.58, "word": " إلها", "probability": 0.6777750651041666}, {"start": 1299.58, "end": 1300.0, "word": " صور", "probability": 0.98486328125}, {"start": 1300.0, "end": 1300.48, "word": " أخرى", "probability": 0.97314453125}, {"start": 1300.48, "end": 1301.16, "word": " متعددة", "probability": 0.9786783854166666}], "temperature": 1.0}, {"id": 49, "seek": 132650, "start": 1302.66, "end": 1326.5, "text": "و هذه الصور هي موجودة في كوريلري واحد ستة عشر اذا النتيجة هذه في ان صور اخرى ل ال Archimedean property ف", "tokens": [2407, 29538, 31767, 13063, 39896, 3714, 29245, 23328, 3660, 8978, 9122, 13063, 26895, 16572, 36764, 24401, 8608, 2655, 3660, 6225, 46309, 1975, 15730, 28239, 31371, 7435, 3660, 29538, 8978, 16472, 20328, 13063, 1975, 34740, 7578, 5296, 2423, 10984, 332, 4858, 282, 4707, 6156], "avg_logprob": -0.42116478424180637, "compression_ratio": 1.2537313432835822, "no_speech_prob": 0.0, "words": [{"start": 1302.66, "end": 1302.9, "word": "و", "probability": 0.76025390625}, {"start": 1302.9, "end": 1303.18, "word": " هذه", "probability": 0.55517578125}, {"start": 1303.18, "end": 1303.58, "word": " الصور", "probability": 0.97705078125}, {"start": 1303.58, "end": 1303.84, "word": " هي", "probability": 0.59423828125}, {"start": 1303.84, "end": 1304.68, "word": " موجودة", "probability": 0.9681396484375}, {"start": 1304.68, "end": 1304.96, "word": " في", "probability": 0.9443359375}, {"start": 1304.96, 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1316.92, "end": 1317.16, "word": " ال", "probability": 0.2252197265625}, {"start": 1317.16, "end": 1318.06, "word": " Archimedean", "probability": 0.7384033203125}, {"start": 1318.06, "end": 1318.76, "word": " property", "probability": 0.7080078125}, {"start": 1318.76, "end": 1326.5, "word": " ف", "probability": 0.87060546875}], "temperature": 1.0}, {"id": 50, "seek": 133976, "start": 1327.84, "end": 1339.76, "text": "Alternative forms يعني صور أخرى لـArchimedean property، let YUZ be positive real numbers، إذن YUZ تنتمي لمجموعة الأعداد الحقيقية المهمة", "tokens": [9171, 2231, 1166, 6422, 37495, 22653, 20328, 13063, 5551, 34740, 7578, 5296, 39184, 10683, 339, 332, 4858, 282, 4707, 12399, 718, 33471, 57, 312, 3353, 957, 3547, 12399, 11933, 8848, 1863, 33471, 57, 6055, 29399, 2304, 1829, 5296, 2304, 7435, 2304, 2407, 27884, 16247, 22488, 18513, 21542, 38436, 4587, 10632, 9673, 16095, 3660], "avg_logprob": -0.3336226835295006, "compression_ratio": 1.1625, "no_speech_prob": 0.0, "words": 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0.337158203125}, {"start": 1787.04, "end": 1787.32, "word": " هو", "probability": 0.986328125}, {"start": 1787.32, "end": 1787.74, "word": " least", "probability": 0.470947265625}, {"start": 1787.74, "end": 1788.26, "word": " element", "probability": 0.96142578125}, {"start": 1788.26, "end": 1788.58, "word": " لل", "probability": 0.80126953125}, {"start": 1788.58, "end": 1788.86, "word": " set", "probability": 0.67431640625}, {"start": 1788.86, "end": 1789.3, "word": " هذه", "probability": 0.779296875}, {"start": 1789.3, "end": 1790.32, "word": " الغير", "probability": 0.947265625}, {"start": 1790.32, "end": 1790.78, "word": " خالية", "probability": 0.98291015625}, {"start": 1790.78, "end": 1791.24, "word": " okay", "probability": 0.452880859375}, {"start": 1791.24, "end": 1792.24, "word": " تمام", "probability": 0.772216796875}], "temperature": 1.0}, {"id": 68, "seek": 181889, "start": 1793.69, "end": 1818.89, "text": "إذا هنا يوجد أنصر nz عدد طبيعي وهذا العدد الطبيعي هو ال least element ل easy طيب الآن هذا أصغر أنصر في الست هذه يعني معناه nz لو طرحت من nz", "tokens": [28814, 15730, 34105, 7251, 29245, 3215, 14739, 9381, 2288, 297, 89, 6225, 3215, 3215, 23032, 21292, 3615, 1829, 37037, 15730, 18863, 3215, 3215, 41950, 21292, 3615, 1829, 31439, 2423, 1935, 4478, 5296, 1858, 23032, 1829, 3555, 6024, 48506, 23758, 5551, 9381, 17082, 2288, 14739, 9381, 2288, 8978, 2423, 14851, 29538, 37495, 22653, 20449, 8315, 3224, 297, 89, 45164, 23032, 2288, 33753, 9154, 297, 89], "avg_logprob": -0.22764423076923077, "compression_ratio": 1.4966887417218544, "no_speech_prob": 0.0, "words": [{"start": 1793.69, "end": 1793.99, "word": "إذا", "probability": 0.48291015625}, {"start": 1793.99, "end": 1794.25, "word": " هنا", "probability": 0.9248046875}, {"start": 1794.25, "end": 1794.99, "word": " يوجد", "probability": 0.97412109375}, {"start": 1794.99, "end": 1795.77, "word": " أنصر", "probability": 0.7918294270833334}, {"start": 1795.77, "end": 1796.19, "word": " nz", 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" طيب", "probability": 0.90673828125}, {"start": 1809.53, "end": 1812.01, "word": " الآن", "probability": 0.6361083984375}, {"start": 1812.01, "end": 1812.33, "word": " هذا", "probability": 0.91552734375}, {"start": 1812.33, "end": 1812.73, "word": " أصغر", "probability": 0.9696044921875}, {"start": 1812.73, "end": 1813.13, "word": " أنصر", "probability": 0.9166666666666666}, {"start": 1813.13, "end": 1813.25, "word": " في", "probability": 0.947265625}, {"start": 1813.25, "end": 1813.63, "word": " الست", "probability": 0.6351318359375}, {"start": 1813.63, "end": 1813.99, "word": " هذه", "probability": 0.5458984375}, {"start": 1813.99, "end": 1814.19, "word": " يعني", "probability": 0.890380859375}, {"start": 1814.19, "end": 1814.75, "word": " معناه", "probability": 0.8984375}, {"start": 1814.75, "end": 1816.15, "word": " nz", "probability": 0.573974609375}, {"start": 1816.15, "end": 1817.35, "word": " لو", "probability": 0.9296875}, {"start": 1817.35, "end": 1817.85, "word": " طرحت", "probability": 0.8522135416666666}, {"start": 1817.85, "end": 1818.15, "word": " من", "probability": 0.98876953125}, {"start": 1818.15, "end": 1818.89, "word": " nz", "probability": 0.778564453125}], "temperature": 1.0}, {"id": 69, "seek": 184554, "start": 1820.12, "end": 1845.54, "text": "طرحت منها واحد فطبعا هذا أصغر من NZ هذا أصغر من NZ صح؟ مظبوط؟ و هذا أصغر أنصر لل set easy هذا أصغر أنصر و هذا أصغر منه إذا هذا الأنصر مش ممكن يكون موجود بال set easy صح؟", "tokens": [9566, 2288, 33753, 9154, 11296, 36764, 24401, 6156, 9566, 3555, 3615, 995, 23758, 5551, 9381, 17082, 2288, 9154, 41089, 23758, 5551, 9381, 17082, 2288, 9154, 41089, 20328, 5016, 22807, 3714, 19913, 3555, 2407, 9566, 22807, 4032, 23758, 5551, 9381, 17082, 2288, 14739, 9381, 2288, 24976, 992, 1858, 23758, 5551, 9381, 17082, 2288, 14739, 9381, 2288, 4032, 23758, 5551, 9381, 17082, 2288, 9154, 3224, 11933, 15730, 23758, 16247, 1863, 9381, 2288, 37893, 3714, 43020, 7251, 30544, 3714, 29245, 23328, 20666, 992, 1858, 20328, 5016, 22807], "avg_logprob": -0.13988970237619736, "compression_ratio": 1.8926174496644295, "no_speech_prob": 0.0, "words": [{"start": 1820.12, "end": 1820.54, "word": "طرحت", "probability": 0.8160807291666666}, {"start": 1820.54, "end": 1820.88, "word": " منها", "probability": 0.989501953125}, {"start": 1820.88, "end": 1821.4, "word": " واحد", "probability": 0.979736328125}, {"start": 1821.4, "end": 1822.94, "word": " فطبعا", "probability": 0.9697265625}, {"start": 1822.94, "end": 1823.28, "word": " هذا", "probability": 0.66552734375}, {"start": 1823.28, "end": 1823.84, "word": " أصغر", "probability": 0.873046875}, {"start": 1823.84, "end": 1824.04, "word": " من", "probability": 0.98974609375}, {"start": 1824.04, "end": 1824.48, "word": " NZ", "probability": 0.351318359375}, {"start": 1824.48, "end": 1825.08, "word": " هذا", "probability": 0.72509765625}, {"start": 1825.08, "end": 1827.0, "word": " أصغر", "probability": 0.9725341796875}, {"start": 1827.0, "end": 1827.24, "word": " من", "probability": 0.99072265625}, {"start": 1827.24, "end": 1827.88, "word": " NZ", "probability": 0.876953125}, {"start": 1827.88, "end": 1829.84, "word": " صح؟", "probability": 0.7893880208333334}, {"start": 1829.84, "end": 1831.74, "word": " مظبوط؟", "probability": 0.8243408203125}, {"start": 1831.74, "end": 1832.58, "word": " و", "probability": 0.45068359375}, {"start": 1832.58, "end": 1832.84, "word": " هذا", "probability": 0.87451171875}, {"start": 1832.84, "end": 1833.3, "word": " أصغر", "probability": 0.9923095703125}, {"start": 1833.3, "end": 1833.84, "word": " أنصر", "probability": 0.8251953125}, {"start": 1833.84, "end": 1834.14, "word": " لل", "probability": 0.68408203125}, {"start": 1834.14, "end": 1834.46, "word": " set", "probability": 0.217041015625}, {"start": 1834.46, "end": 1834.92, "word": " easy", "probability": 0.85888671875}, {"start": 1834.92, "end": 1836.88, "word": " هذا", "probability": 0.93359375}, {"start": 1836.88, "end": 1837.26, "word": " أصغر", 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"end": 2395.55, "text": "احنا أثبتنا سابقا في السابق أثبتنا أنه في كان نظرية أو مثال بتقول أن جذر 2 is not a rational number", "tokens": [39319, 8315, 5551, 12984, 3555, 2655, 8315, 8608, 16758, 4587, 995, 8978, 21136, 16758, 4587, 5551, 12984, 3555, 2655, 8315, 14739, 3224, 8978, 25961, 8717, 19913, 2288, 10632, 34051, 50113, 6027, 39894, 39648, 14739, 10874, 8848, 2288, 568, 307, 406, 257, 15090, 1230], "avg_logprob": -0.1850141981108622, "compression_ratio": 1.2868852459016393, "no_speech_prob": 0.0, "words": [{"start": 2372.67, "end": 2373.33, "word": "احنا", "probability": 0.57501220703125}, {"start": 2373.33, "end": 2373.87, "word": " أثبتنا", "probability": 0.97197265625}, {"start": 2373.87, "end": 2374.71, "word": " سابقا", "probability": 0.96875}, {"start": 2374.71, "end": 2381.53, "word": " في", "probability": 0.76025390625}, {"start": 2381.53, "end": 2382.29, "word": " السابق", "probability": 0.9855143229166666}, {"start": 2382.29, "end": 2383.11, "word": " أثبتنا", 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0.74365234375}, {"start": 2508.61, "end": 2509.11, "word": " نكتفي", "probability": 0.9700927734375}, {"start": 2509.11, "end": 2509.45, "word": " بهذا", "probability": 0.9697265625}, {"start": 2509.45, "end": 2510.01, "word": " القدر", "probability": 0.9892578125}, {"start": 2510.01, "end": 2510.43, "word": " و", "probability": 0.9912109375}, {"start": 2510.43, "end": 2512.09, "word": " نكمل", "probability": 0.9313151041666666}, {"start": 2512.09, "end": 2512.25, "word": " ان", "probability": 0.84814453125}, {"start": 2512.25, "end": 2512.45, "word": " شاء", "probability": 0.987060546875}, {"start": 2512.45, "end": 2512.55, "word": " الله", "probability": 0.9677734375}, {"start": 2512.55, "end": 2512.95, "word": " المرة", "probability": 0.7841796875}, {"start": 2512.95, "end": 2513.23, "word": " الجاية", "probability": 0.7098795572916666}], "temperature": 1.0}], "language": "ar", "language_probability": 1.0, "duration": 2514.489, "duration_after_vad": 2277.2262499999915}
\ No newline at end of file
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/7KfEZYA9kIA_raw.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/7KfEZYA9kIA_raw.srt
new file mode 100644
index 0000000000000000000000000000000000000000..8579b3b6d2387280d319113a30c60ff472a8377d
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/7KfEZYA9kIA_raw.srt
@@ -0,0 +1,1592 @@
+1
+00:00:21,840 --> 00:00:28,120
+المحاضرة اللى فاتت بدينا في عرض بعض ال
+
+2
+00:00:28,120 --> 00:00:32,760
+applications of the soprano property و بعتقد ان
+
+3
+00:00:32,760 --> 00:00:37,680
+احنا اخدنا اول مثال اللى هو المثال هذا مظبوط
+
+4
+00:00:37,680 --> 00:00:40,980
+فقولنا
+
+5
+00:00:40,980 --> 00:00:46,160
+انه المثال هذا لو اخدت اى bounded set bounded
+
+6
+00:00:46,160 --> 00:00:56,150
+above وعرفت المجموع a زاد sبالطريقة هذه فأثبتنا
+
+7
+00:00:56,150 --> 00:01:00,770
+وممكن بسهولة أثبات أن ال supremum للمجموعة الجديدة
+
+8
+00:01:00,770 --> 00:01:09,870
+A plus S بتساوي A plus ال supremum ل S وشوفنا
+
+9
+00:01:09,870 --> 00:01:15,630
+البرهان بالتفصيل المرة اللي فاتت وكان هنا البرهان
+
+10
+00:01:15,630 --> 00:01:18,390
+بعتمد على أن ال 6 اللي bounded above
+
+11
+00:01:32,250 --> 00:01:36,230
+الست اس هى dependent above لإن السوبرمان تباعها
+
+12
+00:01:36,230 --> 00:01:42,940
+exist by the superman propertyوشوفنا بعد هيك أنه
+
+13
+00:01:42,940 --> 00:01:49,660
+ال .. العدد a زائد u بيطلع upper bound للست هذه و
+
+14
+00:01:49,660 --> 00:01:53,500
+بعدين أثبتنا أن هذا العدد هو أصغر upper bound أو
+
+15
+00:01:53,500 --> 00:01:59,320
+supremum للست هذه وبالتالي هيك بنكون أثبتنا أن
+
+16
+00:01:59,320 --> 00:02:03,760
+supremum للست هذه موجود و بيساوي العدد a زائد u
+
+17
+00:02:03,760 --> 00:02:09,420
+اللي هو a زائد supremum S المثال التاني
+
+18
+00:02:16,520 --> 00:02:20,320
+لو أخدت two functions المجال ال domain تبعهم
+
+19
+00:02:20,320 --> 00:02:25,300
+مجموعة D subset من R وكتبت
+
+20
+00:02:25,300 --> 00:02:29,280
+F of D علي أنها مجموعة كل العناصر F of X حيث و X
+
+21
+00:02:29,280 --> 00:02:34,400
+متملة D فال set F of D هذه هي ال range تبع ال
+
+22
+00:02:34,400 --> 00:02:39,120
+function F صح؟ هي المدى تبع ال function F و كذلك
+
+23
+00:02:39,120 --> 00:02:46,000
+ال set G of D هي ال range تبع ال function G
+
+24
+00:02:48,510 --> 00:02:53,250
+فلو فرضنا ان ال set f of d و ال set g of d bounded
+
+25
+00:02:53,250 --> 00:03:01,530
+set R فطبعا حسب ال supremum property المجموعات دول
+
+26
+00:03:01,530 --> 00:03:06,430
+كل واحدة لها supremum كذلك حسب ال infimum property
+
+27
+00:03:07,290 --> 00:03:11,050
+المجمعتين هذول كل واحدة فيهم إلها infimum، الـ
+
+28
+00:03:11,050 --> 00:03:15,350
+infimum تبعهم exist إذا نفرض إن المجمعتين هذول
+
+29
+00:03:15,350 --> 00:03:18,570
+bounded عشان إيه نضمن وجود ال suprem والinfimum
+
+30
+00:03:18,570 --> 00:03:26,450
+لكل واحدة منهم الآن في عندي بدي أبرن حاجة تانية لو
+
+31
+00:03:26,450 --> 00:03:31,930
+كان الفرض f of x أصغر من أو ساوي g of x بتحقق لكل
+
+32
+00:03:31,930 --> 00:03:38,040
+x ديفبطلع ال supremum للمجموعة F of D بطلع أصغر من
+
+33
+00:03:38,040 --> 00:03:44,660
+لو ساوي ال supremum للمجموعة G of D وبرهان هذا
+
+34
+00:03:44,660 --> 00:03:54,220
+البرهان يعني سهل أنا كاتب انه easy exercise لكن
+
+35
+00:03:54,220 --> 00:04:02,780
+ممكن تبرهنه ممكن تبرهنه كل سهولة فهي نكتب ال proof
+
+36
+00:04:06,320 --> 00:04:14,320
+of part one للجزء الأول فخلّينا
+
+37
+00:04:14,320 --> 00:04:19,400
+نثبت fix x
+
+38
+00:04:19,400 --> 00:04:29,400
+ينتمي إلى d ناخد عنصر x ينتمي إلى d عشوائي by
+
+39
+00:04:29,400 --> 00:04:31,240
+hypothesis من الفرض
+
+40
+00:04:33,710 --> 00:04:40,970
+من الفرض أنا عندي f of x أصغر من أو يساوي g of x
+
+41
+00:04:40,970 --> 00:04:52,470
+لل x هذه و لأي x دي صح هذا من الفرض و g of x g of
+
+42
+00:04:52,470 --> 00:05:00,550
+x أصغر من أو يساوي ال supremum لل set g of d
+
+43
+00:05:04,610 --> 00:05:14,410
+طبعا هذا زي ما قلنا exist by supremum property
+
+44
+00:05:14,410 --> 00:05:20,970
+باستخدام خاصية التمام ال
+
+45
+00:05:20,970 --> 00:05:26,910
+supremum .. هذا .. هذا عنصر في set هذا g of x عنصر
+
+46
+00:05:26,910 --> 00:05:32,550
+في set g of d صح؟وهذا upper bound ال suprem ل g of
+
+47
+00:05:32,550 --> 00:05:38,690
+d و هذا عنصر في ال 6 g of d فهذا أكيد أزم أو ساوي
+
+48
+00:05:38,690 --> 00:05:43,610
+ال upper bound لل 6 اللي بينتمي إليها فهذا صحيح
+
+49
+00:05:43,610 --> 00:05:56,610
+الآن هذا صحيح لكل x since x belonged to D was
+
+50
+00:05:56,610 --> 00:05:57,610
+arbitrarily
+
+51
+00:06:03,450 --> 00:06:10,110
+arbitrary ان ان بطلع اندي F of X أصغر من أو ساوي
+
+52
+00:06:10,110 --> 00:06:20,490
+ال supremum ل G of D وهذا صحيح لكل X في D هذا
+
+53
+00:06:20,490 --> 00:06:29,900
+معناه انه العدد هذاهذا العدد أكبر من أو ساوي كل
+
+54
+00:06:29,900 --> 00:06:36,960
+عناصر ال set F of D صح؟ هاي هذا معناه أن ال
+
+55
+00:06:36,960 --> 00:06:47,600
+supremum ل set G of D is an upper bound an upper
+
+56
+00:06:47,600 --> 00:06:50,860
+bound
+
+57
+00:06:50,860 --> 00:06:53,780
+لمين؟
+
+58
+00:06:54,920 --> 00:07:01,100
+of set f of d بصح؟
+
+59
+00:07:01,100 --> 00:07:07,040
+لأن هيكل أنصر f of x في f of d أصغر من أول ساول
+
+60
+00:07:07,040 --> 00:07:18,980
+عدد هذا، صح؟ طيب since ال supremum لset f of d
+
+61
+00:07:18,980 --> 00:07:25,890
+existin r طبعا برضه by supremum property لأن احنا
+
+62
+00:07:25,890 --> 00:07:31,890
+فرضين ان ال set هذه bounded صح فال supremum تبعها
+
+63
+00:07:31,890 --> 00:07:37,110
+موجود الآن ال set هذه ال supremum تبعها موجود
+
+64
+00:07:37,110 --> 00:07:42,750
+والعدد هذا هذا العدد عبارة عن upper bound ل set
+
+65
+00:07:42,750 --> 00:07:46,850
+إذا ما العلاقة بين ال upper bound هذا لل set وال
+
+66
+00:07:46,850 --> 00:07:53,650
+supremum لل setفي واحد اكبر من او ساوى التاني لان
+
+67
+00:07:53,650 --> 00:07:59,770
+بما ان هذا الكلام صحيح نحن نحن نحن نحن نحن نحن نحن
+
+68
+00:07:59,770 --> 00:08:01,050
+نحن نحن نحن نحن نحن نحن نحن نحن نحن نحن نحن نحن
+
+69
+00:08:01,050 --> 00:08:01,350
+نحن نحن نحن نحن نحن نحن نحن نحن نحن نحن نحن نحن
+
+70
+00:08:01,350 --> 00:08:04,050
+نحن نحن نحن نحن نحن نحن نحن نحن نحن نحن نحن نحن
+
+71
+00:08:04,050 --> 00:08:05,880
+نحن نحن نحن نحن نحن نحن نحن نحن نحن نحن نحنهذا
+
+72
+00:08:05,880 --> 00:08:10,820
+أصغر upper bound للست f of d وهذا upper bound للست
+
+73
+00:08:10,820 --> 00:08:15,440
+f of d إذا ال supremum بيطلع أصغر من أو ساوي ال
+
+74
+00:08:15,440 --> 00:08:22,480
+upper bound اللي هو supremum g of d وهو المطلوب
+
+75
+00:08:22,480 --> 00:08:29,800
+وهذا بثبت الجزء الأول okay تمام إذا الجزء الأول مش
+
+76
+00:08:29,800 --> 00:08:33,680
+صعب وهنا أثبتنا واضح
+
+77
+00:08:37,050 --> 00:08:42,310
+برهان الجزء التاني برضه شبيه فيه الجزء التاني، إيش
+
+78
+00:08:42,310 --> 00:08:47,510
+بيقول ليه؟ الفرض، لاحظوا الفرق بين الفرض تبع الجزء
+
+79
+00:08:47,510 --> 00:08:54,910
+التاني والجزء الأول الفرض
+
+80
+00:08:54,910 --> 00:09:00,210
+هنا إن f of x أصغر من أو ساوي g of y لكل x و y في
+
+81
+00:09:00,210 --> 00:09:00,450
+D
+
+82
+00:09:04,010 --> 00:09:09,170
+هذا أشمل وهذا أعمل من هذا وهذا أقوى لتاعتي لاحظوا
+
+83
+00:09:09,170 --> 00:09:14,690
+إنه لو هذا صح فهذا بيطلع صح اللي فور لكن الأكس مش
+
+84
+00:09:14,690 --> 00:09:18,430
+صحيح طيب
+
+85
+00:09:18,430 --> 00:09:22,130
+إذا .. إذا هذا الكلام صحيح فهذا بيقدي إن ال
+
+86
+00:09:22,130 --> 00:09:26,410
+supremum ل F of D بيطلع أصغر من أو ساوى الinfimum
+
+87
+00:09:26,410 --> 00:09:31,110
+لست G of D نشوف
+
+88
+00:09:31,110 --> 00:09:32,710
+ال .. نبرهن الكلام هذا
+
+89
+00:09:50,270 --> 00:10:02,090
+البرهان الجزء التاني البرهان
+
+90
+00:10:02,090 --> 00:10:05,030
+الجزء التاني هذا conditional statement هاي الفرض
+
+91
+00:10:05,030 --> 00:10:11,370
+وهي النتيجة ال conclusion فبنفرض ان الفرض هذا صحيح
+
+92
+00:10:11,370 --> 00:10:23,770
+و بنثبت يثبت يثبت أنصر Y في Dمن الفرض بيطلع عندي f
+
+93
+00:10:23,770 --> 00:10:29,530
+of x أصغر من أو ساوي g of y وهذا صحيح لكل x في دي
+
+94
+00:10:29,530 --> 00:10:38,280
+و ال y ثابت يعني هذا من الفرض صحيح لكل x في ديطيب،
+
+95
+00:10:38,280 --> 00:10:45,040
+الان هذا معناه ان العدد هذا g of y هي في y أنصة
+
+96
+00:10:45,040 --> 00:10:49,600
+ثابت هي أكبر .. هذا العدد أكبر من او ساوي كل ال F
+
+97
+00:10:49,600 --> 00:10:54,100
+of X لكل X دي معناه هذا upper bound لل set F of D
+
+98
+00:10:54,100 --> 00:10:59,020
+الان g of y عبارة عن upper bound لل set F of D من
+
+99
+00:10:59,020 --> 00:11:01,860
+هنا، مظبوط؟ تمام؟
+
+100
+00:11:04,040 --> 00:11:07,840
+وبالتالي ال least upper bound ل F of D بيطلع أصلا
+
+101
+00:11:07,840 --> 00:11:12,080
+أو ساوي ال upper bound ل F of D اللي هو G of Y لأن
+
+102
+00:11:12,080 --> 00:11:13,620
+هذه المتباينة صحيحة
+
+103
+00:11:18,050 --> 00:11:22,770
+اخترناها was arbitrary fixed احنا اخترناها عشوائي
+
+104
+00:11:22,770 --> 00:11:27,470
+arbitrary وثبتناها ان الكلام المتبين هذه الآن صحيح
+
+105
+00:11:27,470 --> 00:11:33,110
+لكل y ان المتبين هذه صحيحة true for every y في D
+
+106
+00:11:33,110 --> 00:11:39,510
+هذا معناه من المتبين هذه ال percentage انه العدد
+
+107
+00:11:39,510 --> 00:11:45,350
+ال supremum ل F of Dهذا عبارة عن lower bound
+
+108
+00:11:45,350 --> 00:11:51,030
+لمجموعة العناصر g of y حيث y ينتمي ل d يعني العدد
+
+109
+00:11:51,030 --> 00:11:58,210
+هذا عبارة عن lower bound للست g of d عظيم صح؟ طيب
+
+110
+00:11:58,210 --> 00:12:04,230
+ال inform ل g of d existو هذا العدد lower bound
+
+111
+00:12:04,230 --> 00:12:08,950
+للست هذه و ال infimum هذا عبارة عن ال greatest
+
+112
+00:12:08,950 --> 00:12:12,970
+lower bound ل G و D إذا ال greatest lower bound
+
+113
+00:12:12,970 --> 00:12:18,810
+دايما بيكون أكبر من أو يساوي أي lower bound إذا ال
+
+114
+00:12:18,810 --> 00:12:23,090
+lower bound هذا أصغر من أو يساوي ال greatest lower
+
+115
+00:12:23,090 --> 00:12:28,610
+bound ل G و D و هذا اللي هو هذا النتيجة اللي احنا
+
+116
+00:12:28,610 --> 00:12:34,800
+عايزين نصل لها okay تمامواضح؟ إذن هذا برهاني جزء
+
+117
+00:12:34,800 --> 00:12:48,220
+التاني الآن في ملاحظة الملاحظة هذه بتقول إنه يعني
+
+118
+00:12:48,220 --> 00:12:56,120
+ممكن طالبة طلعت تسأل أو تستفسر أو تتساءل طب ما هذا
+
+119
+00:12:56,120 --> 00:13:01,400
+الشرط تبين زي هذا مافيش فرق بينهمفاحنا بنقول لأ
+
+120
+00:13:01,400 --> 00:13:05,480
+هذا الشرط التحت أقوى من اللي فوق اللي تحت لو كان
+
+121
+00:13:05,480 --> 00:13:09,160
+التحت صحيح بيقدي للي فوق لكن لو كان اللي فوق صحيح
+
+122
+00:13:09,160 --> 00:13:14,300
+هذا ما بيقدي للي تحت هذا الشرط أقوى من اللي فوق
+
+123
+00:13:14,300 --> 00:13:20,240
+فممكن واحدة فيكم تسأل تقول طب لو احنا أخدنا الفرض
+
+124
+00:13:20,240 --> 00:13:24,840
+هذا لو فرضنا أن هذا الكلام صح هل ممكن نحصل على
+
+125
+00:13:24,840 --> 00:13:30,580
+نتيجة اللي تحتفالإجابة لأ، الإجابة لأ، هذا مش
+
+126
+00:13:30,580 --> 00:13:36,900
+ممكن، إذا ال .. لو شيلنا الفرض هذا و بدلناه بالفرض
+
+127
+00:13:36,900 --> 00:13:41,820
+اللي فوق فالنتيجة هذه لا يمكن نحصل عليها، مش شرط
+
+128
+00:13:41,820 --> 00:13:53,110
+تكونصحيحة او مثال يوضح انه لا يمكن استبدال الفرض
+
+129
+00:13:53,110 --> 00:13:58,610
+تبع الجزء التاني بالفرض تبع الجزء الأول ونحصل نحصل
+
+130
+00:13:58,610 --> 00:14:00,630
+على نتيجة الجزء التاني
+
+131
+00:14:12,790 --> 00:14:16,530
+فناخد على سبيل المثال او counterexample بيسميه في
+
+132
+00:14:16,530 --> 00:14:22,910
+رياضيات لو أخدت f of x بالساوي x تربية دالة تربية
+
+133
+00:14:22,910 --> 00:14:26,830
+و g of x ال identity function و أخدت ال domain
+
+134
+00:14:26,830 --> 00:14:30,950
+المشترك لf و g ال closed unit interval
+
+135
+00:14:34,300 --> 00:14:40,040
+فطبعا بنلاحظ أن f of x اللي هي x تربية لكل x في ال
+
+136
+00:14:40,040 --> 00:14:45,220
+closed unit interval x تربية أصغر من أو ساوى x،
+
+137
+00:14:45,220 --> 00:14:51,180
+مظبوط؟و X بيساوي G of X فهي في عندي ال two
+
+138
+00:14:51,180 --> 00:14:54,460
+functions هدول بالمناسبة ال two functions هدول
+
+139
+00:14:54,460 --> 00:14:59,220
+كلاهم كلاهم bounded bounded below by zero bounded
+
+140
+00:14:59,220 --> 00:15:08,940
+above by ال range تبعهم ال range تبعهم F of D و G
+
+141
+00:15:08,940 --> 00:15:14,900
+of D ك sets كمجموعات بطلوا subset من المجموعة من
+
+142
+00:15:14,900 --> 00:15:20,520
+السفر لواحدوبالتالي كلا هما bounded above by واحد
+
+143
+00:15:20,520 --> 00:15:27,000
+و bounded below by سفره إذن
+
+144
+00:15:27,000 --> 00:15:32,420
+هذه المجموعات هي bounded وهي عند ال function f of
+
+145
+00:15:32,420 --> 00:15:36,860
+x أصغر من أو ساوي g of x لكل x دي هذا الفرض تبع
+
+146
+00:15:36,860 --> 00:15:41,600
+الجزء واحد اللي شوفناه قبل شوية لكن النتيجة تبع
+
+147
+00:15:41,600 --> 00:15:45,560
+الجزء التالي لا تتحقق تعالى نشوف هي ال supremum ل
+
+148
+00:15:45,560 --> 00:15:52,190
+f of dهي مجموعة f of d الواحد
+
+149
+00:15:52,190 --> 00:15:56,430
+أكبر
+
+150
+00:15:56,430 --> 00:16:00,090
+من السفر السفر
+
+151
+00:16:00,090 --> 00:16:05,650
+برضه عبارة عن greatest lower bound أو الانفم من
+
+152
+00:16:05,650 --> 00:16:09,950
+المجموعة هذه واضح ان السفر lower bound للسفر هذه
+
+153
+00:16:09,950 --> 00:16:15,580
+وهو greatest lower boundإذاً هي عند الـ supremum
+
+154
+00:16:15,580 --> 00:16:20,220
+لـ F of D أكبر من الـ infimum لـ G of D وهذا نفي
+
+155
+00:16:20,220 --> 00:16:23,700
+نتيجة نتيجة نتيجة نتيجة نتيجة نتيجة نتيجة نتيجة
+
+156
+00:16:23,700 --> 00:16:24,240
+نتيجة نتيجة نتيجة نتيجة نتيجة نتيجة نتيجة نتيجة
+
+157
+00:16:24,240 --> 00:16:26,120
+نتيجة نتيجة نتيجة نتيجة نتيجة نتيجة نتيجة نتيجة
+
+158
+00:16:26,120 --> 00:16:26,240
+نتيجة نتيجة نتيجة نتيجة نتيجة نتيجة نتيجة نتيجة
+
+159
+00:16:26,240 --> 00:16:26,440
+نتيجة نتيجة نتيجة نتيجة نتيجة نتيجة نتيجة نتيجة
+
+160
+00:16:26,440 --> 00:16:32,560
+نتيجة نتيجة نتيجة نتيجة نتيجة نتيجة نتيجة نتيجة
+
+161
+00:16:49,380 --> 00:16:56,900
+كنتيجة على الـ completeness property في عندي نتيجة
+
+162
+00:16:56,900 --> 00:17:05,420
+كتير مهمة و هنستخدمها كتير معناها اللي هو ال
+
+163
+00:17:05,420 --> 00:17:10,120
+material اللي هناخدها لاحقا اللي هو ال Archimedean
+
+164
+00:17:10,120 --> 00:17:16,220
+property او خاصية Archimedes ايه الخاصية هذه بتقول
+
+165
+00:17:17,950 --> 00:17:23,890
+لأي عدد حقيقي x في عدد طبيعي أكبر منه، أعطيني أي
+
+166
+00:17:23,890 --> 00:17:29,650
+عدد حقيقي x سواء كان سفر أو موجب أو سالب، بقدر
+
+167
+00:17:29,650 --> 00:17:36,970
+أعطيكي عدد طبيعي أكبر منه أو بقدر أوجدلك عدد طبيعي
+
+168
+00:17:36,970 --> 00:17:42,760
+يكون أكبر منهالبرهان تبع النظرية هذه بيعتمد على
+
+169
+00:17:42,760 --> 00:17:47,040
+الـ completeness property فلبرهان ذلك نبدأ بالـ
+
+170
+00:17:47,040 --> 00:17:54,320
+Fix X في R ونثبتها ونعمل برهان بالتناقل نحن عايزين
+
+171
+00:17:54,320 --> 00:17:58,840
+نثبت أنه للـ Fix X اللي احنا ثبتناها يوجد
+
+172
+00:18:01,850 --> 00:18:07,810
+عايزين نثبت العبارة أن العبارة هذه تكون صحيحة يوجد
+
+173
+00:18:07,810 --> 00:18:12,430
+عدد طبيعي أكبر من X فبدا أعمل برهان بالتناقض بدا
+
+174
+00:18:12,430 --> 00:18:17,610
+أفرض أن نفي العبارة هذه هو الصح إذا ن assume ال
+
+175
+00:18:17,610 --> 00:18:21,030
+contrary أن نفي العبارة هذه الصح طب نفي العبارة
+
+176
+00:18:21,030 --> 00:18:27,750
+هذه الصح there exist ما بصير لكل N في Nعكس
+
+177
+00:18:27,750 --> 00:18:32,730
+المتباينة هذه اللي هو n أصغر من أو ساوي x إذن هنا
+
+178
+00:18:32,730 --> 00:18:37,550
+ال contrary أو النفي، نفي النتيجة هذه معناها أن كل
+
+179
+00:18:37,550 --> 00:18:44,610
+الأعداد الطبيعية أصغر من أو ساوي x هذا معناه أن ال
+
+180
+00:18:44,610 --> 00:18:51,230
+x هذا upper bound لست nوبالتالي الـ sit-in إلها
+
+181
+00:18:51,230 --> 00:18:54,850
+upper bound أو bounded above، إذا by the supremum
+
+182
+00:18:54,850 --> 00:19:00,590
+أو completeness of property، الـ sit-in بطلع يوجد
+
+183
+00:19:00,590 --> 00:19:04,970
+إلها supremum، الـ supremum تبعها exist and are،
+
+184
+00:19:04,970 --> 00:19:12,410
+سميه، فلينا نسميه you، فلينا نسميه you، تمام؟ في
+
+185
+00:19:12,410 --> 00:19:19,340
+لمة واحد اتناشلما واحدة اثناء عاش كده بتقول لو كان
+
+186
+00:19:19,340 --> 00:19:28,300
+U أو U بساوي ال superman لست S if and only if لكل
+
+187
+00:19:28,300 --> 00:19:35,920
+epsilon أكبر من الصفر نقدر نلاقي S epsilon في الست
+
+188
+00:19:35,920 --> 00:19:42,460
+S بحيث انه U سالب epsilon أصغر من S epsilon
+
+189
+00:19:45,010 --> 00:19:50,110
+طب اقل انا عندى فيه U بساوية Supremum ل N S بساوية
+
+190
+00:19:50,110 --> 00:19:55,450
+6 N كل الاعداد الطبيعية هى عندى Supremum ل N اللى
+
+191
+00:19:55,450 --> 00:20:01,890
+هو U exist اذا حسب لمّة واحد اتناش لو اخدت ابسلون
+
+192
+00:20:01,890 --> 00:20:06,670
+لو اخدت ابسلون بالساوية واحد هذا عدد موجب اذا لهذا
+
+193
+00:20:06,670 --> 00:20:11,690
+الابسلون بقدر الاقي عدد S epsilon هسمي M هنا بدل S
+
+194
+00:20:11,690 --> 00:20:16,930
+epsilon فى اللمةعدد طبيعي بحيث أنه لما أخد U minus
+
+195
+00:20:16,930 --> 00:20:20,670
+epsilon اللي هو الواحد هذا بيطلع أصغر من S epsilon
+
+196
+00:20:20,670 --> 00:20:25,050
+اللي هو M إذاً هذا نحصل عليه من لمّة واحدة واثنين
+
+197
+00:20:25,050 --> 00:20:30,870
+عشر طيب المتباين هذه ودي واحد نجري واحد علي مين
+
+198
+00:20:30,870 --> 00:20:35,010
+فبيطلع U أصغر من M زياد واحد طيب ال M عدد طبيعي
+
+199
+00:20:35,010 --> 00:20:40,130
+إذاً M زياد واحد عدد طبيعي صح؟ إذاً هذا M زياد
+
+200
+00:20:40,130 --> 00:20:47,360
+واحد عدد طبيعي وأكبر من Uو U قلنا ال U هو ال
+
+201
+00:20:47,360 --> 00:20:50,520
+supremum ل N يعني upper bound بيطلع upper bound ل
+
+202
+00:20:50,520 --> 00:20:55,860
+N فكيف U upper bound ل set N للعداد الطبيعية و في
+
+203
+00:20:55,860 --> 00:20:59,620
+عنصر في العداد الطبيعية أكبر منه لأن هذا بيديني
+
+204
+00:20:59,620 --> 00:21:06,380
+تناقض لكون U هو upper bound ل set للعداد الطبيعية
+
+205
+00:21:06,380 --> 00:21:13,060
+إذا وصلنا إلى تناقض وبالتالي هذا بكمل البرهانةإذا
+
+206
+00:21:13,060 --> 00:21:16,980
+الفرض تبعنا التناقض هذا تقول إن ال assumption
+
+207
+00:21:16,980 --> 00:21:24,720
+تبعنا هذا إن الكلام هذا صح كان خطر إذا الصح نفيه
+
+208
+00:21:24,720 --> 00:21:29,480
+اللي هو المطلوب okay تمام إذا هذه ال Archimedean
+
+209
+00:21:29,480 --> 00:21:35,460
+property هذه ال Archimedean property الآن ال
+
+210
+00:21:35,460 --> 00:21:39,580
+Archimedean property هذه أو خاصية Archimedes إلها
+
+211
+00:21:39,580 --> 00:21:45,520
+صور أخرى متعددةو هذه الصور هي موجودة في كوريلري
+
+212
+00:21:45,520 --> 00:21:50,700
+واحد ستة عشر اذا
+
+213
+00:21:50,700 --> 00:21:58,060
+النتيجة هذه في ان صور اخرى ل ال Archimedean
+
+214
+00:21:58,060 --> 00:22:06,500
+property ف
+
+215
+00:22:07,840 --> 00:22:11,520
+Alternative forms يعني صور أخرى لـArchimedean
+
+216
+00:22:11,520 --> 00:22:16,520
+property، let YUZ be positive real numbers، إذن
+
+217
+00:22:16,520 --> 00:22:19,760
+YUZ تنتمي لمجموعة الأعداد الحقيقية المهمة
+
+218
+00:22:22,550 --> 00:22:28,990
+أول نتيجة يوجد n عدد طبيعي بحيث ان الـ z أصغر من n
+
+219
+00:22:28,990 --> 00:22:35,410
+مضروب في y إذا لو فيندي عددين حقيقين موجبين z وy
+
+220
+00:22:35,410 --> 00:22:39,790
+بقدر ألاقي عدد طبيعي بحيث ان ال z أصغر من n مضروب
+
+221
+00:22:39,790 --> 00:22:49,740
+في y كذلك لأي عدد حقيقي موجب yبقدر ألاقي عدد طبيعي
+
+222
+00:22:49,740 --> 00:22:54,740
+مقلوبه أصغر من العدد الموجب Y طبعا مقلوب العدد
+
+223
+00:22:54,740 --> 00:22:59,220
+الطبيعي دائما موجب كذلك
+
+224
+00:22:59,220 --> 00:23:04,820
+لأي عدد حقيقي موجب Z بقدر ألاقي عدد طبيعي بحيث أن
+
+225
+00:23:04,820 --> 00:23:09,920
+العدد الموجب Z أكبر من أو ساوي N سالب واحد وأصغر
+
+226
+00:23:09,920 --> 00:23:16,770
+من N إذن التلات خواص هدولةكل واحدة منهم بنسميها
+
+227
+00:23:16,770 --> 00:23:20,730
+Archimedean property أو صورة أخرى من ال
+
+228
+00:23:20,730 --> 00:23:25,590
+Archimedean property الجزء
+
+229
+00:23:25,590 --> 00:23:30,250
+الأخير هذا هو عبارة عن مثال وليس ال Archimedean
+
+230
+00:23:30,250 --> 00:23:37,810
+يعني هذا استثناء يعني مجرد set بالساوي ال sequence
+
+231
+00:23:37,810 --> 00:23:44,140
+واحد على nمتتالية العداد الحقيقية 1 على N حيث N
+
+232
+00:23:44,140 --> 00:23:49,540
+عدد طبيعي فال set هذه هنثبت أن ال infimum إلها هو
+
+233
+00:23:49,540 --> 00:23:59,860
+السفر طيب إذا نشوف و نثبت العزاء الأولى الجزء
+
+234
+00:23:59,860 --> 00:24:00,780
+الأول
+
+235
+00:24:06,710 --> 00:24:15,270
+الجزء A لإثبات الجزء A خلّينا نعرف X بساوي Z على Y
+
+236
+00:24:15,270 --> 00:24:19,930
+طبعا Z وY أعداد حقيقية موجبة إذن خارج قسمتهم أعداد
+
+237
+00:24:19,930 --> 00:24:26,090
+موجب إذن هذا عبارة عن عدد حقيقي موجب يعني ال X هذا
+
+238
+00:24:26,090 --> 00:24:33,170
+عبارة عن real number وموجب فحسب ال Archimedean
+
+239
+00:24:33,170 --> 00:24:42,860
+propertyلأي x عدد حقيقي يوجد عدد طبيعي أكبر من الـ
+
+240
+00:24:42,860 --> 00:24:48,000
+x إذا الـ x اللي أنا أخده z على y بقدر ألاقي عدد
+
+241
+00:24:48,000 --> 00:24:53,440
+طبيعي n أكبر منه يعني z على y أصغر من n لو ضربت
+
+242
+00:24:53,440 --> 00:25:01,550
+المتباينة هذه في yy عدد موجب فهيصير عندي z أصغر من
+
+243
+00:25:01,550 --> 00:25:08,110
+n في y وهذه هي النتيجة تبع الجزء الأول okay إذا
+
+244
+00:25:08,110 --> 00:25:13,270
+هيك يكون أثبتنا الجزء الأول واضح لإثبات الجزء
+
+245
+00:25:13,270 --> 00:25:19,410
+التاني لو أخدنا في الجزء الأول لو أخدت z بساوي
+
+246
+00:25:19,410 --> 00:25:30,500
+واحد فهيصير عندي واحد1 أصغر من n في y ال z هذا عدد
+
+247
+00:25:30,500 --> 00:25:35,780
+موجب فلو أخد ال z بالساعة واحد هذا عدد موجب فحسب
+
+248
+00:25:35,780 --> 00:25:41,420
+النتيجة a بيطلع عند z أصغر من n يوجد عدد طبيعي n
+
+249
+00:25:41,420 --> 00:25:48,080
+بحيث ان z أصغر من ny يعني 1 أصغر من ny الآن نضرب
+
+250
+00:25:48,080 --> 00:25:53,910
+في 1 على n 1 على n عدد موجب لو ضربنا الطرفينبالعدد
+
+251
+00:25:53,910 --> 00:25:57,850
+الموجة بواحد علينا بيطلع واحد علينا أصغر من Y وهذا
+
+252
+00:25:57,850 --> 00:26:01,330
+اللي احنا عايزينه تمام ان هذا برهان الجزء التاني
+
+253
+00:26:01,330 --> 00:26:14,310
+لبرهان الجزء التالت الجزء
+
+254
+00:26:14,310 --> 00:26:14,730
+C
+
+255
+00:26:18,400 --> 00:26:23,700
+بنثبت أنه لأي عدد حقيقي موجب Z فيه عدد طبيعي بحيث
+
+256
+00:26:23,700 --> 00:26:30,940
+أن Z محصورة بين N سارب واحد و M تمام نعرف الست EZ
+
+257
+00:26:30,940 --> 00:26:36,380
+على إنها كل الأعداد الطبيعية M اللي بتكون أكبر من
+
+258
+00:26:36,380 --> 00:26:46,880
+Z الآن هذه المجموعة غير خالية لأنه
+
+259
+00:26:51,070 --> 00:26:57,610
+لأن الـ z هذا عدد موجب وبالتالي في الآخر هو عدد
+
+260
+00:26:57,610 --> 00:27:01,950
+حقيقي فby Archimedean property
+
+261
+00:27:10,880 --> 00:27:17,220
+اللي هي 115 رقمها نظرية 115 بتقول أي عدد حقيقي z
+
+262
+00:27:17,220 --> 00:27:26,880
+يوجد عدد .. يوجد عدد طبيعي يوجد m في n بحيث ان z
+
+263
+00:27:26,880 --> 00:27:32,820
+أصغر من n اذا
+
+264
+00:27:32,820 --> 00:27:42,120
+المجموعة هذهعلى الأقل فيها وانصر واحد اللي هو الـ
+
+265
+00:27:42,120 --> 00:27:49,100
+M هذا او خليني اسميه MZ تمام
+
+266
+00:27:49,100 --> 00:27:58,000
+الـ Archimedean property تضمن انه للعدد Z هذا اللي
+
+267
+00:27:58,000 --> 00:28:05,100
+هو يعني احنا فرضين ان العدد موجبالـ set هذه بقدر
+
+268
+00:28:05,100 --> 00:28:10,460
+ألاقي عدد طبيعي MZ أكبر من Z وبالتالي المجموعة هذه
+
+269
+00:28:10,460 --> 00:28:15,580
+تحتوي تحتوي على العنصر هذا على الأقل لأن هذه
+
+270
+00:28:15,580 --> 00:28:22,720
+مجموعة جار خالية واضحة النقطة هذه؟ الآن في خاصية
+
+271
+00:28:22,720 --> 00:28:29,920
+الترتيب أو بنسميها ال will ordering propertyوهذه
+
+272
+00:28:29,920 --> 00:28:34,400
+في الحقيقة بتدرسها في نهاية في أخر chapter في
+
+273
+00:28:34,400 --> 00:28:40,640
+مبادئ رياضيات ال will ordering property بتقول ان
+
+274
+00:28:40,640 --> 00:28:46,240
+every non-empty subset of N has a least element
+
+275
+00:28:46,240 --> 00:28:51,020
+يعني أي مجموعة غير خالية من مجموعة الأعداد
+
+276
+00:28:51,020 --> 00:28:55,880
+الطبيعية لازم اللي جي لها least element لازم يكون
+
+277
+00:28:55,880 --> 00:29:00,520
+لها أصغر عنصريعني خدي انت على الجربة حتى خدي اي
+
+278
+00:29:00,520 --> 00:29:04,060
+مجموعة جزئية من العدالة الطبيعية هتجد ان فيها أنصر
+
+279
+00:29:04,060 --> 00:29:08,620
+فيها هو أصغر أنصر فهذا طبعا حسب ال will ordering
+
+280
+00:29:08,620 --> 00:29:12,880
+property يعني درس المبادئ و انا شخصيا لما بدرس
+
+281
+00:29:12,880 --> 00:29:16,400
+مبادئ بحاول يعني امر عليها او اعطيها حتى لو يعني
+
+282
+00:29:16,400 --> 00:29:21,620
+بصورة مختصرة بقرابش الناس التانية لما بدرسوا
+
+283
+00:29:21,620 --> 00:29:25,340
+المبادئ بعتقد ممكن موصلوش اليها لكن مش مشكلة هاي
+
+284
+00:29:25,340 --> 00:29:26,400
+نحن بنحكيلكم عنها
+
+285
+00:29:29,700 --> 00:29:35,480
+إذا هي عندي هذه عبارة عن subset من مجموعة الأعداد
+
+286
+00:29:35,480 --> 00:29:40,060
+الطبيعية و non-empty إذا لازم يكون في إلها least
+
+287
+00:29:40,060 --> 00:29:45,640
+element إذا بقدر ألاقي NZ في مجموعة الأعداد
+
+288
+00:29:45,640 --> 00:29:49,300
+الطبيعية و هذا ال NZ هو least element لل set هذه
+
+289
+00:29:49,300 --> 00:29:56,530
+الغير خالية okay تمامإذا هنا يوجد أنصر nz عدد
+
+290
+00:29:56,530 --> 00:30:02,390
+طبيعي وهذا العدد الطبيعي هو ال least element ل
+
+291
+00:30:02,390 --> 00:30:09,530
+easy طيب
+
+292
+00:30:09,530 --> 00:30:17,350
+الآن هذا أصغر أنصر في الست هذه يعني معناه nz لو
+
+293
+00:30:17,350 --> 00:30:25,080
+طرحت من nzطرحت منها واحد فطبعا هذا أصغر من NZ هذا
+
+294
+00:30:25,080 --> 00:30:34,920
+أصغر من NZ صح؟ مظبوط؟ و هذا أصغر أنصر لل set easy
+
+295
+00:30:34,920 --> 00:30:41,700
+هذا أصغر أنصر و هذا أصغر منه إذا هذا الأنصر مش
+
+296
+00:30:41,700 --> 00:30:49,690
+ممكن يكون موجود بال set easy صح؟لأن هذا أصغر من
+
+297
+00:30:49,690 --> 00:30:53,370
+أصغر
+
+298
+00:30:53,370 --> 00:30:59,410
+عنصر في ال set طيب،
+
+299
+00:30:59,410 --> 00:31:04,290
+معناه أن هذا نز سالب واحد ما هوش في ez
+
+300
+00:31:09,210 --> 00:31:13,650
+مدين هذا الأنصار مش موجود في set easy هذا هي
+
+301
+00:31:13,650 --> 00:31:21,730
+معناته بيحققش الصفة المميزة لل set easy متى
+
+302
+00:31:21,730 --> 00:31:27,210
+الانصار بيكون موجود هنا إذا بيحقق الصفة هذه أو
+
+303
+00:31:27,210 --> 00:31:30,390
+المتبينة هذه طب إذا كان الأنصار لا ينتمي لل set
+
+304
+00:31:30,390 --> 00:31:36,240
+معناته بيحققشالمتبينة دي بحقق ما فيها إذا هي بحقق
+
+305
+00:31:36,240 --> 00:31:43,740
+ما فيها هاي NZ-1 بدل ما يكون أكبر بصير أصغر من أو
+
+306
+00:31:43,740 --> 00:31:47,900
+يساوي ال Z إذا كون العنصر هذا مش موجود في EZ
+
+307
+00:31:47,900 --> 00:31:56,560
+معناته بيطلع أصغر من أو يساوي ال Z وال Z هو أصغر
+
+308
+00:31:56,560 --> 00:31:59,440
+عنصر لل set EZ
+
+309
+00:32:06,800 --> 00:32:16,820
+ف ال z أصغر من n احنا قلنا انه ال ..
+
+310
+00:32:16,820 --> 00:32:18,760
+او أصغر من ال nz
+
+311
+00:32:44,130 --> 00:32:50,890
+الان زي هذا انصر يعني
+
+312
+00:32:50,890 --> 00:32:57,270
+هذا بينتمي الى الست easy لأنه أصغر عنصر فيها
+
+313
+00:32:57,270 --> 00:33:06,070
+فينتمي إليها فان زي ينتمي ل easy معناته الان زي
+
+314
+00:33:06,070 --> 00:33:11,050
+هذا أكبر من ال z الان زي أكبر من ال z ومن هنا ان
+
+315
+00:33:11,050 --> 00:33:17,910
+زي سلب واحد مش موجود في easyفهو أصغر من أو يساوي
+
+316
+00:33:17,910 --> 00:33:24,290
+ال Z وبالتالي هيك بنكون أثبتنا المتباينة هذه اللي
+
+317
+00:33:24,290 --> 00:33:29,090
+هو اللي احنا عايزينه في الجزء C لأن هيك بنكون
+
+318
+00:33:29,090 --> 00:33:34,420
+كملنا برهان الجزء Cالأقل بالنسبة للجزء الأخير هذا
+
+319
+00:33:34,420 --> 00:33:42,460
+يعني عبارة عن ليس مش alternative form لل
+
+320
+00:33:42,460 --> 00:33:46,180
+Archimedean property ليس صورة أخرى لخاصية
+
+321
+00:33:46,180 --> 00:33:51,500
+Archimedean بس مجرد مثال، مجرد مثال أندي ست و الست
+
+322
+00:33:51,500 --> 00:33:56,290
+هذي boundedbounded above by one bounded below by
+
+323
+00:33:56,290 --> 00:34:02,570
+zero لبرهان
+
+324
+00:34:02,570 --> 00:34:12,350
+ذلك البرهان سهل نشوف
+
+325
+00:34:12,350 --> 00:34:12,950
+البرهان
+
+326
+00:34:29,410 --> 00:34:34,370
+كمان مرة ال set هذه هي عبارة عن .. نكتبها إيش هي
+
+327
+00:34:34,370 --> 00:34:37,710
+ال
+
+328
+00:34:37,710 --> 00:34:44,490
+set is عبارة عن ال set of all واحد على n حيث n is
+
+329
+00:34:44,490 --> 00:34:45,650
+natural number
+
+330
+00:34:51,720 --> 00:34:59,580
+واضح أن السفر أصغر من أو ساوي واحد على N لكل N
+
+331
+00:34:59,580 --> 00:35:11,180
+ينتمي إلى N صح؟ وبالتالي إذا zero is lower lower
+
+332
+00:35:11,180 --> 00:35:22,090
+bound لمين of set S وبالتالي ال infimumإذا it has
+
+333
+00:35:22,090 --> 00:35:25,890
+an infimum by the infimum property الـ infimum
+
+334
+00:35:25,890 --> 00:35:30,630
+property بتقول كل set bounded below بيكون ال في
+
+335
+00:35:30,630 --> 00:35:37,070
+إلها infimum say w بيساوي infimum s إذا هنا say
+
+336
+00:35:37,070 --> 00:35:41,290
+دعنا نسمي ال infimum هذا اللي إحنا ضمنين وجوده
+
+337
+00:35:41,290 --> 00:35:48,760
+باستخدام ال infimum property دعنا نسميه wتمام؟ إذا
+
+338
+00:35:48,760 --> 00:35:55,540
+ال .. ال w هذا هو أكبر .. هو أكبر lower bound لست
+
+339
+00:35:55,540 --> 00:36:02,640
+S و السفر lower bound إذا أكيد ال w أكبر من أوسعه
+
+340
+00:36:02,640 --> 00:36:09,100
+و سفر صح؟ السفر قلنا هايه lower bound لست و ال w
+
+341
+00:36:09,100 --> 00:36:11,960
+هو ال infimum اللي هو أكبر lower bound إذا ال w
+
+342
+00:36:11,960 --> 00:36:16,830
+أكبر من او .. أكبر من أوسعه و سفرطب احنا عايزين
+
+343
+00:36:16,830 --> 00:36:22,630
+نثبت .. احنا عايزين في النهاية نثبت ان الـ W هذا
+
+344
+00:36:22,630 --> 00:36:27,490
+اللي هو الـ infimum بساوء سفر هذا اللي عايزين
+
+345
+00:36:27,490 --> 00:36:33,570
+نثبته انا عندي W أكبر من أو ساوء سفر لكن انا بدي
+
+346
+00:36:33,570 --> 00:36:39,750
+أثبت ان الـ W بساوء سفر، تمام؟
+
+347
+00:36:39,750 --> 00:36:41,510
+فلإثبات ذلك
+
+348
+00:36:47,400 --> 00:36:54,780
+خلّينا ناخد أي إبسلون أكبر من السفر فحسب
+
+349
+00:36:54,780 --> 00:36:59,600
+الـ Archimedean property اللي هو الجزء بي المكافلة
+
+350
+00:36:59,600 --> 00:37:04,640
+Archimedean property لأي عدد موجة بإبسلون بقدر
+
+351
+00:37:04,640 --> 00:37:08,880
+ألاقي عدد طبيعي مقلوب وأصغر من إبسلون، صح؟ هذا
+
+352
+00:37:08,880 --> 00:37:12,000
+الجزء بي من النتيجة
+
+353
+00:37:14,540 --> 00:37:18,960
+إن أنا في عيندي هي 1 على n أصغر من epsilon يوجد
+
+354
+00:37:18,960 --> 00:37:24,760
+انها ده الطبيعي بحيث 1 على n أصغر من epsilon و 1
+
+355
+00:37:24,760 --> 00:37:30,700
+على n هده عنصر ال 1 على n هده عبارة عن عنصر في ال
+
+356
+00:37:30,700 --> 00:37:37,180
+6S و ال W هده lower bound إلها ال W هده هو ال
+
+357
+00:37:37,180 --> 00:37:44,890
+minimum لل 6Sو 1 على N عنصر في S إذا ال W بطلع
+
+358
+00:37:44,890 --> 00:37:48,490
+أصغر من أو ساوي أي عنصر في ال set لأنه lower bound
+
+359
+00:37:48,490 --> 00:37:53,830
+صح؟ و قبل شوية قلنا إن ال W هي U بس نتجنا إن ال W
+
+360
+00:37:53,830 --> 00:37:57,990
+اللي هو ال infimum أكبر من أو ساوي السفر اللي هو
+
+361
+00:37:57,990 --> 00:38:02,190
+lower bound وهذا أكبر lower bound الآن هذه ال
+
+362
+00:38:02,190 --> 00:38:06,850
+epsilon عشوائية إن الكلام هذا صحيح لكل epsilon
+
+363
+00:38:06,850 --> 00:38:13,170
+أكبر من السفرإذا في عندي نظرية واحد تمانية بتقول
+
+364
+00:38:13,170 --> 00:38:19,630
+ليه؟ كانت بتقول إن لو كان ال a عدد غير سالب و أصغر
+
+365
+00:38:19,630 --> 00:38:24,810
+من epsilon لكل epsilon أكبر من السفر فهذا بيقدي إن
+
+366
+00:38:24,810 --> 00:38:33,630
+a بساوي سفر، صح؟ هذه نظرية واحد تمانية، صح؟هي الـ
+
+367
+00:38:33,630 --> 00:38:39,230
+W التي هي الـ A أكبر من أو يساوي سفر وأصغر من
+
+368
+00:38:39,230 --> 00:38:44,590
+إبسلون لكل إبسلون عدد موجة فحسب النظرية هذه بيطلع
+
+369
+00:38:44,590 --> 00:38:50,590
+W بساوي سفر وهذا اللي احنا عايزين نثبته، تمام؟ إذن
+
+370
+00:38:50,590 --> 00:38:56,050
+هذا بيثبت أن الـ infimum للست دي أو لل sequence
+
+371
+00:38:56,050 --> 00:39:03,650
+واحد على N هو السفر، تمام؟وهنا استخدمنا في البرهان
+
+372
+00:39:03,650 --> 00:39:09,010
+الـ Archimedean property الصورة بيه من ال
+
+373
+00:39:09,010 --> 00:39:24,610
+Archimedean property في
+
+374
+00:39:24,610 --> 00:39:27,390
+النظرية هذه احنا أثبتنا قبل هيك
+
+375
+00:39:32,670 --> 00:39:41,530
+احنا أثبتنا سابقا في
+
+376
+00:39:41,530 --> 00:39:51,490
+السابق أثبتنا أنه في كان نظرية أو مثال بتقول أن
+
+377
+00:39:51,490 --> 00:39:55,550
+جذر 2 is not a rational number
+
+378
+00:39:58,290 --> 00:40:04,470
+أو العدد جدر اتنين is irrational نعم مظبوط فطبعا
+
+379
+00:40:04,470 --> 00:40:08,730
+في البرهان هذا اعتمدنا في البرهان على انه جدر
+
+380
+00:40:08,730 --> 00:40:12,850
+اتنين هذا عدد حقيقي يعني exist هو احد العداد
+
+381
+00:40:12,850 --> 00:40:20,950
+الحقيقية وفرضنا عملنا برهان غير مباشر فرضنا انه
+
+382
+00:40:20,950 --> 00:40:26,450
+جدر اتنين ينتمي ل Q او عدد نسبي ووصلنا الى تنظيم
+
+383
+00:40:26,450 --> 00:40:32,380
+تماماليوم بنرجع للوراء شوية و بنقول احنا هنا في
+
+384
+00:40:32,380 --> 00:40:36,220
+النظرية هذه في البرهان او في النظرية هذه افترضنا
+
+385
+00:40:36,220 --> 00:40:42,140
+جدلا او افترضنا مسبقا ان جدر اتنين هذا عدد حقيقي
+
+386
+00:40:42,140 --> 00:40:47,600
+اليوم هنرجع و نثبت ان existence of جدر اتنين يعني
+
+387
+00:40:47,600 --> 00:40:51,720
+جدر اتنين هذا بنثبت ان هو فعلا عدد حقيقي مش عدد
+
+388
+00:40:51,720 --> 00:40:53,040
+اخر مش عدد تقيم
+
+389
+00:40:55,660 --> 00:41:02,360
+فهذا يعني البرهان او
+
+390
+00:41:02,360 --> 00:41:05,560
+نظريها دي بالظبط بتقول انه جذر اتنين وعدد حقيقي
+
+391
+00:41:05,560 --> 00:41:14,760
+يعني يوجد عدد حقيقي موجب x ومربعه هو اتنين okay
+
+392
+00:41:16,030 --> 00:41:20,890
+فبرهان النظرية هذه يعني ممكن شوية طويل لكن موجود
+
+393
+00:41:20,890 --> 00:41:29,250
+عندكم بالتفصيل ويعني انجزة إلى أعزاء ويعني مش صعب
+
+394
+00:41:29,250 --> 00:41:35,490
+أنكم يعني تقرؤوا بمجتمعهم و تفهموه فأرجو أنكم
+
+395
+00:41:35,490 --> 00:41:39,990
+تقرؤوا البرهان و تحاولوا تفهموه و ممكن يعني المرة
+
+396
+00:41:39,990 --> 00:41:45,510
+الجاية ان شاء اللهنسأل نحاول نمر عليه او نحاول
+
+397
+00:41:45,510 --> 00:41:52,090
+نبره نقصر عليه، طبعا؟ اذا نكتفي بهذا القدر و نكمل
+
+398
+00:41:52,090 --> 00:41:53,230
+ان شاء الله المرة الجاية
+
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/8N3n8lL04hg_raw.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/8N3n8lL04hg_raw.srt
new file mode 100644
index 0000000000000000000000000000000000000000..15583f134bed5c8a17e38b71103959e566c7d548
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/8N3n8lL04hg_raw.srt
@@ -0,0 +1,1960 @@
+1
+00:00:21,630 --> 00:00:28,730
+Okay ان شاء الله اليوم هنعمل مناقشة لبعض المسائل
+
+2
+00:00:28,730 --> 00:00:34,230
+في section اتنين تلاتة و اتنين اربعة زي ما وعدناكم
+
+3
+00:00:34,230 --> 00:00:44,690
+سابقا و نشوف بعض الحلول لبعض المسائل المهمة ففي
+
+4
+00:00:44,690 --> 00:00:52,530
+بسألةسؤال خامسة في section اتنين تلاتة بيقول لو في
+
+5
+00:00:52,530 --> 00:00:57,270
+عندي مجموعة غير خالية من الأعداد الحقيقية و
+
+6
+00:00:57,270 --> 00:01:03,550
+bounded below فال infimum ل ال set S هو سالب ال
+
+7
+00:01:03,550 --> 00:01:09,110
+supremum ل سالب S هذا
+
+8
+00:01:09,110 --> 00:01:13,870
+التمرين حالة خاصة من التمرين رقم أربعة في section
+
+9
+00:01:13,870 --> 00:01:21,120
+اتنين أربعةو بالتحديد هو حالة خاصة من الجزء بي من
+
+10
+00:01:21,120 --> 00:01:26,980
+التمرين هذا ففي الجزء بي لو كان بي .. إيش بقول هذا
+
+11
+00:01:26,980 --> 00:01:34,160
+الجزء؟ لو كان بي عدد سالب ف infimum بي S بساوي بي
+
+12
+00:01:34,160 --> 00:01:42,620
+في suprem S فلو أخدت بي بساوي سالب واحد و هذا عدد
+
+13
+00:01:42,620 --> 00:01:50,580
+سالبفبطل عندى infimum infimum
+
+14
+00:01:50,580 --> 00:01:58,780
+سالب s لأ هذا عبارة عن حالة خاصة من الجزء التانى
+
+15
+00:01:58,780 --> 00:02:05,560
+لو أخدنا بيه بساوي سالب واحد في الجزء هذا اللى هنا
+
+16
+00:02:07,490 --> 00:02:14,150
+فبطلع عندي supremum سالب S بيساوي
+
+17
+00:02:14,150 --> 00:02:19,390
+سالب infimum S هاي سالب اضربك سالب واحد سالب
+
+18
+00:02:19,390 --> 00:02:23,390
+infimum S لأن هذا التمرين حالة خاصة من الجزء هذا
+
+19
+00:02:23,390 --> 00:02:30,450
+التاني في الفرع B وبالتالي هذا التمرين تعميم لهذا
+
+20
+00:02:30,450 --> 00:02:37,140
+الجزء ولا جزء تانيو لجزء تاني اللي هو عبارة عن ال
+
+21
+00:02:37,140 --> 00:02:47,140
+supremum او ال infimum ل سالب S بساوي سالب ال
+
+22
+00:02:47,140 --> 00:02:54,240
+supremum ل S هذا تعميم لجزء اللي هان وهذا تعميم
+
+23
+00:02:54,240 --> 00:03:00,920
+لجزء اللي هان و ذلك باخذ by taking B equals سالب
+
+24
+00:03:00,920 --> 00:03:12,510
+واحدخلّينا نبرهن الجزء الأول من الفرع A و الجزء
+
+25
+00:03:12,510 --> 00:03:17,190
+الأول من الفرع B و بالمثل بإمكانكم تبرهن الجزء
+
+26
+00:03:17,190 --> 00:03:22,510
+التاني من ال part A و الجزء التاني من part B
+
+27
+00:03:22,510 --> 00:03:30,890
+فنبرهن الجزء A لبرهان الجزء A اللي
+
+28
+00:03:30,890 --> 00:03:37,490
+هو هذا الجزءفانا عندي a عدد موجب S is bounded
+
+29
+00:03:37,490 --> 00:03:42,390
+وبالتالي bounded below إذا ال info ل S exist سميه
+
+30
+00:03:42,390 --> 00:03:47,110
+W طبعا ال info عبارة عن lower bound ل 6S إذا ال W
+
+31
+00:03:47,110 --> 00:03:53,030
+أزرر من أو ساوي X لكل X S وبالتالي لو ضربت في عدد
+
+32
+00:03:53,030 --> 00:03:57,510
+موجب A فبطلع AW أزرر من أو ساوي A X لكل S هذا
+
+33
+00:03:57,510 --> 00:04:05,230
+معناه إن العدد هذا lower bound ل 6ASانا عايز اثبت
+
+34
+00:04:05,230 --> 00:04:10,670
+ان اي w هذا العدد مش بس lower bound هو اكبر lower
+
+35
+00:04:10,670 --> 00:04:19,690
+bound للست AS فباخد اي let V be any lower bound
+
+36
+00:04:19,690 --> 00:04:27,790
+any lower bound للست AS وبينا
+
+37
+00:04:27,790 --> 00:04:32,710
+نثبت ان هذا ال V أصغر من أو ساوي AW عشان يكون هو
+
+38
+00:04:32,710 --> 00:04:33,390
+ال infimum
+
+39
+00:04:35,910 --> 00:04:43,990
+طيب هذا معناه V lower bound للست AS معناه V أصغر
+
+40
+00:04:43,990 --> 00:04:52,010
+من أوي ساوي A X لكل X في S طيب أنا عندي واحد على A
+
+41
+00:04:52,010 --> 00:04:57,330
+عدد موجب إذا واحد على A عدد موجب فلو ضربت المتباني
+
+42
+00:04:57,330 --> 00:05:00,270
+هذه في العدد الموجب واحد على A اشتريت هنا
+
+43
+00:05:00,270 --> 00:05:07,900
+مابتتغيرش فبصير عندي V على Aأصغر من أو ساوي X لكل
+
+44
+00:05:07,900 --> 00:05:12,300
+XS طب
+
+45
+00:05:12,300 --> 00:05:20,540
+ما هذا معناه أنه العدد ال number V over A is a
+
+46
+00:05:20,540 --> 00:05:25,840
+lower bound لمن؟
+
+47
+00:05:25,840 --> 00:05:30,580
+لل 6S وبالتالي
+
+48
+00:05:30,580 --> 00:05:38,490
+إذا ال infimum .. إذا ال V على Aأصغر من أو ساوي ال
+
+49
+00:05:38,490 --> 00:05:48,090
+infimum للست S صح؟ طب اضربي في A عدد موجب بطلع
+
+50
+00:05:48,090 --> 00:05:58,990
+عندي V أصغر من أو ساوي A في infimum S طب
+
+51
+00:05:58,990 --> 00:06:07,730
+infimum S هذا سمنها W لأن هذا بساوي AWإذن هين
+
+52
+00:06:07,730 --> 00:06:13,790
+أثبتنا إنه العدد AW هذا أبارع ال lower bound للست
+
+53
+00:06:13,790 --> 00:06:20,390
+AS واخدنا أي lower bound للست AS فوجدنا إن ال
+
+54
+00:06:20,390 --> 00:06:27,770
+lower bound هذا أصغر من أو ساوي A في W فهذا معناه
+
+55
+00:06:27,770 --> 00:06:37,630
+إن AW هو ال infimum لمن؟ للست AS كما هوموضح في الـ
+
+56
+00:06:37,630 --> 00:06:44,290
+claim أو في الإدعاء تمام؟ وهذا بثبت الجزء الأول في
+
+57
+00:06:44,290 --> 00:06:51,650
+ال part A هاي infimum AS بساوي A في W اللي هو
+
+58
+00:06:51,650 --> 00:06:58,250
+infimum S إذن هذا بثبت الجزء الأول في الفرع A
+
+59
+00:06:58,250 --> 00:07:01,850
+Similarly بالمثل ممكن
+
+60
+00:07:05,820 --> 00:07:12,760
+بالمثل ممكن نثبت الفرع التاني او
+
+61
+00:07:12,760 --> 00:07:20,060
+الجزء التاني في الفرع A تمام؟ فهسيب هذا جزء لكم
+
+62
+00:07:20,060 --> 00:07:27,840
+لأن هذا مشابه الفرع اللي انا واضح؟ في اي سؤال؟ طيب
+
+63
+00:07:27,840 --> 00:07:30,780
+نحاول نثبت الجزء الأول في الفرع B
+
+64
+00:07:35,110 --> 00:07:42,150
+بنثبت الجزء هذا في الفرق دي لت
+
+65
+00:07:42,150 --> 00:07:53,770
+بأصغر من سفر، عدد حقيقي سالب وأنا عندي ال set ال
+
+66
+00:07:53,770 --> 00:07:58,230
+set since ال set S is bounded
+
+67
+00:08:01,660 --> 00:08:10,440
+إذا الـ infimum w بساوي ال infimum ل S exists in R
+
+68
+00:08:10,440 --> 00:08:13,460
+إذا
+
+69
+00:08:13,460 --> 00:08:18,240
+في عندي أنا ال .. ال infimum ل 6S .. 6S bounded
+
+70
+00:08:18,240 --> 00:08:21,180
+below bounded وبالتالي bounded below إذا by
+
+71
+00:08:21,180 --> 00:08:26,460
+infimum property ال infimum ل S مي W exist
+
+72
+00:08:30,860 --> 00:08:41,580
+هذا معناه .. او هذا بقد .. اذا
+
+73
+00:08:41,580 --> 00:08:46,180
+هذا معناه ان w lower bound ل S و W أصغر من أو ساوي
+
+74
+00:08:46,180 --> 00:08:49,880
+X لكل X في S
+
+75
+00:08:53,000 --> 00:08:58,980
+طيب و أندي أنا ال B عدد سالب فلو ضربنا المتباينة
+
+76
+00:08:58,980 --> 00:09:06,840
+هذه في B عدد سالب فبصير BX أصغر من أو ساوي BW لكل
+
+77
+00:09:06,840 --> 00:09:18,890
+XS صح؟ إذن هذا معناهإنه العدد بي دابليو is an
+
+78
+00:09:18,890 --> 00:09:28,750
+upper is an upper bound لمين للست بي في اس للست بي
+
+79
+00:09:28,750 --> 00:09:33,930
+في اس اللي هي مجموعة كل العناصر بي ضرب اكس بي ضرب
+
+80
+00:09:33,930 --> 00:09:38,570
+اكس حيث اكس ينتمي الاس هذا عبارة عن upper bound
+
+81
+00:09:38,570 --> 00:09:46,570
+طيب الست هذي الست هذي boundedلأن ال set S bounded
+
+82
+00:09:46,570 --> 00:09:51,270
+فضربها تعدد بتظلها bounded وبالتالي bounded above
+
+83
+00:09:51,270 --> 00:09:57,250
+إذا ال .. ال .. إلها superman by superman property
+
+84
+00:09:57,250 --> 00:10:08,990
+ودلتالي إذا ال BW هذا أو ال supermanللست BS هذا
+
+85
+00:10:08,990 --> 00:10:14,330
+عبارة عن ال least upper bound for the set BS هذا
+
+86
+00:10:14,330 --> 00:10:20,270
+بيطلع أصغر من أو ساوي أي upper bound و ليه هو أصغر
+
+87
+00:10:20,270 --> 00:10:28,150
+من أو ساوي ال upper bound BW للست BS طب
+
+88
+00:10:28,150 --> 00:10:29,610
+احنا عايزين نثبت
+
+89
+00:10:32,240 --> 00:10:38,840
+احنا عايزين نثبت ان بي دابليو هي ال supreme لست بي
+
+90
+00:10:38,840 --> 00:10:42,460
+في اس فهين
+
+91
+00:10:42,460 --> 00:10:47,020
+اثبتنا ان العدد بي دابليو هذا upper bound للست هذي
+
+92
+00:10:47,020 --> 00:10:51,240
+بي دابليو هو upper bound للست الاثبات ان هو ال
+
+93
+00:10:51,240 --> 00:10:55,240
+supreme باقي اثبات ان انا لو اخدت اي upper bound
+
+94
+00:10:55,240 --> 00:11:00,400
+للست هذه لازم يطلع اكبر من او يساوي بي دابليو
+
+95
+00:11:04,070 --> 00:11:11,310
+any upper bound
+
+96
+00:11:11,310 --> 00:11:18,490
+of except bs هذا
+
+97
+00:11:18,490 --> 00:11:28,090
+معناه أن b في x أصغر من أوي سوى b لكل xs تمام؟
+
+98
+00:11:29,920 --> 00:11:34,420
+طيب انا عندي بي عدل سالب اذا واحد على بي ايضا عدل
+
+99
+00:11:34,420 --> 00:11:38,960
+سالب فلو ضربت المتباينة هذه في عدل سالب اللي هو
+
+100
+00:11:38,960 --> 00:11:50,040
+واحد على بي فهيطلع عندي بي .. بي على بي أصغر من أو
+
+101
+00:11:50,040 --> 00:11:52,340
+ساوي X لكل X في S
+
+102
+00:11:55,350 --> 00:12:04,150
+هذا معناه ان العدد V على B is a lower bound لمن؟
+
+103
+00:12:04,150 --> 00:12:11,510
+لست S مصبوط صح؟ وبالتالي
+
+104
+00:12:11,510 --> 00:12:17,930
+اذا .. اذا
+
+105
+00:12:17,930 --> 00:12:23,970
+ال V على Bاللي هو lower bound للست S أصغر من أو
+
+106
+00:12:23,970 --> 00:12:28,370
+ساوي ال infimum للست S
+
+107
+00:12:54,340 --> 00:13:06,560
+احنا ايش قاعدين نثبت ال ..
+
+108
+00:13:06,560 --> 00:13:12,960
+يبدو ان انا يعني هنا بثبت الجزء التاني يعنى، يالا
+
+109
+00:13:12,960 --> 00:13:22,410
+من حظكمحاول نثبت الجزء التاني مش الأول فكمان مرة
+
+110
+00:13:22,410 --> 00:13:26,810
+نراجع بي عدد سالم S is bounded وبالتالي bounded
+
+111
+00:13:26,810 --> 00:13:33,650
+below إذن ال inform ل set S موجود وبالتالي
+
+112
+00:13:33,650 --> 00:13:37,630
+المتابعين هذا بتتحقق وبالتالي هذا بتتحقق بعد ما
+
+113
+00:13:37,630 --> 00:13:42,070
+ضربنا في بي عدد سالم إذن بي و طلع upper bound ل
+
+114
+00:13:42,070 --> 00:13:48,410
+set بي S وبالتالي ال supermanللست بي اس بيطلع أصغر
+
+115
+00:13:48,410 --> 00:13:52,510
+من أو ساوي بي دابليو الان بدنا نثبت ان ال بي
+
+116
+00:13:52,510 --> 00:14:00,810
+دابليو هذا هو ال supremum لست بي اس تمام فأخدنا اي
+
+117
+00:14:00,810 --> 00:14:05,550
+upper bound بي .. اي upper bound لست بي اس فوجدنا
+
+118
+00:14:05,550 --> 00:14:09,930
+ان v على بي is a lower bound لست اس وبالتالي v على
+
+119
+00:14:09,930 --> 00:14:14,290
+بي أصغر من أو ساوي ال greatest lower bound لست اس
+
+120
+00:14:17,060 --> 00:14:27,860
+طب لو ضربنا في بي و بي عدد سالب فهيطلع عندي .. إذا
+
+121
+00:14:27,860 --> 00:14:34,940
+لو ضربنا المتباينة هذه في بي عدد سالب فهيطلع عندي
+
+122
+00:14:34,940 --> 00:14:43,120
+اللي هو بي في infimum S هيطلع أصغر من أو ساوي ال
+
+123
+00:14:43,120 --> 00:14:45,120
+V، مظبوط هيك؟
+
+124
+00:14:48,920 --> 00:14:56,120
+طب هذا هذا سمنها w إذا بي في w أصغر من أو ساوي ال
+
+125
+00:14:56,120 --> 00:15:02,100
+b إذا البرهان هذا أثبتنا فيه حاجتين إنه أول شيء
+
+126
+00:15:02,100 --> 00:15:07,540
+العدد بي دابليو هذا upper bound للست بي اس و بعدين
+
+127
+00:15:07,540 --> 00:15:14,350
+أخدنا أي upper boundV أي upper bound لست بي اس طلع
+
+128
+00:15:14,350 --> 00:15:19,910
+ال V هذا أكبر من أو ساوي بي دابليو وبالتالي هذا
+
+129
+00:15:19,910 --> 00:15:29,650
+معناه إذا العدد بي دابليو هو عبارة عن ال supremum
+
+130
+00:15:29,650 --> 00:15:40,970
+ال supremum لست بي في اس لست بي في اسلأن هذا العدد
+
+131
+00:15:40,970 --> 00:15:45,570
+upper bound للست هذه وهو أصغر upper bound أخدنا أي
+
+132
+00:15:45,570 --> 00:15:51,390
+upper bound للست هذه طلع بي دابليو أصغر من أو ساوي
+
+133
+00:15:51,390 --> 00:15:56,050
+إذن بي دابليو هو أصغر upper bound للست هذه والأن
+
+134
+00:15:56,050 --> 00:16:03,410
+بنعود عن w إذن ال b في w اللي هو infimum of s
+
+135
+00:16:03,410 --> 00:16:12,590
+بتطلع بساوي supremum ل b في sوهذا برهين الجزء
+
+136
+00:16:12,590 --> 00:16:18,330
+التاني من الفرع B بالمثل الممكن برهان الجزء الأول
+
+137
+00:16:18,330 --> 00:16:24,850
+من الفرع B فأنا بدأكم إلى كتابة برهين الأجزاء
+
+138
+00:16:24,850 --> 00:16:30,330
+المشابهة هذه تمام؟ إذن هيك بنكون .. يعني أخدنا
+
+139
+00:16:30,330 --> 00:16:37,150
+حلول تقريبا شبه كاملة للتمرين 5 section 2 تلاتةفي
+
+140
+00:16:37,150 --> 00:16:41,530
+عندكم أي أسئلة تانية في ال section اتنين تلاتة او
+
+141
+00:16:41,530 --> 00:16:48,470
+اتنين اربعة؟ في
+
+142
+00:16:48,470 --> 00:16:54,190
+أي أسئلة تانية؟ السؤال عشرة في section اتنين تلاتة
+
+143
+00:17:28,800 --> 00:17:38,060
+سؤال عشرة section اتنين تلاتة ملخص السؤال بيقول S
+
+144
+00:17:38,060 --> 00:17:52,000
+is bounded bounded subset of R و Phi
+
+145
+00:17:52,000 --> 00:17:55,460
+لا يساوي S subset
+
+146
+00:18:00,440 --> 00:18:07,020
+ف ال S0 non-empty subset من S مجموعة جزئية غير
+
+147
+00:18:07,020 --> 00:18:17,280
+خالية من المجموعة S فبدنا نثبت شو برهني ان ال
+
+148
+00:18:17,280 --> 00:18:26,260
+infimum لست S أصغر من أو ساوي ال infimum لست S0
+
+149
+00:18:26,260 --> 00:18:32,540
+أصغر من أو ساوي ال supremumللست S Zero أصغر من لو
+
+150
+00:18:32,540 --> 00:18:41,940
+يساوي ال supremum للست S نشوف
+
+151
+00:18:41,940 --> 00:18:46,860
+البرهان مع بعض برهان سهل وبسيط يعتمد على تعريف ال
+
+152
+00:18:46,860 --> 00:18:52,760
+infimum وعلى تعريف ال supremum طيب
+
+153
+00:18:52,760 --> 00:18:57,900
+أنا عندي المجموعة S since
+
+154
+00:19:00,710 --> 00:19:08,790
+بما أن S مجموعة غير خالية و bounded is bounded
+
+155
+00:19:08,790 --> 00:19:12,990
+then ال
+
+156
+00:19:12,990 --> 00:19:28,810
+infimum لست S exist and supremum لست S both exist
+
+157
+00:19:36,050 --> 00:19:44,310
+بعد الـ infimum property ست اس لإنفمام وكذلك ست اس
+
+158
+00:19:44,310 --> 00:19:52,290
+لسوبرمام هدول موجودين في R طيب
+
+159
+00:19:52,290 --> 00:19:56,150
+أنا عندي السوبرمام
+
+160
+00:19:56,150 --> 00:20:15,640
+للست اس السوبرمام للست اسis an upper bound فهي
+
+161
+00:20:15,640 --> 00:20:25,520
+أيضا it is also an upper bound لأي
+
+162
+00:20:25,520 --> 00:20:31,060
+subset لأي subset S0 من ال 6S
+
+163
+00:20:36,460 --> 00:20:44,900
+و بالتالي and therefore and
+
+164
+00:20:44,900 --> 00:20:52,600
+therefore ال
+
+165
+00:20:52,600 --> 00:20:57,540
+supremum لست S0
+
+166
+00:20:57,540 --> 00:21:01,840
+أصغر من أو ساوي ال supremum لست S
+
+167
+00:21:07,110 --> 00:21:15,710
+كمان مرة ال .. ال 6S هذه ال S0 سبسط من S فأي upper
+
+168
+00:21:15,710 --> 00:21:20,070
+bound ل S هو أيضا upper bound لأي مجموعة جزئية
+
+169
+00:21:20,070 --> 00:21:26,410
+منها طيب ال supremum ل 6S upper bound ل 6S
+
+170
+00:21:26,410 --> 00:21:32,830
+وبالتالي هو upper bound ل 6S0 طيب ال supremum ل S0
+
+171
+00:21:32,830 --> 00:21:39,130
+هذا أصغر upper bound ل S0وهذا upper bound ل S0 إذا
+
+172
+00:21:39,130 --> 00:21:42,550
+أصغر upper bound أصغر من لو ساوي أي upper bound
+
+173
+00:21:42,550 --> 00:21:51,650
+وبالتالي المتباينة هذه صحيحة كذلك by
+
+174
+00:21:51,650 --> 00:21:57,950
+definition حسب التعريفات ال
+
+175
+00:21:57,950 --> 00:22:06,790
+infimumللست S0 أصغر من أو ساوي ال supremum للست S0
+
+176
+00:22:06,790 --> 00:22:10,750
+الست
+
+177
+00:22:10,750 --> 00:22:11,750
+S0 هذه
+
+178
+00:22:15,230 --> 00:22:21,930
+طبعا هذه ال set S0 subset من S و S bounded إلى S0
+
+179
+00:22:21,930 --> 00:22:26,710
+bounded ال infimum ل S0 exist و ال suprem ل S0
+
+180
+00:22:26,710 --> 00:22:32,770
+exist دائما لأي set S0 ال infimum دائما أصغر من أو
+
+181
+00:22:32,770 --> 00:22:39,250
+يساوي ال supremum نعمل رسمة نوضح الكلام هذا
+
+182
+00:22:44,850 --> 00:22:56,850
+نعتبر أن هذه هي الست اس وهي
+
+183
+00:22:56,850 --> 00:23:07,950
+ال .. ال .. ال supremum للست اس وهي ال infimum
+
+184
+00:23:11,090 --> 00:23:17,810
+للـ set S فدائما ال .. دائما
+
+185
+00:23:17,810 --> 00:23:24,050
+ال minimum لأي set هو lower bound لل set وبالتالي
+
+186
+00:23:24,050 --> 00:23:28,950
+أصغر من لو ساوي كل عناصرهاهو عبارة عن lower bound
+
+187
+00:23:28,950 --> 00:23:32,810
+للست ال supreme للست S هو عبارة عن upper bound
+
+188
+00:23:32,810 --> 00:23:37,650
+للست وبالتالي أكبر من أو ساوي كل عناصرها فواضح أن
+
+189
+00:23:37,650 --> 00:23:42,770
+ال infimum للست S لازم يكون أصغر من أو ساوي ال
+
+190
+00:23:42,770 --> 00:23:52,970
+supremum ونفس الشيء لو أخذنا أي مجموعة جزئية سمنها
+
+191
+00:23:52,970 --> 00:23:53,790
+S0
+
+192
+00:23:56,180 --> 00:24:02,200
+يعني هذه المجموعة اسمها S0 فبما أن ال set S
+
+193
+00:24:02,200 --> 00:24:10,400
+bounded إذن S0 bounded وبالتالي ال supremum ل S0
+
+194
+00:24:10,400 --> 00:24:16,220
+دايما أكبر من أو ساوي ال infimum ل S0 بنفس الطريقة
+
+195
+00:24:16,220 --> 00:24:23,710
+إذن هذا دايما .. هذا دايما صحيحعشان احنا نكمل
+
+196
+00:24:23,710 --> 00:24:30,150
+البرهان اذا احنا أثبتنا هذا واضح من التعريفات وهذا
+
+197
+00:24:30,150 --> 00:24:35,150
+الجزء أثبتناه باقي
+
+198
+00:24:35,150 --> 00:24:40,930
+إثبات الجزء الأخير هذا فإذا
+
+199
+00:24:40,930 --> 00:24:45,790
+بنقول finally أخيرا لإثبات الجزء الأخير هذا أنا
+
+200
+00:24:45,790 --> 00:24:49,570
+عندي ال inform ل S is lower bound ل 6S
+
+201
+00:24:52,070 --> 00:24:57,350
+وبالتالي هو lower bound لأي مجموعة جزئية S0 من S
+
+202
+00:24:57,350 --> 00:25:00,890
+وبالتالي
+
+203
+00:25:00,890 --> 00:25:11,770
+إذا ال influence ل S0 هذا
+
+204
+00:25:11,770 --> 00:25:19,180
+أكبر lower bound ل S0 هذا أكبر lower bound ل S0و
+
+205
+00:25:19,180 --> 00:25:25,960
+هذا lower bound ل S0 إذاً هذا بيطلع أكبر من أو
+
+206
+00:25:25,960 --> 00:25:33,500
+ساوي infimum ال 6S هذا lower bound ل 6S0 و هذا
+
+207
+00:25:33,500 --> 00:25:37,820
+أكبر lower bound ل 6S0 إذاً هذا أصغر من أو ساوي
+
+208
+00:25:37,820 --> 00:25:43,700
+هذا و هذا بيكملبرهان المتباينة اللى حاطين عليها
+
+209
+00:25:43,700 --> 00:25:48,380
+علامة استفهام إذا هيك بيكون برهاننا التمرين okay
+
+210
+00:25:48,380 --> 00:25:53,660
+تمام واضح؟
+
+211
+00:25:53,660 --> 00:26:03,660
+فى أسئلة تانية خلنا نحل كمان سؤال إذا بتحبه ممكن
+
+212
+00:26:03,660 --> 00:26:04,900
+نحل كمان سؤال
+
+213
+00:26:08,660 --> 00:26:16,040
+في section اتنين تلاتة برضه؟ اه في اي section؟
+
+214
+00:26:16,040 --> 00:26:21,840
+اتنين تلاتة ولا اتنين اربعة؟ اتنين تلاتة؟ طيب نحل
+
+215
+00:26:21,840 --> 00:26:24,020
+هذا السؤال و بعد هيك يعني نوجد
+
+216
+00:26:43,630 --> 00:26:57,410
+هي السؤال الأحداش سيكشن اتنين تلاتة بنشوف
+
+217
+00:26:57,410 --> 00:27:05,850
+السؤال شو بيقول S
+
+218
+00:27:05,850 --> 00:27:11,530
+subset من R و
+
+219
+00:27:11,530 --> 00:27:25,720
+SS star بساوي ال supremum ل 6S وهذا بينتمي لل 6S
+
+220
+00:27:25,720 --> 00:27:31,040
+belongs to S فإذا
+
+221
+00:27:31,040 --> 00:27:41,140
+كان U لا ينتمي لل 6S إذا كان U لا ينتمي لل 6S شو
+
+222
+00:27:42,390 --> 00:27:49,090
+عايزين نثبت ان ال superman لست
+
+223
+00:27:49,090 --> 00:28:05,890
+S union singleton U بيطلع بيساوي ال superman لست
+
+224
+00:28:05,890 --> 00:28:10,330
+اللي تتكون من أنصرين S star و U
+
+225
+00:28:13,540 --> 00:28:28,400
+where are you؟ طبعا في برهانين للسؤال هذا ال
+
+226
+00:28:28,400 --> 00:28:33,840
+proof one البرهان الأول we
+
+227
+00:28:33,840 --> 00:28:38,580
+use .. we use exercise
+
+228
+00:28:42,560 --> 00:28:51,600
+تسعة section اتنين تلاتة وهذا ال exercise بيقول
+
+229
+00:28:51,600 --> 00:28:59,340
+إذا كانت لو
+
+230
+00:28:59,340 --> 00:29:03,380
+كان a و b bounded
+
+231
+00:29:09,480 --> 00:29:18,660
+فهذا بيقدي ان a union b is bounded and
+
+232
+00:29:18,660 --> 00:29:32,360
+مش هيكوا بس و ال supremum .. ال supremum لإتحاد b
+
+233
+00:29:32,360 --> 00:29:36,980
+بساوي supremum
+
+234
+00:29:39,920 --> 00:29:44,900
+Supermom A وSupermom
+
+235
+00:29:44,900 --> 00:29:51,760
+B إذا
+
+236
+00:29:51,760 --> 00:29:57,440
+هذا تمرين رقم تسعة هناخده نستخدمه فلو استخدمنا هذا
+
+237
+00:29:57,440 --> 00:30:07,700
+التمرين فالنتيجة هذه بتطلع على طول مباشرة إذا
+
+238
+00:30:07,700 --> 00:30:08,540
+هنا take
+
+239
+00:30:11,570 --> 00:30:17,410
+A بساوي S و
+
+240
+00:30:17,410 --> 00:30:25,570
+طبعا هادي ال set bounded ال set هادي bounded و
+
+241
+00:30:25,570 --> 00:30:32,610
+عندي ال set B هاخدها singleton euro و هادي bounded
+
+242
+00:30:32,610 --> 00:30:41,790
+setإذا by exercise 9 a hat b اللي هي ال 6 هذه
+
+243
+00:30:41,790 --> 00:30:47,650
+بتطلع bounded by
+
+244
+00:30:47,650 --> 00:30:56,490
+exercise 9 section 2 3 ال 6 a union singleton u is
+
+245
+00:30:56,490 --> 00:31:00,750
+bounded and
+
+246
+00:31:00,750 --> 00:31:10,540
+مش هيكوا بس ال supremumلـ A اتحاد بالـ 6S union
+
+247
+00:31:10,540 --> 00:31:18,160
+هذا الـ A وهذا الـ Singleton U بتساوي الـ Supremum
+
+248
+00:31:18,160 --> 00:31:22,440
+لـ
+
+249
+00:31:22,440 --> 00:31:32,820
+Supremum A هذا عبارة عن S star و Supremum D هذا
+
+250
+00:31:32,820 --> 00:31:37,830
+عبارة عن Singleton Uأنا عندي set فيها عنصر واحد
+
+251
+00:31:37,830 --> 00:31:42,510
+فال Supreme تبعها هو ال info تبعها هو نفس ال
+
+252
+00:31:42,510 --> 00:31:46,850
+answer يعني هذا واضح من تعريف ال suprem
+
+253
+00:31:54,620 --> 00:31:59,580
+و هذا هو المطلوب اذا هذا تطبيق مباشر على تمرين 9
+
+254
+00:31:59,580 --> 00:32:03,860
+اذا المعناه ان انتوا لازم تحلوا تمرين 9 و هذا
+
+255
+00:32:03,860 --> 00:32:11,260
+التمرين موجود في يعني في رشاد له او hint لحله في
+
+256
+00:32:11,260 --> 00:32:16,680
+خلف .. خلف الكتاب في حل تمرين اللي .. اللي الكتاب
+
+257
+00:32:16,680 --> 00:32:21,280
+بيحاول يعرضها عشان يساعد الطالب نعم تفضلي
+
+258
+00:32:28,890 --> 00:32:37,250
+أه صحيح نعم و
+
+259
+00:32:37,250 --> 00:32:45,170
+في السؤال تسعة و في السؤال إحداش ال 6S
+
+260
+00:32:45,170 --> 00:32:51,010
+من المقطيات bounded صحيح لإنهاحنا فرضين ان S
+
+261
+00:32:51,010 --> 00:32:56,370
+subset من R و ال supremum لل 6S اللي هو S star عدد
+
+262
+00:32:56,370 --> 00:33:06,050
+ينتمي ل S و S subset من R هذا بيقدي ان ال 6S is
+
+263
+00:33:06,050 --> 00:33:12,750
+bounded above على الأقل bounded above تمام؟
+
+264
+00:33:16,370 --> 00:33:22,230
+تمام؟ فلو كانت ال A و ال B bounded above فهيطلع
+
+265
+00:33:22,230 --> 00:33:25,510
+الاتحاد تبعهم bounded above و هذا اللي احنا
+
+266
+00:33:25,510 --> 00:33:30,490
+عايزينه و ال supremum اللي لهم بساوي .. لاتحادهم
+
+267
+00:33:30,490 --> 00:33:37,540
+بساوي الكلام هذا فعلى الأقل .. اه؟و نفس الكلام
+
+268
+00:33:37,540 --> 00:33:41,860
+للانفمام ممكن نثبت حاجة مشابه بالنسبة للانفمام
+
+269
+00:33:41,860 --> 00:33:47,140
+يعني ممكن نثبت ان الانفمام هنا يعني ها and ممكن
+
+270
+00:33:47,140 --> 00:33:58,820
+نضيف انفمام ل a union b بساوي انفمام انف a و انف b
+
+271
+00:34:01,670 --> 00:34:06,630
+فاحنا بس أخدنا .. طبخنا الجزء هذا الجزء بيكون صحيح
+
+272
+00:34:06,630 --> 00:34:13,390
+إذا كانت a و b both are bounded above وبالتالي
+
+273
+00:34:13,390 --> 00:34:16,430
+اتحادهم بيطلع bounded below و ال infimum للاتحاد
+
+274
+00:34:16,430 --> 00:34:23,780
+بيطلع infimum لinfimum المجمعة التانيةفهذا متحقق
+
+275
+00:34:23,780 --> 00:34:28,640
+هنا متحقق ان هاي S star ينتمي ل S وبالتالي عدد
+
+276
+00:34:28,640 --> 00:34:32,420
+حقيقي انها S ال set هذه لها supremum وبالتالي
+
+277
+00:34:32,420 --> 00:34:37,360
+bounded above و single to new ما هي finite set و
+
+278
+00:34:37,360 --> 00:34:41,960
+كل finite set is bounded فهي bounded above و below
+
+279
+00:34:41,960 --> 00:34:47,530
+طبعا وبالتالي ممكن نطبق الجزء هذاهذا برهان برهان
+
+280
+00:34:47,530 --> 00:34:51,790
+تاني ممكن ان احنا نعمل برهان مباشر يعني بلاش
+
+281
+00:34:51,790 --> 00:35:00,970
+نستخدم exercise تسعة تاني
+
+282
+00:35:00,970 --> 00:35:09,310
+ممكن we
+
+283
+00:35:09,310 --> 00:35:13,450
+consider we
+
+284
+00:35:13,450 --> 00:35:15,230
+consider two cases
+
+285
+00:35:18,470 --> 00:35:24,390
+نعتبر حالتين ال S star هذا من المعطيات عدد حقيقي و
+
+286
+00:35:24,390 --> 00:35:31,790
+U عدد حقيقي آخر لا ينتمي ل S فممكن يكون عندي ال U
+
+287
+00:35:31,790 --> 00:35:40,850
+أكبر من أو يساوي S star or ال U أصغر من S star هذا
+
+288
+00:35:40,850 --> 00:35:46,750
+طبعا by trichotomy by trichotomy
+
+289
+00:35:50,710 --> 00:35:58,670
+property من الخاصية الثلاثية U S*) أعداد حقيقية
+
+290
+00:35:58,670 --> 00:36:04,850
+ففي عندي تلت حالات أما U أصغر من S*) أو U أكبر من
+
+291
+00:36:04,850 --> 00:36:10,450
+S*) أو U بساوي S*) هدول حالتين وهذه التالتة
+
+292
+00:36:10,450 --> 00:36:15,950
+فتعالوا في كل حالة نثبت هذا اللي هو المطلوب فإذا
+
+293
+00:36:15,950 --> 00:36:22,180
+في عندي في الحالة الأولىX أقل أو بيساوي من السقر
+
+294
+00:36:22,180 --> 00:36:27,400
+الموجود في ال U أو
+
+295
+00:36:27,400 --> 00:36:33,000
+إيش التانية؟ أو X أقل أو بيساوي ال U X أصغر من أو
+
+296
+00:36:33,000 --> 00:36:38,280
+بيساوي ال U، صح؟ بعدها أنا هقول أكيد إن ال X أقل
+
+297
+00:36:38,280 --> 00:36:45,360
+أو بيساوي من ال .. إن ال X lower bound is lower
+
+298
+00:36:45,360 --> 00:36:45,960
+bound
+
+299
+00:36:49,050 --> 00:37:03,630
+لل set اللي بتتكون من S star و U صح؟ وبالتالي لحظة
+
+300
+00:37:03,630 --> 00:37:09,490
+شوية لو سمحتني اذا
+
+301
+00:37:09,490 --> 00:37:14,830
+ال X lower bound لل set هذي اذا ال infimum
+
+302
+00:37:22,180 --> 00:37:27,840
+الـ X أصغر
+
+303
+00:37:27,840 --> 00:37:36,400
+من أو ساوي الـ infimum ل Sلأ ما هو هذا lower bound
+
+304
+00:37:36,400 --> 00:37:41,960
+ل S star لسنا المجموعة هذه وبالتالي هو أصغر من أو
+
+305
+00:37:41,960 --> 00:37:45,700
+ساوي ال infimum و ال infimum دائما قولنا قبل شوية
+
+306
+00:37:45,700 --> 00:37:51,780
+أصغر من أو ساوي ال supremum لنفس المجموعة لسه
+
+307
+00:37:51,780 --> 00:37:58,160
+متبتيلوا قبل شوية في التمرين السابق صح؟ طيب هيك
+
+308
+00:37:58,160 --> 00:37:59,260
+منكون أثبتنا
+
+309
+00:38:06,750 --> 00:38:17,210
+إذا هذا صحيح since this holds لكل
+
+310
+00:38:17,210 --> 00:38:26,130
+x ينتمي احنا خدنا x عشوائية فهي fix x مظبوط؟ x
+
+311
+00:38:26,130 --> 00:38:33,700
+كانت عنصر عشوائي ف fix x ينتمي ل S unionSingleton
+
+312
+00:38:33,700 --> 00:38:39,260
+U فإذا هذه الأداء صحيح لكل X ينتمي للمجموعة هذه
+
+313
+00:38:39,260 --> 00:38:50,460
+وبالتالي إذا ال supreme ل S star و U is upper
+
+314
+00:38:50,460 --> 00:39:00,300
+bound Upper bound لمن؟ ل 6 S union singleton U
+
+315
+00:39:08,160 --> 00:39:23,180
+مظبوط؟ اذا ال supremum لست S union singleton U لأ
+
+316
+00:39:23,180 --> 00:39:28,280
+مش هيك لأ اذا هذا عبارة عن upper bound لست هذه
+
+317
+00:39:28,280 --> 00:39:34,830
+بنثبت ان هو ال supremumيعني هيك بيطلع هذا .. هذا
+
+318
+00:39:34,830 --> 00:39:40,610
+upper bound ل 6 هذه لأن هذا بيطلع أكبر من أو ساوي
+
+319
+00:39:40,610 --> 00:39:49,610
+.. هذا أصغر من أو ساوي ال supremum ل
+
+320
+00:39:49,610 --> 00:39:57,310
+S star و U احنا بدنا مساوية صح؟فبقدرش أستنتج
+
+321
+00:39:57,310 --> 00:40:03,070
+مساواة هنا تمام؟ أما شو ممكن أما زي ما عملنا في
+
+322
+00:40:03,070 --> 00:40:07,430
+البراهين السابقة ممكن نثبت ال claim ممكن نثبت
+
+323
+00:40:07,430 --> 00:40:13,070
+المساواة كما يليه أنا عندي هذا .. هذا العدد .. هذا
+
+324
+00:40:13,070 --> 00:40:19,270
+العدد عبارة عن upper bound لل set هذهأحنا عايزين
+
+325
+00:40:19,270 --> 00:40:22,970
+نثبت إن هذا مش upper bound هو ال least upper bound
+
+326
+00:40:22,970 --> 00:40:29,330
+إذا ن claim إن ال supremum
+
+327
+00:40:29,330 --> 00:40:36,590
+لست S union لست هذه هو العدد هذا
+
+328
+00:40:49,020 --> 00:41:02,440
+انشوف let V be any upper bound لست S union
+
+329
+00:41:02,440 --> 00:41:11,840
+singleton U هذا بيقدي ان X أصغر من أو بساوي او هذا
+
+330
+00:41:11,840 --> 00:41:12,640
+بيقدي ان
+
+331
+00:41:25,690 --> 00:41:38,530
+هذا بيقدي أن x أصغر من أو يساوي S لكل x في S and
+
+332
+00:41:38,530 --> 00:41:43,990
+x أصغر من أو يساوي لأ
+
+333
+00:41:46,040 --> 00:41:53,780
+عفوا إيش هذا؟ X أصغر من أو ساوي V لكل X في S and U
+
+334
+00:41:53,780 --> 00:41:57,120
+أصغر من أو ساوي V صح؟
+
+335
+00:42:02,420 --> 00:42:05,840
+طيب، معناته هذا upper bound، ال V upper bound للست
+
+336
+00:42:05,840 --> 00:42:13,880
+S إذن ال supremum للست S اللي هو S star بطلع أصغر
+
+337
+00:42:13,880 --> 00:42:22,600
+من أو ساوى V and U أصغر من أو ساوى V معناته إن ال
+
+338
+00:42:22,600 --> 00:42:30,660
+V is upper bound Upper bound لمين؟ للست
+
+339
+00:42:33,070 --> 00:42:39,670
+اللي هي S star و U صح؟ لأن هاي V أكبر من أو يساوي
+
+340
+00:42:39,670 --> 00:42:48,670
+S star و أكبر من أو يساوي ال U فهذا
+
+341
+00:42:48,670 --> 00:42:55,990
+بيقدي إذا ال supremum إذا كان ال V upper bound لل
+
+342
+00:42:55,990 --> 00:43:10,590
+6 هذه فال supremumللست هذي اللي هي S star و U أصغر
+
+343
+00:43:10,590 --> 00:43:17,270
+من أو ساوي ال V هذا أكبر upper bound للست وهذا
+
+344
+00:43:17,270 --> 00:43:21,490
+upper bound لنفس الست لأن أصغر upper bound أصغر من
+
+345
+00:43:21,490 --> 00:43:23,050
+أو ساوي أي upper bound
+
+346
+00:43:26,490 --> 00:43:33,690
+وبالتالي هين أثبتنا .. هين أثبتنا أنه ال .. العدد
+
+347
+00:43:33,690 --> 00:43:40,890
+هذا .. العدد هذا .. هذا العدد أثبتنا حاجتين هذا
+
+348
+00:43:40,890 --> 00:43:46,470
+العدد هيه upper bound لمين لل 6 هذه كذلك في ال
+
+349
+00:43:46,470 --> 00:43:51,410
+claim هذا أثبتنا أنه لو أخدت أي upper bound لل 6
+
+350
+00:43:51,410 --> 00:43:57,370
+هذه وسميته Vفهذا العدد أصغر من أو ساوى D، إذن
+
+351
+00:43:57,370 --> 00:44:04,550
+العدد هذا هو أصغر، إذن العدد هذا هو ال supreme لست
+
+352
+00:44:04,550 --> 00:44:10,750
+هذه، إذن هذا this proves
+
+353
+00:44:10,750 --> 00:44:14,110
+the
+
+354
+00:44:14,110 --> 00:44:21,070
+claim الادعاء اللي احنا حكينا عنه وبالتاليهذا
+
+355
+00:44:21,070 --> 00:44:27,310
+بيكون برهان تاني او برهان اخر وزي مزمرتكم اقترحت
+
+356
+00:44:27,310 --> 00:44:33,670
+مافيش داعي لل cases هنا البرهان التاني مبدأ ب X
+
+357
+00:44:33,670 --> 00:44:43,180
+تنتمي لل set هذه وهنا أثبتنا ان العدد هذاهو ال
+
+358
+00:44:43,180 --> 00:44:48,440
+supremum للست هذه او ال supremum للست هذه اللي هي
+
+359
+00:44:48,440 --> 00:44:52,400
+S إتحاد single to new ال supremum إليها exist
+
+360
+00:44:52,400 --> 00:45:00,900
+موجود و بساوي العدد supremum S star و Uهيو هذا
+
+361
+00:45:00,900 --> 00:45:05,240
+العدد upper bound للست هذه و أي upper bound أخر
+
+362
+00:45:05,240 --> 00:45:10,340
+للست طلع أصغر من .. أكبر من أو يساوي العدد هذا
+
+363
+00:45:10,340 --> 00:45:13,520
+وبالتالي هذا هو أصغر upper bound أو super bound
+
+364
+00:45:13,520 --> 00:45:19,780
+نعم هذي؟
+
+365
+00:45:19,780 --> 00:45:23,180
+اه
+
+366
+00:45:23,180 --> 00:45:24,260
+صح
+
+367
+00:45:32,010 --> 00:45:38,490
+عن؟ بينهم or مش end لأ من تعريف .. من تعريف
+
+368
+00:45:38,490 --> 00:45:43,710
+الاتحاد x ينتمي للاتحاد معناته x ينتمي لل .. او ..
+
+369
+00:45:43,710 --> 00:45:47,130
+مش هيك تعريف الاتحاد؟ اه sorry اه ف or مافيش end
+
+370
+00:45:47,130 --> 00:45:51,330
+ليش ال end؟ معرفة انها or بس احنا استنتجنا .. يعني
+
+371
+00:45:51,330 --> 00:45:54,730
+هنا مكان ال end استنتجنا انها upper bound لكن هنا
+
+372
+00:45:54,730 --> 00:45:57,490
+or يعني مش end عشان نستنتج انها x lower bound
+
+373
+00:46:05,960 --> 00:46:10,580
+صحيح يعني لو كانت x أقل من أم يساوي أس أسطر and x
+
+374
+00:46:10,580 --> 00:46:13,860
+أقل من أم يساوي u فإنت صحيح إحنا نستنتج إنه x
+
+375
+00:46:13,860 --> 00:46:18,340
+lower bound للمجموعة أه صحيح كلامك إذا عشان هيك
+
+376
+00:46:18,340 --> 00:46:25,920
+احنا لازم نحدد هل ال u هو بالتالي كان لازم عشان
+
+377
+00:46:25,920 --> 00:46:32,760
+البرهنة ده فعلا يكون صح كان لازم نفصل حالتين فلو
+
+378
+00:46:32,760 --> 00:46:41,400
+كانت هنا ال uلو كانت ال .. ال S star أصغر من أو
+
+379
+00:46:41,400 --> 00:46:45,420
+يساوي ال U دكتور؟
+
+380
+00:46:45,420 --> 00:46:51,540
+نعم مش X هي أصغر أو يساوي ال supremum لل S أو إن
+
+381
+00:46:51,540 --> 00:46:56,060
+ال X أصغر أو يساوي مجموعة ال U الحالة هي كأنا خبرت
+
+382
+00:46:56,060 --> 00:46:59,460
+إن ال X هتكون أصغر أو يساوي ال supremum يا إما
+
+383
+00:46:59,460 --> 00:47:06,300
+supremum لل S أو supremum لل مجموعة ال Uيعني المهم
+
+384
+00:47:06,300 --> 00:47:14,460
+هي هتطلع الـ Supremum لواحدة من المجموع التاني أنا
+
+385
+00:47:14,460 --> 00:47:19,900
+قبل جملة ال X أزيدور أنا قصدي إن أكتر X أصغر أو
+
+386
+00:47:19,900 --> 00:47:28,380
+بيساوي ال Supremum يعني بشكل مجموحة واحدة X أصغر
+
+387
+00:47:28,380 --> 00:47:35,770
+أو بيساوي ال Supremum لأسطر Star يعني هي اللي هولأ
+
+388
+00:47:35,770 --> 00:47:43,570
+هاد أبراهن S أنها أصغر أو نسبة مجموعة بستار كمه
+
+389
+00:47:43,570 --> 00:47:50,620
+قلو يعني لو حضرتيهم المهم هتطلع لل superأه صح لأن
+
+390
+00:47:50,620 --> 00:47:56,760
+ال suprem هذا أكبر من أو ساوي S star و أكبر من أو
+
+391
+00:47:56,760 --> 00:48:02,960
+ساوي ال U و X أصغر من أو ساوي .. لو كانت ال X أصغر
+
+392
+00:48:02,960 --> 00:48:05,980
+من أو ساوي هذا فهي أكيد أصغر من أو ساوي ال suprem
+
+393
+00:48:05,980 --> 00:48:10,780
+و لو كانت ال X أصغر من أو ساوي ال U فهي أكيد أصغر
+
+394
+00:48:10,780 --> 00:48:12,900
+من أو ساوي ال suprem
+
+395
+00:48:17,590 --> 00:48:26,170
+وبالتالي هذا معناه انه الصحيح
+
+396
+00:48:26,170 --> 00:48:34,450
+ففي الحالة هذه اذا ال supreman لست ال star و you
+
+397
+00:48:34,450 --> 00:48:41,610
+is upper bound upper bound للإتحاد
+
+398
+00:48:44,300 --> 00:48:54,800
+bound of S union single to new لأن
+
+399
+00:48:54,800 --> 00:49:03,260
+هذا fixed ماشي الحال فهذا بحل إشكالية و بعديها
+
+400
+00:49:03,260 --> 00:49:07,380
+بنشطب كل الكلام هذا لأ ما هو هذا الكلام يعني هو
+
+401
+00:49:07,380 --> 00:49:15,430
+تقريبا تفسير ل .. بما أن ال ..هذا مالوش داعي صار
+
+402
+00:49:15,430 --> 00:49:23,350
+هذا مالوش داعي وهذه الخطوة بدل ما نكتبها هنا هذا
+
+403
+00:49:23,350 --> 00:49:27,430
+هي إذا مرة تانية إن أيد البرهان الآن يعني البرهان
+
+404
+00:49:27,430 --> 00:49:33,170
+مافي مشكلة ان شاء الله هاي بنثبت X في الاتحاد تبع
+
+405
+00:49:33,170 --> 00:49:38,990
+المجمعتين هذول الآن X تنتمي للست هذه أو تنتمي للست
+
+406
+00:49:38,990 --> 00:49:52,140
+هذه يعني بتساوي LUوبالتالي ال X تنتمي ل S فهي
+
+407
+00:49:52,140 --> 00:49:56,180
+أصغر من أو ساوي ال supremum ل 6S اللي هو S الصغير
+
+408
+00:49:57,460 --> 00:50:04,020
+أو X أصغر من أو يساوي ال U X بالساوي ال U بتقدي ان
+
+409
+00:50:04,020 --> 00:50:08,900
+X أصغر من أو يساوي ال U الان لو أخدت ال suprem ل S
+
+410
+00:50:08,900 --> 00:50:12,920
+أصغر و U طبعا هذه finite set of real numbers وفي
+
+411
+00:50:12,920 --> 00:50:16,780
+تمرين بيقول لو عندي finite set of real numbers فال
+
+412
+00:50:16,780 --> 00:50:21,390
+suprem تبعها موجودو ينتمي لل set و ال infimum
+
+413
+00:50:21,390 --> 00:50:24,630
+تبعها أيضا موجود و ينتمي ل .. يعني يكون عنصر في ال
+
+414
+00:50:24,630 --> 00:50:28,530
+set هذا أحد التمارين اللي طبعا ما عليناهوش لكن
+
+415
+00:50:28,530 --> 00:50:34,090
+بإمكانكم تثبتوه by induction فهذه finally ال set
+
+416
+00:50:34,090 --> 00:50:37,390
+إذا ال suprem تبعها exist إلا أن هذا ال suprem
+
+417
+00:50:37,390 --> 00:50:41,990
+أكبر من أو ساوي S star وبالتالي أكبر من أو ساوي X
+
+418
+00:50:41,990 --> 00:50:46,790
+و هذا ال suprem أكبر من أو ساوي U
+
+419
+00:50:50,610 --> 00:50:55,450
+وبالتالي أكبر من أو يساوي ال X اللي هي U أكبر من
+
+420
+00:50:55,450 --> 00:51:01,150
+أو ساوي، إذا الأن هذا الكلام صحيح لكل X ينتمي
+
+421
+00:51:01,150 --> 00:51:09,230
+للإتحادهذا العدد الان أكبر من أو ساوي كل عناصر ال
+
+422
+00:51:09,230 --> 00:51:13,350
+6 في الاتحاد فهو upper bound لل 6 هذه فهو upper ان
+
+423
+00:51:13,350 --> 00:51:18,770
+العدد هذا upper bound لل 6 هذه الان أثبتنا ان هذا
+
+424
+00:51:18,770 --> 00:51:23,380
+ال upper bound هو أصغر upper bound للاتحادو هي
+
+425
+00:51:23,380 --> 00:51:29,160
+أخدنا أي upper bound عشوائي للاتحاد طلع هذا ال
+
+426
+00:51:29,160 --> 00:51:33,140
+upper bound العشوائي أكبر من أو ساوي العدد هذا
+
+427
+00:51:33,140 --> 00:51:36,720
+اللي بدنا إياه هو ال supremum إذا هذا العدد هو ال
+
+428
+00:51:36,720 --> 00:51:42,940
+supremum للست هذه تمام؟ okay؟ في أي سؤال تاني؟
+
+429
+00:51:42,940 --> 00:51:51,480
+فخلينا نحللنا كمان سؤالينفي ال .. نحل مثلا خليني
+
+430
+00:51:51,480 --> 00:51:54,300
+انا اختارلكم بعض الأسئلة مدام انتوا يعني شاكلكم
+
+431
+00:51:54,300 --> 00:51:59,300
+الا طبعا اذا حد سائل خليني امسح اللوح الأول و نحل
+
+432
+00:51:59,300 --> 00:52:00,240
+كمان سؤالين
+
+433
+00:52:16,370 --> 00:52:21,990
+يعني قبل شوية ذكرنا التمرين
+
+434
+00:52:21,990 --> 00:52:34,770
+هذا التمرين 12 section 2 3 وهذا التمرين بيقول let
+
+435
+00:52:34,770 --> 00:52:51,380
+S بي .. let S بالساوي X1 إلى XNbe any non
+
+436
+00:52:51,380 --> 00:52:58,260
+-empty finite finite
+
+437
+00:52:58,260 --> 00:53:12,080
+set أو subset من R فبنثبت
+
+438
+00:53:12,080 --> 00:53:14,920
+ان ال show
+
+439
+00:53:17,460 --> 00:53:34,980
+in from S و supreme S ينتمي ل S وكذلك
+
+440
+00:53:34,980 --> 00:53:41,720
+ال supreme ل 6S موجود و هو عنصر في 6S
+
+441
+00:53:52,980 --> 00:53:59,400
+Okay إذا ال finite set تبعتي هذه فرضنا أن عناصرها
+
+442
+00:53:59,400 --> 00:54:06,300
+سمينا عناصرها x1, x2 إلى xn لأن هذه set فيها n
+
+443
+00:54:06,300 --> 00:54:18,540
+elements طيب ممكن نرتب العناصر هذهby rearranging
+
+444
+00:54:18,540 --> 00:54:23,200
+indices
+
+445
+00:54:23,200 --> 00:54:27,220
+if
+
+446
+00:54:27,220 --> 00:54:36,520
+necessary اذا كان ضروري we
+
+447
+00:54:36,520 --> 00:54:50,310
+may and dowe may and do assume that
+
+448
+00:54:50,310 --> 00:54:53,890
+x1
+
+449
+00:54:53,890 --> 00:55:04,950
+less than x2 less than less than xn أنا
+
+450
+00:55:04,950 --> 00:55:13,580
+عندي finite set call it x1 إلى xnممكن ان اعيد
+
+451
+00:55:13,580 --> 00:55:20,620
+ترتيب العناصر هذه هى طبعا عداد حقيقية فممكن ان
+
+452
+00:55:20,620 --> 00:55:26,880
+اعيد .. و طبعا كلهم عناصر مش متساوية فممكن
+
+453
+00:55:26,880 --> 00:55:32,200
+اعيد ترتيب او تسمية العناصر هذه المؤشرات تبعات هذه
+
+454
+00:55:32,200 --> 00:55:38,680
+ممكن اعيد ترتيبها بحيث انه يطلع x1 اصغر من x2 اصغر
+
+455
+00:55:38,680 --> 00:55:44,920
+من x3 او هكذا الاكثرهذا ممكن نعمله ولا لأ؟ ممكن
+
+456
+00:55:44,920 --> 00:55:48,380
+الان
+
+457
+00:55:48,380 --> 00:55:54,640
+تعالوا نثبت claim
+
+458
+00:55:54,640 --> 00:56:01,120
+انا بتدعي ان ال minimum لل set S هيطلع بساوي X
+
+459
+00:56:01,120 --> 00:56:08,200
+واحد وهذا ينتمي ل Sيعني بعد ما رتبت العناصر عملت
+
+460
+00:56:08,200 --> 00:56:12,740
+ordering لهم بالطريقة دي فحثبت أن الinfant plus
+
+461
+00:56:12,740 --> 00:56:18,820
+set S بساوي أصغر عنصر في ال set اللي هو X1 و هذا
+
+462
+00:56:18,820 --> 00:56:29,620
+طبعا ينتمي إلى S طيب لبرهان ذلك clearly واضح
+
+463
+00:56:29,620 --> 00:56:40,900
+أن X1 is a lower boundlower bound لست S نظبط لأن
+
+464
+00:56:40,900 --> 00:56:45,740
+X1 أصغر من أو ساوي كل العناصر اللي في الست فهو
+
+465
+00:56:45,740 --> 00:56:51,000
+واضح انه lower bound الان انا بتثبت انه مش بس
+
+466
+00:56:51,000 --> 00:56:54,400
+lower bound هو ال infimum هو ال greatest lower
+
+467
+00:56:54,400 --> 00:57:01,620
+bound اذا هنا now if W is
+
+468
+00:57:04,400 --> 00:57:16,580
+any lower bound .. any lower bound of S فهذا
+
+469
+00:57:16,580 --> 00:57:25,780
+معناه أن W أصغر من أو يساوي Xi لكل I بيساوي 1 2
+
+470
+00:57:25,780 --> 00:57:29,640
+إلى N صح؟
+
+471
+00:57:30,510 --> 00:57:38,370
+و أصغر من أو ساوي كل عناصرها و بالتالي therefore w
+
+472
+00:57:38,370 --> 00:57:44,970
+أصغر من أو ساوي x واحد لأن x واحد هو واحد من عناصر
+
+473
+00:57:44,970 --> 00:57:54,350
+الست إذا أنا عندي الان x واحد is lower bound للستو
+
+474
+00:57:54,350 --> 00:58:00,190
+أي lower bound للست بيطلع أصغر من أو يساوي x واحد
+
+475
+00:58:00,190 --> 00:58:08,770
+اذا by definition ال x واحد اه او ال infimum للست
+
+476
+00:58:08,770 --> 00:58:16,330
+s exist and بيساوي x واحد تمام؟
+
+477
+00:58:16,330 --> 00:58:22,610
+بالمثل ممكن نثبت ال .. اه هنا similarly
+
+478
+00:58:26,410 --> 00:58:33,190
+similarly show that ان انا هاسيبكم بطريقة مشابعة
+
+479
+00:58:34,440 --> 00:58:39,920
+تثبتوا ال claim التاني وهو ان ال supremum لل set S
+
+480
+00:58:39,920 --> 00:58:47,620
+exist و بساوي XN و طبعا هذا بينتمي لل set S و هو
+
+481
+00:58:47,620 --> 00:58:52,040
+المطلوب okay تمام ان هيك بنكون أثبتنا ان اي finite
+
+482
+00:58:52,040 --> 00:58:56,920
+set لها supremum لها infimum و هدولة بيطلعوا عناصر
+
+483
+00:58:56,920 --> 00:59:01,960
+فيها بالتحديد ال infimum هو ال least element اصغر
+
+484
+00:59:01,960 --> 00:59:07,600
+عنصرفي ال set و ال supremum هو ال greatest element
+
+485
+00:59:07,600 --> 00:59:12,480
+اللي هو أكبر أنصار في ال setهذا طبعا الكلام مش
+
+486
+00:59:12,480 --> 00:59:16,360
+صحيح إذا ال set S كانت infinite هذا بس صحيح في
+
+487
+00:59:16,360 --> 00:59:22,600
+حالة ال finite set إذا ال .. هذا بيكون بيكمل برهان
+
+488
+00:59:22,600 --> 00:59:30,220
+التمرين هذا و بالتالي بنكتفي بحل أو بهذا القدر من
+
+489
+00:59:30,220 --> 00:59:34,260
+حل التمرين و ان شاء الله أسبوع الجاي بنكمل حل
+
+490
+00:59:34,260 --> 00:59:35,400
+تمرين أخرى
+
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/8Xs3EWM1_9g.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/8Xs3EWM1_9g.srt
new file mode 100644
index 0000000000000000000000000000000000000000..e3c2bd791a26adf88a5be726c2058891f65f2110
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/8Xs3EWM1_9g.srt
@@ -0,0 +1,895 @@
+1
+00:00:21,620 --> 00:00:25,660
+طيب نأخذ أمثلة
+
+2
+00:00:25,660 --> 00:00:31,280
+كيف نجيب الـ supremum والـ infimum لمجموعات جزئية
+
+3
+00:00:31,280 --> 00:00:36,300
+من مجموعة الأعداد الحقيقية، فلو أخذت الفترة المغلقة
+
+4
+00:00:36,300 --> 00:00:42,660
+من صفر لواحد، فعايز أفت claim هنا، أدعي أن الـ
+
+5
+00:00:42,660 --> 00:00:48,540
+supremum للست هو واحد، البرهان لذلك حسب تعريف الـ
+
+6
+00:00:48,540 --> 00:00:53,320
+supremum اللي هو least upper bound لازم أثبت شرطين
+
+7
+00:00:53,320 --> 00:00:59,140
+أول شيء الواحد upper bound للـ S، وهذا صحيح، واضح، واحد 
+
+8
+00:00:59,140 --> 00:01:03,860
+is upper bound لمجموعة S لأن الواحد أكبر من أو 
+
+9
+00:01:03,860 --> 00:01:08,930
+يساوي كل العناصر اللي في الفترة، صح؟ إذًا واحد upper 
+
+10
+00:01:08,930 --> 00:01:13,170
+bound، الآن لإثبات أن واحد هو أصغر upper bound الـ
+
+11
+00:01:13,170 --> 00:01:16,950
+supremum، يعني لازم أثبت إنه واحد أصغر من أو يساوي
+
+12
+00:01:16,950 --> 00:01:25,170
+أي upper bound، فلو أخذنا V، V any upper bound، فالـ V
+
+13
+00:01:25,170 --> 00:01:28,310
+أكبر من أو يساوي كل العناصر اللي هنا، من ضمنها
+
+14
+00:01:28,310 --> 00:01:33,530
+الواحد، إذًا الـ V أكبر من أو يساوي الـ واحد، الآن واحد 
+
+15
+00:01:33,530 --> 00:01:38,230
+upper bound والواحد أصغر من أو يساوي أي upper bound
+
+16
+00:01:38,230 --> 00:01:43,910
+V، إذًا الـ واحد هو الـ supremum، إذًا هيك أثبتنا إن 
+
+17
+00:01:43,910 --> 00:01:49,390
+واحد هو الـ supremum، بالمثل ممكن إثبات إن العنصر أو 
+
+18
+00:01:49,390 --> 00:01:54,170
+العدد صفر هو الـ infimum للفترة المغلقة من صفر إلى 
+
+19
+00:01:54,170 --> 00:02:00,850
+واحد، طيب مثال ثاني لو أخذت T هي الفترة المفتوحة من
+
+20
+00:02:00,850 --> 00:02:11,950
+0 لـ 1، فبرضه كمان لو 
+
+21
+00:02:11,950 --> 00:02:18,030
+أخذت T هي الفترة المفتوحة من 0 لـ 1، فممكن إثبات أن
+
+22
+00:02:18,030 --> 00:02:23,970
+الـ supremum لـ T هو 1، واضح إن الواحد upper bound
+
+23
+00:02:23,970 --> 00:02:29,030
+للست، للفترة المفتوحة، لأن واحد أكبر من أو يساوي كل
+
+24
+00:02:29,030 --> 00:02:34,390
+الـ X اللي هنا، هذا واضح، الآن لإثبات أن الواحد هذا
+
+25
+00:02:34,390 --> 00:02:37,310
+هو الـ supremum، في لمّة واحد اتناش خدناها المرة
+
+26
+00:02:37,310 --> 00:02:42,070
+اللي فاتت، بتقول عشان الـ upper bound واحد يكون هو
+
+27
+00:02:42,070 --> 00:02:47,310
+الـ supremum لازم أثبت إنه في شرط لكل ابسلون أكبر 
+
+28
+00:02:47,310 --> 00:02:56,120
+من الصفر يوجد عنصر S، Y في الست S أو T هنا، بحيث إنه
+
+29
+00:02:56,120 --> 00:03:02,300
+واحد سالب الـ epsilon أصغر من S، epsilon، فهنثبت
+
+30
+00:03:02,300 --> 00:03:07,900
+الكلام هذا، إذًا هنا هينبدأ let epsilon أكبر من 
+
+31
+00:03:07,900 --> 00:03:11,940
+الصفر be given، لأن الـ epsilon هذا ممكن يكون أصغر
+
+32
+00:03:11,940 --> 00:03:17,980
+من أو يساوي الواحد أو أكبر من أو أكبر من الواحد
+
+33
+00:03:20,030 --> 00:03:22,970
+الإبسلون هذا عدد موجب، ممكن جدًا يكون أصغر من أو
+
+34
+00:03:22,970 --> 00:03:26,170
+يساوي الواحد أو أكبر من واحد، نأخذ الحالة الأولى، لو
+
+35
+00:03:26,170 --> 00:03:30,770
+إبسلون أصغر من أو يساوي الواحد، فحأخذ S إبسلون، أعرف
+
+36
+00:03:30,770 --> 00:03:36,330
+S إبسلون، واحد سالب إبسلون على اتنين، هذا العدد
+
+37
+00:03:36,330 --> 00:03:41,350
+بيطلع عدد أكبر من صفر وأصغر من واحد، وبالتالي ينتمي
+
+38
+00:03:41,350 --> 00:03:45,510
+للتين، الآن
+
+39
+00:03:45,510 --> 00:03:53,380
+لو أخذت واحد وطرحت منها إبسلون، فهذا بيطلع أصغر يعني
+
+40
+00:03:53,380 --> 00:03:59,840
+لو أخذت واحد وطرحت منها epsilon فهذا أصغر من واحد 
+
+41
+00:03:59,840 --> 00:04:06,500
+سالب epsilon على اتنين، هذا طرحت منه عدد أكبر من هذا
+
+42
+00:04:06,500 --> 00:04:17,080
+لذا هذا أصغر من الثاني، وبعدين ليش يقصر؟ طب 
+
+43
+00:04:17,080 --> 00:04:25,100
+ما هذا هو S إبسلون، هذا هو سإبسلون، إذا
+
+44
+00:04:25,100 --> 00:04:30,160
+في الحالة هذه لأي إبسلون أكبر من الصفر، هين أثبتت 
+
+45
+00:04:30,160 --> 00:04:36,740
+إن يوجد S إبسلون في T، وهذا الـ S إبسلون أكبر من 
+
+46
+00:04:36,740 --> 00:04:40,600
+واحد سالب إبسلون أو واحد سالب إبسلون أصغر من S
+
+47
+00:04:40,600 --> 00:04:47,480
+إبسلون، هذا هو الشرط اللي في لمّة واحد اتناش، هينتقل
+
+48
+00:04:48,090 --> 00:04:52,170
+الحالة الثانية، لو كان إمسنان أكبر من واحد فأكيد
+
+49
+00:04:52,170 --> 00:04:56,050
+واحد سالب إمسنان هيطلع عدد سالب، يعني أصغر من صفر،
+
+50
+00:04:56,050 --> 00:05:01,930
+والـ X هذا .. الـ X هذا لو أخذت أي X في T فأي X في T
+
+51
+00:05:01,930 --> 00:05:06,300
+موجب، أي X في T موجب، إذًا هين أثبتنا في الحالة
+
+52
+00:05:06,300 --> 00:05:13,160
+الثانية إنه لو كان epsilon أكبر من واحد فبطلع مش
+
+53
+00:05:13,160 --> 00:05:18,620
+يوجد S epsilon واحد في T، كل عناصر الـ T بتحقق إنه
+
+54
+00:05:18,620 --> 00:05:24,120
+واحد سالب epsilon أصغر من S أو S epsilon، وبالتالي
+
+55
+00:05:24,120 --> 00:05:28,420
+في كلتا الحالتين الـ both cases الشرط تبع لما واحد 
+
+56
+00:05:28,420 --> 00:05:33,490
+اتناشر تبع الـ supremum اللي بكافئ الـ supremum متحقق
+
+57
+00:05:33,490 --> 00:05:39,810
+وبالتالي واحد هو الـ supremum لـ T، مثال
+
+58
+00:05:39,810 --> 00:05:46,710
+ثالث، احنا شفنا قبل شوية في بداية المحاضرة إن كل
+
+59
+00:05:46,710 --> 00:05:51,510
+عدد حقيقي هو upper bound وكذلك lower bound
+
+60
+00:05:51,510 --> 00:05:57,070
+للمجموعة الخالية Φ، وبناء على ذلك Φ does not
+
+61
+00:05:57,070 --> 00:06:00,730
+have a supremum ولا infimum
+
+62
+00:06:03,600 --> 00:06:14,960
+هي برهان Φ has no .. Φ has no supremum، البرهان
+
+63
+00:06:14,960 --> 00:06:19,380
+proof، assume 
+
+64
+00:06:19,380 --> 00:06:24,240
+that
+
+65
+00:06:24,240 --> 00:06:32,620
+u belong to R is supremum Φ، الـ least upper bound
+
+66
+00:06:32,620 --> 00:06:33,120
+لـ Φ
+
+67
+00:06:40,890 --> 00:06:53,830
+then u سالب واحد أصغر من u، and u سالب واحد هاد عدد
+
+68
+00:06:53,830 --> 00:07:00,610
+حقيقي is upper bound
+
+69
+00:07:00,610 --> 00:07:13,110
+of الـ Φ، كمان مرة نفرض إن U جد U، نفرض
+
+70
+00:07:13,110 --> 00:07:21,590
+إن U جد U جد U بالنمط R وهو Supremum لـ Φ، طيب U
+
+71
+00:07:21,590 --> 00:07:27,000
+سالب واحد أصغر من U، وقبل شوية كنا ملاحظة إن أي عدد
+
+72
+00:07:27,000 --> 00:07:32,440
+حقيقي زي هذا عبارة عن upper bound لـ Φ، فـ K في الـ
+
+73
+00:07:32,440 --> 00:07:37,080
+U .. K في الـ U هو الـ supremum، K في الـ U هو الـ
+
+74
+00:07:37,080 --> 00:07:40,580
+supremum هو أصغر upper bound، وفي upper bound أصغر
+
+75
+00:07:40,580 --> 00:07:47,260
+منه، هذا بدّيني تناقض which
+
+76
+00:07:47,260 --> 00:07:52,340
+.. which is a contradiction
+
+77
+00:07:59,520 --> 00:08:04,320
+إن هذا بدّيني تناقض، وبالتالي هذا إثبات أن الـ Fi
+
+78
+00:08:04,320 --> 00:08:10,700
+مالهاش Supremum، بالمثل ممكن إثبات أن الـ Fi أو 
+
+79
+00:08:10,700 --> 00:08:20,420
+المجموعة الخالية ليس لها Supremum، طيب
+
+80
+00:08:20,420 --> 00:08:22,620
+نيجي للـ completeness property
+
+81
+00:08:29,610 --> 00:08:34,370
+الـ completeness property of R بتنص على إنه كل
+
+82
+00:08:34,370 --> 00:08:40,990
+مجموعة غير خالية .. كل مجموعة غير خالية S من R و
+
+83
+00:08:40,990 --> 00:08:45,010
+bounded above .. و bounded above محدودة من أعلى
+
+84
+00:08:45,010 --> 00:08:50,430
+has supremum، لازم يكون فيه لها supremum، يعني مثال
+
+85
+00:08:50,430 --> 00:08:57,580
+على ذلك لو أخذنا S بسبب الفترة المغلقة 0،1 أو الفترة
+
+86
+00:08:57,580 --> 00:09:04,960
+المفتوحة من صفر، واحد فهي هذي set و bounded above، إذا
+
+87
+00:09:04,960 --> 00:09:10,960
+الـ property بتقولي بتضمنلي، تضمن إن هذي الـ set لها
+
+88
+00:09:10,960 --> 00:09:15,840
+supremum اللي هو الواحد اللي اثبتناه قبل شوية، إذا 
+
+89
+00:09:15,840 --> 00:09:19,700
+الـ property بتضمن وجود supremum، لكن ما بتجيبليها
+
+90
+00:09:19,700 --> 00:09:26,050
+ولا بتقولي إيش هو؟ عشان نجيبه لازم نعمل برهان زي ما
+
+91
+00:09:26,050 --> 00:09:30,310
+شفنا في الأمثلة السابقة، هد هي الـ supremum أو الـ
+
+92
+00:09:30,310 --> 00:09:33,790
+completeness property، خاصية التمام للأعداد 
+
+93
+00:09:33,790 --> 00:09:38,510
+الحقيقية، الآن زي ما قلتلكم قبل هيك في توقع ما بين
+
+94
+00:09:38,510 --> 00:09:42,130
+الـ upper bounds والـ lower bounds، الـ supremums والـ
+
+95
+00:09:42,130 --> 00:09:52,510
+infimums، فالـ .. الـ .. أي خاصية صحيحة للـ supremum
+
+96
+00:09:52,510 --> 00:09:58,170
+بتكون في مقابلها خاصية صحيحة للـ infimum، ففي نتيجة 
+
+97
+00:09:58,170 --> 00:10:03,640
+هنا على completeness property، corollary بنسميها الـ
+
+98
+00:10:03,640 --> 00:10:07,580
+infimum property of R، لإن في supremum property of 
+
+99
+00:10:07,580 --> 00:10:12,260
+R وفي بقبلها infimum property of R، فالـ infimum
+
+100
+00:10:12,260 --> 00:10:16,160
+property of R بتقول إن every non-empty subset S of 
+
+101
+00:10:16,160 --> 00:10:21,160
+R which is bounded below has an infimum، يعني كل
+
+102
+00:10:21,160 --> 00:10:26,440
+مجموعة غير خالية من العداد الحقيقية ومحصورة من
+
+103
+00:10:26,440 --> 00:10:30,460
+أسفل لازم يكون لها infimum أو أكبر حد أدنى
+
+104
+00:10:38,820 --> 00:10:45,060
+وهي البرهان .. نشوف البرهان تبع الـ .. الـ corollary
+
+105
+00:10:45,060 --> 00:10:54,520
+أو النتيجة هذه، بنعرف set .. بنعرف الـ set E على 
+
+106
+00:10:54,520 --> 00:10:59,120
+أنها كل العناصر W اللي بتكون lower bound للمجموعة
+
+107
+00:10:59,120 --> 00:11:06,510
+S، طيب by hypothesis حسب الفرض الـ E مجموعة غير
+
+108
+00:11:06,510 --> 00:11:09,610
+خالية، يعني فيها على الأقل عنصر، ليه؟ لإن احنا
+
+109
+00:11:09,610 --> 00:11:16,090
+فرضين إن المجموعة S، المجموعة S هذه bounded below،
+
+110
+00:11:16,090 --> 00:11:19,710
+يعني إلها lower bound، وبالتالي إذا في على الأقل 
+
+111
+00:11:19,710 --> 00:11:24,350
+عنصر واحد، W في E، إذا الـ E مجموعة غير خالية، 
+
+112
+00:11:24,350 --> 00:11:25,990
+تمام؟ هذا من الفرض
+
+113
+00:11:29,380 --> 00:11:34,720
+كذلك من الفرض أي X في S ثبار عن upper bound لـ E
+
+114
+00:11:34,720 --> 00:11:49,760
+لو كان X ينتمي إلى S فهذا بيقدّي إنه W أصغر من أو 
+
+115
+00:11:49,760 --> 00:11:56,160
+يساوي X لكل W في E
+
+116
+00:12:04,760 --> 00:12:11,300
+ليه هذا الكلام صحيح؟ لأن كل W في E عبارة عن lower
+
+117
+00:12:11,300 --> 00:12:17,300
+bound لـ S، وبما أن W lower bound لـ S فأي عنصر في S
+
+118
+00:12:17,300 --> 00:12:23,480
+بيكون أكبر من أو يساوي الـ lower bound، صح؟ إذًا هذا
+
+119
+00:12:23,480 --> 00:12:28,360
+معناه إن X upper bound، هي X أكبر من أو يساوي كل
+
+120
+00:12:28,360 --> 00:12:33,820
+عناصر الـ E، وبالتالي أي X في S هو عبارة عن 
+
+121
+00:12:40,550 --> 00:12:45,910
+أي x في s هو upper bound للست
+
+122
+00:12:51,680 --> 00:12:57,900
+خاصية التمام، إذا الـ .. الـ set E هذه is bounded
+
+123
+00:12:57,900 --> 00:13:02,580
+above وبالتالي يوجد إلها supremum، الـ supremum تبعها 
+
+124
+00:13:02,580 --> 00:13:08,100
+لو سميته small s exists in R، هذا .. وجود الـ supremum 
+
+125
+00:13:08,100 --> 00:13:14,560
+مضمون باستخدام الـ supremum property، الآن بدنا نثبت إن
+
+126
+00:13:14,560 --> 00:13:21,000
+هذا العدد small s هو الـ infimum، هو الـ infimum
+
+127
+00:13:21,000 --> 00:13:27,100
+للست S، وهيك بنكون كملنا البرهان، إذا الإثبات 
+
+128
+00:13:27,100 --> 00:13:33,580
+للادعاء هذا إن عندي الـ S هنا بساوي supremum E
+
+129
+00:13:33,580 --> 00:13:40,780
+وبالتالي الـ S هذا upper bound لـ E، يعني S أكبر من 
+
+130
+00:13:40,780 --> 00:13:42,340
+أو يساوي كل الـ X في E
+
+131
+00:13:46,050 --> 00:13:52,070
+الآن بناء على المتباينة هذه أو الجملة هذه لإثبات
+
+132
+00:13:52,070 --> 00:13:58,610
+أن S هي الـ infimum لـ Capital S يبقى إثبات إن S
+
+133
+00:13:58,610 --> 00:14:06,830
+عبارة عن lower bound، S is a lower bound of S، ليش
+
+134
+00:14:06,830 --> 00:14:11,350
+هذا يكفي لإثبات إن S هو الـ infimum لـ S؟
+
+135
+00:14:15,610 --> 00:14:20,590
+تعال نشوف ليش هذا يكفي، يكفي
+
+136
+00:14:20,590 --> 00:14:28,850
+إثبات إن الـ S is a lower bound للـ Capital S، يعني بدنا
+
+137
+00:14:28,850 --> 00:14:34,830
+نثبت إن الـ X عفوا
+
+138
+00:14:34,830 --> 00:14:43,410
+الـ S أصغر من أو يساوي كل العناصر Y
+
+139
+00:14:58,200 --> 00:15:03,540
+يعني بدنا نثبت إن S ينتمي
+
+140
+00:15:03,540 --> 00:15:09,980
+للست E يعني 
+
+141
+00:15:09,980 --> 00:15:17,320
+لإثبات أن S is the lower bound of S معناه بدّ أثبت
+
+142
+00:15:17,320 --> 00:15:20,560
+أن S عنصر في E لأن E is the set of all lower
+
+143
+00:15:20,560 --> 00:15:25,380
+bounds of S صح؟ فلو أثبتت أن S تنتمي إلى E
+
+144
+00:15:34,100 --> 00:15:41,300
+فالمفروض هذا معناه أن الـ S .. آه هايه .. لو هذا الـ
+
+145
+00:15:41,300 --> 00:15:47,680
+S .. لو هذا الـ S أثبتت أنه .. لو أثبتت أن الـ S هذا
+
+146
+00:15:47,680 --> 00:15:49,380
+ينتمي إلى E؟
+
+147
+00:15:52,900 --> 00:15:58,420
+فمعناه أن كل العناصر اللي في E أصغر من أو يساوي الـ
+
+148
+00:15:58,420 --> 00:16:04,900
+S طيب كل العناصر X اللي في E هي عبارة عن lower
+
+149
+00:16:04,900 --> 00:16:11,330
+bounds لـ S وإذا كان S موجود في E بيكون أيضًا lower
+
+150
+00:16:11,330 --> 00:16:17,350
+bound لـ S لكن الـ S هذا يتمتع بالخاصية أنه أكبر من
+
+151
+00:16:17,350 --> 00:16:22,970
+أو يساوي كل عناصر الـ set A إذا هو أكبر lower bound
+
+152
+00:16:22,970 --> 00:16:29,560
+يعني هو الـ infimum صح؟ تمام؟ مرة ثانية احنا وصلنا
+
+153
+00:16:29,560 --> 00:16:35,780
+أن الـ X كل العناصر X في E أصغر من أو يساوي S الآن
+
+154
+00:16:35,780 --> 00:16:42,800
+لو أثبتت أن الـ S هذا ينتمي لـ E يعني lower bound لـ
+
+155
+00:16:42,800 --> 00:16:50,130
+S معناه أن الـ S هذا أكبر من أو يساوي كل عناصر الـ E
+
+156
+00:16:50,130 --> 00:16:54,890
+وبالتالي هو أكبر lower
+
+157
+00:16:54,890 --> 00:17:02,450
+bound يعني هو الـ infimum إذا فعلاً يكفي أو يبقى
+
+158
+00:17:02,450 --> 00:17:06,990
+إثبات أن الـ S اسمه الـ S lower bound للـ S فلبرهان
+
+159
+00:17:06,990 --> 00:17:11,770
+ذلك بنعمل برهان بالتناقض افترض أن اللي احنا
+
+160
+00:17:11,770 --> 00:17:18,960
+بنثبته خطأ يعني اسمه الـ S ليس lower bound للـ set S
+
+161
+00:17:18,960 --> 00:17:23,500
+هذا معناه بقدر ألاقي عنصر Y في S وهذا الـ Y أصغر
+
+162
+00:17:23,500 --> 00:17:30,600
+من S لأن S ليس lower bound فهذا بيؤدي .. لاحظوا أن
+
+163
+00:17:30,600 --> 00:17:35,400
+الـ S هو الـ supremum لـ E .. S هو الـ supremum لـ E و
+
+164
+00:17:35,400 --> 00:17:42,980
+Y أصغر منه إذن Y هذا مش ممكن يكون upper bound للـ set
+
+165
+00:17:42,980 --> 00:17:49,920
+E الـ Y أصغر من S و S يساوي supremum E إذا Y مش ممكن
+
+166
+00:17:49,920 --> 00:17:54,740
+يكون upper bound لـ E لأنه ما يجوز هذا يكون upper
+
+167
+00:17:54,740 --> 00:18:00,320
+bound لـ E وهذا أصغر upper bound لـ E صح؟ طيب إذا
+
+168
+00:18:00,320 --> 00:18:05,980
+الـ Y مش ممكن يكون upper bound لـ E إذا بقدر ألاقي X
+
+169
+00:18:05,980 --> 00:18:12,160
+في E وهذا الـ X أكبر من الـ Y هذه المتباينة بتعطيني 
+
+170
+00:18:12,160 --> 00:18:12,840
+تناقض
+
+171
+00:18:16,450 --> 00:18:23,870
+تتناقض مع تعريف الـ set E كيف X تنتمي لـ E كيف الـ X 
+
+172
+00:18:23,870 --> 00:18:29,510
+تنتمي لـ E وفي نفس الوقت X أكبر من عنصر ما اللي 
+
+173
+00:18:29,510 --> 00:18:35,010
+هو Y في S يعني الـ X هذا ليس lower bound هذا تناقض
+
+174
+00:18:35,010 --> 00:18:40,130
+okay إذا نصل إلى تناقض وبالتالي هذا التناقض بيقول
+
+175
+00:18:40,130 --> 00:18:42,990
+لي أن الفرض الفرض تبعنا هذا
+
+176
+00:18:45,580 --> 00:18:50,800
+أن s is not lower bound كان فرض خطأ إذا لازم
+
+177
+00:18:50,800 --> 00:19:01,520
+يكون s lower bound وهذا بيكمل برهان الـ claim تمام؟
+
+178
+00:19:01,520 --> 00:19:08,040
+في
+
+179
+00:19:08,040 --> 00:19:09,500
+الـ section القادم
+
+180
+00:19:12,270 --> 00:19:18,530
+هناخد تطبيقات على الـ supreme property والـ infimum
+
+181
+00:19:18,530 --> 00:19:24,410
+property فالتطبيقات
+
+182
+00:19:24,410 --> 00:19:35,230
+هذه هتكون على شكل أمثلة فمثلاً
+
+183
+00:19:35,230 --> 00:19:43,410
+أول تطبيق لو أخدت أي subset من R و bounded above و
+
+184
+00:19:43,410 --> 00:19:49,510
+A أي عدد حقيقي فبنعرف A زائد S على أنه
+
+185
+00:19:49,510 --> 00:19:54,110
+مجموعة كل العناصر على الصورة A plus X حيث X ينتمي
+
+186
+00:19:54,110 --> 00:20:00,890
+لـ S الآن ممكن أثبت أن الـ supremum للمجموعة هذه هو
+
+187
+00:20:00,890 --> 00:20:04,870
+عبارة عن A زائد الـ supremum لـ S
+
+188
+00:20:07,460 --> 00:20:16,840
+و هذا يعني البرهان مش صعب أيه بسيط وسهل نشوف مع
+
+189
+00:20:16,840 --> 00:20:22,540
+بعض نفترض أن U هو الـ supremum لـ S الـ set S is bounded
+
+190
+00:20:22,540 --> 00:20:28,980
+above، إذن لها supremum هذا مضمون حسب الـ supremum
+
+191
+00:20:28,980 --> 00:20:33,920
+property وبالتالي الـ U هذا اللي هو الـ supremum هو
+
+192
+00:20:33,920 --> 00:20:38,520
+upper bound لـ S إذا U أكبر من أو يساوي كل عناصر الـ
+
+193
+00:20:38,520 --> 00:20:45,800
+S إذا لو ضفت A على الطرفين فبيطلع A زائد X أصغر من
+
+194
+00:20:45,800 --> 00:20:54,270
+أو يساوي A زائد U لكل X في S وبالتالي العدد هذا عبارة
+
+195
+00:20:54,270 --> 00:20:59,830
+عن upper bound لمن؟ لـ set A زائد S اللي عرفناها قبل
+
+196
+00:20:59,830 --> 00:21:04,310
+شوية لأن هذا العدد أكبر من أو يساوي كل عناصر الـ set
+
+197
+00:21:04,310 --> 00:21:08,850
+هذه اللي على الصورة A زائد X لذلك هي اللي أثبتت أن
+
+198
+00:21:08,850 --> 00:21:13,110
+A زائد U is upper bound للـ set هذه لأن نريد أن نثبت
+
+199
+00:21:13,110 --> 00:21:18,510
+أن A زائد U هو أصغر upper bound للـ set هذه فبناخد أي
+
+200
+00:21:18,510 --> 00:21:24,550
+upper bound آخر للـ set A plus S فطبعًا الـ V Upper
+
+201
+00:21:24,550 --> 00:21:30,410
+Bound للـ set هي U أكبر من أو يساوي كل عناصرها الآن
+
+202
+00:21:30,410 --> 00:21:34,430
+انقل الـ A عن ناحية الثانية فبيصير X أصغر من أو يساوي
+
+203
+00:21:34,430 --> 00:21:40,710
+V ناقص A لكل X في S طيب
+
+204
+00:21:40,710 --> 00:21:47,410
+الآن احنا عندنا الـ U هو الـ supremum لـ S الـ U هو الـ
+
+205
+00:21:47,410 --> 00:21:52,800
+supremum لـ S والآن هذا العدد هذا عبارة عن upper
+
+206
+00:21:52,800 --> 00:22:00,200
+bound of S لأن U أكبر من أو يساوي كل عناصر الـ S
+
+207
+00:22:00,200 --> 00:22:07,400
+وهذا أصغر upper bound لـ S إذن الـ supremum بيطلع
+
+208
+00:22:07,400 --> 00:22:13,240
+أصغر من أو يساوي الـ upper bound V ناقص A لـ S إذن
+
+209
+00:22:13,240 --> 00:22:16,080
+بيطلع عندي U أصغر من أو يساوي
+
+210
+00:22:19,910 --> 00:22:26,350
+أن أنا بيطلع عندي U أصغر من أو يساوي V ناقص A ودي A
+
+211
+00:22:26,350 --> 00:22:30,290
+عن ناحية الثانية فبيصير A زائد U أصغر من أو يساوي V
+
+212
+00:22:30,290 --> 00:22:35,870
+إذا هنا أثبتنا حاجتين أول شيء إنه العدد هذا upper
+
+213
+00:22:35,870 --> 00:22:40,590
+bound للـ set هذه أخذنا أي upper bound عشوائي للـ set
+
+214
+00:22:40,590 --> 00:22:47,640
+هذه فبيطلع العدد A زائد U أصغر من أو يساوي أي upper
+
+215
+00:22:47,640 --> 00:22:52,880
+bound لـ set A زائد S إذا من تعريف الـ supremum بيطلع الـ
+
+216
+00:22:52,880 --> 00:23:00,520
+supremum لـ set A زائد S exist وبيساوى A زائد U أن الـ
+
+217
+00:23:00,520 --> 00:23:05,380
+supremum للـ set هذي هو A زائد U وبالتالي وهذا بيساوي
+
+218
+00:23:05,380 --> 00:23:08,720
+A والـ U هي الـ supremum لـ S أننا هيك بنكون أثبتنا
+
+219
+00:23:08,720 --> 00:23:15,900
+أن supremum الـ set A زائد S هو A زائد supremum S،
+
+220
+00:23:15,900 --> 00:23:21,540
+تمام؟ لو كانت الـ set هذي bounded below فممكن أيضًا
+
+221
+00:23:21,540 --> 00:23:26,960
+نثبت أن الـ infimum لـ A زائد S بيساوي A زائد infimum
+
+222
+00:23:26,960 --> 00:23:33,430
+S، تمام؟ طبعًا في أمثلة أخرى هنا ممكن تقرؤوها و
+
+223
+00:23:33,430 --> 00:23:39,650
+تحضروها ونوقف هنا نكتفي بهذا القدر وبنكمل إن شاء
+
+224
+00:23:39,650 --> 00:23:42,170
+الله يوم السبت المحاضرة القادمة
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/8Xs3EWM1_9g_raw.json b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/8Xs3EWM1_9g_raw.json
new file mode 100644
index 0000000000000000000000000000000000000000..23de69e5aea8031e1c333adcd3cf41d545cd99a0
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/8Xs3EWM1_9g_raw.json
@@ -0,0 +1 @@
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supremum لتين مثال تالت احنا شفنا قبل شوية في بداية المحاضرة ان كل عدد حقيقي هو upper bound و كذلك lower bound للمجموع الخالي Phi و بناء على ذلك Phi does not have a supremum ولا infimum", "tokens": [2304, 2655, 5016, 4587, 4587, 46599, 6027, 2655, 6027, 1829, 36764, 24401, 31439, 2423, 23710, 449, 5296, 2655, 9957, 50113, 6027, 6055, 6027, 2655, 1975, 5016, 8315, 13412, 5172, 8315, 12174, 36150, 13412, 2407, 10632, 8978, 4724, 28259, 10632, 9673, 5016, 46958, 25720, 16472, 28242, 6225, 3215, 3215, 11331, 38436, 38436, 31439, 6597, 5472, 4032, 9122, 8848, 23275, 3126, 5472, 5296, 19528, 7435, 2304, 45367, 33962, 6027, 1829, 41435, 4032, 44945, 16606, 15844, 29910, 23275, 41435, 775, 406, 362, 257, 23710, 449, 49429, 1536, 332, 449], "avg_logprob": -0.15939295291900635, "compression_ratio": 1.5311004784688995, "no_speech_prob": 0.0, "words": [{"start": 332.69, "end": 333.49, "word": "متحقق", "probability": 0.96572265625}, {"start": 333.49, "end": 334.21, "word": " وبالتالي", 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423.53, "text": "then u سالب واحد أصغر من u and u سالب واحد هاد عدد حقيقي is upper bound of ال five", "tokens": [19096, 344, 8608, 6027, 3555, 36764, 24401, 5551, 9381, 17082, 2288, 9154, 344, 293, 344, 8608, 6027, 3555, 36764, 24401, 8032, 18513, 6225, 3215, 3215, 11331, 38436, 38436, 307, 6597, 5472, 295, 2423, 1732], "avg_logprob": -0.1779017925262451, "compression_ratio": 1.1938775510204083, "no_speech_prob": 0.0, "words": [{"start": 400.89000000000004, "end": 402.29, "word": "then", "probability": 0.7412109375}, {"start": 402.29, "end": 403.69, "word": " u", "probability": 0.374267578125}, {"start": 403.69, "end": 404.53, "word": " سالب", "probability": 0.7744140625}, {"start": 404.53, "end": 405.07, "word": " واحد", "probability": 0.978759765625}, {"start": 405.07, "end": 405.69, "word": " أصغر", "probability": 0.872802734375}, {"start": 405.69, "end": 405.91, "word": " من", "probability": 0.99365234375}, {"start": 405.91, "end": 406.29, "word": " u", "probability": 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423.53, "word": " five", "probability": 0.52294921875}], "temperature": 1.0}, {"id": 16, "seek": 44315, "start": 425.99, "end": 443.15, "text": "كمان مرة نفرض ان U جد U نفرض ان U جد U جد U بالنمط R و هو Supremum ل Phi طيب U سالب واحد أصغر من U", "tokens": [24793, 7649, 3714, 25720, 8717, 5172, 43042, 16472, 624, 10874, 3215, 624, 8717, 5172, 43042, 16472, 624, 10874, 3215, 624, 10874, 3215, 624, 20666, 1863, 2304, 9566, 497, 4032, 31439, 9141, 2579, 449, 5296, 41435, 23032, 1829, 3555, 624, 8608, 6027, 3555, 36764, 24401, 5551, 9381, 17082, 2288, 9154, 624], "avg_logprob": -0.3844975548632005, "compression_ratio": 1.3043478260869565, "no_speech_prob": 0.0, "words": [{"start": 425.99, "end": 426.43, "word": "كمان", "probability": 0.792724609375}, {"start": 426.43, "end": 426.83, "word": " مرة", "probability": 0.95263671875}, {"start": 426.83, "end": 427.43, "word": " نفرض", "probability": 0.8693033854166666}, {"start": 427.43, "end": 427.65, "word": " ان", "probability": 0.50244140625}, 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472.34, "text": "و قبل شوية كنا ملاحظة ان اي عدد حقيقي زي هذا عبارة عن upper bound لفائي ف K في ال U .. 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supremum أو ال completeness property خاصية التمام للأعداد الحقيقية الآن زي ما قلتلكم قبل هيك في توقع ما بين ال upper bounds و ال lower bounds ال supremums و ال infimums", "tokens": [28814, 1829, 8592, 31439, 22807, 6225, 8592, 7649, 8717, 7435, 1829, 3555, 3224, 5296, 31377, 2304, 8717, 25957, 1211, 4724, 2288, 3224, 7649, 30767, 1829, 19446, 13412, 38688, 8315, 8978, 16247, 2304, 12984, 37977, 21136, 16758, 28671, 8032, 3215, 39896, 2423, 23710, 449, 34051, 2423, 1557, 15264, 4707, 16490, 33546, 10632, 16712, 2304, 10943, 24976, 10721, 22488, 18513, 21542, 38436, 4587, 10632, 6024, 48506, 30767, 1829, 19446, 12174, 1211, 2655, 23275, 2304, 12174, 36150, 39896, 4117, 8978, 33427, 4587, 3615, 19446, 49374, 2423, 6597, 29905, 4032, 2423, 3126, 29905, 2423, 23710, 8099, 4032, 2423, 1536, 332, 8099], "avg_logprob": -0.18000637998386304, "compression_ratio": 1.6359649122807018, "no_speech_prob": 0.0, "words": [{"start": 561.51, "end": 561.89, "word": "إيش", "probability": 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"probability": 0.8765869140625}, {"start": 574.49, "end": 575.73, "word": " الآن", "probability": 0.74560546875}, {"start": 575.73, "end": 575.97, "word": " زي", "probability": 0.9814453125}, {"start": 575.97, "end": 576.03, "word": " ما", "probability": 0.986328125}, {"start": 576.03, "end": 576.53, "word": " قلتلكم", "probability": 0.94873046875}, {"start": 576.53, "end": 576.83, "word": " قبل", "probability": 0.888671875}, {"start": 576.83, "end": 577.25, "word": " هيك", "probability": 0.97314453125}, {"start": 577.25, "end": 577.49, "word": " في", "probability": 0.7939453125}, {"start": 577.49, "end": 578.01, "word": " توقع", "probability": 0.68798828125}, {"start": 578.01, "end": 578.17, "word": " ما", "probability": 0.91650390625}, {"start": 578.17, "end": 578.51, "word": " بين", "probability": 0.9833984375}, {"start": 578.51, "end": 579.35, "word": " ال", "probability": 0.92431640625}, {"start": 579.35, "end": 579.61, "word": " upper", "probability": 0.82666015625}, {"start": 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completeness property corollary", "tokens": [5172, 6027, 4386, 2423, 4386, 1975, 1829, 16490, 33546, 10632, 20328, 5016, 1829, 5016, 3660, 24976, 27756, 39894, 30544, 8978, 4724, 4587, 16758, 1211, 11296, 16490, 33546, 10632, 20328, 5016, 1829, 5016, 3660, 24976, 1536, 332, 449, 6156, 41185, 8717, 31371, 7435, 3660, 34105, 15844, 1557, 15264, 4707, 1181, 1833, 822], "avg_logprob": -0.16466346268470472, "compression_ratio": 1.375, "no_speech_prob": 0.0, "words": [{"start": 585.11, "end": 585.97, "word": "فال", "probability": 0.69482421875}, {"start": 585.97, "end": 586.11, "word": " ..", "probability": 0.7919921875}, {"start": 586.11, "end": 587.67, "word": " ال", "probability": 0.90185546875}, {"start": 587.67, "end": 588.23, "word": " ..", "probability": 0.97802734375}, {"start": 588.23, "end": 589.35, "word": " اي", "probability": 0.648681640625}, {"start": 589.35, "end": 590.11, "word": " خاصية", "probability": 0.96728515625}, {"start": 590.11, "end": 591.95, "word": " صحيحة", 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38, "seek": 103293, "start": 1004.97, "end": 1032.93, "text": "معناته ال S هدى اكبر من او ساوي كل عناصر ال 6E وبالتالي هو اكبر lower bound يعني هو ال infimum اذا فعلا يكفي او يبقى اثبات ان ال S اسمه ال S lower bound لل 6S فلبرهان ذلك بنعمل برهان بالتناقض افرضى انه اللي احنا بنلثبته خطأ", "tokens": [2304, 3615, 8315, 47395, 2423, 318, 8032, 3215, 7578, 1975, 4117, 26890, 9154, 1975, 2407, 8608, 995, 45865, 28242, 18871, 33546, 2288, 2423, 1386, 36, 46599, 6027, 2655, 6027, 1829, 31439, 1975, 4117, 26890, 3126, 5472, 37495, 22653, 31439, 2423, 1536, 332, 449, 1975, 15730, 6156, 3615, 15040, 7251, 4117, 41185, 1975, 2407, 7251, 3555, 4587, 7578, 1975, 12984, 3555, 9307, 16472, 2423, 318, 24525, 2304, 3224, 2423, 318, 3126, 5472, 24976, 1386, 50, 6156, 46152, 2288, 3224, 7649, 29910, 23275, 44945, 25957, 1211, 4724, 2288, 3224, 7649, 20666, 2655, 8315, 4587, 11242, 1975, 5172, 43042, 7578, 16472, 3224, 13672, 1829, 1975, 5016, 8315, 44945, 1211, 12984, 3555, 47395, 16490, 9566, 10721], 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"probability": 0.9921875}, {"start": 1117.83, "end": 1118.55, "word": " وبالتالي", "probability": 0.90107421875}, {"start": 1118.55, "end": 1119.23, "word": " هذا", "probability": 0.9482421875}, {"start": 1119.23, "end": 1119.79, "word": " التناقض", "probability": 0.9921875}, {"start": 1119.79, "end": 1120.13, "word": " بيقول", "probability": 0.9191080729166666}, {"start": 1120.13, "end": 1120.27, "word": " لي", "probability": 0.5966796875}, {"start": 1120.27, "end": 1120.41, "word": " أن", "probability": 0.5966796875}, {"start": 1120.41, "end": 1120.97, "word": " الفرض", "probability": 0.77294921875}, {"start": 1120.97, "end": 1122.05, "word": " الفرض", "probability": 0.820556640625}, {"start": 1122.05, "end": 1122.55, "word": " تبعنا", "probability": 0.9375}, {"start": 1122.55, "end": 1122.99, "word": " هذا", "probability": 0.96923828125}], "temperature": 1.0}, {"id": 42, "seek": 114950, "start": 1125.58, "end": 1149.5, "text": "إن small s is not lower bound كان فرض خطأ إذا لازم يكون s lower bound وهذا بيكمل برهان ال claim تمام؟ في ال section القادم", "tokens": [28814, 1863, 1359, 262, 307, 406, 3126, 5472, 25961, 6156, 43042, 16490, 9566, 10721, 11933, 15730, 5296, 31377, 2304, 7251, 30544, 262, 3126, 5472, 37037, 15730, 4724, 1829, 24793, 1211, 4724, 2288, 3224, 7649, 2423, 3932, 46811, 10943, 22807, 8978, 2423, 3541, 25062, 18513, 2304], "avg_logprob": -0.23963995277881622, "compression_ratio": 1.2681159420289856, "no_speech_prob": 0.0, "words": [{"start": 1125.58, "end": 1125.94, "word": "إن", "probability": 0.6231689453125}, {"start": 1125.94, "end": 1126.48, "word": " small", "probability": 0.6396484375}, {"start": 1126.48, "end": 1127.04, "word": " s", "probability": 0.6513671875}, {"start": 1127.04, "end": 1128.26, "word": " is", "probability": 0.85302734375}, {"start": 1128.26, "end": 1128.6, "word": " not", "probability": 0.958984375}, {"start": 1128.6, "end": 1128.94, "word": " lower", "probability": 0.88623046875}, {"start": 1128.94, "end": 1129.28, "word": " bound", "probability": 0.87646484375}, {"start": 1129.28, "end": 1129.52, "word": " كان", "probability": 0.86669921875}, {"start": 1129.52, "end": 1129.84, "word": " فرض", "probability": 0.6729736328125}, {"start": 1129.84, "end": 1130.18, "word": " خطأ", "probability": 0.9886067708333334}, {"start": 1130.18, "end": 1130.42, "word": " إذا", "probability": 0.52008056640625}, {"start": 1130.42, "end": 1130.8, "word": " لازم", "probability": 0.9347330729166666}, {"start": 1130.8, "end": 1131.08, "word": " يكون", "probability": 0.97705078125}, {"start": 1131.08, "end": 1131.34, "word": " s", "probability": 0.4423828125}, {"start": 1131.34, "end": 1131.66, "word": " lower", "probability": 0.80810546875}, {"start": 1131.66, "end": 1132.1, "word": " bound", "probability": 0.88330078125}, {"start": 1132.1, "end": 1132.48, "word": " وهذا", "probability": 0.84130859375}, {"start": 1132.48, "end": 1133.16, "word": " بيكمل", "probability": 0.824951171875}, {"start": 1133.16, "end": 1134.3, "word": " برهان", "probability": 0.959228515625}, {"start": 1134.3, "end": 1134.44, "word": " ال", "probability": 0.9013671875}, {"start": 1134.44, "end": 1134.78, "word": " claim", "probability": 0.76806640625}, {"start": 1134.78, "end": 1141.52, "word": " تمام؟", "probability": 0.6355794270833334}, {"start": 1141.52, "end": 1148.04, "word": " في", "probability": 0.67578125}, {"start": 1148.04, "end": 1148.18, "word": " ال", "probability": 0.900390625}, {"start": 1148.18, "end": 1148.62, "word": " section", "probability": 0.93408203125}, {"start": 1148.62, "end": 1149.5, "word": " القادم", "probability": 0.9938151041666666}], "temperature": 1.0}, {"id": 43, "seek": 117627, "start": 1152.27, "end": 1176.27, "text": "هناخد تطبيقات على الـ supreme property و ال infame property فالتطبيقات هذه هتكون على شكل أمثلة فمثلا أول تطبيق", "tokens": [3224, 1863, 47283, 3215, 6055, 9566, 21292, 4587, 9307, 15844, 2423, 39184, 27756, 4707, 4032, 2423, 1536, 529, 4707, 6156, 6027, 2655, 9566, 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أي عدد حقيقي فمنعرف A زائد capital S على أنه مجموعة كل العناصر على الصورة A plus X حيث X ينتمي لS الآن ممكن أثبت أن ال supremum للمجموعة هذه هو عبارة عن A زائد ال supremum لS", "tokens": [1211, 2407, 5551, 9778, 3215, 2655, 36632, 25993, 9154, 497, 4032, 37498, 3673, 4032, 316, 36632, 6225, 3215, 3215, 11331, 38436, 38436, 6156, 27842, 3615, 28480, 316, 30767, 16373, 3215, 4238, 318, 15844, 14739, 3224, 3714, 7435, 2304, 2407, 27884, 28242, 18863, 8315, 9381, 2288, 15844, 31767, 13063, 3660, 316, 1804, 1783, 11331, 1829, 12984, 1783, 7251, 29399, 2304, 1829, 5296, 50, 6024, 48506, 3714, 43020, 5551, 12984, 3555, 2655, 14739, 2423, 23710, 449, 5296, 19528, 7435, 2304, 2407, 27884, 29538, 31439, 6225, 3555, 9640, 3660, 18871, 316, 30767, 16373, 3215, 2423, 23710, 449, 5296, 50], "avg_logprob": -0.17992912862718718, "compression_ratio": 1.5395348837209302, "no_speech_prob": 0.0, "words": [{"start": 1177.33, "end": 1177.67, "word": "لو", "probability": 0.81640625}, {"start": 1177.67, 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diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/BdWUrxEOLII_raw.json b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/BdWUrxEOLII_raw.json
new file mode 100644
index 0000000000000000000000000000000000000000..318bb36bb1c563a82e5b64fc466f9ee6fe9b337a
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/BdWUrxEOLII_raw.json
@@ -0,0 +1 @@
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هذا يعني مش يعني سيء ومش غلط فهنبدأ .. احنا هناخد أربع شباتر في المادة هذه لكن الشباتر مش متساوية الشبتر الأول بتحدث عن ال real number system أو نظام الأعداد الحقيقية", "tokens": [8297, 6156, 40228, 22653, 23758, 37495, 22653, 37893, 37495, 22653, 8608, 1829, 38207, 4032, 2304, 8592, 32771, 1211, 9566, 6156, 3224, 1863, 44510, 10721, 4386, 1975, 5016, 8315, 8032, 1863, 47283, 3215, 5551, 25513, 3615, 13412, 3555, 9307, 2288, 8978, 9673, 18513, 3660, 29538, 44381, 25124, 3555, 9307, 2288, 37893, 44650, 3794, 995, 2407, 10632, 25124, 3555, 2655, 2288, 16247, 12610, 39894, 24401, 12984, 18871, 2423, 957, 1230, 1185, 34051, 8717, 19913, 10943, 16247, 22488, 18513, 21542, 38436, 4587, 10632], "avg_logprob": -0.11332947530864197, "compression_ratio": 1.5904255319148937, "no_speech_prob": 0.0, "words": [{"start": 742.48, "end": 743.02, "word": "Okay", "probability": 0.90087890625}, {"start": 743.02, "end": 743.74, "word": " فيعني", "probability": 0.82958984375}, {"start": 743.74, "end": 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new file mode 100644
index 0000000000000000000000000000000000000000..940b63a3bb140242b0d8c36bb621eab2968566c2
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@@ -0,0 +1 @@
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" C", "probability": 0.97607421875}], "temperature": 1.0}, {"id": 8, "seek": 22853, "start": 205.45, "end": 228.53, "text": "وهي البرهان proof to show مثلا ال function fg is continuous at c", "tokens": [2407, 3224, 1829, 2423, 26890, 3224, 7649, 8177, 281, 855, 50113, 15040, 2423, 2445, 283, 70, 307, 10957, 412, 269], "avg_logprob": -0.24125743479955764, "compression_ratio": 0.963855421686747, "no_speech_prob": 0.0, "words": [{"start": 205.45, "end": 206.11, "word": "وهي", "probability": 0.6007486979166666}, {"start": 206.11, "end": 206.75, "word": " البرهان", "probability": 0.9329833984375}, {"start": 206.75, "end": 207.37, "word": " proof", "probability": 0.45458984375}, {"start": 207.37, "end": 218.19, "word": " to", "probability": 0.6875}, {"start": 218.19, "end": 218.75, "word": " show", "probability": 0.97119140625}, {"start": 218.75, "end": 220.69, "word": " مثلا", "probability": 0.9775390625}, {"start": 220.69, "end": 220.89, "word": " ال", "probability": 0.82080078125}, 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"probability": 0.876953125}, {"start": 251.15, "end": 251.45, "word": " دالك", "probability": 0.76220703125}, {"start": 251.45, "end": 252.07, "word": " بتثبت", "probability": 0.97412109375}, {"start": 252.07, "end": 252.29, "word": " ان", "probability": 0.8818359375}, {"start": 252.29, "end": 254.03, "word": " الشرط", "probability": 0.7620442708333334}, {"start": 254.03, "end": 254.27, "word": " هذا", "probability": 0.7431640625}, {"start": 254.27, "end": 254.51, "word": " تبع", "probability": 0.7897135416666666}, {"start": 254.51, "end": 254.99, "word": " الاتصال", "probability": 0.88134765625}, {"start": 254.99, "end": 255.11, "word": " على", "probability": 0.44580078125}, {"start": 255.11, "end": 255.47, "word": " النقطة", "probability": 0.9685872395833334}, {"start": 255.47, "end": 256.11, "word": " بتحقق", "probability": 0.8941650390625}, {"start": 256.11, "end": 257.15, "word": " فتعالى", "probability": 0.933203125}, {"start": 257.15, "end": 257.55, "word": " نشوف", "probability": 0.9954427083333334}, {"start": 257.55, "end": 258.09, "word": " high", "probability": 0.350341796875}, {"start": 258.09, "end": 258.57, "word": " limit", "probability": 0.93310546875}, {"start": 258.57, "end": 260.71, "word": " F", "probability": 0.828125}, {"start": 260.71, "end": 261.13, "word": " ضرب", "probability": 0.96728515625}, {"start": 261.13, "end": 261.49, "word": " G", "probability": 0.95263671875}], "temperature": 1.0}, {"id": 10, "seek": 29213, "start": 263.27, "end": 292.13, "text": "عند X لما X تقول لـC بنثبت أن هذا بيساوي FG عند C فهذا بيساوي limit F of X ضرب G of X as X tends to C هذا من تعريف حصل ضرب اخترانين وهذا بيساوي أنا عندي limit F of X لما X تقول لـC exist", "tokens": [3615, 1863, 3215, 1783, 5296, 15042, 1783, 6055, 39648, 5296, 39184, 34, 44945, 12984, 3555, 2655, 14739, 23758, 4724, 1829, 3794, 995, 45865, 479, 38, 43242, 383, 6156, 3224, 15730, 4724, 1829, 3794, 995, 45865, 4948, 479, 295, 1783, 48812, 25513, 460, 295, 1783, 382, 1783, 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0.6181640625}, {"start": 267.55, "end": 267.89, "word": " هذا", "probability": 0.92041015625}, {"start": 267.89, "end": 268.57, "word": " بيساوي", "probability": 0.73271484375}, {"start": 268.57, "end": 269.41, "word": " FG", "probability": 0.724853515625}, {"start": 269.41, "end": 269.93, "word": " عند", "probability": 0.255615234375}, {"start": 269.93, "end": 270.21, "word": " C", "probability": 0.970703125}, {"start": 270.21, "end": 273.19, "word": " فهذا", "probability": 0.9161783854166666}, {"start": 273.19, "end": 273.81, "word": " بيساوي", "probability": 0.97958984375}, {"start": 273.81, "end": 274.33, "word": " limit", "probability": 0.771484375}, {"start": 274.33, "end": 275.67, "word": " F", "probability": 0.76220703125}, {"start": 275.67, "end": 275.97, "word": " of", "probability": 0.69775390625}, {"start": 275.97, "end": 276.45, "word": " X", "probability": 0.9951171875}, {"start": 276.45, "end": 277.33, "word": " ضرب", "probability": 0.880615234375}, {"start": 277.33, 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{"start": 283.31, "end": 283.95, "word": " اخترانين", "probability": 0.884765625}, {"start": 283.95, "end": 285.49, "word": " وهذا", "probability": 0.771728515625}, {"start": 285.49, "end": 286.29, "word": " بيساوي", "probability": 0.9826171875}, {"start": 286.29, "end": 288.25, "word": " أنا", "probability": 0.42822265625}, {"start": 288.25, "end": 288.61, "word": " عندي", "probability": 0.777099609375}, {"start": 288.61, "end": 288.91, "word": " limit", "probability": 0.97802734375}, {"start": 288.91, "end": 289.25, "word": " F", "probability": 0.94921875}, {"start": 289.25, "end": 289.49, "word": " of", "probability": 0.9638671875}, {"start": 289.49, "end": 289.87, "word": " X", "probability": 0.9970703125}, {"start": 289.87, "end": 290.33, "word": " لما", "probability": 0.97607421875}, {"start": 290.33, "end": 290.63, "word": " X", "probability": 0.98291015625}, {"start": 290.63, "end": 291.11, "word": " تقول", "probability": 0.98876953125}, {"start": 291.11, "end": 291.63, "word": " لـC", "probability": 0.87744140625}, {"start": 291.63, "end": 292.13, "word": " exist", "probability": 0.57666015625}], "temperature": 1.0}, {"id": 11, "seek": 30701, "start": 293.69, "end": 307.01, "text": "و limit ال function g of x لما x تقول ل c exist لأن ال function f continuous عند ال c و ال function g احنا فرضينها continuous عند c", "tokens": [2407, 4948, 2423, 2445, 290, 295, 2031, 5296, 15042, 2031, 6055, 39648, 5296, 269, 2514, 5296, 33456, 2423, 2445, 283, 10957, 43242, 2423, 269, 4032, 2423, 2445, 290, 1975, 5016, 8315, 6156, 43042, 9957, 11296, 10957, 43242, 269], "avg_logprob": -0.31590544871794873, "compression_ratio": 1.4444444444444444, "no_speech_prob": 0.0, "words": [{"start": 293.69, "end": 294.05, "word": "و", "probability": 0.548828125}, {"start": 294.05, "end": 294.65, "word": " limit", "probability": 0.494873046875}, {"start": 294.65, "end": 296.15, "word": " ال", "probability": 0.51708984375}, {"start": 296.15, "end": 296.63, "word": " function", 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" تمام", "probability": 0.90625}, {"start": 350.24, "end": 352.04, "word": " ال", "probability": 0.34619140625}, {"start": 352.04, "end": 352.5, "word": " proof", "probability": 0.50927734375}, {"start": 352.5, "end": 355.26, "word": " ال", "probability": 0.72607421875}, {"start": 355.26, "end": 355.9, "word": " proof", "probability": 0.97705078125}, {"start": 355.9, "end": 357.88, "word": " of", "probability": 0.84814453125}, {"start": 357.88, "end": 359.94, "word": " the", "probability": 0.53369140625}, {"start": 359.94, "end": 360.58, "word": " other", "probability": 0.90625}], "temperature": 1.0}, {"id": 14, "seek": 38830, "start": 365.54, "end": 388.3, "text": "parts is similar مشابه للبرهان اللي احنا لسه ماخدينه يعني لإثبات ان مثلا مجموعة دلتين continuous برضه ممكن اثبات ان limit f زائد g لما x تقول ل c بساوي f زائد g and c", "tokens": [6971, 82, 307, 2531, 37893, 16758, 3224, 24976, 26890, 3224, 7649, 13672, 1829, 1975, 5016, 8315, 5296, 3794, 3224, 3714, 47283, 3215, 9957, 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"word": " احنا", "probability": 0.8572591145833334}, {"start": 372.76, "end": 374.2, "word": " لسه", "probability": 0.955078125}, {"start": 374.2, "end": 375.76, "word": " ماخدينه", "probability": 0.95146484375}, {"start": 375.76, "end": 376.9, "word": " يعني", "probability": 0.739990234375}, {"start": 376.9, "end": 377.5, "word": " لإثبات", "probability": 0.8384765625}, {"start": 377.5, "end": 377.64, "word": " ان", "probability": 0.341552734375}, {"start": 377.64, "end": 377.96, "word": " مثلا", "probability": 0.779541015625}, {"start": 377.96, "end": 378.56, "word": " مجموعة", "probability": 0.89287109375}, {"start": 378.56, "end": 379.18, "word": " دلتين", "probability": 0.76031494140625}, {"start": 379.18, "end": 380.04, "word": " continuous", "probability": 0.9423828125}, {"start": 380.04, "end": 381.68, "word": " برضه", "probability": 0.9812825520833334}, {"start": 381.68, "end": 381.96, "word": " ممكن", "probability": 0.979248046875}, {"start": 381.96, "end": 382.42, "word": " اثبات", "probability": 0.9068603515625}, {"start": 382.42, "end": 382.58, "word": " ان", "probability": 0.8671875}, {"start": 382.58, "end": 382.92, "word": " limit", "probability": 0.833984375}, {"start": 382.92, "end": 383.28, "word": " f", "probability": 0.6767578125}, {"start": 383.28, "end": 383.7, "word": " زائد", "probability": 0.6900227864583334}, {"start": 383.7, "end": 384.02, "word": " g", "probability": 0.95849609375}, {"start": 384.02, "end": 384.38, "word": " لما", "probability": 0.816650390625}, {"start": 384.38, "end": 384.66, "word": " x", "probability": 0.69140625}, {"start": 384.66, "end": 385.02, "word": " تقول", "probability": 0.809814453125}, {"start": 385.02, "end": 385.26, "word": " ل", "probability": 0.35107421875}, {"start": 385.26, "end": 385.54, "word": " c", "probability": 0.509765625}, {"start": 385.54, "end": 386.28, "word": " بساوي", "probability": 0.8333740234375}, {"start": 386.28, "end": 387.08, "word": " f", "probability": 0.91162109375}, {"start": 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"word": " R", "probability": 0.74658203125}, {"start": 1288.22, "end": 1288.82, "word": " لأنه", "probability": 0.80029296875}, {"start": 1288.82, "end": 1289.06, "word": " برضه", "probability": 0.9705403645833334}, {"start": 1289.06, "end": 1289.18, "word": " ال", "probability": 0.95068359375}, {"start": 1289.18, "end": 1289.62, "word": " cosine", "probability": 0.9580078125}, {"start": 1289.62, "end": 1290.5, "word": " ال", "probability": 0.84375}, {"start": 1290.5, "end": 1290.92, "word": " absolute", "probability": 0.9345703125}, {"start": 1290.92, "end": 1291.44, "word": " مجزمة", "probability": 0.8953857421875}, {"start": 1291.44, "end": 1291.7, "word": " منها", "probability": 0.734130859375}, {"start": 1291.7, "end": 1292.12, "word": " واحد", "probability": 0.963134765625}, {"start": 1292.12, "end": 1292.26, "word": " وال", "probability": 0.50537109375}, {"start": 1292.26, "end": 1292.64, "word": " absolute", "probability": 0.8740234375}, {"start": 1292.64, "end": 1293.0, "word": " minimum", "probability": 0.57666015625}, {"start": 1293.0, "end": 1293.42, "word": " سالب", "probability": 0.7840169270833334}, {"start": 1293.42, "end": 1293.62, "word": " واحد", "probability": 0.992919921875}, {"start": 1293.62, "end": 1293.98, "word": " وبالتالي", "probability": 0.91298828125}, {"start": 1293.98, "end": 1294.52, "word": " قيمة", "probability": 0.86962890625}, {"start": 1294.52, "end": 1295.6, "word": " محصورة", "probability": 0.86591796875}, {"start": 1295.6, "end": 1295.76, "word": " بين", "probability": 0.46728515625}, {"start": 1295.76, "end": 1296.14, "word": " سالب", "probability": 0.92919921875}, {"start": 1296.14, "end": 1296.46, "word": " واحد", "probability": 0.990478515625}, {"start": 1296.46, "end": 1296.84, "word": " واحد", "probability": 0.963134765625}, {"start": 1296.84, "end": 1298.54, "word": " الآن", "probability": 0.646240234375}, {"start": 1298.54, "end": 1299.06, "word": " خلينا", "probability": 0.7987060546875}, {"start": 1299.06, "end": 1299.6, "word": " ناخد", "probability": 0.9892578125}, {"start": 1299.6, "end": 1299.86, "word": " ال", "probability": 0.900390625}, {"start": 1299.86, "end": 1300.02, "word": " ..", "probability": 0.90625}], "temperature": 1.0}, {"id": 48, "seek": 130608, "start": 1302.89, "end": 1306.09, "text": "من المعادلة الأخيرة", "tokens": [27842, 9673, 3615, 18513, 37977, 16247, 9778, 48923], "avg_logprob": -0.1474609375, "compression_ratio": 0.8181818181818182, "no_speech_prob": 0.0, "words": [{"start": 1302.89, "end": 1304.17, "word": "من", "probability": 0.6064453125}, {"start": 1304.17, "end": 1305.45, "word": " المعادلة", "probability": 0.9129638671875}, {"start": 1305.45, "end": 1306.09, "word": " الأخيرة", "probability": 0.9523111979166666}], "temperature": 1.0}, {"id": 49, "seek": 133270, "start": 1316.72, "end": 1332.7, "text": "من المعادلة الأخيرة بطلع عندي لو أخدت ال absolute value للطرفين فبطلع عندي absolute sin x minus sin c طبعا هذا الكلام كله صحيح لكل x و c أعداد حقيقية", 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لو", "probability": 0.485107421875}, {"start": 1319.12, "end": 1319.48, "word": " أخدت", "probability": 0.9324951171875}, {"start": 1319.48, "end": 1319.58, "word": " ال", "probability": 0.7373046875}, {"start": 1319.58, "end": 1319.96, "word": " absolute", "probability": 0.83544921875}, {"start": 1319.96, "end": 1320.52, "word": " value", "probability": 0.9912109375}, {"start": 1320.52, "end": 1321.48, "word": " للطرفين", "probability": 0.89404296875}, {"start": 1321.48, "end": 1322.86, "word": " فبطلع", "probability": 0.9251953125}, {"start": 1322.86, "end": 1323.14, "word": " عندي", "probability": 0.938720703125}, {"start": 1323.14, "end": 1323.58, "word": " absolute", "probability": 0.90673828125}, {"start": 1323.58, "end": 1324.22, "word": " sin", "probability": 0.5126953125}, {"start": 1324.22, "end": 1324.62, "word": " x", "probability": 0.5849609375}, {"start": 1324.62, "end": 1325.16, "word": " minus", "probability": 0.84765625}, {"start": 1325.16, "end": 1325.84, "word": " sin", "probability": 0.92041015625}, {"start": 1325.84, "end": 1326.46, "word": " c", "probability": 0.81494140625}, {"start": 1326.46, "end": 1327.62, "word": " طبعا", "probability": 0.967529296875}, {"start": 1327.62, "end": 1327.82, "word": " هذا", "probability": 0.86328125}, {"start": 1327.82, "end": 1328.22, "word": " الكلام", "probability": 0.9287109375}, {"start": 1328.22, "end": 1328.6, "word": " كله", "probability": 0.97802734375}, {"start": 1328.6, "end": 1329.1, "word": " صحيح", "probability": 0.998046875}, {"start": 1329.1, "end": 1329.6, "word": " لكل", "probability": 0.9765625}, {"start": 1329.6, "end": 1330.0, "word": " x", "probability": 0.94873046875}, {"start": 1330.0, "end": 1330.3, "word": " و", "probability": 0.96826171875}, {"start": 1330.3, "end": 1330.62, "word": " c", "probability": 0.49560546875}, {"start": 1330.62, "end": 1331.94, "word": " أعداد", "probability": 0.9580078125}, {"start": 1331.94, "end": 1332.7, "word": " حقيقية", "probability": 0.9681396484375}], "temperature": 1.0}, {"id": 50, "seek": 136335, "start": 1334.59, "end": 1363.35, "text": "فهذا بيطلع بساوي او اصغر من او ساوي اتنين في absolute sin z absolute sin z اصغر من او ساوي absolute z اللي هو نص في absolute x minus z ضرب absolute cosine z اصغر من او ساوي الواحد اصغر من او ساوي الواحد", "tokens": [5172, 3224, 15730, 4724, 1829, 9566, 1211, 3615, 4724, 3794, 995, 45865, 1975, 2407, 1975, 9381, 17082, 2288, 9154, 1975, 2407, 8608, 995, 45865, 1975, 2655, 1863, 9957, 8978, 8236, 3343, 710, 8236, 3343, 710, 1975, 9381, 17082, 2288, 9154, 1975, 2407, 8608, 995, 45865, 8236, 710, 13672, 1829, 31439, 8717, 9381, 8978, 8236, 2031, 3175, 710, 48812, 25513, 8236, 23565, 710, 1975, 9381, 17082, 2288, 9154, 1975, 2407, 8608, 995, 45865, 2423, 14407, 24401, 1975, 9381, 17082, 2288, 9154, 1975, 2407, 8608, 995, 45865, 2423, 14407, 24401], "avg_logprob": -0.15001755216148463, "compression_ratio": 2.143884892086331, "no_speech_prob": 0.0, "words": [{"start": 1334.59, "end": 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"end": 1344.29, "word": " z", "probability": 0.477294921875}, {"start": 1344.29, "end": 1346.01, "word": " absolute", "probability": 0.53515625}, {"start": 1346.01, "end": 1346.65, "word": " sin", "probability": 0.75830078125}, {"start": 1346.65, "end": 1347.05, "word": " z", "probability": 0.97119140625}, {"start": 1347.05, "end": 1347.57, "word": " اصغر", "probability": 0.9510498046875}, {"start": 1347.57, "end": 1347.69, "word": " من", "probability": 0.97802734375}, {"start": 1347.69, "end": 1347.87, "word": " او", "probability": 0.9619140625}, {"start": 1347.87, "end": 1348.23, "word": " ساوي", "probability": 0.9661458333333334}, {"start": 1348.23, "end": 1348.69, "word": " absolute", "probability": 0.78125}, {"start": 1348.69, "end": 1349.21, "word": " z", "probability": 0.83203125}, {"start": 1349.21, "end": 1349.53, "word": " اللي", "probability": 0.83837890625}, {"start": 1349.53, "end": 1349.91, "word": " هو", "probability": 0.98486328125}, {"start": 1349.91, "end": 1351.51, 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3794, 995, 45865, 8236, 2031, 3175, 269, 4032, 9154, 9673, 2655, 3555, 995, 9957, 8032, 8848, 1829, 4724, 9957, 2655, 7435, 16472, 11778, 3224, 5296, 1465, 44650, 36520, 18871, 269, 1392, 22807, 1975, 15730, 281, 855, 3191, 269, 5784, 281, 497], "avg_logprob": -0.3278846153846154, "compression_ratio": 1.303030303030303, "no_speech_prob": 0.0, "words": [{"start": 1365.32, "end": 1365.98, "word": "تمام؟", "probability": 0.5401611328125}, {"start": 1365.98, "end": 1366.12, "word": " و", "probability": 0.52685546875}, {"start": 1366.12, "end": 1366.3, "word": " هذا", "probability": 0.77001953125}, {"start": 1366.3, "end": 1366.9, "word": " صحيح", "probability": 0.9940185546875}, {"start": 1366.9, "end": 1367.58, "word": " لكل", "probability": 0.951904296875}, {"start": 1367.58, "end": 1369.06, "word": " x", "probability": 0.413818359375}, {"start": 1369.06, "end": 1369.42, "word": " و", "probability": 0.434814453125}, {"start": 1369.42, "end": 1369.7, "word": " c", "probability": 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{"id": 55, "seek": 148133, "start": 1466.69, "end": 1481.33, "text": "من delta وانا اختار ال delta تساوي epsilon عشان يطلع absolute الفرق بين f of x وf of z أصغر من epsilon إذا هاي شرط epsilon delta لتعريف ال continuity و النقطة المتحققة", "tokens": [27842, 8289, 4032, 7649, 995, 1975, 46456, 9640, 2423, 8289, 6055, 3794, 995, 45865, 17889, 6225, 8592, 7649, 7251, 9566, 1211, 3615, 8236, 27188, 2288, 4587, 49374, 283, 295, 2031, 4032, 69, 295, 710, 5551, 9381, 17082, 2288, 9154, 17889, 11933, 15730, 8032, 47302, 13412, 2288, 9566, 17889, 8289, 5296, 2655, 3615, 16572, 5172, 2423, 23807, 4032, 28239, 47432, 3660, 9673, 2655, 5016, 4587, 4587, 3660], "avg_logprob": -0.22469683058226286, "compression_ratio": 1.4201183431952662, "no_speech_prob": 0.0, "words": [{"start": 1466.69, "end": 1467.05, "word": "من", "probability": 0.67626953125}, {"start": 1467.05, "end": 1467.53, "word": " delta", "probability": 0.334716796875}, {"start": 1467.53, "end": 1468.13, "word": " وانا", "probability": 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"end": 1628.0, "word": " اتنين", "probability": 0.909423828125}, {"start": 1628.0, "end": 1628.38, "word": " بيطلع", "probability": 0.8552734375}, {"start": 1628.38, "end": 1628.96, "word": " اتنين", "probability": 0.9765625}, {"start": 1628.96, "end": 1630.18, "word": " وهذا", "probability": 0.824462890625}, {"start": 1630.18, "end": 1630.84, "word": " أصغر", "probability": 0.958740234375}, {"start": 1630.84, "end": 1631.38, "word": " من", "probability": 0.986328125}, {"start": 1631.38, "end": 1632.94, "word": " absolute", "probability": 0.462646484375}, {"start": 1632.94, "end": 1633.68, "word": " sine", "probability": 0.5224609375}, {"start": 1633.68, "end": 1633.96, "word": " of", "probability": 0.26220703125}, {"start": 1633.96, "end": 1634.26, "word": " z", "probability": 0.76708984375}, {"start": 1634.26, "end": 1634.88, "word": " أصغر", "probability": 0.8858642578125}, {"start": 1634.88, "end": 1635.1, "word": " من", "probability": 0.91455078125}, {"start": 1635.1, "end": 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{"start": 1651.33, "end": 1651.57, "word": " هذه", "probability": 0.3154296875}, {"start": 1651.57, "end": 1651.95, "word": " مش", "probability": 0.990234375}, {"start": 1651.95, "end": 1652.37, "word": " cosine", "probability": 0.72412109375}, {"start": 1652.37, "end": 1652.65, "word": " هذه", "probability": 0.5}, {"start": 1652.65, "end": 1653.03, "word": " sin", "probability": 0.49853515625}], "temperature": 1.0}, {"id": 63, "seek": 168346, "start": 1654.46, "end": 1683.46, "text": "هذه الـ sine فهي sine الـ z ال absolute value لها أصغر من أو يساوي الواحد وهي كمان sine أو absolute value ل sine ال z أصغر من أو يساوي absolute ال z ال z هنا اللي هو نص في x minus z فبطلع نص في absolute في absolute x minus z بطلع هذا بساوي absolute x minus z", "tokens": [3224, 24192, 2423, 39184, 18609, 6156, 3224, 1829, 18609, 2423, 39184, 710, 2423, 8236, 2158, 5296, 11296, 5551, 9381, 17082, 2288, 9154, 34051, 7251, 3794, 995, 45865, 2423, 14407, 24401, 37037, 1829, 9122, 2304, 7649, 18609, 34051, 8236, 2158, 5296, 18609, 2423, 710, 5551, 9381, 17082, 2288, 9154, 34051, 7251, 3794, 995, 45865, 8236, 2423, 710, 2423, 710, 34105, 13672, 1829, 31439, 8717, 9381, 8978, 2031, 3175, 710, 6156, 3555, 9566, 1211, 3615, 8717, 9381, 8978, 8236, 8978, 8236, 2031, 3175, 710, 4724, 9566, 1211, 3615, 23758, 4724, 3794, 995, 45865, 8236, 2031, 3175, 710], "avg_logprob": -0.21712240219737092, "compression_ratio": 2.0168539325842696, "no_speech_prob": 5.960464477539063e-08, "words": [{"start": 1654.46, "end": 1654.86, "word": "هذه", "probability": 0.4949951171875}, {"start": 1654.86, "end": 1655.0, "word": " الـ", "probability": 0.4671630859375}, {"start": 1655.0, "end": 1655.18, "word": " sine", "probability": 0.305908203125}, {"start": 1655.18, "end": 1656.72, "word": " فهي", "probability": 0.5377604166666666}, {"start": 1656.72, "end": 1657.48, "word": " sine", "probability": 0.375}, {"start": 1657.48, "end": 1657.92, "word": " الـ", "probability": 0.53515625}, {"start": 1657.92, "end": 1658.08, "word": " z", "probability": 0.68505859375}, {"start": 1658.08, "end": 1658.28, "word": " ال", "probability": 0.59423828125}, {"start": 1658.28, "end": 1658.64, "word": " absolute", "probability": 0.57763671875}, {"start": 1658.64, "end": 1659.12, "word": " value", "probability": 0.9873046875}, {"start": 1659.12, "end": 1659.36, "word": " لها", "probability": 0.59521484375}, {"start": 1659.36, "end": 1659.76, "word": " أصغر", "probability": 0.9732666015625}, {"start": 1659.76, "end": 1659.88, "word": " من", "probability": 0.97265625}, {"start": 1659.88, "end": 1660.0, "word": " أو", "probability": 0.87255859375}, {"start": 1660.0, "end": 1660.32, "word": " يساوي", "probability": 0.79931640625}, {"start": 1660.32, "end": 1660.8, "word": " الواحد", "probability": 0.9246419270833334}, {"start": 1660.8, "end": 1661.86, "word": " وهي", "probability": 0.81103515625}, {"start": 1661.86, "end": 1662.3, "word": " كمان", "probability": 0.9269205729166666}, {"start": 1662.3, "end": 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absolute", "probability": 0.927734375}, {"start": 1669.42, "end": 1670.3, "word": " ال", "probability": 0.90234375}, {"start": 1670.3, "end": 1670.62, "word": " z", "probability": 0.947265625}, {"start": 1670.62, "end": 1671.54, "word": " ال", "probability": 0.92236328125}, {"start": 1671.54, "end": 1671.7, "word": " z", "probability": 0.93994140625}, {"start": 1671.7, "end": 1671.96, "word": " هنا", "probability": 0.9716796875}, {"start": 1671.96, "end": 1672.12, "word": " اللي", "probability": 0.96484375}, {"start": 1672.12, "end": 1672.3, "word": " هو", "probability": 0.712890625}, {"start": 1672.3, "end": 1672.74, "word": " نص", "probability": 0.98779296875}, {"start": 1672.74, "end": 1672.94, "word": " في", "probability": 0.8310546875}, {"start": 1672.94, "end": 1673.36, "word": " x", "probability": 0.73095703125}, {"start": 1673.36, "end": 1673.74, "word": " minus", "probability": 0.80224609375}, {"start": 1673.74, "end": 1674.06, "word": " z", "probability": 0.9609375}, {"start": 1674.06, "end": 1675.04, "word": " فبطلع", "probability": 0.94287109375}, {"start": 1675.04, "end": 1675.52, "word": " نص", "probability": 0.990234375}, {"start": 1675.52, "end": 1675.74, "word": " في", "probability": 0.96630859375}, {"start": 1675.74, "end": 1676.56, "word": " absolute", "probability": 0.9697265625}, {"start": 1676.56, "end": 1678.66, "word": " في", "probability": 0.55908203125}, {"start": 1678.66, "end": 1679.2, "word": " absolute", "probability": 0.962890625}, {"start": 1679.2, "end": 1679.76, "word": " x", "probability": 0.974609375}, {"start": 1679.76, "end": 1680.24, "word": " minus", "probability": 0.990234375}, {"start": 1680.24, "end": 1680.62, "word": " z", "probability": 0.93994140625}, {"start": 1680.62, "end": 1681.18, "word": " بطلع", "probability": 0.86737060546875}, {"start": 1681.18, "end": 1681.44, "word": " هذا", "probability": 0.9013671875}, {"start": 1681.44, "end": 1681.84, "word": " بساوي", "probability": 0.83740234375}, {"start": 1681.84, "end": 1682.26, "word": " absolute", "probability": 0.95703125}, {"start": 1682.26, "end": 1682.78, "word": " x", "probability": 0.986328125}, {"start": 1682.78, "end": 1683.14, "word": " minus", "probability": 0.99462890625}, {"start": 1683.14, "end": 1683.46, "word": " z", "probability": 0.9794921875}], "temperature": 1.0}, {"id": 64, "seek": 171387, "start": 1684.63, "end": 1713.87, "text": "و باقي البرهان زي ما عملنا هنا okay تمام لأي epsilon أكبر من السفر choose delta بساوي epsilon ف this delta will work تمام إذا باقي البرهان كما عملنا في حالة ال sine إذا هذا المثال الرابع شوفنا فيه كيف نثبت إن ال cosine function is continuous تمام واضح", "tokens": [2407, 4724, 995, 38436, 2423, 26890, 3224, 7649, 30767, 1829, 19446, 6225, 42213, 8315, 34105, 1392, 46811, 10943, 5296, 10721, 1829, 17889, 5551, 4117, 26890, 9154, 21136, 5172, 2288, 2826, 8289, 4724, 3794, 995, 45865, 17889, 6156, 341, 8289, 486, 589, 46811, 10943, 11933, 15730, 4724, 995, 38436, 2423, 26890, 3224, 7649, 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{"start": 1701.33, "end": 1702.07, "word": " ال", "probability": 0.346923828125}, {"start": 1702.07, "end": 1702.37, "word": " sine", "probability": 0.424072265625}, {"start": 1702.37, "end": 1702.69, "word": " إذا", "probability": 0.895751953125}, {"start": 1702.69, "end": 1703.09, "word": " هذا", "probability": 0.798828125}, {"start": 1703.09, "end": 1704.45, "word": " المثال", "probability": 0.95703125}, {"start": 1704.45, "end": 1705.25, "word": " الرابع", "probability": 0.9871419270833334}, {"start": 1705.25, "end": 1707.89, "word": " شوفنا", "probability": 0.71240234375}, {"start": 1707.89, "end": 1708.15, "word": " فيه", "probability": 0.97119140625}, {"start": 1708.15, "end": 1708.39, "word": " كيف", "probability": 0.97265625}, {"start": 1708.39, "end": 1708.89, "word": " نثبت", "probability": 0.9945068359375}, {"start": 1708.89, "end": 1709.07, "word": " إن", "probability": 0.343017578125}, {"start": 1709.07, "end": 1709.21, "word": " ال", "probability": 0.92578125}, {"start": 1709.21, "end": 1709.95, "word": " cosine", "probability": 0.703125}, {"start": 1709.95, "end": 1710.61, "word": " function", "probability": 0.916015625}, {"start": 1710.61, "end": 1711.69, "word": " is", "probability": 0.89794921875}, {"start": 1711.69, "end": 1712.37, "word": " continuous", "probability": 0.921875}, {"start": 1712.37, "end": 1712.97, "word": " تمام", "probability": 0.9091796875}, {"start": 1712.97, "end": 1713.87, "word": " واضح", "probability": 0.89453125}], "temperature": 1.0}, {"id": 65, "seek": 173880, "start": 1717.34, "end": 1738.8, "text": "الان ممكن اثبات بعد هيك انه ال tangent function tangent x اللي هي بساوي sin x على cos x is continuous", "tokens": [6027, 7649, 3714, 43020, 1975, 12984, 3555, 9307, 39182, 39896, 4117, 16472, 3224, 2423, 27747, 2445, 27747, 2031, 13672, 1829, 39896, 4724, 3794, 995, 45865, 3343, 2031, 15844, 3792, 2031, 307, 10957], "avg_logprob": -0.24621212121212122, "compression_ratio": 1.275229357798165, "no_speech_prob": 0.0, "words": 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{"start": 1731.3, "end": 1731.76, "word": " هي", "probability": 0.89111328125}, {"start": 1731.76, "end": 1733.66, "word": " بساوي", "probability": 0.672119140625}, {"start": 1733.66, "end": 1734.2, "word": " sin", "probability": 0.469970703125}, {"start": 1734.2, "end": 1734.66, "word": " x", "probability": 0.72021484375}, {"start": 1734.66, "end": 1734.96, "word": " على", "probability": 0.323974609375}, {"start": 1734.96, "end": 1735.36, "word": " cos", "probability": 0.6552734375}, {"start": 1735.36, "end": 1736.52, "word": " x", "probability": 0.904296875}, {"start": 1736.52, "end": 1738.04, "word": " is", "probability": 0.83203125}, {"start": 1738.04, "end": 1738.8, "word": " continuous", "probability": 0.9208984375}], "temperature": 1.0}, {"id": 66, "seek": 176797, "start": 1741.89, "end": 1767.97, "text": "الصين مستمر على الار والكوسين مستمر على الار هذه راشيونال فانتشار فانتشار راشيونال فانتشار مستمر على الار ما عدا عند أسفار المخام ما هي أسفار الكوسين المضاعفات الفردية لا πاية 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"word": " on", "probability": 0.96240234375}, {"start": 1953.15, "end": 1954.37, "word": " a", "probability": 0.96728515625}, {"start": 1954.37, "end": 1961.19, "word": " proof", "probability": 0.94482421875}, {"start": 1961.19, "end": 1964.23, "word": " we", "probability": 0.9384765625}, {"start": 1964.23, "end": 1964.85, "word": " use", "probability": 0.8896484375}], "temperature": 1.0}, {"id": 74, "seek": 198990, "start": 1967.24, "end": 1989.9, "text": "we use exercise exercise رقم تلتاش في section أربعة اتنين نرجع لل exercise هذا و نكتبه", "tokens": [826, 764, 5380, 5380, 12602, 4587, 2304, 6055, 1211, 2655, 33599, 8978, 3541, 5551, 25513, 27884, 1975, 2655, 1863, 9957, 8717, 47341, 3615, 24976, 5380, 23758, 4032, 8717, 4117, 2655, 3555, 3224], "avg_logprob": -0.1181936553030303, "compression_ratio": 1.1523809523809523, "no_speech_prob": 0.0, "words": [{"start": 1967.24, "end": 1967.56, "word": "we", "probability": 0.67431640625}, {"start": 1967.56, "end": 1968.26, "word": " use", 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"word": " هذا", "probability": 0.8740234375}, {"start": 1989.06, "end": 1989.22, "word": " و", "probability": 0.99365234375}, {"start": 1989.22, "end": 1989.9, "word": " نكتبه", "probability": 0.95341796875}], "temperature": 1.0}, {"id": 75, "seek": 202147, "start": 1996.29, "end": 2021.47, "text": "الـ exercise هذا بيقول if ال limit ل ال function f of x لما x تقول إلى c exists", "tokens": [6027, 39184, 5380, 23758, 4724, 1829, 39648, 498, 2423, 4948, 5296, 2423, 2445, 283, 295, 2031, 5296, 15042, 2031, 6055, 39648, 30731, 269, 8198], "avg_logprob": -0.21609375, "compression_ratio": 1.09375, "no_speech_prob": 0.0, "words": [{"start": 1996.29, "end": 1996.93, "word": "الـ", "probability": 0.7412109375}, {"start": 1996.93, "end": 1997.41, "word": " exercise", "probability": 0.62353515625}, {"start": 1997.41, "end": 1997.97, "word": " هذا", "probability": 0.84814453125}, {"start": 1997.97, "end": 1998.61, "word": " بيقول", "probability": 0.966796875}, {"start": 1998.61, "end": 2009.47, 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307, 10957, 412, 269, 6055, 29399, 2304, 1829, 5296, 257], "avg_logprob": -0.2403017282485962, "compression_ratio": 1.1041666666666667, "no_speech_prob": 0.0, "words": [{"start": 2123.87, "end": 2124.75, "word": "لبرهان", "probability": 0.755279541015625}, {"start": 2124.75, "end": 2125.11, "word": " الجزء", "probability": 0.9547526041666666}, {"start": 2125.11, "end": 2125.65, "word": " الأول", "probability": 0.90576171875}, {"start": 2125.65, "end": 2130.21, "word": " to", "probability": 0.7021484375}, {"start": 2130.21, "end": 2131.09, "word": " show", "probability": 0.96875}, {"start": 2131.09, "end": 2136.41, "word": " if", "probability": 0.56640625}, {"start": 2136.41, "end": 2138.09, "word": " is", "probability": 0.78662109375}, {"start": 2138.09, "end": 2138.65, "word": " ..", "probability": 0.372314453125}, {"start": 2138.65, "end": 2139.81, "word": " to", "probability": 0.62060546875}, {"start": 2139.81, "end": 2140.57, "word": " show", "probability": 0.97265625}, {"start": 2140.57, "end": 2141.11, "word": " absolute", "probability": 0.91259765625}, {"start": 2141.11, "end": 2141.73, "word": " if", "probability": 0.95068359375}, {"start": 2141.73, "end": 2144.89, "word": " is", "probability": 0.92529296875}, {"start": 2144.89, "end": 2146.09, "word": " continuous", "probability": 0.900390625}, {"start": 2146.09, "end": 2148.97, "word": " at", "probability": 0.9404296875}, {"start": 2148.97, "end": 2150.19, "word": " c", "probability": 0.54736328125}, {"start": 2150.19, "end": 2151.21, "word": " تنتمي", "probability": 0.9581298828125}, {"start": 2151.21, "end": 2151.35, "word": " ل", "probability": 0.7861328125}, {"start": 2151.35, "end": 2151.71, "word": " a", "probability": 0.5517578125}], "temperature": 1.0}, {"id": 80, "seek": 217665, "start": 2163.81, "end": 2176.65, "text": "لدينا اتصالين اتصال اتصال اتصال اتصال اتصال اتصال اتصال اتصال اتصال", "tokens": [1211, 16254, 8315, 1975, 2655, 9381, 6027, 9957, 1975, 2655, 9381, 6027, 1975, 2655, 9381, 6027, 1975, 2655, 9381, 6027, 1975, 2655, 9381, 6027, 1975, 2655, 9381, 6027, 1975, 2655, 9381, 6027, 1975, 2655, 9381, 6027, 1975, 2655, 9381, 6027, 1975, 2655, 9381, 6027], "avg_logprob": -0.5302083121405707, "compression_ratio": 3.3513513513513513, "no_speech_prob": 0.0, "words": [{"start": 2163.81, "end": 2165.21, "word": "لدينا", "probability": 0.64093017578125}, {"start": 2165.21, "end": 2166.05, "word": " اتصالين", "probability": 0.5955078125}, {"start": 2166.05, "end": 2169.35, "word": " اتصال", "probability": 0.65643310546875}, {"start": 2169.35, "end": 2169.73, "word": " اتصال", "probability": 0.657806396484375}, {"start": 2169.73, "end": 2170.75, "word": " اتصال", "probability": 0.78839111328125}, {"start": 2170.75, "end": 2171.03, "word": " اتصال", "probability": 0.799041748046875}, {"start": 2171.03, "end": 2171.79, "word": " اتصال", "probability": 0.793365478515625}, {"start": 2171.79, "end": 2173.59, "word": " اتصال", "probability": 0.790496826171875}, {"start": 2173.59, "end": 2173.59, "word": " اتصال", "probability": 0.791473388671875}, {"start": 2173.59, "end": 2173.71, "word": " اتصال", "probability": 0.795440673828125}, {"start": 2173.71, "end": 2176.65, "word": " اتصال", "probability": 0.801422119140625}], "temperature": 1.0}, {"id": 81, "seek": 220956, "start": 2183.2, "end": 2209.56, "text": "فشوفنا ان لو كانت الـ C ماهياش cluster point فالاتصال عندها بيطلع متحقق اوتوماتيكي شوفنا في التعريف then the continuity of absolute if at C is automatic اوتوماتيكي", "tokens": [5172, 8592, 38688, 8315, 16472, 45164, 25961, 2655, 2423, 39184, 383, 19446, 3224, 1829, 33599, 13630, 935, 6156, 6027, 9307, 9381, 6027, 43242, 11296, 4724, 1829, 9566, 1211, 3615, 44650, 5016, 4587, 4587, 1975, 35473, 20498, 9307, 1829, 4117, 1829, 13412, 38688, 8315, 8978, 16712, 3615, 16572, 5172, 550, 264, 23807, 295, 8236, 498, 412, 383, 307, 12509, 1975, 35473, 20498, 9307, 1829, 4117, 1829], "avg_logprob": -0.20371685538328055, "compression_ratio": 1.4294117647058824, 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"probability": 0.790966796875}, {"start": 2188.36, "end": 2188.94, "word": " متحقق", "probability": 0.95458984375}, {"start": 2188.94, "end": 2189.72, "word": " اوتوماتيكي", "probability": 0.84716796875}, {"start": 2189.72, "end": 2191.64, "word": " شوفنا", "probability": 0.7376302083333334}, {"start": 2191.64, "end": 2191.78, "word": " في", "probability": 0.86865234375}, {"start": 2191.78, "end": 2192.42, "word": " التعريف", "probability": 0.99365234375}, {"start": 2192.42, "end": 2194.76, "word": " then", "probability": 0.387939453125}, {"start": 2194.76, "end": 2196.16, "word": " the", "probability": 0.83154296875}, {"start": 2196.16, "end": 2197.14, "word": " continuity", "probability": 0.8486328125}, {"start": 2197.14, "end": 2200.6, "word": " of", "probability": 0.9697265625}, {"start": 2200.6, "end": 2201.46, "word": " absolute", "probability": 0.908203125}, {"start": 2201.46, "end": 2202.02, "word": " if", "probability": 0.56640625}, {"start": 2202.02, "end": 2203.26, "word": " at", "probability": 0.72314453125}, {"start": 2203.26, "end": 2203.8, "word": " C", "probability": 0.712890625}, {"start": 2203.8, "end": 2204.46, "word": " is", "probability": 0.95751953125}, {"start": 2204.46, "end": 2207.7, "word": " automatic", "probability": 0.62060546875}, {"start": 2207.7, "end": 2209.56, "word": " اوتوماتيكي", "probability": 0.90185546875}], "temperature": 1.0}, {"id": 82, "seek": 223992, "start": 2210.79, "end": 2239.93, "text": "إذا احنا بنهتم بالحالة التانية انه C is a cluster point of A ففي الحالة هذه by exercise تلتاش of section اربعة اتنين", "tokens": [28814, 15730, 1975, 5016, 8315, 44945, 3224, 39237, 20666, 5016, 6027, 3660, 16712, 7649, 10632, 16472, 3224, 383, 307, 257, 13630, 935, 295, 316, 6156, 41185, 21542, 6027, 3660, 29538, 538, 5380, 6055, 1211, 2655, 33599, 295, 3541, 1975, 25513, 27884, 1975, 2655, 1863, 9957], "avg_logprob": -0.21484374676061713, "compression_ratio": 1.2835820895522387, "no_speech_prob": 0.0, "words": [{"start": 2210.79, 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2266.94, "text": "بما أنه limit ل f of x as x tends to c بيساوي c احنا فرضين ان f continuous by continuity of f at c", "tokens": [3555, 15042, 14739, 3224, 4948, 5296, 283, 295, 2031, 382, 2031, 12258, 281, 269, 4724, 1829, 3794, 995, 45865, 269, 1975, 5016, 8315, 6156, 43042, 9957, 16472, 283, 10957, 538, 23807, 295, 283, 412, 269], "avg_logprob": -0.3237847155994839, "compression_ratio": 1.0990990990990992, "no_speech_prob": 0.0, "words": [{"start": 2248.44, "end": 2249.44, "word": "بما", "probability": 0.6519775390625}, {"start": 2249.44, "end": 2250.44, "word": " أنه", "probability": 0.4493408203125}, {"start": 2250.44, "end": 2251.88, "word": " limit", "probability": 0.5166015625}, {"start": 2251.88, "end": 2253.82, "word": " ل", "probability": 0.460205078125}, {"start": 2253.82, "end": 2254.12, "word": " f", "probability": 0.2763671875}, {"start": 2254.12, "end": 2254.38, "word": " of", "probability": 0.358642578125}, {"start": 2254.38, "end": 2254.76, "word": " x", "probability": 0.9267578125}, {"start": 2254.76, "end": 2255.26, "word": " as", "probability": 0.72509765625}, {"start": 2255.26, "end": 2255.62, "word": " x", "probability": 0.97265625}, {"start": 2255.62, "end": 2256.0, "word": " tends", "probability": 0.427734375}, {"start": 2256.0, "end": 2256.16, "word": " to", "probability": 0.98388671875}, {"start": 2256.16, "end": 2256.48, "word": " c", "probability": 0.89892578125}, {"start": 2256.48, "end": 2257.18, "word": " بيساوي", "probability": 0.77451171875}, {"start": 2257.18, "end": 2257.66, "word": " c", "probability": 0.6796875}, {"start": 2257.66, "end": 2258.62, "word": " احنا", "probability": 0.8387044270833334}, {"start": 2258.62, "end": 2259.04, "word": " فرضين", "probability": 0.9718424479166666}, {"start": 2259.04, "end": 2259.16, "word": " ان", "probability": 0.78662109375}, {"start": 2259.16, "end": 2259.38, "word": " f", "probability": 0.82470703125}, {"start": 2259.38, "end": 2260.06, "word": " continuous", "probability": 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"word": " كمان", "probability": 0.95703125}, {"start": 2378.84, "end": 2379.34, "word": " نظرية", "probability": 0.991455078125}, {"start": 2379.34, "end": 2379.74, "word": " أخرى", "probability": 0.97802734375}, {"start": 2379.74, "end": 2380.4, "word": " مشابهة", "probability": 0.9447021484375}, {"start": 2380.4, "end": 2380.6, "word": " زي", "probability": 0.972900390625}, {"start": 2380.6, "end": 2380.9, "word": " هذه", "probability": 0.7197265625}], "temperature": 1.0}, {"id": 88, "seek": 241289, "start": 2383.79, "end": 2412.89, "text": "لكن بدل absolute f ففي عندى هنا let f be function from a to r such that f of x أكبر من أو يساوي سفر لكل x في a يعني هنا ال dialer قيمها غير سالبة فلو كانت f continuous at c فال square root ل f", "tokens": [1211, 19452, 47525, 1211, 8236, 283, 6156, 41185, 43242, 7578, 34105, 718, 283, 312, 2445, 490, 257, 281, 367, 1270, 300, 283, 295, 2031, 5551, 4117, 26890, 9154, 34051, 7251, 3794, 995, 45865, 8608, 5172, 2288, 5296, 28820, 2031, 8978, 257, 37495, 22653, 34105, 2423, 5502, 260, 12174, 32640, 11296, 32771, 13546, 8608, 6027, 49401, 6156, 1211, 2407, 25961, 2655, 283, 10957, 412, 269, 6156, 6027, 3732, 5593, 5296, 283], "avg_logprob": -0.2627640752725198, "compression_ratio": 1.3936170212765957, "no_speech_prob": 0.0, "words": [{"start": 2383.79, "end": 2384.55, "word": "لكن", "probability": 0.6290283203125}, {"start": 2384.55, "end": 2385.61, "word": " بدل", "probability": 0.9296875}, {"start": 2385.61, "end": 2386.11, "word": " absolute", "probability": 0.61474609375}, {"start": 2386.11, "end": 2386.81, "word": " f", "probability": 0.30126953125}, {"start": 2386.81, "end": 2388.79, "word": " ففي", "probability": 0.73583984375}, {"start": 2388.79, "end": 2389.13, "word": " عندى", "probability": 0.49169921875}, {"start": 2389.13, "end": 2389.41, "word": " هنا", "probability": 0.9228515625}, {"start": 2389.41, "end": 2389.77, "word": " let", "probability": 0.7333984375}, {"start": 2389.77, "end": 2390.09, "word": " f", "probability": 0.8603515625}, {"start": 2390.09, "end": 2390.31, "word": " be", "probability": 0.78125}, {"start": 2390.31, "end": 2390.77, "word": " function", "probability": 0.7666015625}, {"start": 2390.77, "end": 2391.23, "word": " from", "probability": 0.87744140625}, {"start": 2391.23, "end": 2391.83, "word": " a", "probability": 0.73388671875}, {"start": 2391.83, "end": 2392.01, "word": " to", "probability": 0.955078125}, {"start": 2392.01, "end": 2392.51, "word": " r", "probability": 0.7314453125}, {"start": 2392.51, "end": 2393.89, "word": " such", "probability": 0.91162109375}, {"start": 2393.89, "end": 2394.39, "word": " that", "probability": 0.9541015625}, {"start": 2394.39, "end": 2394.73, "word": " f", "probability": 0.9033203125}, {"start": 2394.73, "end": 2394.97, "word": " of", "probability": 0.6162109375}, {"start": 2394.97, "end": 2395.41, "word": " x", "probability": 0.92236328125}, {"start": 2395.41, "end": 2396.07, "word": " أكبر", "probability": 0.8912760416666666}, {"start": 2396.07, "end": 2396.29, "word": " من", "probability": 0.91552734375}, {"start": 2396.29, "end": 2396.53, "word": " أو", "probability": 0.80029296875}, {"start": 2396.53, "end": 2396.99, "word": " يساوي", "probability": 0.7525634765625}, {"start": 2396.99, "end": 2397.51, "word": " سفر", "probability": 0.8017578125}, {"start": 2397.51, "end": 2398.89, "word": " لكل", "probability": 0.977294921875}, {"start": 2398.89, "end": 2399.29, "word": " x", "probability": 0.91064453125}, {"start": 2399.29, "end": 2399.49, "word": " في", "probability": 0.76904296875}, {"start": 2399.49, "end": 2399.71, "word": " a", "probability": 0.94140625}, {"start": 2399.71, "end": 2399.95, "word": " يعني", "probability": 0.814697265625}, {"start": 2399.95, "end": 2400.13, "word": " هنا", "probability": 0.80126953125}, {"start": 2400.13, "end": 2400.33, "word": " ال", "probability": 0.56640625}, {"start": 2400.33, "end": 2400.61, "word": " dialer", "probability": 0.28277587890625}, {"start": 2400.61, "end": 2401.09, "word": " قيمها", "probability": 0.7669270833333334}, {"start": 2401.09, "end": 2401.37, "word": " غير", "probability": 0.98681640625}, {"start": 2401.37, "end": 2402.55, "word": " سالبة", "probability": 0.8133138020833334}, {"start": 2402.55, "end": 2405.17, "word": " فلو", "probability": 0.9669596354166666}, {"start": 2405.17, "end": 2407.55, "word": " كانت", "probability": 0.771240234375}, {"start": 2407.55, "end": 2407.73, "word": " f", "probability": 0.6923828125}, {"start": 2407.73, "end": 2408.35, "word": " continuous", "probability": 0.91748046875}, {"start": 2408.35, "end": 2408.81, "word": " at", "probability": 0.87060546875}, {"start": 2408.81, "end": 2409.29, "word": " c", "probability": 0.8798828125}, {"start": 2409.29, "end": 2411.15, "word": " فال", "probability": 0.90185546875}, {"start": 2411.15, "end": 2411.53, "word": " square", "probability": 0.82470703125}, {"start": 2411.53, "end": 2412.21, "word": " root", "probability": 0.94384765625}, {"start": 2412.21, "end": 2412.53, "word": " ل", "probability": 0.94189453125}, {"start": 2412.53, "end": 2412.89, "word": " f", "probability": 0.4775390625}], "temperature": 1.0}, {"id": 89, "seek": 243758, "start": 2414.1, "end": 2437.58, "text": "بطلع continuous at C كذلك لو كانت F continuous on A ف ال square root ل F is continuous على كل ال A و المرة هذه البرهان بستخدم exercise ثاني في section 42 اللي هو exercise 14", "tokens": [3555, 9566, 1211, 3615, 10957, 412, 383, 9122, 8848, 23275, 45164, 25961, 2655, 479, 10957, 322, 316, 6156, 2423, 3732, 5593, 5296, 479, 307, 10957, 15844, 28242, 2423, 316, 4032, 9673, 25720, 29538, 2423, 26890, 3224, 7649, 4724, 14851, 9778, 40448, 5380, 38637, 7649, 1829, 8978, 3541, 14034, 13672, 1829, 31439, 5380, 3499], "avg_logprob": -0.18084490713145998, "compression_ratio": 1.380952380952381, "no_speech_prob": 0.0, "words": [{"start": 2414.1, "end": 2414.66, "word": "بطلع", "probability": 0.88037109375}, {"start": 2414.66, "end": 2415.2, "word": " continuous", "probability": 0.85546875}, {"start": 2415.2, "end": 2415.7, "word": " at", "probability": 0.8857421875}, {"start": 2415.7, "end": 2416.06, "word": " C", "probability": 0.5263671875}, {"start": 2416.06, "end": 2417.5, "word": " كذلك", "probability": 0.9599609375}, {"start": 2417.5, "end": 2417.7, "word": " لو", "probability": 0.93701171875}, {"start": 2417.7, "end": 2418.16, "word": " كانت", "probability": 0.943603515625}, {"start": 2418.16, "end": 2418.38, "word": " F", "probability": 0.78515625}, {"start": 2418.38, "end": 2419.02, "word": " continuous", "probability": 0.921875}, {"start": 2419.02, "end": 2419.54, "word": " on", "probability": 0.91552734375}, {"start": 2419.54, "end": 2419.96, "word": " A", "probability": 0.97021484375}, {"start": 2419.96, "end": 2421.58, "word": " ف", "probability": 0.95947265625}, {"start": 2421.58, "end": 2421.86, "word": " ال", "probability": 0.446044921875}, {"start": 2421.86, "end": 2422.26, "word": " square", "probability": 0.78955078125}, {"start": 2422.26, "end": 2422.72, "word": " root", "probability": 0.92529296875}, {"start": 2422.72, "end": 2422.88, "word": " ل", "probability": 0.89453125}, {"start": 2422.88, "end": 2423.16, "word": " F", "probability": 0.81689453125}, {"start": 2423.16, "end": 2423.48, "word": " is", "probability": 0.86767578125}, {"start": 2423.48, "end": 2424.76, "word": " continuous", "probability": 0.92333984375}, {"start": 2424.76, "end": 2425.22, "word": " على", "probability": 0.88232421875}, {"start": 2425.22, "end": 2425.74, "word": " كل", "probability": 0.931640625}, {"start": 2425.74, "end": 2426.5, "word": " ال", "probability": 0.5810546875}, {"start": 2426.5, "end": 2426.8, "word": " A", "probability": 0.95068359375}, {"start": 2426.8, "end": 2429.76, "word": " و", "probability": 0.470703125}, {"start": 2429.76, "end": 2430.12, "word": " المرة", "probability": 0.935302734375}, {"start": 2430.12, "end": 2430.44, "word": " هذه", "probability": 0.8154296875}, {"start": 2430.44, "end": 2431.1, "word": " البرهان", "probability": 0.8529052734375}, {"start": 2431.1, "end": 2431.88, "word": " بستخدم", "probability": 0.9593505859375}, {"start": 2431.88, "end": 2433.3, "word": " exercise", "probability": 0.88623046875}, {"start": 2433.3, "end": 2434.06, "word": " ثاني", "probability": 0.8889973958333334}, {"start": 2434.06, "end": 2434.2, "word": " في", "probability": 0.8642578125}, {"start": 2434.2, "end": 2434.5, "word": " section", "probability": 0.7587890625}, {"start": 2434.5, "end": 2435.06, "word": " 42", "probability": 0.191162109375}, {"start": 2435.06, "end": 2435.4, "word": " اللي", "probability": 0.796875}, {"start": 2435.4, "end": 2435.62, "word": " هو", "probability": 0.9873046875}, {"start": 2435.62, "end": 2436.84, "word": " exercise", "probability": 0.90185546875}, {"start": 2436.84, "end": 2437.58, "word": " 14", "probability": 0.93310546875}], "temperature": 1.0}, {"id": 90, "seek": 246411, "start": 2438.67, "end": 2464.11, "text": "الـ exercise هذا بيقول إذا كانت ال limit للدالة هذه، يعني C موجودة، then ال limit للـ square .. لل function اللي هي square root of F عند الـ C موجودة وبتساوي ال square root وبتساوي جذر التربيع أيه؟ ال limit لل square root", "tokens": [6027, 39184, 5380, 23758, 4724, 1829, 39648, 11933, 15730, 25961, 2655, 2423, 4948, 24976, 3215, 6027, 3660, 29538, 12399, 37495, 22653, 383, 3714, 29245, 23328, 3660, 12399, 550, 2423, 4948, 24976, 39184, 3732, 4386, 24976, 2445, 13672, 1829, 39896, 3732, 5593, 295, 479, 43242, 2423, 39184, 383, 3714, 29245, 23328, 3660, 46599, 2655, 3794, 995, 45865, 2423, 3732, 5593, 46599, 2655, 3794, 995, 45865, 10874, 8848, 2288, 16712, 2288, 21292, 3615, 36632, 3224, 22807, 2423, 4948, 24976, 3732, 5593], "avg_logprob": -0.2962890651077032, "compression_ratio": 1.6806282722513088, "no_speech_prob": 0.0, "words": [{"start": 2438.67, "end": 2438.93, "word": "الـ", "probability": 0.5711669921875}, {"start": 2438.93, "end": 2439.33, "word": " exercise", "probability": 0.3984375}, {"start": 2439.33, "end": 2439.63, "word": " هذا", "probability": 0.55078125}, {"start": 2439.63, "end": 2439.91, "word": " بيقول", "probability": 0.8719075520833334}, {"start": 2439.91, "end": 2440.09, "word": " إذا", "probability": 0.770263671875}, {"start": 2440.09, "end": 2440.47, "word": " كانت", "probability": 0.974365234375}, {"start": 2440.47, "end": 2440.59, "word": " ال", "probability": 0.8017578125}, {"start": 2440.59, "end": 2440.81, "word": " limit", "probability": 0.67138671875}, {"start": 2440.81, "end": 2441.47, "word": " للدالة", "probability": 0.8846435546875}, {"start": 2441.47, "end": 2441.85, "word": " هذه،", "probability": 0.4771728515625}, {"start": 2441.85, "end": 2442.03, "word": " يعني", "probability": 0.814697265625}, {"start": 2442.03, "end": 2442.35, "word": " C", "probability": 0.2978515625}, {"start": 2442.35, "end": 2443.75, "word": " موجودة،", "probability": 0.873583984375}, {"start": 2443.75, "end": 2444.25, "word": " then", "probability": 0.8017578125}, {"start": 2444.25, "end": 2444.51, "word": " ال", "probability": 0.7294921875}, {"start": 2444.51, "end": 2444.89, "word": " limit", "probability": 0.92822265625}, {"start": 2444.89, "end": 2447.05, "word": " للـ", "probability": 0.42022705078125}, {"start": 2447.05, "end": 2447.45, "word": " square", "probability": 0.427490234375}, {"start": 2447.45, "end": 2447.63, "word": " ..", "probability": 0.4599609375}, {"start": 2447.63, "end": 2447.77, "word": " لل", "probability": 0.9599609375}, {"start": 2447.77, "end": 2448.21, "word": " function", "probability": 0.79736328125}, {"start": 2448.21, "end": 2448.53, "word": " اللي", "probability": 0.893798828125}, {"start": 2448.53, "end": 2448.69, "word": " هي", "probability": 0.9404296875}, {"start": 2448.69, "end": 2449.03, "word": " square", "probability": 0.8095703125}, {"start": 2449.03, "end": 2449.41, "word": " root", "probability": 0.9169921875}, {"start": 2449.41, "end": 2449.59, "word": " of", "probability": 0.98779296875}, {"start": 2449.59, "end": 2450.01, "word": " F", "probability": 0.8935546875}, {"start": 2450.01, "end": 2451.91, "word": " عند", "probability": 0.409423828125}, {"start": 2451.91, "end": 2452.55, "word": " الـ", "probability": 0.715087890625}, {"start": 2452.55, "end": 2452.81, "word": " C", "probability": 0.75927734375}, {"start": 2452.81, "end": 2454.09, "word": " موجودة", "probability": 0.9896240234375}, {"start": 2454.09, "end": 2455.13, "word": " وبتساوي", "probability": 0.82607421875}, {"start": 2455.13, "end": 2455.87, "word": " ال", "probability": 0.9248046875}, {"start": 2455.87, "end": 2456.33, "word": " square", "probability": 0.91796875}, {"start": 2456.33, "end": 2456.97, "word": " root", "probability": 0.9267578125}, {"start": 2456.97, "end": 2459.15, "word": " وبتساوي", "probability": 0.9017578125}, {"start": 2459.15, "end": 2459.63, "word": " جذر", "probability": 0.8321940104166666}, {"start": 2459.63, "end": 2460.33, "word": " التربيع", "probability": 0.781494140625}, {"start": 2460.33, "end": 2462.09, "word": " أيه؟", "probability": 0.6874186197916666}, {"start": 2462.09, "end": 2462.37, "word": " ال", "probability": 0.515625}, {"start": 2462.37, "end": 2462.91, "word": " limit", "probability": 0.97509765625}, {"start": 2462.91, "end": 2463.39, "word": " لل", "probability": 0.29833984375}, {"start": 2463.39, "end": 2463.71, "word": " square", "probability": 0.81689453125}, {"start": 2463.71, "end": 2464.11, "word": " root", "probability": 0.908203125}], "temperature": 1.0}, {"id": 91, "seek": 247575, "start": 2465.35, "end": 2475.75, "text": "يعني بمعنى اخر انا ممكن ابدل ال limit مع ال square root و البرهان زي برهان النظرية السابقة", "tokens": [40228, 22653, 4724, 2304, 3615, 1863, 7578, 1975, 34740, 1975, 8315, 3714, 43020, 48127, 3215, 1211, 2423, 4948, 20449, 2423, 3732, 5593, 4032, 29739, 2288, 3224, 7649, 30767, 1829, 4724, 2288, 3224, 7649, 28239, 19913, 2288, 10632, 21136, 16758, 28671], "avg_logprob": -0.15129573461486073, "compression_ratio": 1.2982456140350878, "no_speech_prob": 0.0, "words": [{"start": 2465.35, "end": 2465.77, "word": "يعني", "probability": 0.928466796875}, {"start": 2465.77, "end": 2466.13, "word": " بمعنى", "probability": 0.976953125}, {"start": 2466.13, "end": 2466.53, "word": " اخر", "probability": 0.6500244140625}, {"start": 2466.53, "end": 2466.79, "word": " انا", "probability": 0.934326171875}, {"start": 2466.79, "end": 2467.03, "word": " ممكن", "probability": 0.987060546875}, {"start": 2467.03, "end": 2467.65, "word": " ابدل", "probability": 0.8289388020833334}, {"start": 2467.65, "end": 2468.07, "word": " ال", "probability": 0.78173828125}, {"start": 2468.07, "end": 2468.93, "word": " limit", "probability": 0.90771484375}, {"start": 2468.93, "end": 2469.15, "word": " مع", "probability": 0.97119140625}, {"start": 2469.15, "end": 2469.29, "word": " ال", "probability": 0.97412109375}, {"start": 2469.29, "end": 2469.53, "word": " square", 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نبرهنهم", "probability": 0.845751953125}, {"start": 2668.11, "end": 2671.91, "word": " فلو", "probability": 0.9627278645833334}, {"start": 2671.91, "end": 2672.23, "word": " بدنا", "probability": 0.796630859375}, {"start": 2672.23, "end": 2672.83, "word": " نبرهن", "probability": 0.9892578125}, {"start": 2672.83, "end": 2673.31, "word": " مثلا", "probability": 0.98046875}], "temperature": 1.0}, {"id": 99, "seek": 270307, "start": 2674.65, "end": 2703.07, "text": "الجزء الأخير هذا فممكن نستخدم ال sequential criterion يعني مثلا ال proof", "tokens": [6027, 7435, 11622, 38207, 16247, 9778, 13546, 23758, 6156, 2304, 43020, 8717, 14851, 9778, 40448, 2423, 42881, 46691, 37495, 22653, 50113, 15040, 2423, 8177], "avg_logprob": -0.09, "compression_ratio": 1.028301886792453, "no_speech_prob": 0.0, "words": [{"start": 2674.65, "end": 2675.33, "word": "الجزء", "probability": 0.98486328125}, {"start": 2675.33, "end": 2675.87, "word": " الأخير", "probability": 0.8658854166666666}, {"start": 2675.87, "end": 2676.37, "word": " هذا", "probability": 0.8818359375}, {"start": 2676.37, "end": 2692.75, "word": " فممكن", "probability": 0.908203125}, {"start": 2692.75, "end": 2695.29, "word": " نستخدم", "probability": 0.9415283203125}, {"start": 2695.29, "end": 2696.19, "word": " ال", "probability": 0.90283203125}, {"start": 2696.19, "end": 2696.77, "word": " sequential", "probability": 0.8427734375}, {"start": 2696.77, "end": 2697.61, "word": " criterion", "probability": 0.9833984375}, {"start": 2697.61, "end": 2702.03, "word": " يعني", "probability": 0.93408203125}, {"start": 2702.03, "end": 2702.39, "word": " مثلا", "probability": 0.9765625}, {"start": 2702.39, "end": 2702.67, "word": " ال", "probability": 0.5859375}, {"start": 2702.67, "end": 2703.07, "word": " proof", "probability": 0.97412109375}], "temperature": 1.0}, {"id": 100, "seek": 272992, "start": 2706.12, "end": 2729.92, "text": "of exercise أربعة طعش section أربعة اتنين we use sequential criterion", "tokens": [2670, 5380, 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"probability": 0.7810546875}, {"start": 2997.71, "end": 2998.19, "word": " عندي", "probability": 0.861572265625}, {"start": 2998.19, "end": 2999.51, "word": " ال", "probability": 0.78564453125}, {"start": 2999.51, "end": 2999.95, "word": " limit", "probability": 0.95556640625}, {"start": 2999.95, "end": 3002.45, "word": " وهذه", "probability": 0.668212890625}, {"start": 3002.45, "end": 3002.91, "word": " عبارة", "probability": 0.9918212890625}, {"start": 3002.91, "end": 3003.39, "word": " عن", "probability": 0.99755859375}, {"start": 3003.39, "end": 3004.55, "word": " limit", "probability": 0.73583984375}], "temperature": 1.0}, {"id": 111, "seek": 303501, "start": 3007.53, "end": 3035.01, "text": "للـ square root ل F of XN لما N تقول infinity اذا انا بدأت ب XN sequence contained in A ونهايتها C فطلع نهايت نهايت صورتها صورة ال sequence موجودة وبساوي ال square root ل L موجودة وبالتالي therefore by sequential", "tokens": [1211, 1211, 39184, 3732, 5593, 5296, 479, 295, 1783, 45, 5296, 15042, 426, 6055, 39648, 13202, 1975, 15730, 1975, 8315, 47525, 10721, 2655, 4724, 1783, 45, 8310, 16212, 294, 316, 4032, 1863, 11296, 36081, 11296, 383, 6156, 9566, 1211, 3615, 8717, 11296, 36081, 8717, 11296, 36081, 20328, 13063, 2655, 11296, 20328, 13063, 3660, 2423, 8310, 3714, 29245, 23328, 3660, 4032, 3555, 3794, 995, 45865, 2423, 3732, 5593, 5296, 441, 3714, 29245, 23328, 3660, 46599, 6027, 2655, 6027, 1829, 4412, 538, 42881], "avg_logprob": -0.2844893234532054, "compression_ratio": 1.563157894736842, "no_speech_prob": 0.0, "words": [{"start": 3007.53, "end": 3008.05, "word": "للـ", "probability": 0.3449300130208333}, {"start": 3008.05, "end": 3008.27, "word": " square", "probability": 0.2294921875}, {"start": 3008.27, "end": 3008.83, "word": " root", "probability": 0.87890625}, {"start": 3008.83, "end": 3009.03, "word": " ل", "probability": 0.828125}, {"start": 3009.03, "end": 3009.43, "word": " F", "probability": 0.356689453125}, {"start": 3009.43, "end": 3010.05, "word": " of", "probability": 0.80908203125}, {"start": 3010.05, "end": 3010.85, "word": " XN", "probability": 0.5357666015625}, {"start": 3010.85, "end": 3012.03, "word": " لما", "probability": 0.894775390625}, {"start": 3012.03, "end": 3012.33, "word": " N", "probability": 0.84765625}, {"start": 3012.33, "end": 3012.73, "word": " تقول", "probability": 0.818115234375}, {"start": 3012.73, "end": 3013.41, "word": " infinity", "probability": 0.208251953125}, {"start": 3013.41, "end": 3015.03, "word": " اذا", "probability": 0.6356201171875}, {"start": 3015.03, "end": 3015.27, "word": " انا", "probability": 0.6668701171875}, {"start": 3015.27, "end": 3015.85, "word": " بدأت", "probability": 0.8846028645833334}, {"start": 3015.85, "end": 3016.01, "word": " ب", "probability": 0.7744140625}, {"start": 3016.01, "end": 3016.63, "word": " XN", "probability": 0.5916748046875}, {"start": 3016.63, "end": 3017.33, "word": " sequence", "probability": 0.82080078125}, {"start": 3017.33, "end": 3017.87, "word": " contained", "probability": 0.58642578125}, {"start": 3017.87, "end": 3018.13, "word": " in", "probability": 0.93359375}, {"start": 3018.13, "end": 3018.39, "word": " A", "probability": 0.96435546875}, {"start": 3018.39, "end": 3019.29, "word": " ونهايتها", "probability": 0.86640625}, {"start": 3019.29, "end": 3019.65, "word": " C", "probability": 0.90234375}, {"start": 3019.65, "end": 3020.97, "word": " فطلع", "probability": 0.9718017578125}, {"start": 3020.97, "end": 3021.89, "word": " نهايت", "probability": 0.799072265625}, {"start": 3021.89, "end": 3025.33, "word": " نهايت", "probability": 0.7262369791666666}, {"start": 3025.33, "end": 3026.31, "word": " صورتها", "probability": 0.98291015625}, {"start": 3026.31, "end": 3026.93, "word": " صورة", "probability": 0.9324544270833334}, {"start": 3026.93, "end": 3027.09, "word": " ال", "probability": 0.95068359375}, {"start": 3027.09, "end": 3027.45, "word": " sequence", "probability": 0.8173828125}, {"start": 3027.45, "end": 3028.57, "word": " موجودة", "probability": 0.9827880859375}, {"start": 3028.57, "end": 3029.79, "word": " وبساوي", "probability": 0.642724609375}, {"start": 3029.79, "end": 3029.93, "word": " ال", "probability": 0.77392578125}, {"start": 3029.93, "end": 3030.25, "word": " square", "probability": 0.77099609375}, {"start": 3030.25, "end": 3030.69, "word": " root", "probability": 0.923828125}, {"start": 3030.69, "end": 3030.87, "word": " ل", "probability": 0.97705078125}, {"start": 3030.87, "end": 3031.17, "word": " L", "probability": 0.70947265625}, {"start": 3031.17, "end": 3032.13, "word": " موجودة", "probability": 0.981201171875}, {"start": 3032.13, "end": 3033.11, "word": " وبالتالي", "probability": 0.9212890625}, {"start": 3033.11, "end": 3033.61, "word": " therefore", "probability": 0.485595703125}, {"start": 3033.61, "end": 3034.15, "word": " by", "probability": 0.8984375}, {"start": 3034.15, "end": 3035.01, "word": " sequential", "probability": 0.95947265625}], "temperature": 1.0}, {"id": 112, "seek": 306382, "start": 3039.06, "end": 3063.82, "text": "criterion ال limit لل square root ل F of X لما X تقول إلى C بساوي exist و بساوي ال square root ل F and C أو اللي هو اللي بساوي .. لأ بساوي اللي هو", "tokens": [10757, 1681, 313, 2423, 4948, 24976, 3732, 5593, 5296, 479, 295, 1783, 5296, 15042, 1783, 6055, 39648, 30731, 383, 4724, 3794, 995, 45865, 2514, 4032, 4724, 3794, 995, 45865, 2423, 3732, 5593, 5296, 479, 293, 383, 34051, 13672, 1829, 31439, 13672, 1829, 4724, 3794, 995, 45865, 4386, 5296, 10721, 4724, 3794, 995, 45865, 13672, 1829, 31439], "avg_logprob": -0.2802905659926565, "compression_ratio": 1.5891472868217054, "no_speech_prob": 0.0, "words": [{"start": 3039.0599999999995, "end": 3039.8999999999996, "word": "criterion", "probability": 0.671966552734375}, {"start": 3039.8999999999996, "end": 3040.74, "word": " ال", "probability": 0.1719970703125}, {"start": 3040.74, "end": 3041.18, "word": " limit", "probability": 0.55322265625}, {"start": 3041.18, "end": 3043.1, "word": " لل", "probability": 0.58447265625}, {"start": 3043.1, "end": 3043.46, "word": " square", "probability": 0.705078125}, {"start": 3043.46, "end": 3043.86, "word": " root", "probability": 0.90673828125}, {"start": 3043.86, "end": 3044.02, "word": " ل", "probability": 0.78857421875}, {"start": 3044.02, "end": 3044.36, "word": " F", "probability": 0.443359375}, {"start": 3044.36, "end": 3045.96, "word": " of", "probability": 0.8671875}, {"start": 3045.96, "end": 3046.36, "word": " X", "probability": 0.904296875}, {"start": 3046.36, "end": 3046.72, "word": " لما", "probability": 0.831298828125}, {"start": 3046.72, "end": 3047.08, "word": " X", "probability": 0.95751953125}, {"start": 3047.08, "end": 3047.62, "word": " تقول", "probability": 0.9052734375}, {"start": 3047.62, "end": 3048.08, "word": " إلى", "probability": 0.48583984375}, {"start": 3048.08, "end": 3048.62, "word": " C", "probability": 0.9326171875}, {"start": 3048.62, "end": 3050.18, "word": " بساوي", "probability": 0.843017578125}, {"start": 3050.18, "end": 3052.28, "word": " exist", "probability": 0.73291015625}, {"start": 3052.28, "end": 3053.76, "word": " و", "probability": 0.470458984375}, {"start": 3053.76, "end": 3054.26, "word": " بساوي", "probability": 0.8648681640625}, {"start": 3054.26, "end": 3054.4, "word": " ال", "probability": 0.64697265625}, {"start": 3054.4, "end": 3054.74, "word": " square", "probability": 0.90966796875}, {"start": 3054.74, "end": 3055.16, "word": " root", "probability": 0.92578125}, {"start": 3055.16, "end": 3055.36, "word": " ل", "probability": 0.9697265625}, {"start": 3055.36, "end": 3055.78, "word": " F", "probability": 0.95654296875}, {"start": 3055.78, "end": 3057.34, "word": " and", "probability": 0.92138671875}, {"start": 3057.34, "end": 3057.84, "word": " C", "probability": 0.96044921875}, {"start": 3057.84, "end": 3060.98, "word": " أو", "probability": 0.52587890625}, {"start": 3060.98, "end": 3061.16, "word": " اللي", "probability": 0.941162109375}, {"start": 3061.16, "end": 3061.42, "word": " هو", "probability": 0.986328125}, {"start": 3061.42, "end": 3061.82, "word": " اللي", "probability": 0.92578125}, {"start": 3061.82, "end": 3062.44, "word": " بساوي", "probability": 0.875732421875}, {"start": 3062.44, "end": 3062.56, "word": " ..", "probability": 0.2467041015625}, {"start": 3062.56, "end": 3062.82, "word": " لأ", "probability": 0.901611328125}, {"start": 3062.82, "end": 3063.36, "word": " بساوي", "probability": 0.9051513671875}, {"start": 3063.36, "end": 3063.5, "word": " اللي", "probability": 0.985595703125}, {"start": 3063.5, "end": 3063.82, "word": " هو", "probability": 0.99365234375}], "temperature": 1.0}, {"id": 113, "seek": 308310, "start": 3069.8, "end": 3083.1, "text": "السكوير روت ال L اللي هو برضه اللي هو نعم نعم", "tokens": [6027, 3794, 4117, 2407, 13546, 12602, 35473, 2423, 441, 13672, 1829, 31439, 4724, 43042, 3224, 13672, 1829, 31439, 8717, 25957, 8717, 25957], "avg_logprob": -0.36854618269464245, "compression_ratio": 1.196969696969697, "no_speech_prob": 0.0, "words": [{"start": 3069.8, "end": 3070.58, "word": "السكوير", "probability": 0.70166015625}, {"start": 3070.58, "end": 3070.88, "word": " روت", "probability": 0.904052734375}, {"start": 3070.88, "end": 3071.04, "word": " ال", "probability": 0.5908203125}, {"start": 3071.04, "end": 3071.28, "word": " L", "probability": 0.82421875}, {"start": 3071.28, "end": 3073.12, "word": " اللي", "probability": 0.787109375}, {"start": 3073.12, "end": 3073.34, "word": " هو", "probability": 0.9921875}, {"start": 3073.34, "end": 3074.02, "word": " برضه", "probability": 0.9733072916666666}, {"start": 3074.02, "end": 3076.54, "word": " اللي", "probability": 0.54205322265625}, {"start": 3076.54, "end": 3080.62, "word": " هو", "probability": 0.40234375}, {"start": 3080.62, "end": 3082.72, "word": " نعم", "probability": 0.7017822265625}, {"start": 3082.72, "end": 3083.1, "word": " نعم", "probability": 0.9794921875}], "temperature": 1.0}, {"id": 114, "seek": 311232, "start": 3091.48, "end": 3112.32, "text": "يعني هاد ممكن هاد يسميها L من الأول فإذا بطلع عندي ال square root function لها limit، limit عن سي موجودة بساوي square root ل L إذا هاد بكمل البرهن بالمثل ممكن نبرهن exercise اللي جابله 13", "tokens": [40228, 22653, 8032, 18513, 3714, 43020, 8032, 18513, 7251, 38251, 1829, 11296, 441, 9154, 16247, 12610, 6156, 28814, 15730, 4724, 9566, 1211, 3615, 18871, 16254, 2423, 3732, 5593, 2445, 5296, 11296, 4948, 12399, 4948, 18871, 8608, 1829, 3714, 29245, 23328, 3660, 4724, 3794, 995, 45865, 3732, 5593, 5296, 441, 11933, 15730, 8032, 18513, 4724, 24793, 1211, 2423, 26890, 3224, 1863, 20666, 2304, 12984, 1211, 3714, 43020, 8717, 26890, 3224, 1863, 5380, 13672, 1829, 10874, 16758, 43761, 3705], "avg_logprob": -0.251802873535034, "compression_ratio": 1.5157894736842106, "no_speech_prob": 0.0, "words": [{"start": 3091.48, "end": 3091.86, "word": "يعني", "probability": 0.699462890625}, {"start": 3091.86, "end": 3092.1, "word": " هاد", "probability": 0.24505615234375}, {"start": 3092.1, "end": 3092.46, "word": " ممكن", "probability": 0.91796875}, {"start": 3092.46, "end": 3092.76, "word": " هاد", "probability": 0.699951171875}, {"start": 3092.76, "end": 3093.42, "word": " يسميها", "probability": 0.9658203125}, {"start": 3093.42, "end": 3093.74, "word": " L", "probability": 0.65966796875}, {"start": 3093.74, "end": 3094.72, "word": " من", "probability": 0.84716796875}, {"start": 3094.72, "end": 3095.12, "word": " الأول", "probability": 0.951904296875}, {"start": 3095.12, "end": 3096.46, "word": " فإذا", "probability": 0.833984375}, {"start": 3096.46, "end": 3096.98, "word": " بطلع", "probability": 0.7957763671875}, {"start": 3096.98, "end": 3097.5, "word": " عندي", "probability": 0.6602783203125}, {"start": 3097.5, "end": 3097.96, "word": " ال", "probability": 0.65185546875}, {"start": 3097.96, "end": 3098.18, "word": " square", "probability": 0.68017578125}, {"start": 3098.18, "end": 3098.42, "word": " root", "probability": 0.8623046875}, {"start": 3098.42, "end": 3098.86, "word": " function", "probability": 0.8916015625}, {"start": 3098.86, "end": 3099.18, "word": " لها", "probability": 0.7283935546875}, {"start": 3099.18, "end": 3099.84, "word": " limit،", "probability": 0.575927734375}, {"start": 3099.84, "end": 3100.18, "word": " limit", "probability": 0.80078125}, {"start": 3100.18, "end": 3100.6, "word": " عن", "probability": 0.24365234375}, {"start": 3100.6, "end": 3100.94, "word": " سي", "probability": 0.790771484375}, {"start": 3100.94, "end": 3101.56, "word": " موجودة", "probability": 0.9630126953125}, {"start": 3101.56, "end": 3102.02, "word": " بساوي", "probability": 0.82568359375}, {"start": 3102.02, "end": 3102.7, "word": " square", "probability": 0.74560546875}, {"start": 3102.7, "end": 3102.96, "word": " root", "probability": 0.93994140625}, {"start": 3102.96, "end": 3103.12, "word": " ل", "probability": 0.93359375}, {"start": 3103.12, "end": 3103.32, "word": " L", "probability": 0.75732421875}, {"start": 3103.32, "end": 3104.16, "word": " إذا", "probability": 0.695556640625}, {"start": 3104.16, "end": 3104.44, "word": " هاد", "probability": 0.814208984375}, {"start": 3104.44, "end": 3105.12, "word": " بكمل", "probability": 0.9254557291666666}, {"start": 3105.12, "end": 3106.26, "word": " البرهن", "probability": 0.880126953125}, {"start": 3106.26, "end": 3108.7, "word": " بالمثل", "probability": 0.9505615234375}, {"start": 3108.7, "end": 3109.04, "word": " ممكن", "probability": 0.989501953125}, {"start": 3109.04, "end": 3109.68, "word": " نبرهن", "probability": 0.85223388671875}, {"start": 3109.68, "end": 3110.58, "word": " exercise", "probability": 0.78564453125}, {"start": 3110.58, "end": 3111.58, "word": " اللي", "probability": 0.91552734375}, {"start": 3111.58, "end": 3111.94, "word": " جابله", "probability": 0.97216796875}, {"start": 3111.94, "end": 3112.32, "word": " 13", "probability": 0.5283203125}], "temperature": 1.0}, {"id": 115, "seek": 312807, "start": 3117.01, "end": 3128.07, "text": "فحاولوا يكونوا تبرهنوا exercise 13 بنفس الطريقة، في أي سؤال أو استفسار؟ okay إذا المرة الجاية بال campbell", "tokens": [5172, 5016, 995, 12610, 14407, 7251, 30544, 14407, 6055, 26890, 3224, 1863, 14407, 5380, 3705, 44945, 36178, 41950, 16572, 28671, 12399, 8978, 36632, 8608, 33604, 6027, 34051, 44713, 36178, 9640, 22807, 1392, 11933, 15730, 9673, 25720, 25724, 995, 10632, 20666, 2255, 7100], "avg_logprob": -0.3631904042044351, "compression_ratio": 1.2357142857142858, "no_speech_prob": 0.0, "words": [{"start": 3117.01, "end": 3117.81, "word": "فحاولوا", "probability": 0.91640625}, {"start": 3117.81, "end": 3118.03, "word": " يكونوا", "probability": 0.5903727213541666}, {"start": 3118.03, "end": 3118.57, "word": " تبرهنوا", "probability": 0.596142578125}, {"start": 3118.57, "end": 3119.07, "word": " exercise", "probability": 0.31298828125}, {"start": 3119.07, "end": 3119.79, "word": " 13", "probability": 0.7216796875}, {"start": 3119.79, "end": 3120.49, "word": " بنفس", "probability": 0.976318359375}, {"start": 3120.49, "end": 3121.53, "word": " الطريقة،", "probability": 0.826416015625}, {"start": 3121.53, "end": 3122.27, "word": " في", "probability": 0.8427734375}, {"start": 3122.27, "end": 3122.43, "word": " أي", "probability": 0.63037109375}, {"start": 3122.43, "end": 3122.79, "word": " سؤال", "probability": 0.9871419270833334}, {"start": 3122.79, "end": 3122.97, "word": " أو", "probability": 0.73681640625}, {"start": 3122.97, "end": 3125.07, "word": " استفسار؟", "probability": 0.9443359375}, {"start": 3125.07, "end": 3126.27, "word": " okay", "probability": 0.623046875}, {"start": 3126.27, "end": 3126.63, "word": " إذا", "probability": 0.589599609375}, {"start": 3126.63, "end": 3127.17, "word": " المرة", "probability": 0.867431640625}, {"start": 3127.17, "end": 3127.47, "word": " الجاية", "probability": 0.544921875}, {"start": 3127.47, "end": 3127.57, "word": " بال", "probability": 0.8173828125}, {"start": 3127.57, "end": 3128.07, "word": " campbell", "probability": 0.5234375}], "temperature": 1.0}], "language": "ar", "language_probability": 1.0, "duration": 3128.70325, "duration_after_vad": 2778.01749999999}
\ No newline at end of file
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/Ehj01gka7EU_postprocess.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/Ehj01gka7EU_postprocess.srt
new file mode 100644
index 0000000000000000000000000000000000000000..01bc4bdff2ebe09d17ddfc5ba51c05f92b5d3e6a
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/Ehj01gka7EU_postprocess.srt
@@ -0,0 +1,1740 @@
+1
+00:00:19,740 --> 00:00:27,020
+بسم الله الرحمن الرحيم هنواصل اليوم تغطية section
+
+2
+00:00:27,020 --> 00:00:34,550
+5-3اللي بتعلق ب .. موضوع ال continuous functions
+
+3
+00:00:34,550 --> 00:00:40,590
+على ال intervals على الفترات احنا بدينا ال ..
+
+4
+00:00:40,590 --> 00:00:46,690
+بدينا الموضوع هذا المحاضرة السابقة و كان اخر نظرية
+
+5
+00:00:46,690 --> 00:00:50,890
+أخدناها اللي هي ال maximum .. maximum minimum
+
+6
+00:00:50,890 --> 00:00:56,870
+theorem نعود نستذكر بس نظرية الأخيرة هذه ال
+
+7
+00:00:56,870 --> 00:00:57,670
+maximum
+
+8
+00:01:12,090 --> 00:01:21,410
+ال maximum minimum minimum
+
+9
+00:01:21,410 --> 00:01:28,050
+theorem يقول
+
+10
+00:01:28,050 --> 00:01:36,050
+إذا كانت if I is a closed and bounded interval is
+
+11
+00:01:36,050 --> 00:01:40,290
+closed and bounded
+
+12
+00:01:46,350 --> 00:01:56,730
+وإذا كانت العملية من I إلى R مستمرة
+
+13
+00:01:56,730 --> 00:02:01,270
+على
+
+14
+00:02:01,270 --> 00:02:02,470
+الفترة I
+
+15
+00:02:07,590 --> 00:02:20,690
+there exist x lower star و x upper star عناصر في I
+
+16
+00:02:20,690 --> 00:02:22,070
+بحيث انه
+
+17
+00:02:24,550 --> 00:02:32,550
+f of x lower star بساوي ال minimum ل range ال
+
+18
+00:02:32,550 --> 00:02:41,450
+function f and f of x super star بساوي ال supremum
+
+19
+00:02:41,450 --> 00:02:49,410
+ل range ال function f وبالتالي هذه بسميها ال
+
+20
+00:02:49,410 --> 00:02:52,930
+absolute maximum
+
+21
+00:02:54,540 --> 00:03:03,820
+value و القيمة هتبسميها ال absolute minimum
+
+22
+00:03:03,820 --> 00:03:06,900
+value
+
+23
+00:03:06,900 --> 00:03:15,920
+لل function f على الفترة I طبعا okay okay في اليوم
+
+24
+00:03:15,920 --> 00:03:22,580
+هناخد نظريات برضه خاصة باتصال الدوال على الفترات
+
+25
+00:03:23,810 --> 00:03:30,090
+فأول نظرية هتكون location location
+
+26
+00:03:30,090 --> 00:03:36,970
+of roots theorem
+
+27
+00:03:36,970 --> 00:03:45,570
+نظرية تحديد ال roots فنفس
+
+28
+00:03:45,570 --> 00:03:46,790
+الحاجة let
+
+29
+00:03:49,750 --> 00:03:57,890
+I be closed and bounded interval على الصورة AB and
+
+30
+00:03:57,890 --> 00:04:06,190
+let f be function from I to R be continuous
+
+31
+00:04:06,190 --> 00:04:09,790
+function
+
+32
+00:04:09,790 --> 00:04:15,390
+على الفترة المغلقة والمحدودة I if
+
+33
+00:04:17,630 --> 00:04:29,370
+لو كان f of a أصغر من صفر أصغر من f of b او f of b
+
+34
+00:04:29,370 --> 00:04:38,610
+أصغر من صفر أصغر من f of a then
+
+35
+00:04:38,610 --> 00:04:48,980
+there exist c ينتميللفترة المفتوحة من a إلى b بحيث
+
+36
+00:04:48,980 --> 00:04:57,540
+أن f of c بيساوي سفر فالنظرية
+
+37
+00:04:57,540 --> 00:05:08,100
+هذه ممكن أنلخصها بالرسمة التالية محاور
+
+38
+00:05:08,100 --> 00:05:13,280
+الأحداثيات وممكن يكون في ending حاجة زي هذه
+
+39
+00:05:22,650 --> 00:05:28,930
+فهي function هذه عبارة عن ال graph y بساوي f of x
+
+40
+00:05:28,930 --> 00:05:37,750
+ال function هذه متصلة على الفترة المغلطة من a ل d
+
+41
+00:05:37,750 --> 00:05:42,890
+وهي
+
+42
+00:05:42,890 --> 00:05:51,450
+عندي f of a أصغر من سفر وهي عندي
+
+43
+00:05:59,190 --> 00:06:02,510
+النظرية بتقول لو كان في اندرالا متصلة زي هذه على
+
+44
+00:06:02,510 --> 00:06:07,830
+فترة مغلقة من a لb وكان f of a أصغر من الصفر و
+
+45
+00:06:07,830 --> 00:06:16,370
+الصفر أصغر من f of bلابد ان نجد نقطة C بين A وB
+
+46
+00:06:16,370 --> 00:06:21,030
+بحيث ان قيمة الـ function عندها بالساوي سفر و واضح
+
+47
+00:06:21,030 --> 00:06:26,270
+ان نقطة C هي قيمة الـ function عندها بالساوي سفر
+
+48
+00:06:26,270 --> 00:06:30,830
+ممكن برضه يكون العكس يعني الملحانة هذا يكون شكله
+
+49
+00:06:30,830 --> 00:06:31,430
+زي هيك
+
+50
+00:06:35,680 --> 00:06:41,700
+فيكون يعني عندي هنا ال F of B هي السالة بقى وهي
+
+51
+00:06:41,700 --> 00:06:46,580
+عند ال A فال F of B هي الموجة بقى برضه نفس النتيجة
+
+52
+00:06:46,580 --> 00:06:47,620
+okay تمام؟
+
+53
+00:06:55,410 --> 00:06:59,790
+البرهان النظرية هذه يعني it's زي ما بيقولوا it's
+
+54
+00:06:59,790 --> 00:07:06,630
+quite technical يعني فيه شوية تفاصيل تقنية زاد انه
+
+55
+00:07:06,630 --> 00:07:13,090
+طويل شوية فاحنا عشان بصدر نهاية الفصل مابناش ناخد
+
+56
+00:07:13,090 --> 00:07:16,330
+.. ناخد .. ناخد في البرهين الطويلة فحسيبكم تقراوا
+
+57
+00:07:16,330 --> 00:07:19,530
+البرهان اذا see the textbook
+
+58
+00:07:25,030 --> 00:07:32,130
+إذا الممكن بدؤوكم يمكن تقرؤوا البرهان من الكتاب و
+
+59
+00:07:32,130 --> 00:07:36,770
+تحاولوا تفهموه طبعا البرهان طويل مابنجوبش طبعا
+
+60
+00:07:36,770 --> 00:07:41,990
+البرهين طويلة جه هدف الامتحانات okay فهذا بالنسبة
+
+61
+00:07:41,990 --> 00:07:46,930
+للبرهان الآن هاي مثال مثلا مثال example
+
+62
+00:07:54,050 --> 00:07:58,270
+Show that the
+
+63
+00:07:58,270 --> 00:08:03,470
+equation f
+
+64
+00:08:03,470 --> 00:08:11,510
+of x بتساوي x في e أُس x سالب اتنين بتساوي سفر has
+
+65
+00:08:11,510 --> 00:08:14,210
+a root
+
+66
+00:08:20,420 --> 00:08:29,980
+in الـ interval من سفر لواحد لنثبت
+
+67
+00:08:29,980 --> 00:08:35,200
+ان المعادلة f of x بالساوي سفر عشان f of x بالساوي
+
+68
+00:08:35,200 --> 00:08:43,100
+الدالة هذه لها جدر يعني بنقدر اللاجم اي
+
+69
+00:08:43,100 --> 00:08:54,460
+هذا يعني showان يوجد C ينتمي للفترة المغلقة من سفر
+
+70
+00:08:54,460 --> 00:09:01,900
+لواحد بحيث انه اخه C مساره سفر ففي الحالة اللي
+
+71
+00:09:01,900 --> 00:09:07,360
+بنقول انه C root جدر للمعادلة او C zero لل
+
+72
+00:09:07,360 --> 00:09:14,900
+function F فبنرفبت الكلام هذا فحسب النظرية هذه
+
+73
+00:09:27,630 --> 00:09:35,370
+F of X بساوي X في E to X سالب اتنين is continuous
+
+74
+00:09:35,370 --> 00:09:40,650
+متصلة على الفترة المغلقة من سفر لواحد
+
+75
+00:09:47,890 --> 00:09:51,510
+لأن X في E to X هي دالة متصلة اتراحى منها ثابت
+
+76
+00:09:51,510 --> 00:09:56,750
+دالة متصلة على R كذلك
+
+77
+00:09:56,750 --> 00:10:06,460
+انا عندي F of سفر بساوي سالب اتنين اصغر من سفرو F
+
+78
+00:10:06,460 --> 00:10:15,300
+of واحد بالساوي E ثاند اتنين وال E معروف انه عدد
+
+79
+00:10:15,300 --> 00:10:21,100
+اكبر من اتنين فهذا اكبر من ساكنة اذا هاي شروط ال
+
+80
+00:10:21,100 --> 00:10:28,500
+location of roots ال theorem كلها متحققة hence by
+
+81
+00:10:28,500 --> 00:10:33,700
+location of roots theorem
+
+82
+00:10:36,360 --> 00:10:42,300
+يوجد C أنتمي للفترة المغلقة من ستة إلى واحد بحيث
+
+83
+00:10:42,300 --> 00:10:55,640
+انه F of C بساوي سفر اذا هنا اثبتنا ان C is a root
+
+84
+00:10:55,640 --> 00:11:01,400
+of equation F of X
+
+85
+00:11:04,080 --> 00:11:09,640
+بساوي X في E أس X minus اتنين بساوي سفر وهو
+
+86
+00:11:09,640 --> 00:11:16,360
+المطلوب اذا هنا اثبتنا ان فعلا المعادلة هذه لها
+
+87
+00:11:16,360 --> 00:11:22,720
+جذر في الفترة هذا الجذر يقع هو عدد C يقع في الفترة
+
+88
+00:11:22,720 --> 00:11:28,180
+من سفر لوحده عدد من سفر لوحده طبعا ممكن هذا العدد
+
+89
+00:11:28,180 --> 00:11:35,450
+C نعمله تقريب إلى أقرب يعنيبحيث يكون النسبة الخطأ
+
+90
+00:11:35,450 --> 00:11:41,470
+من القيمة الحقيقية تبقى تكون أقل من واحد على ألف
+
+91
+00:11:41,470 --> 00:11:46,210
+أو واحد على مية أو واحد على عشر ألف فالكتاب الموضح
+
+92
+00:11:46,210 --> 00:11:51,610
+لكم هي هنا في المثال كيف نجيب تقريب نحصل العدد
+
+93
+00:11:51,610 --> 00:11:55,730
+سيرة بحيث نطلع تقريبا قريب من القيمة الحقيقية
+
+94
+00:11:55,730 --> 00:11:59,030
+والفرق بينها ومن القيمة الحقيقية واللي هنجيبها في
+
+95
+00:11:59,030 --> 00:12:05,240
+المثال تكون أقل من واحد على ألف أوش زيهافممكن تقرأ
+
+96
+00:12:05,240 --> 00:12:09,960
+و تشوف الكلام هذا في الكتاب لكن احنا اللي بهمنا ان
+
+97
+00:12:09,960 --> 00:12:15,460
+ال equation هذه ضمننا انه في لها root في الفترة
+
+98
+00:12:15,460 --> 00:12:18,740
+هذه حسب ال location of roots في الفترة الباقية
+
+99
+00:12:18,740 --> 00:12:24,520
+كانت تخلي ال root هذا يعني تجيبله قيمة قريبة جدا
+
+100
+00:12:24,520 --> 00:12:30,020
+من القيمة الحقيقية هذه مجرد يعني تفاصيل حسابية
+
+101
+00:12:30,020 --> 00:12:36,320
+okay فحاولوا تقراوها من الكتاب لو سمحتالان هذه
+
+102
+00:12:36,320 --> 00:12:47,540
+النظرية بتقود الى نظرية تانية وهي
+
+103
+00:12:47,540 --> 00:12:55,380
+Bolzano's
+
+104
+00:12:55,380 --> 00:12:57,140
+intermediate
+
+105
+00:13:04,990 --> 00:13:25,730
+value theorem let
+
+106
+00:13:25,730 --> 00:13:29,210
+I be any interval
+
+107
+00:13:36,870 --> 00:13:50,310
+and if from I to R be continuous على
+
+108
+00:13:50,310 --> 00:14:00,830
+الفترة I إذا كان A و B أعداد في الفترة I and
+
+109
+00:14:03,830 --> 00:14:16,170
+K عدد حقيقي such that F of A أصغر من K أصغر من F
+
+110
+00:14:16,170 --> 00:14:20,150
+of B then
+
+111
+00:14:20,150 --> 00:14:31,610
+النتيجة أنه يوجد C ينتمي للفترة I وهذا العدد C يقع
+
+112
+00:14:31,610 --> 00:14:32,010
+بين
+
+113
+00:14:38,340 --> 00:14:48,280
+between a and b such that بحيث ان f and c تطلع
+
+114
+00:14:48,280 --> 00:14:56,720
+بالساوية قيمة k لنعمل
+
+115
+00:14:56,720 --> 00:14:58,740
+رسمة قبل أن أظهر المظهر
+
+116
+00:15:17,560 --> 00:15:37,280
+فممكن يكون في عندي function زي هذه مثلا فهي
+
+117
+00:15:37,280 --> 00:15:43,320
+في عندي فترة I ال dialer معرفة و متصل عليها
+
+118
+00:15:45,810 --> 00:15:52,110
+يعني هذه الفترة من هنا إلى هنا I وممكن يكون في
+
+119
+00:15:52,110 --> 00:15:59,510
+عندي أعداد A وB فممكن يكون مثلا هذه ال A وهذه ال B
+
+120
+00:15:59,510 --> 00:16:04,630
+فهذه
+
+121
+00:16:04,630 --> 00:16:10,450
+F of A فهذه
+
+122
+00:16:10,450 --> 00:16:11,430
+F of A
+
+123
+00:16:16,610 --> 00:16:22,070
+وهي F of B فلو
+
+124
+00:16:22,070 --> 00:16:25,290
+كان
+
+125
+00:16:25,290 --> 00:16:38,180
+K عدد بين F of A و F of B فهي F of B وهي F of Aف K
+
+126
+00:16:38,180 --> 00:16:45,220
+عدد بين F of A و F of B فلهذا العدد نقدر نلاقي C
+
+127
+00:16:45,220 --> 00:16:49,420
+عدد C عدد
+
+128
+00:16:49,420 --> 00:16:53,960
+C بين A و B وبالتالي ينتمي للفترة I
+
+129
+00:16:57,560 --> 00:17:06,760
+إذا C بين A وB وينتمي للفترة I بحيث إن صورة C
+
+130
+00:17:06,760 --> 00:17:12,380
+هي صورة الـ C بساوي العدد P هذا هو بولزانو
+
+131
+00:17:12,380 --> 00:17:16,560
+intermediate value theorem نظرية القيمة الوسيطية
+
+132
+00:17:16,560 --> 00:17:22,740
+نظرية القيمة الوسيطية لبولزانو مرهانة نظرية هذه مش
+
+133
+00:17:22,740 --> 00:17:23,880
+صعبة سهل
+
+134
+00:17:43,340 --> 00:17:48,440
+Proof البرهان بعتمد على ال maximum minimum theorem
+
+135
+00:17:48,440 --> 00:17:55,160
+وعلى اللي هو location of roads theorem ففي عندي
+
+136
+00:17:55,160 --> 00:17:58,740
+هنا حلتين لاحظوا أن a و b أعداد دراوي
+
+137
+00:18:19,180 --> 00:18:25,800
+النتيجة بتكون واضحة لو كان a بساوي b ف f of a
+
+138
+00:18:25,800 --> 00:18:31,470
+بتطلع بساوي f of bوبالتالي اي k بين f of a وf of b
+
+139
+00:18:31,470 --> 00:18:35,690
+هيساوي واحدة منهم وبالتالي ال k بيساوي f of a خد
+
+140
+00:18:35,690 --> 00:18:40,790
+ال c بيساوي a او b فالنتيجة ايه واضح بدهية يعني
+
+141
+00:18:40,790 --> 00:18:49,030
+متحققات القائمة so assume ان
+
+142
+00:18:49,030 --> 00:18:52,630
+a لايساوي b then
+
+143
+00:18:54,390 --> 00:18:58,750
+by tricotomy property إذا كان في عددين بيسويش بعض
+
+144
+00:18:58,750 --> 00:19:06,610
+فبطلع a أصغر من b or b أصغر من a فناخد الحالة
+
+145
+00:19:06,610 --> 00:19:14,850
+الأولى case one لو كان a أصغر من b ففي الحالة هذه
+
+146
+00:19:21,390 --> 00:19:29,810
+لو كان ال a أصغر من b فبدي أعرف define
+
+147
+00:19:29,810 --> 00:19:39,130
+في الحالة هذه define g of x علي أنها الدالة
+
+148
+00:19:39,130 --> 00:19:43,990
+اللي هي بالساوي f
+
+149
+00:19:43,990 --> 00:19:47,990
+of x minus
+
+150
+00:19:47,990 --> 00:19:48,470
+k
+
+151
+00:19:51,460 --> 00:19:56,340
+فطبعا الـ function g الـ function f متصل على
+
+152
+00:19:56,340 --> 00:20:01,680
+الفترة I هو متصل على الفترة المغلقة من a إلى b
+
+153
+00:20:01,680 --> 00:20:07,080
+اللي هي جزء من الفترة I فالـ function g اللي
+
+154
+00:20:07,080 --> 00:20:11,760
+بتساوي f ثالث ثابت مثلها متصل على نفس الفترة اذا g
+
+155
+00:20:11,760 --> 00:20:18,450
+is continuous على الفترة المغلقة من a إلى bاللي هي
+
+156
+00:20:18,450 --> 00:20:22,130
+بالمناسبة مجموعة جزئية من I لأن ال A و ال B
+
+157
+00:20:22,130 --> 00:20:26,530
+موجودين في I و
+
+158
+00:20:26,530 --> 00:20:35,210
+كذلك لاحظوا أن G of A بساوي F of A minus K وهذا من
+
+159
+00:20:35,210 --> 00:20:44,570
+هنا من الفرض هذا بيطلع أصغر من سفروهذا أصغر من F
+
+160
+00:20:44,570 --> 00:20:52,070
+of B minus K F of B minus K بيطلع عموجة اللي هو
+
+161
+00:20:52,070 --> 00:20:58,110
+بساوي G of B اذا هاي شروط ال location of roots ال
+
+162
+00:20:58,110 --> 00:21:01,990
+theorem كلها متحققة هي اندي فانش جي متصلة على فترة
+
+163
+00:21:01,990 --> 00:21:06,560
+مغلقة ومحدودةوقيمة الـ G عند الـ left endpoint
+
+164
+00:21:06,560 --> 00:21:11,980
+سالبة وقيمة الـ G عند ال right endpoint موجبة and
+
+165
+00:21:11,980 --> 00:21:16,220
+then by then
+
+166
+00:21:16,220 --> 00:21:28,020
+by location of roots theorem يوجد
+
+167
+00:21:28,020 --> 00:21:37,570
+C ينتميللفترة I يعني ينتمي يوجد C ينتمي للفترة
+
+168
+00:21:37,570 --> 00:21:46,150
+المطوحة من A وB اللي هي subset من I بحيث انه صورة
+
+169
+00:21:46,150 --> 00:21:54,170
+الـ C عندها بساوي سفر لكن انا عندي G of C من تعريف
+
+170
+00:21:54,170 --> 00:22:02,490
+ال function GG of C بساوي F of C negative K حل
+
+171
+00:22:02,490 --> 00:22:09,850
+المعادلة هذه في F of C فبطلع F of C بساوي K كما هو
+
+172
+00:22:09,850 --> 00:22:15,270
+مطلوب زي ما هو مطلوب ان هيك بتكون برهانة نظرية بس
+
+173
+00:22:15,270 --> 00:22:20,540
+A في الحالة اللي فيها بتكون A أصغر من Bيبقى ندرين
+
+174
+00:22:20,540 --> 00:22:25,300
+النظرية في الحالة التالية case 2 اللي فيها ال b
+
+175
+00:22:25,300 --> 00:22:29,920
+أصغر من a ففي
+
+176
+00:22:29,920 --> 00:22:37,080
+الحالة هذه خلّيني أعرف المرة هذه function h على
+
+177
+00:22:37,080 --> 00:22:45,820
+أنها بتساوي k minus f of x فواضح clearly
+
+178
+00:22:48,290 --> 00:22:57,210
+واضح ان الـ H زيها زي ال F متصلة is continuous على
+
+179
+00:22:57,210 --> 00:23:06,690
+الفترة المغلقة والمحدودة من A ل B and H
+
+180
+00:23:06,690 --> 00:23:15,910
+of A بيساوي K minus K minus F of A بيطلع سالب K
+
+181
+00:23:15,910 --> 00:23:23,460
+minus F of Aومن الفرب هذا بيطلع سالب وهذا أصغر من
+
+182
+00:23:23,460 --> 00:23:36,300
+k minus f of b اللي هو بيطلع h of b كذلك كي لو
+
+183
+00:23:36,300 --> 00:23:41,600
+طرحت من ال k f of b فبيطلع سالب فرق إذا الأن في
+
+184
+00:23:41,600 --> 00:23:45,200
+اندي function h continuous على فترة مغلقة ومحدودة
+
+185
+00:23:45,970 --> 00:23:49,610
+وقيمتها عند الـ left endpoint سالبة وعند ال right
+
+186
+00:23:49,610 --> 00:23:58,170
+point موجبة اذا كل شروط ال location of roots في
+
+187
+00:23:58,170 --> 00:24:04,550
+المحققة so by
+
+188
+00:24:04,550 --> 00:24:12,790
+locationof roots theorem يوجد
+
+189
+00:24:12,790 --> 00:24:23,950
+C ينتمي الى الفترة مظبوط هيك؟ كده كده كده كده كده
+
+190
+00:24:23,950 --> 00:24:30,130
+كده كده كده كده
+
+191
+00:24:30,130 --> 00:24:31,150
+كده كده كده كده كده
+
+192
+00:24:36,660 --> 00:24:43,600
+هك صح K سالب F of B بطلع سالب و هنا هاد المفروض
+
+193
+00:24:43,600 --> 00:24:53,120
+تكون A و هاد A صحيح، بظبط، صح، اذا H of A اللي هي
+
+194
+00:24:53,120 --> 00:24:58,180
+K minus F of A هي K اطرح منها F of A بطلع موجة
+
+195
+00:24:58,940 --> 00:25:03,460
+بينما K سالم F of B بيطلع سالم، مظبوط هيك، إذا U
+
+196
+00:25:03,460 --> 00:25:10,920
+جان C بين B وA وهي طبعا فترة contained in R بحيث
+
+197
+00:25:10,920 --> 00:25:19,420
+انه H of C بيساوي سفر، لكن H of C من تعريفها هي
+
+198
+00:25:19,420 --> 00:25:24,400
+عبارة عن K minus F of C وبالتالي هذا بيقدر حل
+
+199
+00:25:24,400 --> 00:25:30,190
+المعادلة هذه في F of Cفبطلع F of C بساوي K وهو
+
+200
+00:25:30,190 --> 00:25:35,010
+المطلوب إذا في الحالتين أثبتنا أن يوجد C في الفترة
+
+201
+00:25:35,010 --> 00:25:43,510
+I بين A وB وقيمتها عند C بساوي LK إذا نليك بيكون
+
+202
+00:25:43,510 --> 00:25:47,930
+برهاننا Bolzano's Intermediate Value
+
+203
+00:25:55,030 --> 00:26:03,170
+الان هذه النظرية في عليها نتيجة مهمة
+
+204
+00:26:20,170 --> 00:26:26,910
+let I بساوي closed and bounded interval and if the
+
+205
+00:26:26,910 --> 00:26:37,070
+function from I to R be continuous واتصلة على
+
+206
+00:26:37,070 --> 00:26:42,590
+الفترة I تمام؟
+
+207
+00:26:42,590 --> 00:26:45,910
+لو كان
+
+208
+00:26:48,570 --> 00:27:01,130
+فك عدد حقيقي satisfies
+
+209
+00:27:01,130 --> 00:27:08,250
+بيحقق الشرط التالي انه ك .. العدد ك هذا أكبر من أو
+
+210
+00:27:08,250 --> 00:27:16,090
+ساوي ال infimum لست f of I اللي هو range ال Fاللي
+
+211
+00:27:16,090 --> 00:27:20,590
+هي القيمة الصغيرة المطلقة ل F على I وأصغر من أوسعه
+
+212
+00:27:20,590 --> 00:27:24,890
+ال supremum ل range ال F اللي هي ال absolute
+
+213
+00:27:24,890 --> 00:27:29,850
+maximum value ل ال function F على I ففي الحالة هذه
+
+214
+00:27:29,850 --> 00:27:42,260
+من نقدر نلاقي C there existC ينتمي للفترة I بحيث
+
+215
+00:27:42,260 --> 00:27:51,400
+ان F of C بيساوي العدد K وبرهان
+
+216
+00:27:51,400 --> 00:28:00,440
+النظرية هذه سهل By
+
+217
+00:28:00,440 --> 00:28:05,440
+maximum minimum theorem
+
+218
+00:28:11,040 --> 00:28:14,040
+الـ maximum minimum theorem بتقول لو كان في أندي
+
+219
+00:28:14,040 --> 00:28:18,640
+function مفتصلة على فترة مغلقة ومحدودة فال
+
+220
+00:28:18,640 --> 00:28:24,020
+function هذه بتأخذ قيمها العظمى المطلقة وقيمتها
+
+221
+00:28:24,020 --> 00:28:29,360
+العظمى المطلقة وقيمتها العظمى المطلقة على الفترة I
+
+222
+00:28:29,360 --> 00:28:34,920
+يعني في أعداد في الفترة I أندها ال function بتاخد
+
+223
+00:28:34,920 --> 00:28:37,760
+قيمتها العظمى المطلقة وقيمتها العظمى المطلقة
+
+224
+00:28:40,910 --> 00:28:51,330
+إذاً there exist x lower star و x super star عناصر
+
+225
+00:28:51,330 --> 00:29:00,870
+في I بحيث أن ال F of x lower star بساول infimum
+
+226
+00:29:01,750 --> 00:29:10,230
+لسيت f of i and f of x super star بيساوي ال
+
+227
+00:29:10,230 --> 00:29:15,050
+supremum لسيت
+
+228
+00:29:15,050 --> 00:29:26,270
+f of i تمام
+
+229
+00:29:26,270 --> 00:29:27,750
+hence
+
+230
+00:29:31,910 --> 00:29:41,670
+by حسب ال hypothesis ال hypothesis star من الفرض
+
+231
+00:29:41,670 --> 00:29:44,750
+ال star اللي هو احنا فرضين انه ال key عدد key هذا
+
+232
+00:29:44,750 --> 00:29:55,670
+بحق المتباينة يعني we have لديناالـ k أكبر من أو
+
+233
+00:29:55,670 --> 00:30:06,970
+ساوي f of x lower star أصغر من أو ساوي f
+
+234
+00:30:06,970 --> 00:30:15,370
+of upper star و
+
+235
+00:30:15,370 --> 00:30:21,630
+ال function and if is continuousعلى الفترة المغلقة
+
+236
+00:30:21,630 --> 00:30:33,690
+من x lower star ل x super star او
+
+237
+00:30:33,690 --> 00:30:41,650
+لعكس ممكن يكونوا متبادلة تانية او x super star x
+
+238
+00:30:41,650 --> 00:30:46,320
+lower starتعتمد على مين اللي أصغر من التانية إذا
+
+239
+00:30:46,320 --> 00:30:50,760
+كانت هذه أصغر من هذه فهذه تطلع فترة داخل I و F
+
+240
+00:30:50,760 --> 00:30:54,340
+continuous على I ايضا continuous على أي فترة جزئية
+
+241
+00:30:54,340 --> 00:30:58,060
+منها وإذا كان ال X Superstar أصغر من X Lower Star
+
+242
+00:30:58,060 --> 00:30:59,380
+فمناخد الفترة أيضا
+
+243
+00:31:03,410 --> 00:31:08,850
+شروط بولزانو فيروس تراسي فيرم هاي في عندي نقطتين A
+
+244
+00:31:08,850 --> 00:31:16,070
+و B بينتموا للفترة I و F continuous على I
+
+245
+00:31:28,770 --> 00:31:35,190
+وعندي a و b بينتموا للفترة I وعندي K أكبر من أو
+
+246
+00:31:35,190 --> 00:31:47,070
+ساوي F of A أصغر من أو ساوي F of B so by Bolzano's
+
+247
+00:31:57,180 --> 00:32:09,500
+يوجد C ينتمي للفترة I بين X
+
+248
+00:32:09,500 --> 00:32:21,000
+lower star و X super starبحيث ان f of c بساوي
+
+249
+00:32:21,000 --> 00:32:25,580
+العدد k وهذا
+
+250
+00:32:25,580 --> 00:32:30,820
+اللي بدنا نقيله يعني اثبتنا يوجد c ينتمي للفترة I
+
+251
+00:32:30,820 --> 00:32:38,800
+وصورة c بساوي العدد k وهو المطلوب اذا هذه النتيجة
+
+252
+00:32:38,800 --> 00:32:44,440
+على بلزانو intermediate valley theoremبرهنها بكل
+
+253
+00:32:44,440 --> 00:32:51,920
+بساطة وبكل أريحية واضح البرهان في أي استفسار أن
+
+254
+00:32:51,920 --> 00:32:55,420
+البرهان هنا تبع النظرية هذه بعتمد على maximum
+
+255
+00:32:55,420 --> 00:32:59,680
+minimum maximum minimum theorem نظرية القيام
+
+256
+00:32:59,680 --> 00:33:03,360
+القصوى نخدناها المحاضرة اللي فاتت وعلى Bolzano
+
+257
+00:33:03,360 --> 00:33:11,040
+intermediate value theorem okay
+
+258
+00:33:11,040 --> 00:33:11,480
+تمام
+
+259
+00:33:15,690 --> 00:33:23,230
+طيب ال ..
+
+260
+00:33:23,230 --> 00:33:29,290
+ناخد نظرية
+
+261
+00:33:29,290 --> 00:33:38,950
+يمكن
+
+262
+00:33:38,950 --> 00:33:43,350
+ما نحتاجش هدول نمسح
+
+263
+00:33:43,350 --> 00:33:44,010
+اللوح هذا
+
+264
+00:34:03,530 --> 00:34:11,490
+فيرم let I بساوي closed bounded interval be closed
+
+265
+00:34:11,490 --> 00:34:25,090
+and bounded closed and bounded interval and let f
+
+266
+00:34:25,090 --> 00:34:49,240
+from I to Rدي continuous متصلة على الفترة I then
+
+267
+00:34:49,240 --> 00:34:57,280
+النتيجة انه ال set او ال rangeالـ range للـ
+
+268
+00:34:57,280 --> 00:35:08,740
+function I is a closed and bounded closed and
+
+269
+00:35:08,740 --> 00:35:17,460
+bounded interval that
+
+270
+00:35:17,460 --> 00:35:18,920
+is هذا يعني
+
+271
+00:35:21,810 --> 00:35:26,930
+هذا يعني .. يعني النص او نتيجة النظرية دي من كلها
+
+272
+00:35:26,930 --> 00:35:33,850
+خصها في عبارة واحدة وهي انه a continuous function
+
+273
+00:35:33,850 --> 00:35:44,230
+a continuous function preserves .. preserves
+
+274
+00:35:44,230 --> 00:35:49,610
+بتحافظ closed
+
+275
+00:35:51,530 --> 00:35:56,510
+and bounded intervals
+
+276
+00:35:56,510 --> 00:36:00,870
+الدوال
+
+277
+00:36:00,870 --> 00:36:05,130
+المتصلة بتحافظ على ال closed و ال bounded interval
+
+278
+00:36:05,130 --> 00:36:10,350
+يعني ال function f بتاخد I اللي هي closed bounded
+
+279
+00:36:10,350 --> 00:36:13,610
+interval بتعطيني صلتها closed bounded interval
+
+280
+00:36:13,610 --> 00:36:19,410
+زيها من نفس الصنف من نفس النوع لبرهان ذلك
+
+281
+00:36:29,060 --> 00:36:44,440
+ف let M بساوي الالفمن ل range ال F و
+
+282
+00:36:44,440 --> 00:36:50,720
+capital M بساوي ال superman ل range ال F
+
+283
+00:36:56,430 --> 00:37:11,770
+بOTH M AND N EXIST IN R BY MAXIMUM
+
+284
+00:37:11,770 --> 00:37:15,290
+MINIMUM THEOREM
+
+285
+00:37:19,900 --> 00:37:25,120
+نظرية القيام القصوى بتقول إنه إذا كانت f function
+
+286
+00:37:25,120 --> 00:37:30,180
+متصة على closed bounded interval فال .. ال .. ال
+
+287
+00:37:30,180 --> 00:37:34,560
+function إلها قيمة صغيرة مطلقة و إلها قيمة أضمة
+
+288
+00:37:34,560 --> 00:37:39,300
+مطلقة سمها قيمة صغيرة المطلقة M و قيمة الأضمة
+
+289
+00:37:39,300 --> 00:37:44,760
+المطلقة capital M تمام؟
+
+290
+00:37:44,760 --> 00:37:47,580
+clearly
+
+291
+00:37:54,210 --> 00:38:02,370
+F of X أكبر من أو ساوي M أصغر من أو ساوي م لكل X
+
+292
+00:38:02,370 --> 00:38:10,630
+في I قيمة الدالة عند أي X في المجال تبعها أصغر من
+
+293
+00:38:10,630 --> 00:38:15,090
+أو ساوي قيمة العظمى المطلقة و أكبر في نفس المجال
+
+294
+00:38:15,090 --> 00:38:17,370
+أكبر من أو ساوي قيمة الصغر المطلقة
+
+295
+00:38:21,460 --> 00:38:26,040
+فهذا بيقدي which
+
+296
+00:38:26,040 --> 00:38:40,440
+implies هذا بيقدي انه ال .. انه f of I contained
+
+297
+00:38:40,440 --> 00:38:47,680
+في الفترة المغلقة من small m لcapital Mالمتبادلة
+
+298
+00:38:47,680 --> 00:38:53,400
+الأخيرة هذه تثبت أن ال set هذه subset من هذه لأنه
+
+299
+00:38:53,400 --> 00:38:59,880
+خدوا أي عنصر هنا فأي عنصر هنا عبارة عن f of x for
+
+300
+00:38:59,880 --> 00:39:07,100
+some x ينتمي ل I صح فأي f of x for some x ينتمي ل
+
+301
+00:39:07,100 --> 00:39:12,730
+I هيمحصور من small m وcapital M وبالتالي ينتمي
+
+302
+00:39:12,730 --> 00:39:16,610
+للفترة المغلقة هذه، لأن كل أنصر أنا هو أنصر في
+
+303
+00:39:16,610 --> 00:39:22,490
+الفترة المغلقة، لأن هذا الاحتواء صحيح، تمام؟ الان
+
+304
+00:39:22,490 --> 00:39:24,770
+احنا بنثبت المساواة
+
+305
+00:39:30,090 --> 00:39:36,190
+إن ال range لل function f بساوي كل الفترة المغلقة
+
+306
+00:39:36,190 --> 00:39:44,150
+من small m لcapital M فلإثبات
+
+307
+00:39:44,150 --> 00:39:55,350
+ذلك هي عندي أنا to prove this it
+
+308
+00:39:55,350 --> 00:39:56,090
+remains
+
+309
+00:39:59,390 --> 00:40:07,930
+it remains to show يبقى اثبات دا في اثبات ان احنا
+
+310
+00:40:07,930 --> 00:40:12,370
+لثبت الاحتواء المعاكس the reverse inclusion
+
+311
+00:40:18,550 --> 00:40:28,670
+إذا يبقى إثبات إن الفترة المغلقة من small m to
+
+312
+00:40:28,670 --> 00:40:35,970
+capital M contained in F of I فكيف نثبت إحنا set
+
+313
+00:40:35,970 --> 00:40:41,570
+subset من الأخرى نسميه
+
+314
+00:40:41,570 --> 00:40:45,950
+برهان بإيه بتتبع العناصر يعني بناخد أنصر في
+
+315
+00:40:45,950 --> 00:40:49,540
+المجموعة الأولىنثبت العناصر في المجموعة التانية
+
+316
+00:40:49,540 --> 00:40:53,080
+هذا بيسموه في رياضيات الـ chasing of elements
+
+317
+00:40:53,080 --> 00:41:03,880
+argument برهان بتطبع العناصر فقالت why تنتمي
+
+318
+00:41:03,880 --> 00:41:12,580
+للفترة المغلقة من small m لcapital M طيب هذا
+
+319
+00:41:12,580 --> 00:41:19,540
+بيقدّيالـ y أكبر من أو ساوي small m أصغر من أو
+
+320
+00:41:19,540 --> 00:41:31,720
+ساوي capital M وهذا عبارة عن الـ infimum لـ range
+
+321
+00:41:31,720 --> 00:41:39,460
+الـ function f وهذا بساوي الـ supremum لـ range
+
+322
+00:41:39,460 --> 00:41:40,500
+الـ function f
+
+323
+00:41:47,540 --> 00:41:57,540
+وعندي ال .. إذا حسب ال .. ال corollary تبع النظرية
+
+324
+00:41:57,540 --> 00:42:04,060
+هذه فإن عندي ال function if continuous على الفترة
+
+325
+00:42:04,060 --> 00:42:10,520
+المغلقة a,b فعندي if continuous على الفترة المغلقة
+
+326
+00:42:10,520 --> 00:42:18,800
+a,bوعندي k اللي هو y عدد محصور بين ال infimum ل f
+
+327
+00:42:18,800 --> 00:42:28,520
+of i و ال suprem ل f of i by
+
+328
+00:42:28,520 --> 00:42:36,090
+above corollaryالقرن اللي لـ Bolzano Intermediate
+
+329
+00:42:36,090 --> 00:42:43,390
+Value Theorem يقول إن يوجد C ينتمي للفترة I بحيث
+
+330
+00:42:43,390 --> 00:42:50,090
+أن F of C بساوي
+
+331
+00:42:50,090 --> 00:42:58,170
+العدد Y اللي هو قابل الـK في نص النظريةالـ C ينتمي
+
+332
+00:42:58,170 --> 00:43:05,170
+لـ I إذاً F of C تنتمي لـ F للست F of I إذا هاني
+
+333
+00:43:05,170 --> 00:43:09,410
+بدأت بـ Y ينتمي للفترة المغلقة طلع Y ينتمي لـ F of
+
+334
+00:43:09,410 --> 00:43:17,570
+I Therefore Hence هيك
+
+335
+00:43:17,570 --> 00:43:20,750
+منكون أثباتنا أن الفترة المغلقة من small m
+
+336
+00:43:20,750 --> 00:43:31,990
+لcapital M is containedفي ال set f of i هذا
+
+337
+00:43:31,990 --> 00:43:37,490
+ببرهن ال claim و النظرية لأن هيك بيكون برهننا ال
+
+338
+00:43:37,490 --> 00:43:42,010
+claim وبالتالي برهننا النظرية لأن هيك هي اثبتت ان
+
+339
+00:43:42,010 --> 00:43:45,950
+ال image ل ال closed bounded interval I طلعت
+
+340
+00:43:45,950 --> 00:43:49,970
+closed bounded interval صح و هو المطلوب
+
+341
+00:43:53,870 --> 00:44:00,730
+Okay واضح البرهان؟ في أي استفسار على البرهان؟
+
+342
+00:44:00,730 --> 00:44:08,030
+في هنا تحذير warning تحذير
+
+343
+00:44:08,030 --> 00:44:15,070
+in
+
+344
+00:44:15,070 --> 00:44:30,000
+the above theorem we hadF of I التي هي F للفترة
+
+345
+00:44:30,000 --> 00:44:35,320
+المغلقة من A لB طلعت
+
+346
+00:44:35,320 --> 00:44:39,900
+بالساوي الفترة المغلقة من small m لcapital M حيث
+
+347
+00:44:39,900 --> 00:44:43,980
+small m is the absolute minimum value وcapital M
+
+348
+00:44:43,980 --> 00:44:47,140
+is the absolute maximum value of the function F on
+
+349
+00:44:47,140 --> 00:44:54,980
+the interval Iو هذا ليس بالضرورة مش شرط هذه الفترة
+
+350
+00:44:54,980 --> 00:45:03,750
+تكون الفترة من F of A ل F of Bهذه الفترة ماحدش جال
+
+351
+00:45:03,750 --> 00:45:08,370
+او مقدر يزم انها الفترة المغلقة من F of A لF of B
+
+352
+00:45:08,370 --> 00:45:13,750
+هذا مش صحيح okay النظرية ما بتقولي الكلام هذا
+
+353
+00:45:13,750 --> 00:45:18,490
+بتقولي الكلام هذا فقط هذا غلط مش شرط ال image
+
+354
+00:45:18,490 --> 00:45:23,050
+للفترة I بالساوي الفترة المغلقة من F of A لF of B
+
+355
+00:45:23,050 --> 00:45:30,640
+فاخدوا بالكم من ايه من التحذير هذاOkay إذا هين
+
+356
+00:45:30,640 --> 00:45:34,720
+أثبتنا إن لو كانت ال function تبعتي متصلة على فترة
+
+357
+00:45:34,720 --> 00:45:39,420
+مغلقة أو محدودة فصورتها بتطلع مغلقة أو محدودة
+
+358
+00:45:39,420 --> 00:45:45,280
+وبالتالي ال function preserves ال .. ال .. ال
+
+359
+00:45:45,280 --> 00:45:50,640
+intervals طيب
+
+360
+00:45:50,640 --> 00:45:53,940
+ال .. النظرية دي إلها تعميم
+
+361
+00:46:03,890 --> 00:46:10,730
+preservation of intervals
+
+362
+00:46:10,730 --> 00:46:14,770
+theorem لو
+
+363
+00:46:14,770 --> 00:46:21,050
+كانت الفترة let I be any interval مش شرط تكون ..
+
+364
+00:46:21,050 --> 00:46:28,070
+مش شرط تكون close about it .. be any interval and
+
+365
+00:46:28,070 --> 00:46:43,200
+letإذا من I إلى R يكون مستمر على الفترة I ثم
+
+366
+00:46:43,200 --> 00:46:49,260
+ستة F من I هي عرفة
+
+367
+00:46:53,620 --> 00:46:57,220
+النظرية هذه بتقول لو كانت f دالة متصلة المجال
+
+368
+00:46:57,220 --> 00:47:03,960
+تبعها أي فترة مغلقة، محدودة، مش محدودة، half-open،
+
+369
+00:47:03,960 --> 00:47:06,580
+open-half-open interval اللي لقاش، أي لوحة من ال
+
+370
+00:47:06,580 --> 00:47:11,820
+intervals اللي شفناهم في جبتر واحد فصورتها أيضا
+
+371
+00:47:11,820 --> 00:47:15,500
+لازم تطلع interval وبالتالي ال continuous function
+
+372
+00:47:15,500 --> 00:47:19,880
+بتحافظ على الفترة، على الفترات يعني بتاكن فترة في
+
+373
+00:47:19,880 --> 00:47:24,780
+مجالها بتعطيل صورتها فترةالفترة هذه ما بنعرفش كيف
+
+374
+00:47:24,780 --> 00:47:29,620
+نوعها لكن اللي بنقدر نزّمه في النظرية السابقة أنه
+
+375
+00:47:29,620 --> 00:47:33,300
+لو كانت الفترة I هذه closed bounded فصورتها هتطلع
+
+376
+00:47:33,300 --> 00:47:36,960
+closed bounded أما لو كانت من نوع أخر فصورتها مش
+
+377
+00:47:36,960 --> 00:47:42,200
+شرط تكون من نفس النوع ماحدش جال الكلام هذا فلبرهان
+
+378
+00:47:42,200 --> 00:47:47,780
+ذلك لبرهان
+
+379
+00:47:47,780 --> 00:47:48,260
+ذلك
+
+380
+00:47:53,440 --> 00:48:01,040
+خلّينا ناخد let alpha و beta belong to except f of
+
+381
+00:48:01,040 --> 00:48:14,580
+I with alpha أصغر من beta خلّينا
+
+382
+00:48:14,580 --> 00:48:15,640
+نستذكر بس
+
+383
+00:48:22,460 --> 00:48:27,560
+في نظرية أخدناها قبل هيك ال theorem اتنين خمسة
+
+384
+00:48:27,560 --> 00:48:44,300
+واحد بتقول if S subset of R contains at least two
+
+385
+00:48:44,300 --> 00:48:48,500
+elements and satisfies
+
+386
+00:48:52,530 --> 00:48:58,810
+Satisfies الخاصية واحد إن لو كان X و Y تلوي ل S و
+
+387
+00:48:58,810 --> 00:49:04,850
+X أصغر من Y هذا بيقدي إن الفترة من X إلى Y
+
+388
+00:49:04,850 --> 00:49:10,670
+contained in S then
+
+389
+00:49:10,670 --> 00:49:13,790
+set S is an interval
+
+390
+00:49:17,430 --> 00:49:19,470
+إن ان هذه النظرية أخدناها في ال chapter .. في ال
+
+391
+00:49:19,470 --> 00:49:23,830
+chapter الأولاري بتقول لو كان في عندي set subset
+
+392
+00:49:23,830 --> 00:49:29,520
+من R فيها على الأقل أنصر Lوبتحقق ال set هذه بتحقق
+
+393
+00:49:29,520 --> 00:49:33,580
+الخاصية واحد property one انه لأي x و y في ال set
+
+394
+00:49:33,580 --> 00:49:39,300
+و x أصغر من y الفترة من x ل y بتكون موجودة داخل ال
+
+395
+00:49:39,300 --> 00:49:43,880
+set في الحالة هذه ال set نفسها S تطلع interval اذا
+
+396
+00:49:43,880 --> 00:49:50,820
+انا بدي اثبت to show طيب
+
+397
+00:49:50,820 --> 00:49:51,780
+انا عندي
+
+398
+00:49:54,280 --> 00:50:00,060
+هذه أخدت نقطتين في ال set هذه هي ال set ال set S
+
+399
+00:50:00,060 --> 00:50:04,600
+هذه أخدت نقطتين و Alpha أصغر من Beta و بتثبت أنها
+
+400
+00:50:04,600 --> 00:50:08,700
+بتحقق الخاصية واحد عشان أثبت أنها interval أنا
+
+401
+00:50:08,700 --> 00:50:13,160
+عندي Alpha و Beta تنتمي ل F of I لأن Alpha بتساوي
+
+402
+00:50:13,160 --> 00:50:22,570
+F of Afor some a تنتمي إلى I و Beta بساوي F of B
+
+403
+00:50:22,570 --> 00:50:30,330
+for some B تنتمي إلى I وبالتالي
+
+404
+00:50:30,330 --> 00:50:36,950
+..
+
+405
+00:50:36,950 --> 00:50:39,790
+بالتالي ..
+
+406
+00:50:47,200 --> 00:50:59,180
+انا عندي ال bolzanova طيب طيب to show to
+
+407
+00:50:59,180 --> 00:51:09,420
+show f of I is an interval we
+
+408
+00:51:09,420 --> 00:51:20,550
+need to showان الset f of i satisfies property
+
+409
+00:51:20,550 --> 00:51:23,830
+واحد
+
+410
+00:51:23,830 --> 00:51:33,430
+of theorem اتنين خمسة واحد فهي
+
+411
+00:51:33,430 --> 00:51:36,830
+عندي alpha و beta تنتمي ل f of i و alpha أصغر من
+
+412
+00:51:36,830 --> 00:51:40,690
+beta ف
+
+413
+00:51:40,690 --> 00:51:42,290
+to show
+
+414
+00:51:44,880 --> 00:51:56,020
+الفترة من Alpha إلى Beta content in F of I let K
+
+415
+00:51:56,020 --> 00:52:00,300
+ينتمي إلى الفترة المغلقة من Alpha إلى Beta
+
+416
+00:52:04,220 --> 00:52:11,680
+أكبر من أو يساوي alpha هي بيساوي f of a وأصغر من
+
+417
+00:52:11,680 --> 00:52:21,460
+أو يساوي beta هي بيساوي f of b وبالتالي so by
+
+418
+00:52:21,460 --> 00:52:26,440
+Bolzano's
+
+419
+00:52:26,440 --> 00:52:31,120
+intermediate
+
+420
+00:52:31,120 --> 00:52:32,940
+value theorem
+
+421
+00:52:35,620 --> 00:52:48,960
+يوجد K عفوا يوجد C ينتمي إلى I between Alpha
+
+422
+00:52:48,960 --> 00:52:59,400
+و Beta بحيث ان F of C بساوي K او
+
+423
+00:52:59,400 --> 00:53:08,640
+K بساوي F of Cطبما ال C تنتمي ل I إذا F of C تنتمي
+
+424
+00:53:08,640 --> 00:53:13,380
+ل F of I إذا
+
+425
+00:53:13,380 --> 00:53:18,200
+هاني أثبتت إنه كل K في الفترة المغلقة من Alpha إلى
+
+426
+00:53:18,200 --> 00:53:25,850
+Beta طلع ينتمي لفترة F of I وبالتالي إذابطلع عند
+
+427
+00:53:25,850 --> 00:53:31,270
+الفترة المغلقة من Alpha إلى Beta ال subset من F of
+
+428
+00:53:31,270 --> 00:53:38,830
+I وبالتالي إذا ال set F of I بتحقق ال property
+
+429
+00:53:38,830 --> 00:53:46,810
+واحد إذا by theorem .. by theorem اتنين خمسة واحد
+
+430
+00:53:46,810 --> 00:53:53,650
+ال set F of I بتطلع interval is an interval
+
+431
+00:53:56,670 --> 00:54:03,570
+و هذا بيكمل النظرية اذا هذا بيكمل البرهان هيك
+
+432
+00:54:03,570 --> 00:54:10,630
+بنكون خلصنا ال section خمسة تلاتة و باقي عننا
+
+433
+00:54:10,630 --> 00:54:16,190
+section خمسة أربعة هناخده في المحاضرة الجاية نحاول
+
+434
+00:54:16,190 --> 00:54:24,130
+نشوف زمنا نخلصه ولا لأفال .. شكرا لحصن إصراعكم و
+
+435
+00:54:24,130 --> 00:54:26,910
+يعطيكم العافية و نشوفكم ان شاء الله المرة الجاية
+
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/Ehj01gka7EU_raw.json b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/Ehj01gka7EU_raw.json
new file mode 100644
index 0000000000000000000000000000000000000000..de5dc93cee275c11870c85a824d4757b9e011098
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/Ehj01gka7EU_raw.json
@@ -0,0 +1 @@
+{"segments": [{"id": 1, "seek": 2780, "start": 19.74, "end": 27.8, "text": "بسم الله الرحمن الرحيم هنواصل اليوم تغطية section 5-3", "tokens": [3555, 38251, 21984, 34892, 5016, 27842, 34892, 5016, 32640, 8032, 1863, 2407, 33546, 1211, 45595, 20498, 6055, 17082, 9566, 10632, 3541, 1025, 12, 18], "avg_logprob": -0.127265625, "compression_ratio": 1.1, "no_speech_prob": 0.0, "words": [{"start": 19.740000000000002, "end": 21.14, "word": "بسم", "probability": 0.852783203125}, {"start": 21.14, "end": 21.26, "word": " الله", "probability": 0.97802734375}, {"start": 21.26, "end": 21.6, "word": " الرحمن", "probability": 0.9645182291666666}, {"start": 21.6, "end": 22.28, "word": " الرحيم", "probability": 0.99072265625}, {"start": 22.28, "end": 24.04, "word": " هنواصل", "probability": 0.9375}, {"start": 24.04, "end": 25.16, "word": " اليوم", "probability": 0.961181640625}, {"start": 25.16, "end": 26.26, "word": " تغطية", "probability": 0.9322509765625}, {"start": 26.26, "end": 27.02, "word": " section", "probability": 0.6435546875}, {"start": 27.02, "end": 27.32, "word": " 5", "probability": 0.69384765625}, {"start": 27.32, "end": 27.8, "word": "-3", "probability": 0.553985595703125}], "temperature": 1.0}, {"id": 2, "seek": 5767, "start": 28.89, "end": 57.67, "text": "اللي بتعلق ب .. موضوع ال continuous functions على ال intervals على الفترات احنا بدينا ال .. بدينا الموضوع هذا المحاضرة السابقة و كان اخر نظرية أخدناها اللي هي ال maximum .. maximum minimum theorem نعود نستذكر بس نظرية الأخيرة هذه ال maximum", "tokens": [6027, 20292, 39894, 30241, 4587, 4724, 4386, 3714, 2407, 11242, 45367, 2423, 10957, 6828, 15844, 2423, 26651, 15844, 27188, 2655, 2288, 9307, 1975, 5016, 8315, 47525, 1829, 8315, 2423, 4386, 47525, 1829, 8315, 9673, 2407, 11242, 45367, 23758, 9673, 5016, 46958, 25720, 21136, 16758, 28671, 4032, 25961, 1975, 34740, 8717, 19913, 2288, 10632, 5551, 9778, 3215, 8315, 11296, 13672, 1829, 39896, 2423, 6674, 4386, 6674, 7285, 20904, 8717, 3615, 23328, 8717, 14851, 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"end": 553.94, "word": " فحسب", "probability": 0.9278971354166666}, {"start": 553.94, "end": 554.52, "word": " النظرية", "probability": 0.947998046875}, {"start": 554.52, "end": 554.9, "word": " هذه", "probability": 0.91552734375}], "temperature": 1.0}, {"id": 21, "seek": 58065, "start": 567.63, "end": 580.65, "text": "F of X بساوي X في E to X سالب اتنين is continuous متصلة على الفترة المغلقة من سفر لواحد", "tokens": [37, 295, 1783, 4724, 3794, 995, 45865, 1783, 8978, 462, 281, 1783, 8608, 6027, 3555, 1975, 2655, 1863, 9957, 307, 10957, 44650, 36520, 3660, 15844, 27188, 2655, 25720, 9673, 17082, 1211, 28671, 9154, 8608, 5172, 2288, 5296, 14407, 24401], "avg_logprob": -0.30039062798023225, "compression_ratio": 1.1551724137931034, "no_speech_prob": 0.0, "words": [{"start": 567.63, "end": 568.61, "word": "F", "probability": 0.06671142578125}, {"start": 568.61, "end": 569.59, "word": " of", "probability": 0.1768798828125}, {"start": 569.59, "end": 570.05, "word": " X", "probability": 0.82275390625}, {"start": 570.05, "end": 571.03, "word": " بساوي", "probability": 0.662689208984375}, {"start": 571.03, "end": 572.05, "word": " X", "probability": 0.83642578125}, {"start": 572.05, "end": 572.35, "word": " في", "probability": 0.83203125}, {"start": 572.35, "end": 572.59, "word": " E", "probability": 0.8896484375}, {"start": 572.59, "end": 572.75, "word": " to", "probability": 0.76123046875}, {"start": 572.75, "end": 573.05, "word": " X", "probability": 0.9951171875}, {"start": 573.05, "end": 573.49, "word": " سالب", "probability": 0.8504231770833334}, {"start": 573.49, "end": 574.03, "word": " اتنين", "probability": 0.9373779296875}, {"start": 574.03, "end": 574.47, "word": " is", "probability": 0.73095703125}, {"start": 574.47, "end": 575.37, "word": " continuous", "probability": 0.91357421875}, {"start": 575.37, "end": 578.01, "word": " متصلة", "probability": 0.8038736979166666}, {"start": 578.01, "end": 578.23, "word": " على", "probability": 0.91064453125}, {"start": 578.23, "end": 578.77, "word": " الفترة", "probability": 0.9737955729166666}, {"start": 578.77, "end": 579.39, "word": " المغلقة", "probability": 0.96484375}, {"start": 579.39, "end": 579.71, "word": " من", "probability": 0.984375}, {"start": 579.71, "end": 580.13, "word": " سفر", "probability": 0.7542317708333334}, {"start": 580.13, "end": 580.65, "word": " لواحد", "probability": 0.6062825520833334}], "temperature": 1.0}, {"id": 22, "seek": 60337, "start": 587.89, "end": 603.37, "text": "لأن X في E to X هي دالة متصلة اتراحى منها ثابت دالة متصلة على R كذلك انا عندي F of سفر بساوي سالب اتنين اصغر من سفر", "tokens": [1211, 33456, 1783, 8978, 462, 281, 1783, 39896, 11778, 6027, 3660, 44650, 36520, 3660, 1975, 2655, 23557, 5016, 7578, 9154, 11296, 38637, 16758, 2655, 11778, 6027, 3660, 44650, 36520, 3660, 15844, 497, 9122, 8848, 23275, 1975, 8315, 18871, 16254, 479, 295, 8608, 5172, 2288, 4724, 3794, 995, 45865, 8608, 6027, 3555, 1975, 2655, 1863, 9957, 1975, 9381, 17082, 2288, 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"probability": 0.949951171875}, {"start": 591.07, "end": 591.51, "word": " ثابت", "probability": 0.9884440104166666}, {"start": 591.51, "end": 591.83, "word": " دالة", "probability": 0.8487955729166666}, {"start": 591.83, "end": 592.35, "word": " متصلة", "probability": 0.9441731770833334}, {"start": 592.35, "end": 592.47, "word": " على", "probability": 0.91748046875}, {"start": 592.47, "end": 592.77, "word": " R", "probability": 0.890625}, {"start": 592.77, "end": 596.75, "word": " كذلك", "probability": 0.736328125}, {"start": 596.75, "end": 596.95, "word": " انا", "probability": 0.760986328125}, {"start": 596.95, "end": 597.39, "word": " عندي", "probability": 0.837646484375}, {"start": 597.39, "end": 597.85, "word": " F", "probability": 0.89404296875}, {"start": 597.85, "end": 598.45, "word": " of", "probability": 0.78857421875}, {"start": 598.45, "end": 599.07, "word": " سفر", "probability": 0.7124430338541666}, {"start": 599.07, "end": 601.09, "word": " بساوي", "probability": 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613.06, "end": 613.34, "word": " E", "probability": 0.5400390625}, {"start": 613.34, "end": 614.74, "word": " معروف", "probability": 0.7277018229166666}, {"start": 614.74, "end": 614.96, "word": " انه", "probability": 0.6165771484375}, {"start": 614.96, "end": 615.3, "word": " عدد", "probability": 0.95361328125}, {"start": 615.3, "end": 615.68, "word": " اكبر", "probability": 0.9593098958333334}, {"start": 615.68, "end": 615.9, "word": " من", "probability": 0.96923828125}, {"start": 615.9, "end": 616.26, "word": " اتنين", "probability": 0.9742431640625}, {"start": 616.26, "end": 616.7, "word": " فهذا", "probability": 0.9401041666666666}, {"start": 616.7, "end": 617.16, "word": " اكبر", "probability": 0.9278971354166666}, {"start": 617.16, "end": 617.38, "word": " من", "probability": 0.970703125}, {"start": 617.38, "end": 618.06, "word": " ساكنة", "probability": 0.743896484375}, {"start": 618.06, "end": 619.24, "word": " اذا", "probability": 0.6512451171875}, {"start": 619.24, "end": 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of roots ال theorem كلها متحققة هي اندي فانش جي متصلة على فترة مغلقة ومحدودة", "tokens": [2407, 3224, 15730, 5551, 9381, 17082, 2288, 9154, 479, 295, 363, 3175, 591, 479, 295, 363, 3175, 591, 4724, 1829, 9566, 1211, 3615, 6225, 2304, 29245, 3660, 13672, 1829, 31439, 4724, 3794, 995, 45865, 460, 295, 363, 1975, 15730, 8032, 47302, 13412, 32887, 9566, 2423, 4914, 295, 10669, 2423, 20904, 28242, 11296, 44650, 5016, 4587, 28671, 39896, 16472, 16254, 6156, 7649, 8592, 10874, 1829, 44650, 36520, 3660, 15844, 6156, 2655, 25720, 3714, 17082, 1211, 28671, 4032, 2304, 24401, 23328, 3660], "avg_logprob": -0.2683256290577076, "compression_ratio": 1.425531914893617, "no_speech_prob": 0.0, "words": [{"start": 1241.71, "end": 1242.55, "word": "وهذا", "probability": 0.7277018229166666}, {"start": 1242.55, "end": 1243.19, "word": " أصغر", "probability": 0.9423828125}, {"start": 1243.19, "end": 1243.57, "word": " من", "probability": 0.96826171875}, {"start": 1243.57, "end": 1244.57, "word": " F", 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1310.45, "word": " لكن", "probability": 0.77587890625}, {"start": 1310.45, "end": 1310.77, "word": " انا", "probability": 0.7252197265625}, {"start": 1310.77, "end": 1311.35, "word": " عندي", "probability": 0.798583984375}, {"start": 1311.35, "end": 1313.01, "word": " G", "probability": 0.7021484375}, {"start": 1313.01, "end": 1313.23, "word": " of", "probability": 0.69775390625}, {"start": 1313.23, "end": 1313.53, "word": " C", "probability": 0.966796875}, {"start": 1313.53, "end": 1313.77, "word": " من", "probability": 0.984375}, {"start": 1313.77, "end": 1314.17, "word": " تعريف", "probability": 0.9884440104166666}, {"start": 1314.17, "end": 1314.31, "word": " ال", "probability": 0.95849609375}, {"start": 1314.31, "end": 1314.63, "word": " function", "probability": 0.646484375}, {"start": 1314.63, "end": 1315.05, "word": " G", "probability": 0.8837890625}], "temperature": 1.0}, {"id": 48, "seek": 133875, "start": 1316.87, "end": 1338.75, "text": "G of C بساوي F of C negative K حل المعادلة هذه في F of C فبطلع F of C بساوي K كما هو مطلوب زي ما هو مطلوب ان هيك بتكون برهانة نظرية بس A في الحالة اللي فيها بتكون A أصغر من B", "tokens": [38, 295, 383, 4724, 3794, 995, 45865, 479, 295, 383, 3671, 591, 11331, 1211, 9673, 3615, 18513, 37977, 29538, 8978, 479, 295, 383, 6156, 3555, 9566, 1211, 3615, 479, 295, 383, 4724, 3794, 995, 45865, 591, 9122, 15042, 31439, 3714, 9566, 1211, 37746, 30767, 1829, 19446, 31439, 3714, 9566, 1211, 37746, 16472, 39896, 4117, 39894, 30544, 4724, 2288, 3224, 7649, 3660, 8717, 19913, 2288, 10632, 4724, 3794, 316, 8978, 21542, 6027, 3660, 13672, 1829, 8978, 11296, 39894, 30544, 316, 5551, 9381, 17082, 2288, 9154, 363], "avg_logprob": -0.1922238349568012, "compression_ratio": 1.6176470588235294, "no_speech_prob": 0.0, "words": [{"start": 1316.87, "end": 1317.43, "word": "G", "probability": 0.128173828125}, {"start": 1317.43, "end": 1317.69, "word": " of", "probability": 0.17724609375}, {"start": 1317.69, "end": 1317.87, "word": " C", 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{"start": 1324.85, "end": 1325.17, "word": " C", "probability": 0.9931640625}, {"start": 1325.17, "end": 1325.95, "word": " فبطلع", "probability": 0.8912109375}, {"start": 1325.95, "end": 1327.27, "word": " F", "probability": 0.74267578125}, {"start": 1327.27, "end": 1327.53, "word": " of", "probability": 0.97998046875}, {"start": 1327.53, "end": 1327.85, "word": " C", "probability": 0.9931640625}, {"start": 1327.85, "end": 1328.75, "word": " بساوي", "probability": 0.9488525390625}, {"start": 1328.75, "end": 1329.23, "word": " K", "probability": 0.88427734375}, {"start": 1329.23, "end": 1329.63, "word": " كما", "probability": 0.965576171875}, {"start": 1329.63, "end": 1329.85, "word": " هو", "probability": 0.9892578125}, {"start": 1329.85, "end": 1330.43, "word": " مطلوب", "probability": 0.86767578125}, {"start": 1330.43, "end": 1331.81, "word": " زي", "probability": 0.6912841796875}, {"start": 1331.81, "end": 1331.93, "word": " ما", "probability": 0.943359375}, {"start": 1331.93, "end": 1332.13, "word": " هو", "probability": 0.99365234375}, {"start": 1332.13, "end": 1332.97, "word": " مطلوب", "probability": 0.9725341796875}, {"start": 1332.97, "end": 1333.47, "word": " ان", "probability": 0.30322265625}, {"start": 1333.47, "end": 1333.71, "word": " هيك", "probability": 0.5616455078125}, {"start": 1333.71, "end": 1333.99, "word": " بتكون", "probability": 0.7301025390625}, {"start": 1333.99, "end": 1334.45, "word": " برهانة", "probability": 0.8990234375}, {"start": 1334.45, "end": 1334.89, "word": " نظرية", "probability": 0.9833984375}, {"start": 1334.89, "end": 1335.27, "word": " بس", "probability": 0.96484375}, {"start": 1335.27, "end": 1335.53, "word": " A", "probability": 0.7412109375}, {"start": 1335.53, "end": 1336.19, "word": " في", "probability": 0.771484375}, {"start": 1336.19, "end": 1336.51, "word": " الحالة", "probability": 0.8772786458333334}, {"start": 1336.51, "end": 1336.63, "word": " اللي", "probability": 0.885009765625}, {"start": 1336.63, 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{"id": 53, "seek": 147115, "start": 1447.03, "end": 1471.15, "text": "of roots theorem يوجد C ينتمي الى الفترة مظبوط هيك؟ كده كده كده كده كده كده كده كده كده كده كده كده كده كده", "tokens": [2670, 10669, 20904, 7251, 29245, 3215, 383, 7251, 29399, 2304, 1829, 2423, 7578, 27188, 2655, 25720, 3714, 19913, 3555, 2407, 9566, 39896, 4117, 22807, 9122, 3215, 3224, 9122, 3215, 3224, 9122, 3215, 3224, 9122, 3215, 3224, 9122, 3215, 3224, 9122, 3215, 3224, 9122, 3215, 3224, 9122, 3215, 3224, 9122, 3215, 3224, 9122, 3215, 3224, 9122, 3215, 3224, 9122, 3215, 3224, 9122, 3215, 3224, 9122, 3215, 3224], "avg_logprob": -0.3325559612530381, "compression_ratio": 1.9775280898876404, "no_speech_prob": 0.0, "words": [{"start": 1447.03, "end": 1447.49, "word": "of", "probability": 0.6923828125}, {"start": 1447.49, "end": 1448.09, "word": " roots", "probability": 0.86767578125}, {"start": 1448.09, "end": 1449.57, "word": " theorem", "probability": 0.79541015625}, {"start": 1449.57, "end": 1452.79, "word": " 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من", "probability": 0.10931396484375}, {"start": 1650.21, "end": 1652.37, "word": " نقدر", "probability": 0.9049479166666666}, {"start": 1652.37, "end": 1653.05, "word": " نلاقي", "probability": 0.9482421875}, {"start": 1653.05, "end": 1655.29, "word": " C", "probability": 0.66162109375}, {"start": 1655.29, "end": 1655.81, "word": " there", "probability": 0.56640625}, {"start": 1655.81, "end": 1656.31, "word": " exist", "probability": 0.61083984375}], "temperature": 1.0}, {"id": 61, "seek": 168544, "start": 1658.22, "end": 1685.44, "text": "C ينتمي للفترة I بحيث ان F of C بيساوي العدد K وبرهان النظرية هذه سهل By maximum minimum theorem", "tokens": [34, 7251, 29399, 2304, 1829, 24976, 5172, 2655, 25720, 286, 4724, 5016, 1829, 12984, 16472, 479, 295, 383, 4724, 1829, 3794, 995, 45865, 18863, 3215, 3215, 591, 4032, 26890, 3224, 7649, 28239, 19913, 2288, 10632, 29538, 8608, 3224, 1211, 3146, 6674, 7285, 20904], "avg_logprob": -0.22425426271828738, "compression_ratio": 1.144, 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"temperature": 1.0}, {"id": 63, "seek": 174087, "start": 1720.91, "end": 1740.87, "text": "إذاً there exist x lower star و x super star عناصر في I بحيث أن ال F of x lower star بساول infimum", "tokens": [28814, 15730, 14111, 456, 2514, 2031, 3126, 3543, 4032, 2031, 1687, 3543, 18871, 33546, 2288, 8978, 286, 4724, 5016, 1829, 12984, 14739, 2423, 479, 295, 2031, 3126, 3543, 4724, 3794, 995, 12610, 1536, 332, 449], "avg_logprob": -0.3632812491721577, "compression_ratio": 1.1081081081081081, "no_speech_prob": 0.0, "words": [{"start": 1720.9100000000003, "end": 1722.2700000000002, "word": "إذاً", "probability": 0.4298909505208333}, {"start": 1722.2700000000002, "end": 1723.63, "word": " there", "probability": 0.468017578125}, {"start": 1723.63, "end": 1724.29, "word": " exist", "probability": 0.77685546875}, {"start": 1724.29, "end": 1726.77, "word": " x", "probability": 0.72314453125}, {"start": 1726.77, "end": 1727.25, "word": " lower", "probability": 0.58544921875}, {"start": 1727.25, 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0.81494140625}, {"start": 2388.35, "end": 2388.87, "word": " prove", "probability": 0.93994140625}, {"start": 2388.87, "end": 2391.25, "word": " this", "probability": 0.9521484375}, {"start": 2391.25, "end": 2395.35, "word": " it", "probability": 0.92529296875}, {"start": 2395.35, "end": 2396.09, "word": " remains", "probability": 0.8623046875}], "temperature": 1.0}, {"id": 88, "seek": 241237, "start": 2399.39, "end": 2412.37, "text": "it remains to show يبقى اثبات دا في اثبات ان احنا لثبت الاحتواء المعاكس the reverse inclusion", "tokens": [270, 7023, 281, 855, 7251, 3555, 4587, 7578, 1975, 12984, 3555, 9307, 11778, 995, 8978, 1975, 12984, 3555, 9307, 16472, 1975, 5016, 8315, 5296, 12984, 3555, 2655, 42963, 33753, 2407, 16606, 9673, 3615, 995, 4117, 3794, 264, 9943, 15874], "avg_logprob": -0.24687499254941941, "compression_ratio": 1.1929824561403508, "no_speech_prob": 0.0, "words": [{"start": 2399.39, "end": 2399.77, "word": "it", "probability": 0.10693359375}, {"start": 2399.77, "end": 2400.39, "word": " remains", "probability": 0.89111328125}, {"start": 2400.39, "end": 2400.67, "word": " to", "probability": 0.95751953125}, {"start": 2400.67, "end": 2401.15, "word": " show", "probability": 0.896484375}, {"start": 2401.15, "end": 2403.99, "word": " يبقى", "probability": 0.896484375}, {"start": 2403.99, "end": 2404.67, "word": " اثبات", "probability": 0.8450927734375}, {"start": 2404.67, "end": 2406.19, "word": " دا", "probability": 0.4658203125}, {"start": 2406.19, "end": 2406.31, "word": " في", "probability": 0.59521484375}, {"start": 2406.31, "end": 2406.89, "word": " اثبات", "probability": 0.9639892578125}, {"start": 2406.89, "end": 2407.73, "word": " ان", "probability": 0.7294921875}, {"start": 2407.73, "end": 2407.93, "word": " احنا", "probability": 0.9547526041666666}, {"start": 2407.93, "end": 2408.43, "word": " لثبت", "probability": 0.882080078125}, {"start": 2408.43, "end": 2409.01, "word": " الاحتواء", "probability": 0.885498046875}, {"start": 2409.01, "end": 2409.81, "word": " المعاكس", "probability": 0.976953125}, {"start": 2409.81, "end": 2411.09, "word": " the", "probability": 0.1673583984375}, {"start": 2411.09, "end": 2411.65, "word": " reverse", "probability": 0.88916015625}, {"start": 2411.65, "end": 2412.37, "word": " inclusion", "probability": 0.92578125}], "temperature": 1.0}, {"id": 89, "seek": 244685, "start": 2418.55, "end": 2446.85, "text": "إذا يبقى إثبات إن الفترة المغلقة من small m to capital M contained in F of I فكيف نثبت إحنا set subset من الأخرى نسميه برهان بإيه بتتبع العناصر يعني بناخد أنصر في المجموعة الأولى", "tokens": [28814, 15730, 7251, 3555, 4587, 7578, 11933, 12984, 3555, 9307, 36145, 27188, 2655, 25720, 9673, 17082, 1211, 28671, 9154, 1359, 275, 281, 4238, 376, 16212, 294, 479, 295, 286, 6156, 4117, 33911, 8717, 12984, 3555, 2655, 11933, 5016, 8315, 992, 25993, 9154, 16247, 34740, 7578, 8717, 38251, 1829, 3224, 4724, 2288, 3224, 7649, 4724, 28814, 1829, 3224, 39894, 2655, 3555, 3615, 18863, 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"end": 2629.97, "text": "في ال set f of i هذا ببرهن ال claim و النظرية لأن هيك بيكون برهننا ال claim وبالتالي برهننا النظرية لأن هيك هي اثبتت ان ال image ل ال closed bounded interval I طلعت closed bounded interval صح و هو المطلوب", "tokens": [41185, 2423, 992, 283, 295, 741, 23758, 4724, 26890, 3224, 1863, 2423, 3932, 4032, 28239, 19913, 2288, 10632, 5296, 33456, 39896, 4117, 4724, 1829, 30544, 4724, 2288, 3224, 1863, 8315, 2423, 3932, 46599, 6027, 2655, 6027, 1829, 4724, 2288, 3224, 1863, 8315, 28239, 19913, 2288, 10632, 5296, 33456, 39896, 4117, 39896, 1975, 12984, 3555, 2655, 2655, 16472, 2423, 3256, 5296, 2423, 5395, 37498, 15035, 286, 23032, 1211, 34268, 5395, 37498, 15035, 20328, 5016, 4032, 31439, 9673, 9566, 1211, 37746], "avg_logprob": -0.3046875040978193, "compression_ratio": 1.78698224852071, "no_speech_prob": 0.0, "words": [{"start": 2605.31, "end": 2605.73, "word": "في", "probability": 0.0948486328125}, {"start": 2605.73, "end": 2606.73, "word": " ال", "probability": 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0.4029541015625}, {"start": 2616.55, "end": 2616.83, "word": " بيكون", "probability": 0.75390625}, {"start": 2616.83, "end": 2617.39, "word": " برهننا", "probability": 0.82265625}, {"start": 2617.39, "end": 2617.49, "word": " ال", "probability": 0.962890625}, {"start": 2617.49, "end": 2617.81, "word": " claim", "probability": 0.9736328125}, {"start": 2617.81, "end": 2619.59, "word": " وبالتالي", "probability": 0.8828125}, {"start": 2619.59, "end": 2620.09, "word": " برهننا", "probability": 0.9583984375}, {"start": 2620.09, "end": 2620.55, "word": " النظرية", "probability": 0.9765625}, {"start": 2620.55, "end": 2620.83, "word": " لأن", "probability": 0.75439453125}, {"start": 2620.83, "end": 2621.15, "word": " هيك", "probability": 0.948486328125}, {"start": 2621.15, "end": 2621.29, "word": " هي", "probability": 0.322265625}, {"start": 2621.29, "end": 2621.85, "word": " اثبتت", "probability": 0.8736328125}, {"start": 2621.85, "end": 2622.01, "word": " ان", "probability": 0.53271484375}, {"start": 2622.01, "end": 2622.19, "word": " ال", "probability": 0.91357421875}, {"start": 2622.19, "end": 2622.59, "word": " image", "probability": 0.9375}, {"start": 2622.59, "end": 2623.69, "word": " ل", "probability": 0.5771484375}, {"start": 2623.69, "end": 2623.91, "word": " ال", "probability": 0.348876953125}, {"start": 2623.91, "end": 2624.25, "word": " closed", "probability": 0.60009765625}, {"start": 2624.25, "end": 2624.65, "word": " bounded", "probability": 0.94677734375}, {"start": 2624.65, "end": 2625.23, "word": " interval", "probability": 0.9892578125}, {"start": 2625.23, "end": 2625.51, "word": " I", "probability": 0.66455078125}, {"start": 2625.51, "end": 2625.95, "word": " طلعت", "probability": 0.9197591145833334}, {"start": 2625.95, "end": 2626.53, "word": " closed", "probability": 0.82177734375}, {"start": 2626.53, "end": 2627.05, "word": " bounded", "probability": 0.94775390625}, {"start": 2627.05, "end": 2627.57, "word": " interval", "probability": 0.98828125}, {"start": 2627.57, "end": 2628.25, "word": " صح", "probability": 0.697509765625}, {"start": 2628.25, "end": 2629.35, "word": " و", "probability": 0.273193359375}, {"start": 2629.35, "end": 2629.51, "word": " هو", "probability": 0.896484375}, {"start": 2629.51, "end": 2629.97, "word": " المطلوب", "probability": 0.86151123046875}], "temperature": 1.0}, {"id": 97, "seek": 266291, "start": 2633.87, "end": 2662.91, "text": "Okay واضح البرهان؟ في أي استفسار على البرهان؟ في هنا تحذير warning تحذير in the above theorem we had", "tokens": [8297, 4032, 46958, 5016, 2423, 26890, 3224, 7649, 22807, 8978, 36632, 44713, 36178, 9640, 15844, 2423, 26890, 3224, 7649, 22807, 8978, 34105, 6055, 5016, 8848, 13546, 9164, 6055, 5016, 8848, 13546, 294, 264, 3673, 20904, 321, 632], "avg_logprob": -0.15768915140315107, "compression_ratio": 1.2521008403361344, "no_speech_prob": 0.0, "words": [{"start": 2633.87, "end": 2634.29, "word": "Okay", "probability": 0.375732421875}, {"start": 2634.29, "end": 2634.73, "word": " واضح", "probability": 0.9578450520833334}, {"start": 2634.73, "end": 2636.17, "word": " البرهان؟", "probability": 0.777392578125}, {"start": 2636.17, "end": 2636.29, "word": " في", "probability": 0.75146484375}, {"start": 2636.29, "end": 2636.39, "word": " أي", "probability": 0.58544921875}, {"start": 2636.39, "end": 2636.99, "word": " استفسار", "probability": 0.9474283854166666}, {"start": 2636.99, "end": 2637.13, "word": " على", "probability": 0.486083984375}, {"start": 2637.13, "end": 2640.73, "word": " البرهان؟", "probability": 0.95107421875}, {"start": 2640.73, "end": 2642.01, "word": " في", "probability": 0.82177734375}, {"start": 2642.01, "end": 2642.25, "word": " هنا", "probability": 0.94482421875}, {"start": 2642.25, "end": 2642.99, "word": " تحذير", "probability": 0.947021484375}, {"start": 2642.99, "end": 2643.73, "word": " warning", "probability": 0.79931640625}, {"start": 2643.73, "end": 2648.03, "word": " تحذير", "probability": 0.960205078125}, {"start": 2648.03, "end": 2655.07, "word": " in", "probability": 0.8525390625}, {"start": 2655.07, "end": 2656.17, "word": " the", "probability": 0.93505859375}, {"start": 2656.17, "end": 2656.91, "word": " above", "probability": 0.966796875}, {"start": 2656.91, "end": 2659.31, "word": " theorem", "probability": 0.94140625}, {"start": 2659.31, "end": 2662.09, "word": " we", "probability": 0.8876953125}, {"start": 2662.09, "end": 2662.91, "word": " had", "probability": 0.9375}], "temperature": 1.0}, {"id": 98, "seek": 268810, "start": 2664.66, "end": 2688.1, "text": "F of I التي هي F للفترة المغلقة من A لB طلعت بالساوي الفترة المغلقة من small m لcapital M حيث small m is the absolute minimum value وcapital M is the absolute maximum value of the function F on the interval I", "tokens": [37, 295, 286, 38392, 39896, 479, 24976, 5172, 2655, 25720, 9673, 17082, 1211, 28671, 9154, 316, 5296, 33, 23032, 1211, 34268, 20666, 3794, 995, 45865, 27188, 2655, 25720, 9673, 17082, 1211, 28671, 9154, 1359, 275, 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"probability": 0.95703125}, {"start": 2670.9, "end": 2671.28, "word": " A", "probability": 0.6826171875}, {"start": 2671.28, "end": 2671.86, "word": " لB", "probability": 0.4876708984375}, {"start": 2671.86, "end": 2675.32, "word": " طلعت", "probability": 0.8041178385416666}, {"start": 2675.32, "end": 2675.9, "word": " بالساوي", "probability": 0.743896484375}, {"start": 2675.9, "end": 2676.4, "word": " الفترة", "probability": 0.95751953125}, {"start": 2676.4, "end": 2677.04, "word": " المغلقة", "probability": 0.9825439453125}, {"start": 2677.04, "end": 2677.3, "word": " من", "probability": 0.9677734375}, {"start": 2677.3, "end": 2677.8, "word": " small", "probability": 0.3251953125}, {"start": 2677.8, "end": 2678.1, "word": " m", "probability": 0.84375}, {"start": 2678.1, "end": 2678.78, "word": " لcapital", "probability": 0.8854166666666666}, {"start": 2678.78, "end": 2679.1, "word": " M", "probability": 0.98095703125}, {"start": 2679.1, "end": 2679.9, "word": " حيث", "probability": 0.9583333333333334}, {"start": 2679.9, "end": 2680.24, "word": " small", "probability": 0.8515625}, {"start": 2680.24, "end": 2680.62, "word": " m", "probability": 0.9287109375}, {"start": 2680.62, "end": 2680.86, "word": " is", "probability": 0.85693359375}, {"start": 2680.86, "end": 2681.0, "word": " the", "probability": 0.76611328125}, {"start": 2681.0, "end": 2681.42, "word": " absolute", "probability": 0.90673828125}, {"start": 2681.42, "end": 2681.8, "word": " minimum", "probability": 0.90625}, {"start": 2681.8, "end": 2682.4, "word": " value", "probability": 0.98974609375}, {"start": 2682.4, "end": 2683.76, "word": " وcapital", "probability": 0.8440755208333334}, {"start": 2683.76, "end": 2683.98, "word": " M", "probability": 0.982421875}, {"start": 2683.98, "end": 2684.22, "word": " is", "probability": 0.94580078125}, {"start": 2684.22, "end": 2684.36, "word": " the", "probability": 0.8505859375}, {"start": 2684.36, "end": 2684.72, "word": " absolute", "probability": 0.89501953125}, {"start": 2684.72, "end": 2685.16, "word": " maximum", "probability": 0.93505859375}, {"start": 2685.16, "end": 2685.68, "word": " value", "probability": 0.97705078125}, {"start": 2685.68, "end": 2685.94, "word": " of", "probability": 0.95703125}, {"start": 2685.94, "end": 2686.1, "word": " the", "probability": 0.85595703125}, {"start": 2686.1, "end": 2686.5, "word": " function", "probability": 0.9736328125}, {"start": 2686.5, "end": 2686.86, "word": " F", "probability": 0.849609375}, {"start": 2686.86, "end": 2687.14, "word": " on", "probability": 0.8916015625}, {"start": 2687.14, "end": 2687.3, "word": " the", "probability": 0.89208984375}, {"start": 2687.3, "end": 2687.72, "word": " interval", "probability": 0.974609375}, {"start": 2687.72, "end": 2688.1, "word": " I", "probability": 0.9765625}], "temperature": 1.0}, {"id": 99, "seek": 270050, "start": 2689.96, "end": 2700.5, "text": "و هذا ليس بالضرورة مش شرط هذه الفترة تكون الفترة من F of A ل F of B", "tokens": 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ال image للفترة I بالساوي الفترة المغلقة من F of A لF of B فاخدوا بالكم من ايه من التحذير هذا", "tokens": [3224, 24192, 27188, 2655, 25720, 19446, 24401, 8592, 10874, 6027, 1975, 2407, 3714, 28543, 2288, 7251, 11622, 2304, 16472, 11296, 27188, 2655, 25720, 9673, 17082, 1211, 28671, 9154, 479, 295, 316, 5296, 37, 295, 363, 23758, 37893, 20328, 5016, 1829, 5016, 1392, 28239, 19913, 2288, 10632, 19446, 39894, 39648, 1829, 2423, 28820, 10943, 23758, 39894, 39648, 1829, 2423, 28820, 10943, 23758, 6156, 47432, 23758, 32771, 1211, 9566, 37893, 13412, 2288, 9566, 2423, 3256, 24976, 5172, 2655, 25720, 286, 20666, 3794, 995, 45865, 27188, 2655, 25720, 9673, 17082, 1211, 28671, 9154, 479, 295, 316, 5296, 37, 295, 363, 6156, 47283, 3215, 14407, 20666, 24793, 9154, 1975, 1829, 3224, 9154, 16712, 5016, 8848, 13546, 23758], "avg_logprob": -0.16858552095659993, "compression_ratio": 1.9903846153846154, "no_speech_prob": 0.0, "words": [{"start": 2701.91, "end": 2702.29, "word": "هذه", "probability": 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3246.25, "end": 3246.61, "word": " section", "probability": 0.9189453125}, {"start": 3246.61, "end": 3247.17, "word": " خمسة", "probability": 0.951904296875}, {"start": 3247.17, "end": 3247.85, "word": " تلاتة", "probability": 0.9151611328125}, {"start": 3247.85, "end": 3249.09, "word": " و", "probability": 0.96435546875}, {"start": 3249.09, "end": 3250.31, "word": " باقي", "probability": 0.68994140625}, {"start": 3250.31, "end": 3250.63, "word": " عننا", "probability": 0.6572265625}, {"start": 3250.63, "end": 3251.11, "word": " section", "probability": 0.84033203125}, {"start": 3251.11, "end": 3252.85, "word": " خمسة", "probability": 0.9674072265625}, {"start": 3252.85, "end": 3253.17, "word": " أربعة", "probability": 0.6904296875}, {"start": 3253.17, "end": 3253.67, "word": " هناخده", "probability": 0.92490234375}, {"start": 3253.67, "end": 3253.77, "word": " في", "probability": 0.9111328125}, {"start": 3253.77, "end": 3254.27, "word": " المحاضرة", "probability": 0.979736328125}, {"start": 3254.27, "end": 3255.59, "word": " الجاية", "probability": 0.7799479166666666}, {"start": 3255.59, "end": 3256.19, "word": " نحاول", "probability": 0.86541748046875}, {"start": 3256.19, "end": 3256.49, "word": " نشوف", "probability": 0.9694010416666666}, {"start": 3256.49, "end": 3256.91, "word": " زمنا", "probability": 0.6673177083333334}, {"start": 3256.91, "end": 3257.53, "word": " نخلصه", "probability": 0.816357421875}, {"start": 3257.53, "end": 3257.67, "word": " ولا", "probability": 0.78662109375}, {"start": 3257.67, "end": 3257.93, "word": " لأ", "probability": 0.770751953125}], "temperature": 1.0}, {"id": 121, "seek": 326691, "start": 3259.35, "end": 3266.91, "text": "فال .. شكرا لحصن إصراعكم و يعطيكم العافية و نشوفكم ان شاء الله المرة الجاية", "tokens": [5172, 6027, 4386, 13412, 4117, 23557, 5296, 5016, 9381, 1863, 11933, 9381, 23557, 3615, 24793, 4032, 37495, 9566, 1829, 24793, 18863, 31845, 10632, 4032, 8717, 8592, 38688, 24793, 16472, 13412, 16606, 21984, 9673, 25720, 25724, 995, 10632], "avg_logprob": -0.2168996714447674, "compression_ratio": 1.3535353535353536, "no_speech_prob": 0.0, "words": [{"start": 3259.35, "end": 3260.15, "word": "فال", "probability": 0.958984375}, {"start": 3260.15, "end": 3260.53, "word": " ..", "probability": 0.47265625}, {"start": 3260.53, "end": 3261.09, "word": " شكرا", "probability": 0.9571940104166666}, {"start": 3261.09, "end": 3262.87, "word": " لحصن", "probability": 0.757568359375}, {"start": 3262.87, "end": 3263.69, "word": " إصراعكم", "probability": 0.721044921875}, {"start": 3263.69, "end": 3264.13, "word": " و", "probability": 0.97509765625}, {"start": 3264.13, "end": 3264.83, "word": " يعطيكم", "probability": 0.9300537109375}, {"start": 3264.83, "end": 3265.31, "word": " العافية", "probability": 0.9744466145833334}, {"start": 3265.31, "end": 3265.47, "word": " و", "probability": 0.41064453125}, {"start": 3265.47, "end": 3265.89, "word": " نشوفكم", "probability": 0.9462890625}, {"start": 3265.89, "end": 3266.03, "word": " ان", "probability": 0.71484375}, {"start": 3266.03, "end": 3266.17, "word": " شاء", "probability": 0.9921875}, {"start": 3266.17, "end": 3266.27, "word": " الله", "probability": 0.96484375}, {"start": 3266.27, "end": 3266.59, "word": " المرة", "probability": 0.85205078125}, {"start": 3266.59, "end": 3266.91, "word": " الجاية", "probability": 0.6692708333333334}], "temperature": 1.0}], "language": "ar", "language_probability": 1.0, "duration": 3267.7675, "duration_after_vad": 2940.1022187499902}
\ No newline at end of file
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/Ejs4dHLsIvo_postprocess.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/Ejs4dHLsIvo_postprocess.srt
new file mode 100644
index 0000000000000000000000000000000000000000..169d3e413349bec5a1670103c92801c4a80ac876
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/Ejs4dHLsIvo_postprocess.srt
@@ -0,0 +1,1548 @@
+1
+00:00:20,920 --> 00:00:24,640
+بسم الله الرحمن الرحيم هنبدأ في المحاضرة هذه
+
+2
+00:00:24,640 --> 00:00:29,600
+chapter جديد وهو chapter أربعة في الكتاب المقرر
+
+3
+00:00:29,600 --> 00:00:35,280
+عنوان ال chapter limits of functions و هنبدأ أول
+
+4
+00:00:35,280 --> 00:00:39,540
+section في ال chapter هذا و برضه عنوان ال section
+
+5
+00:00:39,540 --> 00:00:44,180
+الأول هو نفس عنوان ال chapter limits of functions
+
+6
+00:00:44,180 --> 00:00:52,780
+فقبل ما نعرف limit of a functionبدنا نتعرف على
+
+7
+00:00:52,780 --> 00:01:00,060
+مصطلح جديد وهو cluster point of a set نقطة تراكم
+
+8
+00:01:00,060 --> 00:01:04,780
+ال cluster point أو بعض الكتب بيسموها accumulation
+
+9
+00:01:04,780 --> 00:01:12,120
+point و كتب أخرى بيسميها limit point فلو في عندي
+
+10
+00:01:12,120 --> 00:01:18,200
+set A subset من R set of real numbers و C real
+
+11
+00:01:18,200 --> 00:01:23,190
+number فال real number هذابنسميه cluster point
+
+12
+00:01:23,190 --> 00:01:28,030
+للست a if and only if the following condition is
+
+13
+00:01:28,030 --> 00:01:33,770
+satisfied for every delta عدد موجب نقدر نجد x
+
+14
+00:01:33,770 --> 00:01:39,650
+ينتمي إلى المجموعة a و ال x هذه مختلفة عن النقطة c
+
+15
+00:01:39,650 --> 00:01:45,330
+بحيث ان المسافة بين x و c تكون أصغر من delta هذا
+
+16
+00:01:45,330 --> 00:01:49,410
+الشرط هذا
+
+17
+00:01:49,410 --> 00:01:54,020
+الشرطis equivalent to saying بكافئ ان انا اقول
+
+18
+00:01:54,020 --> 00:01:59,160
+every delta neighborhood every delta neighborhood
+
+19
+00:01:59,160 --> 00:02:03,940
+لنقطة c اللى هو الفترة المفتوحة اللى مركزها c ونص
+
+20
+00:02:03,940 --> 00:02:11,500
+قطرة delta every delta neighborhoodof c يتقاطع مع
+
+21
+00:02:11,500 --> 00:02:18,200
+المجموعة a في نقطة واحدة على الأقل x مختلفة عن ال
+
+22
+00:02:18,200 --> 00:02:25,720
+c يعني بمعنى أخر بقدر ألاقي في التقاطع هذا نقطة x
+
+23
+00:02:25,720 --> 00:02:32,440
+يعني التقاطع هذا لا يساوي five okay
+
+24
+00:02:32,440 --> 00:02:37,990
+كمان مرةالنقطة C هذه بتكون cluster point للمجموعة
+
+25
+00:02:37,990 --> 00:02:44,370
+A إذا أي delta neighborhood للنقطة C بيتخاطع مع
+
+26
+00:02:44,370 --> 00:02:52,690
+المجموعة A في نقطة X مختلفة عن الـC بس لازم
+
+27
+00:02:52,690 --> 00:02:58,060
+كل delta neighborhood لـCيتقاطع مع المجموعة A في
+
+28
+00:02:58,060 --> 00:03:02,720
+نقطة X مختلفة عن الـC طب عشان اثبت ان الـC ليست
+
+29
+00:03:02,720 --> 00:03:09,000
+cluster point بنفي الشرط هذا يكفي ان اقول there
+
+30
+00:03:09,000 --> 00:03:14,520
+exist بدل for every delta او every delta
+
+31
+00:03:14,520 --> 00:03:18,860
+neighborhoodيكفى ان اجيب there exists delta
+
+32
+00:03:18,860 --> 00:03:24,720
+neighborhood واحد ل C و التقاطع هذا بساوي فاي يعني
+
+33
+00:03:24,720 --> 00:03:30,540
+بحيث ان ال delta neighborhood لا يتقاطع مع اي
+
+34
+00:03:30,540 --> 00:03:39,180
+مشيول منها C بالمرة ناخد نظرية الأول ال definition
+
+35
+00:03:39,180 --> 00:03:43,180
+هذا بيكافئ النظرية التانية بتقول ان
+
+36
+00:03:46,060 --> 00:03:53,140
+الـ condition هذا تبع التعريف بكافئ شرط تاني اذا
+
+37
+00:03:53,140 --> 00:04:06,140
+هنا let A subset من R و C real number C
+
+38
+00:04:06,140 --> 00:04:16,740
+is a cluster is a cluster pointof the set A if and
+
+39
+00:04:16,740 --> 00:04:22,400
+only if the following condition is satisfied there
+
+40
+00:04:22,400 --> 00:04:26,600
+exist a
+
+41
+00:04:26,600 --> 00:04:38,640
+sequence a n contained in A وكل عناصرها مختلفة
+
+42
+00:04:38,640 --> 00:04:52,650
+عن ال Csuch that limit a n بساوي c اذا هذا الشرط
+
+43
+00:04:52,650 --> 00:04:59,310
+بكافئ الشرط اللي هناك الشرط هذا او اللي بكافه
+
+44
+00:04:59,310 --> 00:05:06,290
+فلبرهان ذلك اذا كمان مرة انا عشان اثبت ان c is a
+
+45
+00:05:06,290 --> 00:05:10,610
+cluster point للمجموع يعني يكفي ان اثبت ان يوجد
+
+46
+00:05:12,310 --> 00:05:17,950
+سيكوانس في المجموعة A وكل على سرها مختلفة لاتساوي
+
+47
+00:05:17,950 --> 00:05:23,370
+C وانهيتها بالساوي لعدد C فتعالى نبرهن النظرية هذه
+
+48
+00:05:23,370 --> 00:05:28,850
+نبرهن ال only if part الأول فال only if part يعني
+
+49
+00:05:28,850 --> 00:05:36,910
+ال assumption assume ان C is a cluster is
+
+50
+00:05:36,910 --> 00:05:39,230
+a cluster point
+
+51
+00:05:40,700 --> 00:05:49,140
+of a then
+
+52
+00:05:49,140 --> 00:06:00,080
+for every n ينتمي إلى n لكل عدد طبيعي n take delta
+
+53
+00:06:00,080 --> 00:06:08,110
+بساوي واحد على n عدد موجببما انه C is a cluster
+
+54
+00:06:08,110 --> 00:06:11,230
+point ل A then by definition of a cluster point
+
+55
+00:06:11,230 --> 00:06:14,410
+then
+
+56
+00:06:14,410 --> 00:06:23,770
+by definition there exist a N ينتمي إلى A مختلف عن
+
+57
+00:06:23,770 --> 00:06:28,210
+ال C such that
+
+58
+00:06:28,210 --> 00:06:31,070
+ال ..
+
+59
+00:06:34,910 --> 00:06:42,950
+الـ AN هذا ينتمي للـ Delta neighborhood للـ
+
+60
+00:06:42,950 --> 00:06:56,090
+C وطبعا ينتمي إلى A negative C هعمل
+
+61
+00:06:56,090 --> 00:07:02,810
+التعريف هذالما انه الـ C is a cluster point لأي
+
+62
+00:07:02,810 --> 00:07:09,410
+Delta أكبر من السفر خد Delta بساوي واحد على N لكل
+
+63
+00:07:09,410 --> 00:07:13,290
+عدد طبيعي N هذا بيطلع عدد موجب لذلك لـ Delta بساوي
+
+64
+00:07:13,290 --> 00:07:18,470
+واحد على N بقدر ألاجني عنصر X اللي هو ساميه An
+
+65
+00:07:18,470 --> 00:07:24,470
+يعتمد على الـ Delta وهذا العنصر موجود في A مختلف
+
+66
+00:07:24,470 --> 00:07:30,570
+عن الـ Cو أيضا موجود في الـ delta neighborhood لـ
+
+67
+00:07:30,570 --> 00:07:38,370
+C طيب
+
+68
+00:07:38,370 --> 00:07:42,610
+إذا و
+
+69
+00:07:42,610 --> 00:07:51,230
+واضح هنا من ال AN ينتمي ل ال
+
+70
+00:07:51,230 --> 00:07:56,150
+AN ينتمي ل DN ال
+
+71
+00:07:56,150 --> 00:08:01,970
+deltaأو الـ 1 على N neighborhood للـ C اللي هو C
+
+72
+00:08:01,970 --> 00:08:09,750
+سالب واحد على N C موجب واحد على N وهذا صحيح لكل N
+
+73
+00:08:09,750 --> 00:08:16,270
+بيقدي انه بيقدي
+
+74
+00:08:16,270 --> 00:08:20,890
+انه C سالم A N أو
+
+75
+00:08:24,630 --> 00:08:31,950
+absolute a n سالب c أصغر من واحد على n وهذا صحيح
+
+76
+00:08:31,950 --> 00:08:36,370
+لكل n هزبوت؟
+
+77
+00:08:36,370 --> 00:08:40,750
+هاي a n أكبر من c سالب واحد على n أصغر من c زاد
+
+78
+00:08:40,750 --> 00:08:45,310
+واحد على n هذا معناه absolute a n minus c أصغر من
+
+79
+00:08:45,310 --> 00:08:48,790
+واحد على n لكل n هذا صحيح لكل n
+
+80
+00:08:52,390 --> 00:08:56,990
+إذا هاي فيها sequence إذا هاي أثبتنا مايوجد
+
+81
+00:08:56,990 --> 00:09:09,350
+sequence إذا am is a sequence in a وكل حدودها
+
+82
+00:09:09,350 --> 00:09:16,610
+مختلفة عن ال c and by theorem اتنين اربعة الشهيرة
+
+83
+00:09:18,560 --> 00:09:21,560
+أنا عندي الـ absolute value هذي أصغر من واحد على N
+
+84
+00:09:21,560 --> 00:09:26,480
+لكل N limit واحد على N بالساوي سفر خد C بالساوي
+
+85
+00:09:26,480 --> 00:09:33,760
+واحد عدن موجب لأن بيطلع limit ال sequence A N as N
+
+86
+00:09:33,760 --> 00:09:39,300
+tends to infinity بساوي S C إذن هين أثبتنا إنه لو
+
+87
+00:09:39,300 --> 00:09:44,920
+كانت C cluster point فأثبتنا إنه يوجد sequence A N
+
+88
+00:09:44,920 --> 00:09:51,380
+في المجموعة Aوكل عناصرها مختلفة عن الـ C ونهايتها
+
+89
+00:09:51,380 --> 00:09:59,080
+بساوي الـ C إذا هذا بثبت جزء ال only if part الآن
+
+90
+00:09:59,080 --> 00:10:04,220
+لثبت العكس لإثبات
+
+91
+00:10:04,220 --> 00:10:04,920
+العكس
+
+92
+00:10:21,910 --> 00:10:28,090
+assume أن الشرط اللي حصل بتتحقق
+
+93
+00:10:28,090 --> 00:10:41,250
+assume ال condition اللي حصل holds يعني بتتحقق to
+
+94
+00:10:41,250 --> 00:10:48,890
+show c is a cluster point of
+
+95
+00:10:48,890 --> 00:11:03,360
+a letdelta أكبر من السفر is given since
+
+96
+00:11:03,360 --> 00:11:15,160
+by star من الشرط star لدي limit لان بساوي c و
+
+97
+00:11:15,160 --> 00:11:19,260
+delta أكبر من السفر is given إذا من تعريف delta
+
+98
+00:11:19,260 --> 00:11:25,560
+capital Nللـ limit لأي delta أو إبسلون عدد موجة
+
+99
+00:11:25,560 --> 00:11:31,320
+there exist n يعتمد على delta natural number دحيث
+
+100
+00:11:31,320 --> 00:11:40,140
+أنه لكل n أكبر من أو سوى capital N هذا بيقدي أنه
+
+101
+00:11:40,140 --> 00:11:47,000
+absolute a n minus c أصغر من delta
+
+102
+00:11:50,570 --> 00:11:57,870
+إذاً هذا بيقدّي إنه هيعندي am
+
+103
+00:12:02,990 --> 00:12:08,910
+طبعا هدف قلبي أن a n أصغر
+
+104
+00:12:08,910 --> 00:12:15,370
+من c زائد delta أكبر من c negative delta يعني a n
+
+105
+00:12:15,370 --> 00:12:22,630
+تنتمي إلى v delta of c وتنتمي طبعا من ال condition
+
+106
+00:12:22,630 --> 00:12:32,780
+star تنتمي إلى a difference cو هذا صحيح لكل n أكبر
+
+107
+00:12:32,780 --> 00:12:38,660
+من أو ساوي capital N إذا هاي كل delta neighborhood
+
+108
+00:12:38,660 --> 00:12:50,240
+ل C بيتقاطع مع A في عدد لانهائي من النقاط المختلفة
+
+109
+00:12:50,240 --> 00:12:57,420
+عن ال C وبالتالي الشرط تبع ال definition هيتحققو
+
+110
+00:12:57,420 --> 00:13:05,660
+then by definition .. by definition C is a cluster
+
+111
+00:13:05,660 --> 00:13:14,000
+point of the set A و هدا بكمل ال F part and
+
+112
+00:13:14,000 --> 00:13:19,140
+therefore completes the proof of the theorem okay
+
+113
+00:13:19,140 --> 00:13:20,000
+تمام؟
+
+114
+00:13:25,140 --> 00:13:27,540
+Fine خلّينا ناخد بعض الأمثلة
+
+115
+00:13:49,070 --> 00:14:05,410
+تشير إلى أن كل X ينتمي إلى مقفل مقفل مقفل
+
+116
+00:14:05,410 --> 00:14:16,770
+مقفل
+
+117
+00:14:23,120 --> 00:14:29,460
+of set A1 بساوي الفترة
+
+118
+00:14:29,460 --> 00:14:40,780
+المفتوحة من سفر إلى واحد من
+
+119
+00:14:40,780 --> 00:14:45,460
+هنا يثبت إن كل X الفترة المغلقة من سفر إلى واحد هي
+
+120
+00:14:45,460 --> 00:14:50,480
+cluster point للفترة المفتوحة من سفر إلى واحد
+
+121
+00:15:01,360 --> 00:15:09,580
+ففي الأول بدي أثبت أنه كل نقطة داخل الفترة المغلقة
+
+122
+00:15:09,580 --> 00:15:14,460
+هي cluster point للمجموع عادي و بعدين في المرحلة
+
+123
+00:15:14,460 --> 00:15:19,840
+التانية حتة من نقاط الأطراف اللي هي 01 تطلع أيضا
+
+124
+00:15:19,840 --> 00:15:26,620
+cluster point لست A1 فنشوف مع بعض ان ال claim 1
+
+125
+00:15:34,010 --> 00:15:45,190
+بنثبت ان كل X ينتمي للفترة المفتوحة is a cluster
+
+126
+00:15:45,190 --> 00:15:50,410
+point of
+
+127
+00:15:50,410 --> 00:15:58,730
+set A1 اللي هي الفترة المفتوحة نفسها لبرهان ذلك to
+
+128
+00:15:58,730 --> 00:16:09,390
+see thisالبرهن ذلك fix ينتمي
+
+129
+00:16:09,390 --> 00:16:17,470
+للفترة المفتوحة ونثبت ان cluster point للمجموعة a1
+
+130
+00:16:17,470 --> 00:16:29,940
+اذا fix x and let delta أكبر من السفر be givenنبدأ
+
+131
+00:16:29,940 --> 00:16:35,900
+بالـ delta أكبر من 0 ونثبت أن كل delta
+
+132
+00:16:35,900 --> 00:16:42,820
+neighborhood للنقطة X بيتقاطع مع المجموعة A1 في
+
+133
+00:16:42,820 --> 00:16:49,400
+نقطة مختلفة عن X وبالتالي الـ X هتطلع cluster
+
+134
+00:16:49,400 --> 00:16:55,960
+point حسب التعريف طيب نحن
+
+135
+00:16:55,960 --> 00:16:56,920
+لدينا اتصالين
+
+136
+00:17:01,270 --> 00:17:07,090
+two cases حالتين الحالة
+
+137
+00:17:07,090 --> 00:17:10,270
+الأولى ان ال delta هذه اللى انا اخترتها العشوائية
+
+138
+00:17:10,270 --> 00:17:17,310
+ال delta اللى انا اخترتها ممكن تكون اصغر من ..
+
+139
+00:17:17,310 --> 00:17:23,590
+طبعا موجبة هي ممكن تكون اصغر من واحد ف in this
+
+140
+00:17:23,590 --> 00:17:27,930
+case in
+
+141
+00:17:27,930 --> 00:17:30,050
+this case في هذه الحالة
+
+142
+00:17:35,000 --> 00:17:49,440
+لدينا in this
+
+143
+00:17:49,440 --> 00:17:58,680
+case الـ delta neighborhood ل X اللي هو X minus
+
+144
+00:17:58,680 --> 00:18:06,650
+Delta X موجة Deltaبنلاحظ انه تقاطع مع المجموعة ايه
+
+145
+00:18:06,650 --> 00:18:14,050
+اللي هي الفترة اللي مفتوحة من سفر لواحد بطلع
+
+146
+00:18:14,050 --> 00:18:23,080
+واحد من الخيارات التاليةاما الفترة المفتوحة x
+
+147
+00:18:23,080 --> 00:18:28,820
+negative delta x positive delta او الفترة المفتوحة
+
+148
+00:18:28,820 --> 00:18:36,420
+من صفر الى اكس positive delta او الفترة
+
+149
+00:18:36,420 --> 00:18:40,340
+المفتوحة من x negative delta الى واحد او الفترة
+
+150
+00:18:40,340 --> 00:18:42,920
+المفتوحة من صفر الى واحد
+
+151
+00:18:48,470 --> 00:18:55,310
+حسب قيمة الـ Delta يعني أنا
+
+152
+00:18:55,310 --> 00:19:03,030
+عندي الفترة المفتوحة من 0 إلى 1 هذه الفترة
+
+153
+00:19:03,030 --> 00:19:12,350
+المفتوحة تبعتي اللي هي A1 هذا ال set A1 و أنا عندي
+
+154
+00:19:12,350 --> 00:19:20,080
+ال Delta عدد موجب أصغر من 1والـ X هذه تنتمي .. الـ
+
+155
+00:19:20,080 --> 00:19:27,300
+X هذه نقطة ما في الفترة
+
+156
+00:19:27,300 --> 00:19:38,320
+fixed number بين 0 و 1 الآن
+
+157
+00:19:38,320 --> 00:19:41,160
+أنا عندي الـ delta neighborhood لـ X هذا هو ممكن
+
+158
+00:19:41,160 --> 00:19:46,510
+يكون زي هيك شكله وبالتالي تقاطعتقاطعه مع الفترة
+
+159
+00:19:46,510 --> 00:19:53,310
+المفتوحة هو نفسه، صح؟ إذا كان زي هيك أو ممكن يكون
+
+160
+00:19:53,310 --> 00:19:57,250
+الـ delta neighborhood لل X يكون شكله زي هيك
+
+161
+00:19:57,250 --> 00:20:01,430
+وبالتالي
+
+162
+00:20:01,430 --> 00:20:06,150
+تقاطعه مع الفترة .. مع الست واحد، هيكون الفترة
+
+163
+00:20:06,150 --> 00:20:10,830
+المفتوحة من صفر إلى X زي الـ delta اللي هي التانية
+
+164
+00:20:10,830 --> 00:20:17,310
+يعنيصح وممكن يكون ال
+
+165
+00:20:17,310 --> 00:20:21,810
+delta neighborhood ال x تكون جريبة من الواحد زي
+
+166
+00:20:21,810 --> 00:20:26,410
+هيك و ال delta neighborhood حوالين ال x يكون زي
+
+167
+00:20:26,410 --> 00:20:34,190
+هيك شكله هاي x negative delta x positive delta
+
+168
+00:20:35,200 --> 00:20:39,440
+وبالتالي تقاطع مع الفترة من صفر إلى واحد هيعطيني
+
+169
+00:20:39,440 --> 00:20:44,700
+الجزء هذا اللي هو فترة مفتوحة من X سالب Delta إلى
+
+170
+00:20:44,700 --> 00:20:48,360
+واحد وممكن
+
+171
+00:20:48,360 --> 00:20:56,170
+ال Delta لبرهود ال X تكون جريبة من المنتصفوالـ
+
+172
+00:20:56,170 --> 00:20:59,250
+Delta تكون قريبة من الواحد قيمتها أصغر من واحد لكن
+
+173
+00:20:59,250 --> 00:21:04,250
+قريبة من واحد وبالتالي الـ Delta neighborhood لل X
+
+174
+00:21:04,250 --> 00:21:09,150
+يكون زي هيك وبالتالي تقاطعه مع المجموعة A واحد
+
+175
+00:21:09,150 --> 00:21:12,970
+بيطلع المجموعة A واحد نفسها، صحيح؟ إن هذه كل
+
+176
+00:21:12,970 --> 00:21:20,590
+احتمالات وفي كل الأحوال التقاطع هذا بيطلع infinite
+
+177
+00:21:20,590 --> 00:21:22,090
+is infinite
+
+178
+00:21:25,670 --> 00:21:29,890
+تقاطع المجمعتين هذول بيطلع فترة و الفترة اي فترة
+
+179
+00:21:29,890 --> 00:21:33,330
+مفتوحة ال cardinal number تبعها بيساوي ال cardinal
+
+180
+00:21:33,330 --> 00:21:36,310
+number تبع ال real numbers اللي هي uncountable set
+
+181
+00:21:36,310 --> 00:21:41,270
+وبالتالي infinite إذا التقاطع هذا infinite وهذا
+
+182
+00:21:41,270 --> 00:21:46,410
+بيقدي ان ال V
+
+183
+00:21:46,410 --> 00:21:57,490
+Delta of X تقاطعالـ a1 منزوعة منها الـ x هيطلع
+
+184
+00:21:57,490 --> 00:22:03,390
+بالتأكيد لا يساوي في لأن التقاطة هذا بيطلع
+
+185
+00:22:03,390 --> 00:22:07,630
+infinite وبالتالي هيك بيكون أثبتنا أن كل delta
+
+186
+00:22:07,630 --> 00:22:13,370
+neighborhood ل x بيتقاطع مع a1 في نقطة مختلفة عن x
+
+187
+00:22:23,000 --> 00:22:31,840
+الحالة التانية case two ان ال delta هذه تكون اكبر
+
+188
+00:22:31,840 --> 00:22:36,670
+من اوسع واحدبرضه فى الحاله دى بنا نثبت انه كل
+
+189
+00:22:36,670 --> 00:22:41,170
+delta neighborhood ال X بتقاطع مع A واحد فى نقطه
+
+190
+00:22:41,170 --> 00:22:49,730
+مختلفه عن ال X نشوف مع بعض in this case in
+
+191
+00:22:49,730 --> 00:22:57,310
+this case ال X negative او ال negative delta X
+
+192
+00:22:57,310 --> 00:23:01,310
+موجب delta هذا اللى هو ال delta neighborhood ل X
+
+193
+00:23:01,310 --> 00:23:06,620
+تقاطعالمجموعة A1 اللي هي الفترة المفتوحة من 0 إلى
+
+194
+00:23:06,620 --> 00:23:17,040
+1 هيطلع بساوي الفترة المفتوحة من 0 إلى 1 لأن الـ
+
+195
+00:23:17,040 --> 00:23:24,780
+Delta هنا أكبر من أو ساوي الواحد يعني هي عندي S 0
+
+196
+00:23:24,780 --> 00:23:36,180
+إلى 1 هذه اللي هي المجموعة A وهي Xنقطة ما داخل
+
+197
+00:23:36,180 --> 00:23:42,300
+الفترة فلما يكون x زايد ال delta لما تكون ال delta
+
+198
+00:23:42,300 --> 00:23:49,160
+تبعتي أكبر من واحد فx زايد ال delta هتكون هنا و x
+
+199
+00:23:49,160 --> 00:23:55,880
+سالب ال delta بالتأكيد هتكون هنا وبالتالي ال delta
+
+200
+00:23:55,880 --> 00:24:01,640
+neighborhood ل x هيحتويالمجموعة a واحد وبالتالي
+
+201
+00:24:01,640 --> 00:24:06,760
+تقاطع معاها تطلع المجموعة a واحد وهذا is infinite
+
+202
+00:24:06,760 --> 00:24:11,680
+وبالتالي
+
+203
+00:24:11,680 --> 00:24:18,060
+اذا ال delta neighborhood هذا تقاطع الفترة
+
+204
+00:24:18,060 --> 00:24:20,460
+المفتوحة minus x
+
+205
+00:24:23,650 --> 00:24:29,910
+لا أكيد بتأكيد لا يساوي five okay تمام اذا في
+
+206
+00:24:29,910 --> 00:24:34,470
+الحالتين ال condition تبع ال cluster point تتحقق
+
+207
+00:24:34,470 --> 00:24:44,730
+therefore by definition x is cluster point is
+
+208
+00:24:44,730 --> 00:24:50,780
+cluster point ofالست ا واحد اللي هي الفترة
+
+209
+00:24:50,780 --> 00:24:56,280
+المفتوحة من صفر الى واحد طبعا إذا هذا بثبت ال
+
+210
+00:24:56,280 --> 00:25:09,140
+claim الأولاني طبعا الآن هثبت claim تاني ال claim
+
+211
+00:25:09,140 --> 00:25:09,880
+التاني
+
+212
+00:25:16,660 --> 00:25:25,180
+النقطة 0 is a cluster point
+
+213
+00:25:25,180 --> 00:25:36,460
+of set A1 الفترة مفتوحة من 0 إلى 1 لبرهان
+
+214
+00:25:36,460 --> 00:25:37,020
+ذلك
+
+215
+00:25:47,440 --> 00:25:55,860
+to see this let نبدأ let delta أكبر من السفر be
+
+216
+00:25:55,860 --> 00:26:03,120
+given فهنا
+
+217
+00:26:03,120 --> 00:26:11,380
+لأي delta ال delta
+
+218
+00:26:11,380 --> 00:26:16,140
+neighborhood للسفر اللي هو هيطلع
+
+219
+00:26:18,570 --> 00:26:28,630
+سالب دلتا زاد سفر وموجب دلتا زاد سفر فتقاطع هذا مع
+
+220
+00:26:28,630 --> 00:26:35,150
+الفترة المفتوحة من سفر إلى واحد بساوي
+
+221
+00:26:36,900 --> 00:26:44,360
+في خيارين إما الفترة المفتوحة من سفر إلى دلتا إذا
+
+222
+00:26:44,360 --> 00:26:52,680
+كانت ال delta أصغر من واحد طبعا أكبر من سفر وبساوي
+
+223
+00:26:52,680 --> 00:26:57,600
+الفترة المفتوحة من سفر إلى واحد إذا كان ال delta
+
+224
+00:26:57,600 --> 00:27:04,220
+أكبر من أو ساوي الواحد زي ما شوفنا في برهان الكلام
+
+225
+00:27:04,220 --> 00:27:12,240
+الأولانيمظبوط هاي الاندي الفترة من سفر إلى واحد
+
+226
+00:27:12,240 --> 00:27:19,520
+هذه المجموعة A1 وهي
+
+227
+00:27:19,520 --> 00:27:27,700
+X نقطة .. لأ هاي السفر بالدفتر أن السفر cluster
+
+228
+00:27:27,700 --> 00:27:34,660
+point للمجموعة A1فأخدت أي delta أكبر من السفر الان
+
+229
+00:27:34,660 --> 00:27:38,260
+ال delta هذه لو كانت ال delta هذه اذا هي سالب
+
+230
+00:27:38,260 --> 00:27:42,720
+delta موجب delta لو كانت ال delta هذه أصغر من واحد
+
+231
+00:27:42,720 --> 00:27:47,120
+فتقاطع ال delta neighborhood مع ال a واحد هيكون
+
+232
+00:27:47,120 --> 00:27:51,140
+الجزء هذا اللي هو الفترة المفتوحة من سفر ل delta
+
+233
+00:27:51,140 --> 00:27:57,100
+صح؟ و لو كانت ال delta هذه أكبر من واحد لو كانت ال
+
+234
+00:27:57,100 --> 00:27:58,940
+delta هذه أكبر من واحد
+
+235
+00:28:01,680 --> 00:28:11,060
+فال .. ف delta هتكون هاي delta أكبر من واحد و سالف
+
+236
+00:28:11,060 --> 00:28:15,120
+delta هتكون هان و بالتالي ال delta لبرهود هذا
+
+237
+00:28:15,120 --> 00:28:24,220
+تقاطع مع a واحد بيساوي a واحد مظبوط صح؟ تمام؟ و في
+
+238
+00:28:24,220 --> 00:28:27,840
+كل الأحوال التقاطع هذا بيطلع infinite is infinite
+
+239
+00:28:27,840 --> 00:28:31,000
+infinite set لأنه open interval
+
+240
+00:28:38,080 --> 00:28:46,040
+تقاطة a-a1 هو
+
+241
+00:28:46,040 --> 00:28:53,940
+نفس تقاطة a1
+
+242
+00:29:00,690 --> 00:29:03,830
+إذن هي اللي أثبتت إن كل delta neighborhood للسفر
+
+243
+00:29:03,830 --> 00:29:09,590
+يتقاطع مع المجموعة A1 في نقطة مختلفة عن السفر في
+
+244
+00:29:09,590 --> 00:29:14,070
+حقيقة الأمر في حقيقة الأمر كل delta neighborhood
+
+245
+00:29:14,070 --> 00:29:19,250
+للسفر بتقاطع مع A1 في عدد لانهائي من النقاط اللي
+
+246
+00:29:19,250 --> 00:29:24,350
+موجودة في A1 ومختلفة عن السفر إذن by definition
+
+247
+00:29:24,350 --> 00:29:27,610
+zero is a cluster
+
+248
+00:29:30,120 --> 00:29:36,540
+point of A1 اللي هي الفترة مفتوحة من صفر لواحد
+
+249
+00:29:36,540 --> 00:29:43,720
+يبقى لإكمال البرهان يمكن أن يظهر الكليل التالت
+
+250
+00:29:43,720 --> 00:29:55,560
+باقي أثبت أن الواحد is a cluster point of set A1
+
+251
+00:29:55,560 --> 00:30:02,300
+اللي هي الفترة مفتوحة من صفر لواحدوبرهان ال claim
+
+252
+00:30:02,300 --> 00:30:07,360
+التالت زي .. similar لبرهان ال claim التالت إذا
+
+253
+00:30:07,360 --> 00:30:19,280
+هنا the proof its proof is similar to
+
+254
+00:30:19,280 --> 00:30:20,260
+claim to
+
+255
+00:30:23,650 --> 00:30:27,990
+فحاسيبكم انتوا تكتبوا وبالتالي هيك بالكون أثبتنا
+
+256
+00:30:27,990 --> 00:30:32,250
+ان كل نقطة في الفترة المغلطة سواء كانت نقطة الطرف
+
+257
+00:30:32,250 --> 00:30:37,570
+اللي هي 0 او 1 او نقطة داخلية interior point نقطة
+
+258
+00:30:37,570 --> 00:30:41,290
+داخل الفترة المغلطة كل النقاط هذه بتطلع cluster
+
+259
+00:30:41,290 --> 00:30:47,970
+points لمجموعة A1 اللي هي الفترة المفتوحة تمام؟
+
+260
+00:30:52,450 --> 00:31:05,790
+بالمثل ممكن إثبات إن
+
+261
+00:31:05,790 --> 00:31:09,250
+كل نقطة في الفترة المغلقة is cluster point
+
+262
+00:31:09,250 --> 00:31:16,990
+للمجموعة A2 اللي هي الفترة المغلقة من 0 إلى 1
+
+263
+00:31:16,990 --> 00:31:20,390
+والبرهان
+
+264
+00:31:20,390 --> 00:31:28,770
+هو نفسهبنعمل three claims وفي كل برهان هيكون الفرق
+
+265
+00:31:28,770 --> 00:31:35,430
+انه عندي انا a بدل a 1 هيكون a 2 فهتكون اللي هو
+
+266
+00:31:35,430 --> 00:31:41,370
+الفترات هذه فترة مغلقة من سفر إلى واحد وبالتالي
+
+267
+00:31:41,370 --> 00:31:47,910
+بيصير هذه الفترة من هنا مغلقة عند السفر ومغلقة عند
+
+268
+00:31:47,910 --> 00:31:54,070
+السفر ومغلقة عند الواحدو هكذا نفس البرهان نسخ لسق
+
+269
+00:31:54,070 --> 00:31:59,790
+مع التعديلات البسيطة ان اي واحد الآن أصبحت بدل ما
+
+270
+00:31:59,790 --> 00:32:03,270
+كانت فترة مفتوحة من صفر لواحد أصبحت فترة مغلقة
+
+271
+00:32:03,270 --> 00:32:06,830
+وبالتالي في التقاطعات الحسابات نغلق اللي هو
+
+272
+00:32:06,830 --> 00:32:13,590
+الفترات and الحاجة المطلوبة okay وبالتالي نفس
+
+273
+00:32:13,590 --> 00:32:18,910
+البرهان will go through هيمشي بالتمام والكمالOkay
+
+274
+00:32:18,910 --> 00:32:24,270
+تمام؟ إذا هذا البرهان مشابه لبرهان المثال الأول
+
+275
+00:32:24,270 --> 00:32:34,010
+ناخد كمان مثال آخر مثال
+
+276
+00:32:34,010 --> 00:32:43,770
+تالت every every
+
+277
+00:32:43,770 --> 00:32:44,310
+finite
+
+278
+00:32:46,970 --> 00:32:59,350
+set A contained in R has no
+
+279
+00:32:59,350 --> 00:33:03,070
+cluster
+
+280
+00:33:03,070 --> 00:33:09,730
+points
+
+281
+00:33:09,730 --> 00:33:19,970
+كل finite set مالهاش ولا cluster pointوالبرهان سهل
+
+282
+00:33:19,970 --> 00:33:24,950
+proof say
+
+283
+00:33:24,950 --> 00:33:35,030
+دعنا ال set a نسمي عناصرها a1, a2 الى an هدف مش
+
+284
+00:33:35,030 --> 00:33:39,950
+هدف finite set اذا عناصرهم ممكن اعملهم list a1, a2
+
+285
+00:33:39,950 --> 00:33:47,530
+الى anو ممكن اعملهم order ارتبهم حسب المؤشر تبعهم
+
+286
+00:33:47,530 --> 00:33:57,450
+يعني a1 اصغر من a2 اصغر من a3 اصغر من اصغر من am
+
+287
+00:33:57,450 --> 00:34:03,570
+ممكن نعمل كلمة من هذا ولا لأ ممكن by the ordering
+
+288
+00:34:03,570 --> 00:34:04,370
+principle
+
+289
+00:34:11,360 --> 00:34:20,780
+أو باستخدام ال ordering تبع ال real numbers إذا
+
+290
+00:34:20,780 --> 00:34:30,880
+هي عندي ال 6A تبعتي هي خط العداد وهي A1 وهي A2 وهي
+
+291
+00:34:30,880 --> 00:34:36,160
+A3 مش شرط المسافة بين كل أنصار والتاني تكون
+
+292
+00:34:36,160 --> 00:34:46,230
+متساوية وهكذا إلىأخر عنصر AN ف
+
+293
+00:34:46,230 --> 00:34:49,910
+fix X
+
+294
+00:34:49,910 --> 00:35:01,190
+ينتمي إلى R و بد أثبت أن claim X is not a cluster
+
+295
+00:35:01,190 --> 00:35:01,910
+point
+
+296
+00:35:09,790 --> 00:35:12,850
+بالتالي المجموع A ليس لديه اي cluster point
+
+297
+00:35:12,850 --> 00:35:16,770
+بالتالي المجموع A ليس لديه اي cluster point
+
+298
+00:35:16,770 --> 00:35:16,790
+بالتالي المجموع A ليس لديه اي cluster point
+
+299
+00:35:16,790 --> 00:35:17,530
+بالتالي المجموع A ليس لديه اي cluster point
+
+300
+00:35:17,530 --> 00:35:17,750
+بالتالي المجموع A ليس لديه اي cluster point
+
+301
+00:35:17,750 --> 00:35:18,430
+بالتالي المجموع A ليس لديه اي cluster point
+
+302
+00:35:18,430 --> 00:35:19,070
+بالتالي المجموع A ليس لديه اي cluster point
+
+303
+00:35:19,070 --> 00:35:19,690
+بالتالي المجموع A ليس لديه اي cluster point
+
+304
+00:35:19,690 --> 00:35:25,170
+بالتالي المجموع A ليس لديه اي cluster point
+
+305
+00:35:25,170 --> 00:35:28,870
+بالتالي المجموع A ليس لديه اي cluster point
+
+306
+00:35:28,870 --> 00:35:33,950
+بالتالي المجم
+
+307
+00:35:37,690 --> 00:35:44,410
+إما X تنتمي إلى A أو X تنتمي إلى الـ complement
+
+308
+00:35:44,410 --> 00:35:50,530
+يعني لا تنتمي إلى A صح؟
+
+309
+00:35:50,530 --> 00:35:59,870
+ففي الحالة الأولى case واحد أثر من X تنتمي إلى A
+
+310
+00:35:59,870 --> 00:36:03,970
+وبالدفع تثبت إن X ليست cluster point
+
+311
+00:36:10,180 --> 00:36:18,560
+say x بساوي a m for some m أكبر من أو ساوي واحد
+
+312
+00:36:18,560 --> 00:36:24,180
+أصغر من أو ساوي مش هذا ال x موجود في a و a على
+
+313
+00:36:24,180 --> 00:36:29,460
+سرها a واحد إلى a n إذا هذا ال x هو a m for some m
+
+314
+00:36:29,460 --> 00:36:36,300
+بين واحد و n طيب let
+
+315
+00:36:37,980 --> 00:36:48,860
+delta بساوي نص المسافة ال minimum المسافة
+
+316
+00:36:48,860 --> 00:37:04,760
+بين am minus am minus واحد وam زائد واحد minus am
+
+317
+00:37:11,460 --> 00:37:21,160
+يعني هاي ال X هاي ال M هاي AM وهاي AM زاد واحد
+
+318
+00:37:21,160 --> 00:37:30,960
+والأنصر اللي جاب لها AM minus واحد احنا قلنا ال X
+
+319
+00:37:30,960 --> 00:37:39,850
+سبعتي ال X سبعتي هي ال AM الآن باخدالمسافة هذه
+
+320
+00:37:39,850 --> 00:37:46,030
+اللي هي بين a m زي دول اللي هي المسافة هذه و باخد
+
+321
+00:37:46,030 --> 00:37:51,150
+المسافة هذه بين a m و a m سالب واحد لازم واحدة
+
+322
+00:37:51,150 --> 00:37:55,730
+تكون أصغر من أو يساوي التانية باخدها ال minimum ال
+
+323
+00:37:55,730 --> 00:37:57,730
+minimum .. ال minimum بين المسافتين .. الأصغر بين
+
+324
+00:37:57,730 --> 00:38:02,650
+المسافتين هدول و باخد نصها و باخد نصها بسميها
+
+325
+00:38:02,650 --> 00:38:08,720
+delta فنص .. لو قلنا الأصغرلو قلنا مثلا الأصغر
+
+326
+00:38:08,720 --> 00:38:15,400
+اللي هي هذه فنص الدلتا هذا هي فإذا ال delta هتكون
+
+327
+00:38:15,400 --> 00:38:21,040
+المسافة هذه و بكوّن delta neighborhood حوالين ال X
+
+328
+00:38:21,040 --> 00:38:29,720
+الأن ال delta neighborhood هذا then verify
+
+329
+00:38:29,720 --> 00:38:34,860
+ممكنكم تتحققوا verify that
+
+330
+00:38:37,340 --> 00:38:43,600
+الـ Delta neighborhood V Delta ل A M اللي هي ال X
+
+331
+00:38:43,600 --> 00:38:46,760
+تقاطع
+
+332
+00:38:46,760 --> 00:38:55,280
+ال set A فاي منزوعة منها A M هيطلع بساوي الفاي
+
+333
+00:38:55,280 --> 00:39:01,980
+مافيش تقاطع بينهم وبالتالي therefore by definition
+
+334
+00:39:04,480 --> 00:39:13,200
+ام اكس بساوي ام is not a
+
+335
+00:39:13,200 --> 00:39:22,360
+cluster point of set A لأن عشان تكونما تكونيش
+
+336
+00:39:22,360 --> 00:39:28,040
+cluster point لمجموعة A لازم اثبت انه يوجد there
+
+337
+00:39:28,040 --> 00:39:35,840
+exist delta neighborhood لل X تبعت اللي هي AM بحيث
+
+338
+00:39:35,840 --> 00:39:42,980
+انه ال delta neighborhood هذا مايتقطعش مع ال 6A في
+
+339
+00:39:42,980 --> 00:39:50,440
+اي نقطة مختلفة عن النقطة X وهذا حصلOkay تمام اذا
+
+340
+00:39:50,440 --> 00:39:56,140
+هذا في حالة لما ال X تكون موجودة في A الحالة
+
+341
+00:39:56,140 --> 00:40:03,340
+التانية ان ال case 2 case
+
+342
+00:40:03,340 --> 00:40:13,040
+2 ان ال X لا تنتمي ال ال set A ففي الحالة هذه
+
+343
+00:40:14,910 --> 00:40:20,590
+معناته x مابستويش ولا واحد من العناصر هذه فالحالة
+
+344
+00:40:20,590 --> 00:40:26,690
+هذه ممكن اجزها الى تلت حالات الحالة
+
+345
+00:40:26,690 --> 00:40:32,750
+الأولى ان ال x تبعتي تكون اصغر من a واحدوبالتالي
+
+346
+00:40:32,750 --> 00:40:37,010
+واضح ان المسافة بين X و A واحد كبيرة وباخد نص
+
+347
+00:40:37,010 --> 00:40:43,930
+المسافة ديلتا إذا هيوجد دلتا نبرود ل X ومابتقطعش
+
+348
+00:40:43,930 --> 00:40:47,650
+مع المجموعة A بالمرة وبالتالي X is not cluster
+
+349
+00:40:47,650 --> 00:40:54,250
+point ممكن الحالة التانية أن X تكون أكبر منالـ AM
+
+350
+00:40:54,250 --> 00:40:59,470
+برضه باخد المسافة دي بجيسها و باخد نصها على انه
+
+351
+00:40:59,470 --> 00:41:03,670
+Delta و بكون Delta neighborhood حوالين ال X هذا ال
+
+352
+00:41:03,670 --> 00:41:06,530
+Delta neighborhood واضح انه مابتخطعش مع ال set A
+
+353
+00:41:06,530 --> 00:41:11,910
+بالمرة وبالتالي إذا X في الحالة دي ليست cluster
+
+354
+00:41:11,910 --> 00:41:20,190
+point الحالة التالتة ان X تكون واقعة بين عنصرين من
+
+355
+00:41:20,190 --> 00:41:25,170
+عناصر ال setفباخد اللي هو المسافة الأصغر من
+
+356
+00:41:25,170 --> 00:41:29,310
+المسافتين هدول و هي تكون هادى و باخد نصها delta و
+
+357
+00:41:29,310 --> 00:41:32,890
+بكون delta neighborhood حواليها هذا ال delta
+
+358
+00:41:32,890 --> 00:41:36,510
+neighborhood بتخطعش مع المجموعه هاد المرة وبالتالي
+
+359
+00:41:36,510 --> 00:41:38,990
+حسب التعريف x ليست cluster point
+
+360
+00:41:43,100 --> 00:41:46,360
+ما احنا قلنا اذا كانت X تنتمي لأيه هي برهانة MA لا
+
+361
+00:41:46,360 --> 00:41:52,640
+تنتمي لأ تنتمي اه ليش قدماها بين تنتمي و لا تنتمي
+
+362
+00:41:52,640 --> 00:41:58,420
+ما هي ال X ما تنتميش لأيه فممكن تكون بين عنصرين من
+
+363
+00:41:58,420 --> 00:42:03,360
+عنصرهم هي تنتمي ل R ما تنتميش لأيه فممكن تكون
+
+364
+00:42:03,360 --> 00:42:09,640
+موجودة بين A2 و A3 صح او بين A1 و A2 او بين A3 او
+
+365
+00:42:09,640 --> 00:42:16,530
+A و هكذاأو ممكن تكون ال X على يمين ال AN أو حالة
+
+366
+00:42:16,530 --> 00:42:20,190
+تالتة X تكون على يسار ال A واحد وشوفنا في كل
+
+367
+00:42:20,190 --> 00:42:24,310
+الحالات هذه التلاتة أنه بقدر ألاقي delta
+
+368
+00:42:24,310 --> 00:42:27,990
+neighborhood حوالين ال X لا يتقاطع مع المجموعة A
+
+369
+00:42:27,990 --> 00:42:32,330
+بالمرة وبالتالي X ليست cluster point إذا الكلام
+
+370
+00:42:32,330 --> 00:42:37,110
+هذا واضح حاولوا تكتبوه بطريقة يعني منطقية okay
+
+371
+00:42:37,110 --> 00:42:37,810
+تمام؟
+
+372
+00:42:42,120 --> 00:42:46,900
+أنا متخيلة ال A عبارة عن set دائر مثلا أنا هيك
+
+373
+00:42:46,900 --> 00:42:50,340
+متخيلة و أنه مثلا ال cluster point هي عبارة عن
+
+374
+00:42:50,340 --> 00:42:54,820
+نقطة .. لأ ال A ماتتخيليش ال A عند ال set ال A هي
+
+375
+00:42:54,820 --> 00:43:01,140
+جزء من الأعداد الحقيقية subset من R و R خطلازم
+
+376
+00:43:01,140 --> 00:43:04,200
+يعني تتخيل الحاجات زمان على ال interior point و ال
+
+377
+00:43:04,200 --> 00:43:08,260
+boundary هذا في ال topology حاجة تانية هي زي ات
+
+378
+00:43:08,260 --> 00:43:11,140
+شبهها يعني ممكن نفهمها بهذا الطريقة بس ممكن اه
+
+379
+00:43:11,140 --> 00:43:13,860
+ممكن بس احنا هنا على ال real line يعني خليني احنا
+
+380
+00:43:13,860 --> 00:43:18,820
+نتقيد بال sets اللي موجودة على ال real line اما هو
+
+381
+00:43:18,820 --> 00:43:24,080
+طبعا في تعني من الكلام هذا في حاجات اعم و فرغات
+
+382
+00:43:24,080 --> 00:43:28,550
+اعم من ال ..ال real number اللي هو ال topological
+
+383
+00:43:28,550 --> 00:43:36,030
+spaces و هذا موضوع طبعا متشعب و بده يعني تدرس ال
+
+384
+00:43:36,030 --> 00:43:40,530
+topology عشان تفهم كل شيء okay فى اي اسئلة تانى؟
+
+385
+00:43:40,530 --> 00:43:44,430
+okay لنكتفي بهذا القدر و ان شاء الله بنكمل
+
+386
+00:43:44,430 --> 00:43:52,670
+المحاضرة الجاية الموضوع و بنخش بتعريف ال limit لل
+
+387
+00:43:52,670 --> 00:43:53,110
+functions
+
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/Ejs4dHLsIvo_raw.json b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/Ejs4dHLsIvo_raw.json
new file mode 100644
index 0000000000000000000000000000000000000000..88f61c80e96e2a58665bb57b2e55878dd81c6920
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/Ejs4dHLsIvo_raw.json
@@ -0,0 +1 @@
+{"segments": [{"id": 1, "seek": 4910, "start": 20.92, "end": 49.1, "text": "بسم الله الرحمن الرحيم هنبدأ في المحاضرة هذه chapter جديد وهو chapter أربعة في الكتاب المقرر عنوان ال chapter limits of functions و هنبدأ أول section في ال chapter هذا و برضه عنوان ال section الأول هو نفس عنوان ال chapter limits of functions فقبل ما نعرف limit of a function", "tokens": [3555, 38251, 21984, 34892, 5016, 27842, 34892, 5016, 32640, 8032, 1863, 44510, 10721, 8978, 9673, 5016, 46958, 25720, 29538, 7187, 10874, 16254, 3215, 37037, 2407, 7187, 5551, 25513, 27884, 8978, 33251, 2655, 16758, 9673, 4587, 2288, 2288, 18871, 2407, 7649, 2423, 7187, 10406, 295, 6828, 4032, 8032, 1863, 44510, 10721, 5551, 12610, 3541, 8978, 2423, 7187, 23758, 4032, 4724, 43042, 3224, 18871, 2407, 7649, 2423, 3541, 16247, 12610, 31439, 8717, 36178, 18871, 2407, 7649, 2423, 7187, 10406, 295, 6828, 6156, 4587, 36150, 19446, 8717, 3615, 28480, 4948, 295, 257, 2445], "avg_logprob": -0.1200206078343339, "compression_ratio": 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عن ال c يعني بمعنى أخر بقدر ألاقي في التقاطع هذا نقطة x يعني التقاطع هذا لا يساوي five okay كمان مرة", "tokens": [2670, 269, 7251, 2655, 4587, 41193, 3615, 20449, 9673, 7435, 2304, 2407, 27884, 257, 8978, 8717, 47432, 3660, 36764, 24401, 3660, 15844, 16247, 4587, 1211, 2031, 3714, 46456, 46538, 3660, 18871, 2423, 269, 37495, 22653, 4724, 2304, 3615, 1863, 7578, 5551, 34740, 4724, 28543, 2288, 5551, 15040, 38436, 8978, 16712, 4587, 41193, 3615, 23758, 8717, 47432, 3660, 2031, 37495, 22653, 16712, 4587, 41193, 3615, 23758, 20193, 7251, 3794, 995, 45865, 1732, 1392, 9122, 2304, 7649, 3714, 25720], "avg_logprob": -0.10446715077910668, "compression_ratio": 1.6402439024390243, "no_speech_prob": 2.384185791015625e-07, "words": [{"start": 128.94, "end": 129.22, "word": "of", "probability": 0.299560546875}, {"start": 129.22, "end": 129.76, "word": " c", "probability": 0.386474609375}, {"start": 129.76, "end": 131.3, "word": " يتقاطع", "probability": 0.9873046875}, {"start": 131.3, "end": 131.5, 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0.62744140625}, {"start": 138.46, "end": 139.5, "word": " يعني", "probability": 0.899658203125}, {"start": 139.5, "end": 140.06, "word": " بمعنى", "probability": 0.990625}, {"start": 140.06, "end": 140.56, "word": " أخر", "probability": 0.7344970703125}, {"start": 140.56, "end": 141.62, "word": " بقدر", "probability": 0.9576822916666666}, {"start": 141.62, "end": 142.14, "word": " ألاقي", "probability": 0.8382161458333334}, {"start": 142.14, "end": 143.16, "word": " في", "probability": 0.97314453125}, {"start": 143.16, "end": 143.96, "word": " التقاطع", "probability": 0.990966796875}, {"start": 143.96, "end": 144.4, "word": " هذا", "probability": 0.94140625}, {"start": 144.4, "end": 145.44, "word": " نقطة", "probability": 0.9817708333333334}, {"start": 145.44, "end": 145.72, "word": " x", "probability": 0.9326171875}, {"start": 145.72, "end": 145.92, "word": " يعني", "probability": 0.8525390625}, {"start": 145.92, "end": 146.56, "word": " التقاطع", "probability": 0.992919921875}, {"start": 146.56, "end": 146.9, "word": " هذا", "probability": 0.95556640625}, {"start": 146.9, "end": 147.14, "word": " لا", "probability": 0.80615234375}, {"start": 147.14, "end": 147.98, "word": " يساوي", "probability": 0.9227294921875}, {"start": 147.98, "end": 148.76, "word": " five", "probability": 0.72705078125}, {"start": 148.76, "end": 152.44, "word": " okay", "probability": 0.74755859375}, {"start": 152.44, "end": 152.86, "word": " كمان", "probability": 0.951171875}, {"start": 152.86, "end": 153.2, "word": " مرة", "probability": 0.98974609375}], "temperature": 1.0}, {"id": 6, "seek": 17435, "start": 154.79, "end": 174.35, "text": "النقطة C هذه بتكون cluster point للمجموعة A إذا أي delta neighborhood للنقطة C بيتخاطع مع المجموعة A في نقطة X مختلفة عن الـC بس لازم كل delta neighborhood لـC", "tokens": [6027, 1863, 47432, 3660, 383, 29538, 39894, 30544, 13630, 935, 5296, 19528, 7435, 2304, 2407, 27884, 316, 11933, 15730, 36632, 8289, 7630, 24976, 1863, 47432, 3660, 383, 4724, 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"end": 167.19, "word": " X", "probability": 0.94287109375}, {"start": 167.19, "end": 167.85, "word": " مختلفة", "probability": 0.9730224609375}, {"start": 167.85, "end": 167.99, "word": " عن", "probability": 0.9765625}, {"start": 167.99, "end": 168.35, "word": " الـC", "probability": 0.6227213541666666}, {"start": 168.35, "end": 169.07, "word": " بس", "probability": 0.705810546875}, {"start": 169.07, "end": 172.69, "word": " لازم", "probability": 0.9733072916666666}, {"start": 172.69, "end": 173.01, "word": " كل", "probability": 0.98828125}, {"start": 173.01, "end": 173.39, "word": " delta", "probability": 0.8486328125}, {"start": 173.39, "end": 173.81, "word": " neighborhood", "probability": 0.90380859375}, {"start": 173.81, "end": 174.35, "word": " لـC", "probability": 0.7823893229166666}], "temperature": 1.0}, {"id": 7, "seek": 19498, "start": 175.4, "end": 194.98, "text": "يتقاطع مع المجموعة A في نقطة X مختلفة عن الـC طب عشان اثبت ان الـC ليست cluster point بنفي الشرط هذا يكفي ان اقول there exist بدل for every delta او every delta neighborhood", "tokens": [1829, 2655, 4587, 41193, 3615, 20449, 9673, 7435, 2304, 2407, 27884, 316, 8978, 8717, 47432, 3660, 1783, 3714, 46456, 46538, 3660, 18871, 2423, 39184, 34, 23032, 3555, 6225, 8592, 7649, 1975, 12984, 3555, 2655, 16472, 2423, 39184, 34, 32239, 14851, 13630, 935, 44945, 41185, 25124, 2288, 9566, 23758, 7251, 4117, 41185, 16472, 1975, 39648, 456, 2514, 47525, 1211, 337, 633, 8289, 1975, 2407, 633, 8289, 7630], "avg_logprob": -0.20102611851336352, "compression_ratio": 1.3846153846153846, "no_speech_prob": 0.0, "words": [{"start": 175.39999999999998, "end": 176.32, "word": "يتقاطع", "probability": 0.818408203125}, {"start": 176.32, "end": 176.64, "word": " مع", "probability": 0.87109375}, {"start": 176.64, "end": 177.48, "word": " المجموعة", "probability": 0.90771484375}, {"start": 177.48, "end": 177.76, "word": " A", "probability": 0.476318359375}, {"start": 177.76, "end": 178.06, "word": " في", "probability": 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{"start": 183.72, "end": 184.42, "word": " point", "probability": 0.98388671875}, {"start": 184.42, "end": 185.36, "word": " بنفي", "probability": 0.74365234375}, {"start": 185.36, "end": 185.8, "word": " الشرط", "probability": 0.93994140625}, {"start": 185.8, "end": 186.24, "word": " هذا", "probability": 0.92919921875}, {"start": 186.24, "end": 188.06, "word": " يكفي", "probability": 0.8619791666666666}, {"start": 188.06, "end": 188.22, "word": " ان", "probability": 0.5029296875}, {"start": 188.22, "end": 188.7, "word": " اقول", "probability": 0.956298828125}, {"start": 188.7, "end": 189.0, "word": " there", "probability": 0.7333984375}, {"start": 189.0, "end": 189.62, "word": " exist", "probability": 0.642578125}, {"start": 189.62, "end": 190.76, "word": " بدل", "probability": 0.95068359375}, {"start": 190.76, "end": 191.22, "word": " for", "probability": 0.93115234375}, {"start": 191.22, "end": 191.72, "word": " every", "probability": 0.8115234375}, {"start": 191.72, "end": 192.18, 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neighborhood", "probability": 0.8916015625}, {"start": 208.44, "end": 208.74, "word": " لا", "probability": 0.88037109375}, {"start": 208.74, "end": 209.6, "word": " يتقاطع", "probability": 0.95244140625}, {"start": 209.6, "end": 209.94, "word": " مع", "probability": 0.87939453125}, {"start": 209.94, "end": 210.54, "word": " اي", "probability": 0.561279296875}, {"start": 210.54, "end": 211.8, "word": " مشيول", "probability": 0.8426106770833334}, {"start": 211.8, "end": 212.16, "word": " منها", "probability": 0.97998046875}, {"start": 212.16, "end": 212.32, "word": " C", "probability": 0.830078125}, {"start": 212.32, "end": 212.84, "word": " بالمرة", "probability": 0.7766927083333334}, {"start": 212.84, "end": 215.58, "word": " ناخد", "probability": 0.9729817708333334}, {"start": 215.58, "end": 216.98, "word": " نظرية", "probability": 0.944580078125}, {"start": 216.98, "end": 217.6, "word": " الأول", "probability": 0.782958984375}, {"start": 217.6, "end": 218.72, "word": " ال", 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"probability": 0.98681640625}, {"start": 525.17, "end": 525.31, "word": " من", "probability": 0.99462890625}, {"start": 525.31, "end": 525.65, "word": " واحد", "probability": 0.990966796875}, {"start": 525.65, "end": 525.83, "word": " على", "probability": 0.921875}, {"start": 525.83, "end": 526.19, "word": " n", "probability": 0.92236328125}, {"start": 526.19, "end": 527.15, "word": " لكل", "probability": 0.94287109375}, {"start": 527.15, "end": 527.41, "word": " n", "probability": 0.94921875}, {"start": 527.41, "end": 527.63, "word": " هذا", "probability": 0.533203125}, {"start": 527.63, "end": 528.07, "word": " صحيح", "probability": 0.9962158203125}, {"start": 528.07, "end": 528.49, "word": " لكل", "probability": 0.9853515625}, {"start": 528.49, "end": 528.79, "word": " n", "probability": 0.951171875}], "temperature": 1.0}, {"id": 20, "seek": 55661, "start": 532.39, "end": 556.61, "text": "إذا هاي فيها sequence إذا هاي أثبتنا مايوجد sequence إذا am is a sequence in a وكل حدودها مختلفة عن ال c and by theorem اتنين اربعة الشهيرة", "tokens": [28814, 15730, 8032, 47302, 8978, 11296, 8310, 11933, 15730, 8032, 47302, 5551, 12984, 3555, 2655, 8315, 19446, 1829, 29245, 3215, 8310, 11933, 15730, 669, 307, 257, 8310, 294, 257, 4032, 28820, 11331, 3215, 23328, 11296, 3714, 46456, 46538, 3660, 18871, 2423, 269, 293, 538, 20904, 1975, 2655, 1863, 9957, 1975, 25513, 27884, 25124, 3224, 48923], "avg_logprob": -0.33677453973463606, "compression_ratio": 1.4206896551724137, "no_speech_prob": 0.0, "words": [{"start": 532.39, "end": 532.75, "word": "إذا", "probability": 0.5218505859375}, {"start": 532.75, "end": 532.95, "word": " هاي", "probability": 0.5963134765625}, {"start": 532.95, "end": 533.19, "word": " فيها", "probability": 0.623779296875}, {"start": 533.19, "end": 533.61, "word": " sequence", "probability": 0.2027587890625}, {"start": 533.61, "end": 535.51, "word": " إذا", "probability": 0.6297607421875}, {"start": 535.51, "end": 535.85, "word": " هاي", "probability": 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واحد عدن موجب لأن بيطلع limit ال sequence A N as N tends to infinity بساوي S C إذن هين أثبتنا إنه لو كانت C cluster point فأثبتنا إنه يوجد sequence A N في المجموعة A", "tokens": [10721, 8315, 18871, 16254, 2423, 39184, 8236, 2158, 8032, 8848, 1829, 5551, 9381, 17082, 2288, 9154, 36764, 24401, 15844, 426, 5296, 28820, 426, 4948, 36764, 24401, 15844, 426, 20666, 3794, 995, 45865, 8608, 5172, 2288, 16490, 3215, 383, 20666, 3794, 995, 45865, 36764, 24401, 6225, 3215, 1863, 3714, 29245, 3555, 5296, 33456, 4724, 1829, 9566, 1211, 3615, 4948, 2423, 8310, 316, 426, 382, 426, 12258, 281, 13202, 4724, 3794, 995, 45865, 318, 383, 11933, 8848, 1863, 8032, 9957, 5551, 12984, 3555, 2655, 8315, 36145, 3224, 45164, 25961, 2655, 383, 13630, 935, 6156, 10721, 12984, 3555, 2655, 8315, 36145, 3224, 7251, 29245, 3215, 8310, 316, 426, 8978, 9673, 7435, 2304, 2407, 27884, 316], "avg_logprob": -0.2712942393480149, "compression_ratio": 1.6182572614107884, "no_speech_prob": 0.0, "words": [{"start": 558.56, 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"end": 576.1, "word": " بساوي", "probability": 0.8883056640625}, {"start": 576.1, "end": 576.44, "word": " S", "probability": 0.2413330078125}, {"start": 576.44, "end": 577.08, "word": " C", "probability": 0.83642578125}, {"start": 577.08, "end": 577.44, "word": " إذن", "probability": 0.6790364583333334}, {"start": 577.44, "end": 577.64, "word": " هين", "probability": 0.4549560546875}, {"start": 577.64, "end": 578.24, "word": " أثبتنا", "probability": 0.9724609375}, {"start": 578.24, "end": 579.12, "word": " إنه", "probability": 0.57568359375}, {"start": 579.12, "end": 579.3, "word": " لو", "probability": 0.98876953125}, {"start": 579.3, "end": 579.78, "word": " كانت", "probability": 0.947021484375}, {"start": 579.78, "end": 580.04, "word": " C", "probability": 0.89501953125}, {"start": 580.04, "end": 580.58, "word": " cluster", "probability": 0.89208984375}, {"start": 580.58, "end": 581.16, "word": " point", "probability": 0.95947265625}, {"start": 581.16, "end": 583.1, "word": " فأثبتنا", "probability": 0.9703776041666666}, {"start": 583.1, "end": 583.36, "word": " إنه", "probability": 0.6650390625}, {"start": 583.36, "end": 583.7, "word": " يوجد", "probability": 0.9912109375}, {"start": 583.7, "end": 584.1, "word": " sequence", "probability": 0.970703125}, {"start": 584.1, "end": 584.58, "word": " A", "probability": 0.8828125}, {"start": 584.58, "end": 584.92, "word": " N", "probability": 0.9375}, {"start": 584.92, "end": 585.8, "word": " في", "probability": 0.96435546875}, {"start": 585.8, "end": 586.58, "word": " المجموعة", "probability": 0.9251953125}, {"start": 586.58, "end": 586.86, "word": " A", "probability": 0.94287109375}], "temperature": 1.0}, {"id": 22, "seek": 60491, "start": 587.84, "end": 604.92, "text": "وكل عناصرها مختلفة عن الـ C ونهايتها بساوي الـ C إذا هذا بثبت جزء ال only if part الآن لثبت العكس لإثبات العكس", "tokens": [2407, 28820, 18871, 33546, 2288, 11296, 3714, 46456, 46538, 3660, 18871, 2423, 39184, 383, 4032, 1863, 11296, 36081, 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"word": " to", "probability": 0.767578125}, {"start": 641.25, "end": 641.63, "word": " show", "probability": 0.9599609375}, {"start": 641.63, "end": 642.19, "word": " c", "probability": 0.404541015625}, {"start": 642.19, "end": 642.81, "word": " is", "probability": 0.91064453125}, {"start": 642.81, "end": 643.01, "word": " a", "probability": 0.890625}, {"start": 643.01, "end": 643.41, "word": " cluster", "probability": 0.923828125}, {"start": 643.41, "end": 644.77, "word": " point", "probability": 0.9853515625}, {"start": 644.77, "end": 648.89, "word": " of", "probability": 0.84375}, {"start": 648.89, "end": 649.21, "word": " a", "probability": 0.6962890625}, {"start": 649.21, "end": 649.67, "word": " let", "probability": 0.69580078125}], "temperature": 1.0}, {"id": 24, "seek": 68012, "start": 651.86, "end": 680.12, "text": "delta أكبر من السفر is given since by star من الشرط star لدي limit لان بساوي c و delta أكبر من السفر is given إذا من تعريف delta capital N", "tokens": [18105, 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دحيث أنه لكل n أكبر من أو سوى capital N هذا بيقدي أنه absolute a n minus c أصغر من delta", "tokens": [1211, 1211, 39184, 4948, 5296, 10721, 1829, 8289, 34051, 11933, 3555, 3794, 1211, 11536, 6225, 3215, 3215, 3714, 29245, 3660, 456, 2514, 297, 7251, 34268, 2304, 3215, 15844, 8289, 3303, 1230, 11778, 5016, 1829, 12984, 14739, 3224, 5296, 28820, 297, 5551, 4117, 26890, 9154, 34051, 8608, 2407, 7578, 4238, 426, 23758, 4724, 1829, 4587, 16254, 14739, 3224, 8236, 257, 297, 3175, 269, 5551, 9381, 17082, 2288, 9154, 8289], "avg_logprob": -0.3043478226316148, "compression_ratio": 1.3204419889502763, "no_speech_prob": 0.0, "words": [{"start": 681.28, "end": 681.78, "word": "للـ", "probability": 0.584228515625}, {"start": 681.78, "end": 682.08, "word": " limit", "probability": 0.81689453125}, {"start": 682.08, "end": 683.66, "word": " لأي", "probability": 0.9093424479166666}, {"start": 683.66, "end": 684.1, "word": " delta", "probability": 0.440673828125}, {"start": 684.1, "end": 684.32, "word": 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"word": "إذاً", "probability": 0.54833984375}, {"start": 711.31, "end": 711.79, "word": " هذا", "probability": 0.87109375}, {"start": 711.79, "end": 712.81, "word": " بيقدّي", "probability": 0.74150390625}, {"start": 712.81, "end": 714.67, "word": " إنه", "probability": 0.66162109375}, {"start": 714.67, "end": 717.19, "word": " هيعندي", "probability": 0.7392578125}, {"start": 717.19, "end": 717.87, "word": " am", "probability": 0.1650390625}], "temperature": 1.0}, {"id": 27, "seek": 74701, "start": 722.99, "end": 747.01, "text": "طبعا هدف قلبي أن a n أصغر من c زائد delta أكبر من c negative delta يعني a n تنتمي إلى v delta of c وتنتمي طبعا من ال condition star تنتمي إلى a difference c", "tokens": [9566, 3555, 3615, 995, 8032, 3215, 5172, 12174, 1211, 21292, 14739, 257, 297, 5551, 9381, 17082, 2288, 9154, 269, 30767, 16373, 3215, 8289, 5551, 4117, 26890, 9154, 269, 3671, 8289, 37495, 22653, 257, 297, 6055, 29399, 2304, 1829, 30731, 371, 8289, 295, 269, 34683, 29399, 2304, 1829, 23032, 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difference", "probability": 0.8193359375}, {"start": 746.43, "end": 747.01, "word": " c", "probability": 0.7841796875}], "temperature": 1.0}, {"id": 28, "seek": 77640, "start": 749.18, "end": 776.4, "text": "و هذا صحيح لكل n أكبر من أو ساوي capital N إذا هاي كل delta neighborhood ل C بيتقاطع مع A في عدد لانهائي من النقاط المختلفة عن ال C وبالتالي الشرط تبع ال definition هيتحقق", "tokens": [2407, 23758, 20328, 5016, 1829, 5016, 5296, 28820, 297, 5551, 4117, 26890, 9154, 34051, 8608, 995, 45865, 4238, 426, 11933, 15730, 8032, 47302, 28242, 8289, 7630, 5296, 383, 4724, 36081, 4587, 41193, 3615, 20449, 316, 8978, 6225, 3215, 3215, 5296, 7649, 3224, 16373, 1829, 9154, 28239, 4587, 41193, 9673, 46456, 46538, 3660, 18871, 2423, 383, 46599, 6027, 2655, 6027, 1829, 25124, 2288, 9566, 6055, 3555, 3615, 2423, 7123, 39896, 2655, 5016, 4587, 4587], "avg_logprob": -0.19172297659757975, "compression_ratio": 1.3958333333333333, "no_speech_prob": 0.0, "words": [{"start": 749.18, "end": 749.38, "word": 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point of the set A و هدا بكمل ال F part and therefore completes the proof of the theorem okay تمام؟", "tokens": [2407, 550, 538, 7123, 4386, 538, 7123, 383, 307, 257, 13630, 935, 295, 264, 992, 316, 4032, 8032, 28259, 4724, 24793, 1211, 2423, 479, 644, 293, 4412, 36362, 264, 8177, 295, 264, 20904, 1392, 46811, 10943, 22807], "avg_logprob": -0.3301809328167062, "compression_ratio": 1.2352941176470589, "no_speech_prob": 0.0, "words": [{"start": 777.18, "end": 777.42, "word": "و", "probability": 0.5615234375}, {"start": 777.42, "end": 777.72, "word": " then", "probability": 0.052337646484375}, {"start": 777.72, "end": 778.68, "word": " by", "probability": 0.90966796875}, {"start": 778.68, "end": 779.36, "word": " definition", "probability": 0.90185546875}, {"start": 779.36, "end": 779.74, "word": " ..", "probability": 0.353759765625}, {"start": 779.74, "end": 781.98, "word": " by", "probability": 0.8623046875}, {"start": 781.98, "end": 782.7, "word": " definition", "probability": 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0.8250325520833334}], "temperature": 1.0}, {"id": 30, "seek": 80754, "start": 805.14, "end": 807.54, "text": "Fine خلّينا ناخد بعض الأمثلة", "tokens": [41415, 16490, 1211, 11703, 9957, 995, 8717, 47283, 3215, 45030, 11242, 16247, 2304, 12984, 37977], "avg_logprob": -0.18225097563117743, "compression_ratio": 0.8727272727272727, "no_speech_prob": 0.0, "words": [{"start": 805.14, "end": 805.44, "word": "Fine", "probability": 0.320068359375}, {"start": 805.44, "end": 805.84, "word": " خلّينا", "probability": 0.755517578125}, {"start": 805.84, "end": 806.42, "word": " ناخد", "probability": 0.9947916666666666}, {"start": 806.42, "end": 806.9, "word": " بعض", "probability": 0.98486328125}, {"start": 806.9, "end": 807.54, "word": " الأمثلة", "probability": 0.9617919921875}], "temperature": 1.0}, {"id": 31, "seek": 85677, "start": 829.07, "end": 856.77, "text": "تشير إلى أن كل X ينتمي إلى مقفل مقفل مقفل مقفل", "tokens": [2655, 8592, 13546, 30731, 14739, 28242, 1783, 7251, 29399, 2304, 1829, 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881.36, "word": " إن", "probability": 0.285400390625}, {"start": 881.36, "end": 881.76, "word": " كل", "probability": 0.92431640625}, {"start": 881.76, "end": 882.1, "word": " X", "probability": 0.576171875}, {"start": 882.1, "end": 882.66, "word": " الفترة", "probability": 0.9265950520833334}, {"start": 882.66, "end": 883.18, "word": " المغلقة", "probability": 0.9755859375}, {"start": 883.18, "end": 883.42, "word": " من", "probability": 0.9833984375}, {"start": 883.42, "end": 883.72, "word": " سفر", "probability": 0.9803059895833334}, {"start": 883.72, "end": 883.88, "word": " إلى", "probability": 0.98046875}, {"start": 883.88, "end": 884.32, "word": " واحد", "probability": 0.992919921875}, {"start": 884.32, "end": 885.46, "word": " هي", "probability": 0.8876953125}, {"start": 885.46, "end": 886.0, "word": " cluster", "probability": 0.93798828125}, {"start": 886.0, "end": 886.48, "word": " point", "probability": 0.9736328125}, {"start": 886.48, "end": 887.14, "word": " للفترة", "probability": 0.9190673828125}, {"start": 887.14, "end": 888.12, "word": " المفتوحة", "probability": 0.994873046875}, {"start": 888.12, "end": 889.08, "word": " من", "probability": 0.9501953125}, {"start": 889.08, "end": 889.72, "word": " سفر", "probability": 0.9900716145833334}, {"start": 889.72, "end": 889.94, "word": " إلى", "probability": 0.98095703125}, {"start": 889.94, "end": 890.48, "word": " واحد", "probability": 0.992431640625}], "temperature": 1.0}, {"id": 33, "seek": 92662, "start": 901.36, "end": 926.62, "text": "ففي الأول بدي أثبت أنه كل نقطة داخل الفترة المغلقة هي cluster point للمجموع عادي و بعدين في المرحلة التانية حتة من نقاط الأطراف اللي هي 01 تطلع أيضا cluster point لست A1 فنشوف مع بعض ان ال claim 1", "tokens": [5172, 41185, 16247, 12610, 4724, 16254, 5551, 12984, 3555, 2655, 14739, 3224, 28242, 8717, 47432, 3660, 11778, 47283, 1211, 27188, 2655, 25720, 9673, 17082, 1211, 28671, 39896, 13630, 935, 5296, 19528, 7435, 2304, 45367, 6225, 995, 16254, 4032, 39182, 9957, 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"probability": 0.370849609375}, {"start": 921.46, "end": 922.6, "word": " A1", "probability": 0.823974609375}, {"start": 922.6, "end": 924.04, "word": " فنشوف", "probability": 0.9556884765625}, {"start": 924.04, "end": 924.22, "word": " مع", "probability": 0.99560546875}, {"start": 924.22, "end": 924.7, "word": " بعض", "probability": 0.987060546875}, {"start": 924.7, "end": 925.62, "word": " ان", "probability": 0.312255859375}, {"start": 925.62, "end": 925.84, "word": " ال", "probability": 0.87744140625}, {"start": 925.84, "end": 926.18, "word": " claim", "probability": 0.92041015625}, {"start": 926.18, "end": 926.62, "word": " 1", "probability": 0.57861328125}], "temperature": 1.0}, {"id": 34, "seek": 95957, "start": 934.01, "end": 959.57, "text": "بنثبت ان كل X ينتمي للفترة المفتوحة is a cluster point of set A1 اللي هي الفترة المفتوحة نفسها لبرهان ذلك to see this", "tokens": [3555, 1863, 12984, 3555, 2655, 16472, 28242, 1783, 7251, 29399, 2304, 1829, 24976, 5172, 2655, 25720, 9673, 5172, 2655, 2407, 5016, 3660, 307, 257, 13630, 935, 295, 992, 316, 16, 13672, 1829, 39896, 27188, 2655, 25720, 9673, 5172, 2655, 2407, 5016, 3660, 8717, 36178, 11296, 5296, 26890, 3224, 7649, 29910, 23275, 281, 536, 341], "avg_logprob": -0.22471591505137356, "compression_ratio": 1.335820895522388, "no_speech_prob": 0.0, "words": [{"start": 934.01, "end": 934.79, "word": "بنثبت", "probability": 0.76279296875}, {"start": 934.79, "end": 934.99, "word": " ان", "probability": 0.61181640625}, {"start": 934.99, "end": 935.65, "word": " كل", "probability": 0.6220703125}, {"start": 935.65, "end": 938.63, "word": " X", "probability": 0.377197265625}, {"start": 938.63, "end": 939.75, "word": " ينتمي", "probability": 0.87841796875}, {"start": 939.75, "end": 941.45, "word": " للفترة", "probability": 0.8533935546875}, {"start": 941.45, "end": 942.59, "word": " المفتوحة", "probability": 0.9762369791666666}, {"start": 942.59, "end": 944.45, "word": " is", "probability": 0.180908203125}, {"start": 944.45, "end": 944.77, "word": " a", "probability": 0.888671875}, {"start": 944.77, "end": 945.19, "word": " cluster", "probability": 0.92138671875}, {"start": 945.19, "end": 945.95, "word": " point", "probability": 0.97314453125}, {"start": 945.95, "end": 950.41, "word": " of", "probability": 0.55517578125}, {"start": 950.41, "end": 951.25, "word": " set", "probability": 0.366943359375}, {"start": 951.25, "end": 952.13, "word": " A1", "probability": 0.91259765625}, {"start": 952.13, "end": 953.45, "word": " اللي", "probability": 0.6453857421875}, {"start": 953.45, "end": 953.57, "word": " هي", "probability": 0.84423828125}, {"start": 953.57, "end": 953.97, "word": " الفترة", "probability": 0.9772135416666666}, {"start": 953.97, "end": 954.51, "word": " المفتوحة", "probability": 0.994384765625}, {"start": 954.51, "end": 956.01, "word": " نفسها", "probability": 0.9816080729166666}, {"start": 956.01, "end": 957.95, "word": " لبرهان", "probability": 0.75640869140625}, {"start": 957.95, "end": 958.49, "word": " ذلك", "probability": 0.993896484375}, {"start": 958.49, "end": 958.73, "word": " to", "probability": 0.53515625}, {"start": 958.73, "end": 959.07, "word": " see", "probability": 0.92822265625}, {"start": 959.07, "end": 959.57, "word": " this", "probability": 0.9619140625}], "temperature": 1.0}, {"id": 35, "seek": 98767, "start": 962.03, "end": 987.67, "text": "البرهن ذلك fix ينتمي للفترة المفتوحة ونثبت ان cluster point للمجموعة a1 اذا fix x and let delta أكبر من السفر be given", "tokens": [6027, 26890, 3224, 1863, 29910, 23275, 3191, 7251, 29399, 2304, 1829, 24976, 5172, 2655, 25720, 9673, 5172, 2655, 2407, 5016, 3660, 4032, 1863, 12984, 3555, 2655, 16472, 13630, 935, 5296, 19528, 7435, 2304, 2407, 27884, 257, 16, 1975, 15730, 3191, 2031, 293, 718, 8289, 5551, 4117, 26890, 9154, 21136, 5172, 2288, 312, 2212], "avg_logprob": -0.3284143524037467, "compression_ratio": 1.2152777777777777, "no_speech_prob": 0.0, "words": [{"start": 962.03, "end": 962.73, "word": 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neighborhood للنقطة X بيتقاطع مع المجموعة A1 في نقطة مختلفة عن X وبالتالي الـ X هتطلع cluster point حسب التعريف طيب نحن لدينا اتصالين", "tokens": [1863, 44510, 10721, 20666, 39184, 8289, 5551, 4117, 26890, 9154, 1958, 4032, 1863, 12984, 3555, 2655, 14739, 28242, 8289, 7630, 24976, 1863, 47432, 3660, 1783, 4724, 36081, 4587, 41193, 3615, 20449, 9673, 7435, 2304, 2407, 27884, 316, 16, 8978, 8717, 47432, 3660, 3714, 46456, 46538, 3660, 18871, 1783, 46599, 6027, 2655, 6027, 1829, 2423, 39184, 1783, 8032, 2655, 9566, 1211, 3615, 13630, 935, 11331, 35457, 16712, 3615, 16572, 5172, 23032, 1829, 3555, 8717, 5016, 1863, 5296, 16254, 8315, 1975, 2655, 9381, 6027, 9957], "avg_logprob": -0.26413691930827643, "compression_ratio": 1.4120603015075377, "no_speech_prob": 5.960464477539063e-08, "words": [{"start": 989.36, "end": 989.94, "word": "نبدأ", "probability": 0.89501953125}, {"start": 989.94, "end": 990.06, "word": " بالـ", "probability": 0.490966796875}, {"start": 990.06, "end": 990.4, "word": 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0.90869140625}, {"start": 1001.22, "end": 1001.92, "word": " المجموعة", "probability": 0.9427734375}, {"start": 1001.92, "end": 1002.62, "word": " A1", "probability": 0.759765625}, {"start": 1002.62, "end": 1002.82, "word": " في", "probability": 0.923828125}, {"start": 1002.82, "end": 1003.18, "word": " نقطة", "probability": 0.9861653645833334}, {"start": 1003.18, "end": 1003.7, "word": " مختلفة", "probability": 0.9881591796875}, {"start": 1003.7, "end": 1003.86, "word": " عن", "probability": 0.955078125}, {"start": 1003.86, "end": 1004.18, "word": " X", "probability": 0.95751953125}, {"start": 1004.18, "end": 1006.18, "word": " وبالتالي", "probability": 0.91806640625}, {"start": 1006.18, "end": 1007.28, "word": " الـ", "probability": 0.54248046875}, {"start": 1007.28, "end": 1007.46, "word": " X", "probability": 0.92578125}, {"start": 1007.46, "end": 1008.12, "word": " هتطلع", "probability": 0.951171875}, {"start": 1008.12, "end": 1009.4, "word": " cluster", "probability": 0.85595703125}, {"start": 1009.4, "end": 1009.94, "word": " point", "probability": 0.96044921875}, {"start": 1009.94, "end": 1010.3, "word": " حسب", "probability": 0.939453125}, {"start": 1010.3, "end": 1010.96, "word": " التعريف", "probability": 0.990234375}, {"start": 1010.96, "end": 1012.1, "word": " طيب", "probability": 0.8023274739583334}, {"start": 1012.1, "end": 1015.96, "word": " نحن", "probability": 0.70703125}, {"start": 1015.96, "end": 1016.18, "word": " لدينا", "probability": 0.8095703125}, {"start": 1016.18, "end": 1016.92, "word": " اتصالين", "probability": 0.5939453125}], "temperature": 1.0}, {"id": 37, "seek": 105005, "start": 1021.27, "end": 1050.05, "text": "two cases حالتين الحالة الأولى ان ال delta هذه اللى انا اخترتها العشوائية ال delta اللى انا اخترتها ممكن تكون اصغر من .. طبعا موجبة هي ممكن تكون اصغر من واحد ف in this case in this case في هذه الحالة", "tokens": [20534, 3331, 11331, 6027, 2655, 9957, 21542, 6027, 3660, 16247, 12610, 7578, 16472, 2423, 8289, 29538, 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"probability": 0.982421875}, {"start": 1034.67, "end": 1035.97, "word": " ممكن", "probability": 0.882568359375}, {"start": 1035.97, "end": 1036.51, "word": " تكون", "probability": 0.992431640625}, {"start": 1036.51, "end": 1037.05, "word": " اصغر", "probability": 0.92578125}, {"start": 1037.05, "end": 1037.19, "word": " من", "probability": 0.91796875}, {"start": 1037.19, "end": 1037.31, "word": " ..", "probability": 0.1962890625}, {"start": 1037.31, "end": 1037.67, "word": " طبعا", "probability": 0.94921875}, {"start": 1037.67, "end": 1038.15, "word": " موجبة", "probability": 0.72900390625}, {"start": 1038.15, "end": 1038.49, "word": " هي", "probability": 0.7490234375}, {"start": 1038.49, "end": 1039.59, "word": " ممكن", "probability": 0.968017578125}, {"start": 1039.59, "end": 1040.09, "word": " تكون", "probability": 0.992919921875}, {"start": 1040.09, "end": 1040.65, "word": " اصغر", "probability": 0.978271484375}, {"start": 1040.65, "end": 1040.83, "word": " من", "probability": 0.99755859375}, {"start": 1040.83, "end": 1041.27, "word": " واحد", "probability": 0.976806640625}, {"start": 1041.27, "end": 1042.91, "word": " ف", "probability": 0.83203125}, {"start": 1042.91, "end": 1043.23, "word": " in", "probability": 0.9072265625}, {"start": 1043.23, "end": 1043.59, "word": " this", "probability": 0.9716796875}, {"start": 1043.59, "end": 1044.15, "word": " case", "probability": 0.94677734375}, {"start": 1044.15, "end": 1047.93, "word": " in", "probability": 0.53759765625}, {"start": 1047.93, "end": 1048.29, "word": " this", "probability": 0.9697265625}, {"start": 1048.29, "end": 1048.79, "word": " case", "probability": 0.9482421875}, {"start": 1048.79, "end": 1049.07, "word": " في", "probability": 0.70068359375}, {"start": 1049.07, "end": 1049.43, "word": " هذه", "probability": 0.9169921875}, {"start": 1049.43, "end": 1050.05, "word": " الحالة", "probability": 0.9947916666666666}], "temperature": 1.0}, {"id": 38, "seek": 108054, "start": 1055.0, "end": 1080.54, "text": "لدينا in this case الـ delta neighborhood ل X اللي هو X minus Delta X موجة Delta", "tokens": [1211, 16254, 8315, 294, 341, 1389, 2423, 39184, 8289, 7630, 5296, 1783, 13672, 1829, 31439, 1783, 3175, 18183, 1783, 3714, 29245, 3660, 18183], "avg_logprob": -0.3136393204331398, "compression_ratio": 1.064516129032258, "no_speech_prob": 0.0, "words": [{"start": 1055.0, "end": 1056.2, "word": "لدينا", "probability": 0.7679036458333334}, {"start": 1056.2, "end": 1057.4, "word": " in", "probability": 0.44970703125}, {"start": 1057.4, "end": 1069.44, "word": " this", "probability": 0.96875}, {"start": 1069.44, "end": 1069.98, "word": " case", "probability": 0.93408203125}, {"start": 1069.98, "end": 1071.76, "word": " الـ", "probability": 0.634521484375}, {"start": 1071.76, "end": 1072.5, "word": " delta", "probability": 0.6806640625}, {"start": 1072.5, "end": 1074.62, "word": " neighborhood", "probability": 0.716796875}, {"start": 1074.62, "end": 1075.2, "word": " ل", "probability": 0.8896484375}, {"start": 1075.2, "end": 1075.94, "word": " X", "probability": 0.321533203125}, {"start": 1075.94, "end": 1077.4, "word": " اللي", "probability": 0.814697265625}, {"start": 1077.4, "end": 1077.68, "word": " هو", "probability": 0.98193359375}, {"start": 1077.68, "end": 1078.08, "word": " X", "probability": 0.83935546875}, {"start": 1078.08, "end": 1078.68, "word": " minus", "probability": 0.62158203125}, {"start": 1078.68, "end": 1079.2, "word": " Delta", "probability": 0.419189453125}, {"start": 1079.2, "end": 1079.6, "word": " X", "probability": 0.93994140625}, {"start": 1079.6, "end": 1080.14, "word": " موجة", "probability": 0.7200520833333334}, {"start": 1080.14, "end": 1080.54, "word": " Delta", "probability": 0.74951171875}], "temperature": 1.0}, {"id": 39, "seek": 109753, "start": 1081.89, "end": 1097.53, "text": "بنلاحظ انه تقاطع مع المجموعة ايه اللي هي الفترة اللي مفتوحة من سفر لواحد بطلع واحد من الخيارات التالية", "tokens": [3555, 1863, 15040, 5016, 19913, 16472, 3224, 6055, 4587, 41193, 3615, 20449, 9673, 7435, 2304, 2407, 27884, 1975, 1829, 3224, 13672, 1829, 39896, 27188, 2655, 25720, 13672, 1829, 3714, 5172, 2655, 2407, 5016, 3660, 9154, 8608, 5172, 2288, 5296, 14407, 24401, 4724, 9566, 1211, 3615, 36764, 24401, 9154, 33962, 1829, 9640, 9307, 16712, 6027, 10632], "avg_logprob": -0.17954798814441478, "compression_ratio": 1.5897435897435896, "no_speech_prob": 0.0, "words": [{"start": 1081.89, "end": 1082.45, "word": "بنلاحظ", "probability": 0.816064453125}, {"start": 1082.45, "end": 1082.75, "word": " انه", "probability": 0.6614990234375}, {"start": 1082.75, "end": 1083.63, "word": " تقاطع", "probability": 0.8160400390625}, {"start": 1083.63, "end": 1084.39, "word": " مع", "probability": 0.93408203125}, {"start": 1084.39, "end": 1086.29, "word": " المجموعة", "probability": 0.8798828125}, {"start": 1086.29, "end": 1086.65, "word": " ايه", "probability": 0.7962239583333334}, {"start": 1086.65, "end": 1086.73, "word": " اللي", 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"word": " التالية", "probability": 0.9895833333333334}], "temperature": 1.0}, {"id": 40, "seek": 112292, "start": 1098.24, "end": 1122.92, "text": "اما الفترة المفتوحة x negative delta x positive delta او الفترة المفتوحة من صفر الى اكس positive delta او الفترة المفتوحة من x negative delta الى واحد او الفترة المفتوحة من صفر الى واحد", "tokens": [995, 15042, 2423, 5172, 2655, 25720, 9673, 5172, 2655, 2407, 5016, 3660, 2031, 3671, 8289, 2031, 3353, 8289, 1975, 2407, 27188, 2655, 25720, 9673, 5172, 2655, 2407, 5016, 3660, 9154, 20328, 5172, 2288, 2423, 7578, 1975, 4117, 3794, 3353, 8289, 1975, 2407, 2423, 5172, 2655, 25720, 9673, 5172, 2655, 2407, 5016, 3660, 9154, 2031, 3671, 8289, 2423, 7578, 36764, 24401, 1975, 2407, 27188, 2655, 25720, 9673, 5172, 2655, 2407, 5016, 3660, 9154, 20328, 5172, 2288, 2423, 7578, 36764, 24401], "avg_logprob": -0.20937499813735486, "compression_ratio": 2.737864077669903, "no_speech_prob": 0.0, "words": [{"start": 1098.24, "end": 1098.7, "word": "اما", "probability": 0.4854736328125}, {"start": 1098.7, "end": 1100.84, "word": " الفترة", "probability": 0.71380615234375}, {"start": 1100.84, "end": 1101.62, "word": " المفتوحة", "probability": 0.9755045572916666}, {"start": 1101.62, "end": 1103.08, "word": " x", "probability": 0.255859375}, {"start": 1103.08, "end": 1103.56, "word": " negative", "probability": 0.25048828125}, {"start": 1103.56, "end": 1104.08, "word": " delta", "probability": 0.54541015625}, {"start": 1104.08, "end": 1104.44, "word": " x", "probability": 0.595703125}, {"start": 1104.44, "end": 1105.08, "word": " positive", "probability": 0.92236328125}, {"start": 1105.08, "end": 1105.74, "word": " delta", "probability": 0.94873046875}, {"start": 1105.74, "end": 1107.5, "word": " او", "probability": 0.850341796875}, {"start": 1107.5, "end": 1108.12, "word": " الفترة", "probability": 0.9816080729166666}, {"start": 1108.12, "end": 1108.82, "word": " المفتوحة", "probability": 0.991455078125}, {"start": 1108.82, "end": 1109.02, "word": " من", "probability": 0.97119140625}, {"start": 1109.02, "end": 1109.48, "word": " صفر", "probability": 0.731201171875}, {"start": 1109.48, "end": 1109.78, "word": " الى", "probability": 0.676513671875}, {"start": 1109.78, "end": 1110.18, "word": " اكس", "probability": 0.7076822916666666}, {"start": 1110.18, "end": 1110.6, "word": " positive", "probability": 0.7412109375}, {"start": 1110.6, "end": 1111.12, "word": " delta", "probability": 0.68115234375}, {"start": 1111.12, "end": 1113.26, "word": " او", "probability": 0.86865234375}, {"start": 1113.26, "end": 1116.42, "word": " الفترة", "probability": 0.7864990234375}, {"start": 1116.42, "end": 1117.02, "word": " المفتوحة", "probability": 0.9923502604166666}, {"start": 1117.02, "end": 1117.16, "word": " من", "probability": 0.97900390625}, {"start": 1117.16, "end": 1117.46, "word": " x", "probability": 0.393798828125}, {"start": 1117.46, "end": 1117.8, "word": " negative", "probability": 0.9482421875}, {"start": 1117.8, 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بيكون", "probability": 0.7057291666666666}, {"start": 1326.37, "end": 1326.83, "word": " أثبتنا", "probability": 0.88330078125}, {"start": 1326.83, "end": 1326.99, "word": " أن", "probability": 0.5}, {"start": 1326.99, "end": 1327.29, "word": " كل", "probability": 0.88427734375}, {"start": 1327.29, "end": 1327.63, "word": " delta", "probability": 0.6552734375}, {"start": 1327.63, "end": 1328.01, "word": " neighborhood", "probability": 0.30859375}, {"start": 1328.01, "end": 1328.27, "word": " ل", "probability": 0.6982421875}, {"start": 1328.27, "end": 1328.55, "word": " x", "probability": 0.357666015625}, {"start": 1328.55, "end": 1329.77, "word": " بيتقاطع", "probability": 0.9154296875}, {"start": 1329.77, "end": 1329.97, "word": " مع", "probability": 0.9580078125}, {"start": 1329.97, "end": 1330.59, "word": " a1", "probability": 0.97802734375}, {"start": 1330.59, "end": 1330.83, "word": " في", "probability": 0.9775390625}, {"start": 1330.83, "end": 1331.47, "word": " نقطة", "probability": 0.9763997395833334}, {"start": 1331.47, "end": 1332.59, "word": " مختلفة", "probability": 0.984375}, {"start": 1332.59, "end": 1332.91, "word": " عن", "probability": 0.97509765625}, {"start": 1332.91, "end": 1333.37, "word": " x", "probability": 0.9462890625}], "temperature": 1.0}, {"id": 49, "seek": 135316, "start": 1343.0, "end": 1353.16, "text": "الحالة التانية case two ان ال delta هذه تكون اكبر من اوسع واحد", "tokens": [6027, 5016, 6027, 3660, 16712, 7649, 10632, 1389, 732, 16472, 2423, 8289, 29538, 6055, 30544, 1975, 4117, 26890, 9154, 1975, 2407, 3794, 3615, 36764, 24401], "avg_logprob": -0.4146634672696774, "compression_ratio": 1.1494252873563218, "no_speech_prob": 0.0, "words": [{"start": 1343.0, "end": 1344.08, "word": "الحالة", "probability": 0.62548828125}, {"start": 1344.08, "end": 1344.62, "word": " التانية", "probability": 0.94580078125}, {"start": 1344.62, "end": 1347.48, "word": " case", "probability": 0.60791015625}, {"start": 1347.48, "end": 1347.94, "word": " two", "probability": 0.450439453125}, {"start": 1347.94, "end": 1349.44, "word": " ان", "probability": 0.297119140625}, {"start": 1349.44, "end": 1349.66, "word": " ال", "probability": 0.53564453125}, {"start": 1349.66, "end": 1350.14, "word": " delta", "probability": 0.5810546875}, {"start": 1350.14, "end": 1350.62, "word": " هذه", "probability": 0.395263671875}, {"start": 1350.62, "end": 1351.2, "word": " تكون", "probability": 0.921630859375}, {"start": 1351.2, "end": 1351.84, "word": " اكبر", "probability": 0.8605143229166666}, {"start": 1351.84, "end": 1352.08, "word": " من", "probability": 0.9794921875}, {"start": 1352.08, "end": 1352.7, "word": " اوسع", "probability": 0.53326416015625}, {"start": 1352.7, "end": 1353.16, "word": " واحد", "probability": 0.9755859375}], "temperature": 1.0}, {"id": 50, "seek": 138207, "start": 1354.39, "end": 1382.07, "text": "برضه فى الحاله دى بنا نثبت انه كل delta neighborhood ال X بتقاطع مع A واحد فى نقطه مختلفه عن ال X نشوف مع بعض in this case in this case ال X negative او ال negative delta X موجب delta هذا اللى هو ال delta neighborhood ل X تقاطع", "tokens": [26890, 11242, 3224, 6156, 7578, 21542, 6027, 3224, 11778, 7578, 4724, 8315, 8717, 12984, 3555, 2655, 16472, 3224, 28242, 8289, 7630, 2423, 1783, 39894, 4587, 41193, 3615, 20449, 316, 36764, 24401, 6156, 7578, 8717, 47432, 3224, 3714, 46456, 46538, 3224, 18871, 2423, 1783, 8717, 8592, 38688, 20449, 45030, 11242, 294, 341, 1389, 294, 341, 1389, 2423, 1783, 3671, 1975, 2407, 2423, 3671, 8289, 1783, 3714, 29245, 3555, 8289, 23758, 13672, 7578, 31439, 2423, 8289, 7630, 5296, 1783, 6055, 4587, 41193, 3615], "avg_logprob": -0.24866615199461217, "compression_ratio": 1.6789473684210525, "no_speech_prob": 0.0, "words": [{"start": 1354.39, "end": 1354.87, "word": "برضه", "probability": 0.8065592447916666}, {"start": 1354.87, "end": 1354.97, "word": " فى", "probability": 0.52197265625}, {"start": 1354.97, "end": 1355.31, "word": " الحاله", "probability": 0.826171875}, 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1359.59, "end": 1360.15, "word": " واحد", "probability": 0.701904296875}, {"start": 1360.15, "end": 1360.93, "word": " فى", "probability": 0.97119140625}, {"start": 1360.93, "end": 1361.17, "word": " نقطه", "probability": 0.88671875}, {"start": 1361.17, "end": 1361.75, "word": " مختلفه", "probability": 0.9466552734375}, {"start": 1361.75, "end": 1361.95, "word": " عن", "probability": 0.9892578125}, {"start": 1361.95, "end": 1362.11, "word": " ال", "probability": 0.93505859375}, {"start": 1362.11, "end": 1362.33, "word": " X", "probability": 0.9833984375}, {"start": 1362.33, "end": 1362.71, "word": " نشوف", "probability": 0.8564453125}, {"start": 1362.71, "end": 1362.89, "word": " مع", "probability": 0.990234375}, {"start": 1362.89, "end": 1363.39, "word": " بعض", "probability": 0.944580078125}, {"start": 1363.39, "end": 1364.75, "word": " in", "probability": 0.640625}, {"start": 1364.75, "end": 1365.09, "word": " this", "probability": 0.97314453125}, {"start": 1365.09, "end": 1365.61, "word": " case", "probability": 0.9482421875}, {"start": 1365.61, "end": 1369.73, "word": " in", "probability": 0.4951171875}, {"start": 1369.73, "end": 1370.03, "word": " this", "probability": 0.96826171875}, {"start": 1370.03, "end": 1370.53, "word": " case", "probability": 0.94677734375}, {"start": 1370.53, "end": 1371.47, "word": " ال", "probability": 0.9384765625}, {"start": 1371.47, "end": 1372.13, "word": " X", "probability": 0.951171875}, {"start": 1372.13, "end": 1372.81, "word": " negative", "probability": 0.916015625}, {"start": 1372.81, "end": 1373.31, "word": " او", "probability": 0.7340087890625}, {"start": 1373.31, "end": 1373.67, "word": " ال", "probability": 0.7548828125}, {"start": 1373.67, "end": 1376.39, "word": " negative", "probability": 0.86181640625}, {"start": 1376.39, "end": 1376.97, "word": " delta", "probability": 0.72607421875}, {"start": 1376.97, "end": 1377.31, "word": " X", "probability": 0.90380859375}, {"start": 1377.31, "end": 1377.81, "word": " موجب", "probability": 0.7027994791666666}, {"start": 1377.81, "end": 1378.25, "word": " delta", "probability": 0.53271484375}, {"start": 1378.25, "end": 1379.63, "word": " هذا", "probability": 0.69482421875}, {"start": 1379.63, "end": 1379.77, "word": " اللى", "probability": 0.85888671875}, {"start": 1379.77, "end": 1379.91, "word": " هو", "probability": 0.95556640625}, {"start": 1379.91, "end": 1380.03, "word": " ال", "probability": 0.89013671875}, {"start": 1380.03, "end": 1380.27, "word": " delta", "probability": 0.87109375}, {"start": 1380.27, "end": 1380.83, "word": " neighborhood", "probability": 0.92626953125}, {"start": 1380.83, "end": 1381.05, "word": " ل", "probability": 0.84765625}, {"start": 1381.05, "end": 1381.31, "word": " X", "probability": 0.77392578125}, {"start": 1381.31, "end": 1382.07, "word": " تقاطع", "probability": 0.982666015625}], "temperature": 1.0}, {"id": 51, "seek": 141234, "start": 1383.0, "end": 1412.34, "text": "المجموعة A1 اللي هي الفترة المفتوحة من 0 إلى 1 هيطلع بساوي الفترة المفتوحة من 0 إلى 1 لأن الـ Delta هنا أكبر من أو ساوي الواحد يعني هي عندي S 0 إلى 1 هذه اللي هي المجموعة A وهي X", "tokens": [45340, 7435, 2304, 2407, 27884, 316, 16, 13672, 1829, 39896, 27188, 2655, 25720, 9673, 5172, 2655, 2407, 5016, 3660, 9154, 1958, 30731, 502, 39896, 9566, 1211, 3615, 4724, 3794, 995, 45865, 27188, 2655, 25720, 9673, 5172, 2655, 2407, 5016, 3660, 9154, 1958, 30731, 502, 5296, 33456, 2423, 39184, 18183, 34105, 5551, 4117, 26890, 9154, 34051, 8608, 995, 45865, 2423, 14407, 24401, 37495, 22653, 39896, 18871, 16254, 318, 1958, 30731, 502, 29538, 13672, 1829, 39896, 9673, 7435, 2304, 2407, 27884, 316, 37037, 1829, 1783], "avg_logprob": -0.17354910838462057, "compression_ratio": 1.910828025477707, "no_speech_prob": 0.0, "words": [{"start": 1383.0, "end": 1383.88, "word": "المجموعة", "probability": 0.87314453125}, {"start": 1383.88, "end": 1384.54, "word": " A1", "probability": 0.5059814453125}, {"start": 1384.54, "end": 1384.8, "word": " اللي", 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"probability": 0.98681640625}, {"start": 1393.02, "end": 1393.42, "word": " 0", "probability": 0.9833984375}, {"start": 1393.42, "end": 1393.76, "word": " إلى", "probability": 0.9765625}, {"start": 1393.76, "end": 1394.26, "word": " 1", "probability": 0.986328125}, {"start": 1394.26, "end": 1396.82, "word": " لأن", "probability": 0.8876953125}, {"start": 1396.82, "end": 1397.04, "word": " الـ", "probability": 0.594482421875}, {"start": 1397.04, "end": 1397.28, "word": " Delta", "probability": 0.392578125}, {"start": 1397.28, "end": 1397.58, "word": " هنا", "probability": 0.97119140625}, {"start": 1397.58, "end": 1398.18, "word": " أكبر", "probability": 0.9674479166666666}, {"start": 1398.18, "end": 1398.44, "word": " من", "probability": 0.85986328125}, {"start": 1398.44, "end": 1398.66, "word": " أو", "probability": 0.95751953125}, {"start": 1398.66, "end": 1399.1, "word": " ساوي", "probability": 0.8763020833333334}, {"start": 1399.1, "end": 1399.84, "word": " الواحد", "probability": 0.772705078125}, {"start": 1399.84, "end": 1401.98, "word": " يعني", "probability": 0.87060546875}, {"start": 1401.98, "end": 1402.2, "word": " هي", "probability": 0.332275390625}, {"start": 1402.2, "end": 1402.58, "word": " عندي", "probability": 0.944091796875}, {"start": 1402.58, "end": 1403.0, "word": " S", "probability": 0.63916015625}, {"start": 1403.0, "end": 1404.78, "word": " 0", "probability": 0.6171875}, {"start": 1404.78, "end": 1405.14, "word": " إلى", "probability": 0.990234375}, {"start": 1405.14, "end": 1405.68, "word": " 1", "probability": 0.9912109375}, {"start": 1405.68, "end": 1407.54, "word": " هذه", "probability": 0.615234375}, {"start": 1407.54, "end": 1407.72, "word": " اللي", "probability": 0.97900390625}, {"start": 1407.72, "end": 1407.96, "word": " هي", "probability": 0.96630859375}, {"start": 1407.96, "end": 1408.92, "word": " المجموعة", "probability": 0.96474609375}, {"start": 1408.92, "end": 1410.04, "word": " A", "probability": 0.9638671875}, {"start": 1410.04, "end": 1411.84, "word": " وهي", "probability": 0.772705078125}, {"start": 1411.84, "end": 1412.34, "word": " X", "probability": 0.90380859375}], "temperature": 1.0}, {"id": 52, "seek": 143846, "start": 1414.76, "end": 1438.46, "text": "نقطة ما داخل الفترة فلما يكون x زايد ال delta لما تكون ال delta تبعتي أكبر من واحد فx زايد ال delta هتكون هنا و x سالب ال delta بالتأكيد هتكون هنا وبالتالي ال delta neighborhood ل x هيحتوي", "tokens": [1863, 47432, 3660, 19446, 11778, 47283, 1211, 27188, 2655, 25720, 6156, 1211, 15042, 7251, 30544, 2031, 30767, 995, 25708, 2423, 8289, 5296, 15042, 6055, 30544, 2423, 8289, 6055, 3555, 3615, 31371, 5551, 4117, 26890, 9154, 36764, 24401, 6156, 87, 30767, 995, 25708, 2423, 8289, 8032, 2655, 30544, 34105, 4032, 2031, 8608, 6027, 3555, 2423, 8289, 20666, 2655, 10721, 4117, 25708, 8032, 2655, 30544, 34105, 46599, 6027, 2655, 6027, 1829, 2423, 8289, 7630, 5296, 2031, 8032, 1829, 33753, 45865], "avg_logprob": -0.20569619762746594, "compression_ratio": 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"word": " تكون", "probability": 0.981689453125}, {"start": 1421.92, "end": 1422.06, "word": " ال", "probability": 0.94091796875}, {"start": 1422.06, "end": 1422.3, "word": " delta", "probability": 0.9482421875}, {"start": 1422.3, "end": 1422.88, "word": " تبعتي", "probability": 0.9210205078125}, {"start": 1422.88, "end": 1423.3, "word": " أكبر", "probability": 0.8346354166666666}, {"start": 1423.3, "end": 1423.48, "word": " من", "probability": 0.98828125}, {"start": 1423.48, "end": 1423.92, "word": " واحد", "probability": 0.925048828125}, {"start": 1423.92, "end": 1425.26, "word": " فx", "probability": 0.6966552734375}, {"start": 1425.26, "end": 1425.6, "word": " زايد", "probability": 0.9308268229166666}, {"start": 1425.6, "end": 1425.76, "word": " ال", "probability": 0.72021484375}, {"start": 1425.76, "end": 1425.98, "word": " delta", "probability": 0.94677734375}, {"start": 1425.98, "end": 1426.42, "word": " هتكون", "probability": 0.9498697916666666}, {"start": 1426.42, "end": 1426.7, "word": " هنا", "probability": 0.99365234375}, {"start": 1426.7, "end": 1428.84, "word": " و", "probability": 0.82421875}, {"start": 1428.84, "end": 1429.16, "word": " x", "probability": 0.66943359375}, {"start": 1429.16, "end": 1429.56, "word": " سالب", "probability": 0.95751953125}, {"start": 1429.56, "end": 1429.66, "word": " ال", "probability": 0.62744140625}, {"start": 1429.66, "end": 1429.92, "word": " delta", "probability": 0.974609375}, {"start": 1429.92, "end": 1430.58, "word": " بالتأكيد", "probability": 0.96240234375}, {"start": 1430.58, "end": 1431.14, "word": " هتكون", "probability": 0.978515625}, {"start": 1431.14, "end": 1431.5, "word": " هنا", "probability": 0.99560546875}, {"start": 1431.5, "end": 1434.24, "word": " وبالتالي", "probability": 0.873193359375}, {"start": 1434.24, "end": 1435.6, "word": " ال", "probability": 0.931640625}, {"start": 1435.6, "end": 1435.88, "word": " delta", "probability": 0.96533203125}, {"start": 1435.88, "end": 1436.34, "word": " neighborhood", "probability": 0.153564453125}, {"start": 1436.34, "end": 1436.58, "word": " ل", "probability": 0.89599609375}, {"start": 1436.58, "end": 1436.96, "word": " x", "probability": 0.564453125}, {"start": 1436.96, "end": 1438.46, "word": " هيحتوي", "probability": 0.83514404296875}], "temperature": 1.0}, {"id": 53, "seek": 146046, "start": 1439.72, "end": 1460.46, "text": "المجموعة a واحد وبالتالي تقاطع معاها تطلع المجموعة a واحد وهذا is infinite وبالتالي اذا ال delta neighborhood هذا تقاطع الفترة المفتوحة minus x", "tokens": [6027, 2304, 7435, 2304, 2407, 27884, 257, 36764, 24401, 46599, 6027, 2655, 6027, 1829, 6055, 4587, 41193, 3615, 20449, 995, 11296, 6055, 9566, 1211, 3615, 9673, 7435, 2304, 2407, 27884, 257, 36764, 24401, 37037, 15730, 307, 13785, 46599, 6027, 2655, 6027, 1829, 1975, 15730, 2423, 8289, 7630, 23758, 6055, 4587, 41193, 3615, 27188, 2655, 25720, 9673, 5172, 2655, 2407, 5016, 3660, 3175, 2031], "avg_logprob": -0.26098632998764515, "compression_ratio": 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{"start": 1445.56, "end": 1446.02, "word": " is", "probability": 0.68896484375}, {"start": 1446.02, "end": 1446.76, "word": " infinite", "probability": 0.92138671875}, {"start": 1446.76, "end": 1451.68, "word": " وبالتالي", "probability": 0.885009765625}, {"start": 1451.68, "end": 1452.48, "word": " اذا", "probability": 0.5606689453125}, {"start": 1452.48, "end": 1452.72, "word": " ال", "probability": 0.9248046875}, {"start": 1452.72, "end": 1453.02, "word": " delta", "probability": 0.64208984375}, {"start": 1453.02, "end": 1453.54, "word": " neighborhood", "probability": 0.6552734375}, {"start": 1453.54, "end": 1454.62, "word": " هذا", "probability": 0.73193359375}, {"start": 1454.62, "end": 1457.6, "word": " تقاطع", "probability": 0.97314453125}, {"start": 1457.6, "end": 1458.06, "word": " الفترة", "probability": 0.9879557291666666}, {"start": 1458.06, "end": 1458.86, "word": " المفتوحة", "probability": 0.9871419270833334}, {"start": 1458.86, "end": 1459.46, "word": " minus", "probability": 0.916015625}, {"start": 1459.46, "end": 1460.46, "word": " x", "probability": 0.70166015625}], "temperature": 1.0}, {"id": 54, "seek": 148647, "start": 1463.65, "end": 1486.47, "text": "لا أكيد بتأكيد لا يساوي five okay تمام اذا في الحالتين ال condition تبع ال cluster point تتحقق therefore by definition x is cluster point is cluster point of", "tokens": [15040, 5551, 4117, 25708, 39894, 10721, 4117, 25708, 20193, 7251, 3794, 995, 45865, 1732, 1392, 46811, 10943, 1975, 15730, 8978, 21542, 6027, 2655, 9957, 2423, 4188, 6055, 3555, 3615, 2423, 13630, 935, 6055, 2655, 5016, 4587, 4587, 4412, 538, 7123, 2031, 307, 13630, 935, 307, 13630, 935, 295], "avg_logprob": -0.19212371962411062, "compression_ratio": 1.4335664335664335, "no_speech_prob": 0.0, "words": [{"start": 1463.65, "end": 1463.99, "word": "لا", "probability": 0.775390625}, {"start": 1463.99, "end": 1464.47, "word": " أكيد", "probability": 0.83544921875}, {"start": 1464.47, "end": 1465.19, "word": " بتأكيد", "probability": 0.817138671875}, {"start": 1465.19, "end": 1465.39, "word": " لا", "probability": 0.818359375}, {"start": 1465.39, "end": 1466.11, "word": " يساوي", "probability": 0.955322265625}, {"start": 1466.11, "end": 1466.59, "word": " five", "probability": 0.305419921875}, {"start": 1466.59, "end": 1467.55, "word": " okay", "probability": 0.384033203125}, {"start": 1467.55, "end": 1468.55, "word": " تمام", "probability": 0.939697265625}, {"start": 1468.55, "end": 1469.45, "word": " اذا", "probability": 0.5943603515625}, {"start": 1469.45, "end": 1469.91, "word": " في", "probability": 0.90966796875}, {"start": 1469.91, "end": 1470.63, "word": " الحالتين", "probability": 0.9822998046875}, {"start": 1470.63, "end": 1471.95, "word": " ال", "probability": 0.92333984375}, {"start": 1471.95, "end": 1472.45, "word": " condition", "probability": 0.98388671875}, {"start": 1472.45, "end": 1472.83, "word": " تبع", "probability": 0.95556640625}, {"start": 1472.83, "end": 1472.95, "word": " ال", "probability": 0.97607421875}, {"start": 1472.95, "end": 1473.31, "word": " cluster", "probability": 0.98388671875}, {"start": 1473.31, "end": 1473.77, "word": " point", "probability": 0.982421875}, {"start": 1473.77, "end": 1474.47, "word": " تتحقق", "probability": 0.91328125}, {"start": 1474.47, "end": 1475.63, "word": " therefore", "probability": 0.728515625}, {"start": 1475.63, "end": 1476.51, "word": " by", "probability": 0.96044921875}, {"start": 1476.51, "end": 1477.47, "word": " definition", "probability": 0.91259765625}, {"start": 1477.47, "end": 1480.17, "word": " x", "probability": 0.5224609375}, {"start": 1480.17, "end": 1480.89, "word": " is", "probability": 0.955078125}, {"start": 1480.89, "end": 1481.47, "word": " cluster", "probability": 0.603515625}, {"start": 1481.47, "end": 1482.21, "word": " point", "probability": 0.98095703125}, {"start": 1482.21, "end": 1484.73, "word": " is", "probability": 0.68896484375}, {"start": 1484.73, "end": 1485.25, "word": " cluster", "probability": 0.76806640625}, {"start": 1485.25, "end": 1485.97, "word": " point", "probability": 0.97509765625}, {"start": 1485.97, "end": 1486.47, "word": " of", "probability": 0.986328125}], "temperature": 1.0}, {"id": 55, "seek": 150988, "start": 1488.14, "end": 1509.88, "text": "الست ا واحد اللي هي الفترة المفتوحة من صفر الى واحد طبعا إذا هذا بثبت ال claim الأولاني طبعا الآن هثبت claim تاني ال claim التاني", "tokens": [6027, 14851, 1975, 36764, 24401, 13672, 1829, 39896, 27188, 2655, 25720, 9673, 5172, 2655, 2407, 5016, 3660, 9154, 20328, 5172, 2288, 2423, 7578, 36764, 24401, 23032, 3555, 3615, 995, 11933, 15730, 23758, 4724, 12984, 3555, 2655, 2423, 3932, 16247, 12610, 7649, 1829, 23032, 3555, 3615, 995, 6024, 48506, 8032, 12984, 3555, 2655, 3932, 6055, 7649, 1829, 2423, 3932, 16712, 7649, 1829], "avg_logprob": -0.2986391124225432, "compression_ratio": 1.6641221374045803, "no_speech_prob": 0.0, "words": [{"start": 1488.14, "end": 1488.9, "word": "الست", "probability": 0.5623779296875}, {"start": 1488.9, "end": 1489.22, "word": " ا", "probability": 0.44091796875}, {"start": 1489.22, "end": 1489.86, "word": " واحد", "probability": 0.7064208984375}, {"start": 1489.86, "end": 1490.04, "word": " اللي", "probability": 0.369384765625}, {"start": 1490.04, "end": 1490.26, "word": " هي", "probability": 0.7119140625}, {"start": 1490.26, "end": 1490.78, "word": " الفترة", "probability": 0.9840494791666666}, {"start": 1490.78, "end": 1491.52, "word": " المفتوحة", "probability": 0.9710286458333334}, {"start": 1491.52, "end": 1491.68, "word": " من", "probability": 0.91845703125}, {"start": 1491.68, "end": 1492.28, "word": " صفر", "probability": 0.7335611979166666}, {"start": 1492.28, "end": 1492.96, "word": " الى", "probability": 0.69580078125}, {"start": 1492.96, "end": 1493.34, "word": " واحد", "probability": 0.9716796875}, {"start": 1493.34, "end": 1494.14, "word": " طبعا", "probability": 0.7601318359375}, {"start": 1494.14, "end": 1495.44, "word": " إذا", "probability": 0.4791259765625}, {"start": 1495.44, "end": 1495.72, "word": " هذا", "probability": 0.65673828125}, {"start": 1495.72, "end": 1496.16, "word": " بثبت", "probability": 0.926513671875}, {"start": 1496.16, "end": 1496.28, "word": " ال", "probability": 0.94580078125}, {"start": 1496.28, "end": 1496.56, "word": " claim", "probability": 0.55322265625}, {"start": 1496.56, "end": 1497.6, "word": " الأولاني", "probability": 0.697784423828125}, {"start": 1497.6, "end": 1498.18, "word": " طبعا", "probability": 0.9359130859375}, {"start": 1498.18, "end": 1499.24, "word": " الآن", "probability": 0.6473388671875}, {"start": 1499.24, "end": 1500.0, "word": " هثبت", "probability": 0.910888671875}, {"start": 1500.0, "end": 1500.78, "word": " claim", "probability": 0.85400390625}, {"start": 1500.78, "end": 1502.12, "word": " تاني", "probability": 0.9248046875}, {"start": 1502.12, "end": 1504.88, "word": " ال", "probability": 0.81494140625}, {"start": 1504.88, "end": 1509.14, "word": " claim", "probability": 0.64453125}, {"start": 1509.14, "end": 1509.88, "word": " التاني", "probability": 0.9658203125}], "temperature": 1.0}, {"id": 56, "seek": 153702, "start": 1516.66, "end": 1537.02, "text": "النقطة 0 is a cluster point of set A1 الفترة مفتوحة من 0 إلى 1 لبرهان ذلك", "tokens": [6027, 1863, 47432, 3660, 1958, 307, 257, 13630, 935, 295, 992, 316, 16, 27188, 2655, 25720, 3714, 5172, 2655, 2407, 5016, 3660, 9154, 1958, 30731, 502, 5296, 26890, 3224, 7649, 29910, 23275], "avg_logprob": -0.3397253787878788, "compression_ratio": 1.0396039603960396, "no_speech_prob": 0.0, "words": [{"start": 1516.66, "end": 1517.9, "word": "النقطة", "probability": 0.758056640625}, {"start": 1517.9, "end": 1518.46, "word": " 0", "probability": 0.11688232421875}, {"start": 1518.46, "end": 1520.46, "word": " is", "probability": 0.197265625}, {"start": 1520.46, "end": 1520.8, "word": " a", "probability": 0.806640625}, {"start": 1520.8, "end": 1521.24, "word": " cluster", "probability": 0.88525390625}, {"start": 1521.24, "end": 1525.18, "word": " point", "probability": 0.97021484375}, {"start": 1525.18, "end": 1525.82, "word": " of", "probability": 0.767578125}, {"start": 1525.82, "end": 1527.1, "word": " set", "probability": 0.75390625}, {"start": 1527.1, "end": 1527.86, "word": " A1", "probability": 0.8251953125}, {"start": 1527.86, "end": 1528.64, "word": " الفترة", "probability": 0.7599283854166666}, {"start": 1528.64, "end": 1529.3, "word": " مفتوحة", "probability": 0.939453125}, {"start": 1529.3, "end": 1529.42, "word": " من", "probability": 0.83447265625}, {"start": 1529.42, "end": 1529.9, "word": " 0", "probability": 0.49658203125}, {"start": 1529.9, "end": 1530.96, "word": " إلى", "probability": 0.8583984375}, {"start": 1530.96, "end": 1531.36, "word": " 1", "probability": 0.95458984375}, {"start": 1531.36, "end": 1536.46, "word": " لبرهان", "probability": 0.7813720703125}, {"start": 1536.46, "end": 1537.02, "word": " ذلك", "probability": 0.98095703125}], "temperature": 1.0}, {"id": 57, "seek": 157614, "start": 1547.44, "end": 1576.14, "text": "to see this let نبدأ let delta أكبر من السفر be given فهنا لأي delta ال delta neighborhood للسفر اللي هو هيطلع", "tokens": [1353, 536, 341, 718, 8717, 44510, 10721, 718, 8289, 5551, 4117, 26890, 9154, 21136, 5172, 2288, 312, 2212, 6156, 3224, 8315, 5296, 10721, 1829, 8289, 2423, 8289, 7630, 24976, 3794, 5172, 2288, 13672, 1829, 31439, 39896, 9566, 1211, 3615], "avg_logprob": -0.2341796848922968, "compression_ratio": 1.2396694214876034, "no_speech_prob": 0.0, "words": [{"start": 1547.44, "end": 1547.68, "word": "to", "probability": 0.1624755859375}, {"start": 1547.68, "end": 1548.02, "word": " see", "probability": 0.859375}, {"start": 1548.02, "end": 1548.62, "word": " this", "probability": 0.9443359375}, {"start": 1548.62, "end": 1550.88, "word": " let", "probability": 0.5478515625}, {"start": 1550.88, "end": 1551.9, "word": " نبدأ", "probability": 0.7897135416666666}, {"start": 1551.9, "end": 1553.7, "word": " let", "probability": 0.7490234375}, {"start": 1553.7, "end": 1554.32, "word": " delta", "probability": 0.7294921875}, {"start": 1554.32, "end": 1554.82, "word": " أكبر", "probability": 0.9191080729166666}, {"start": 1554.82, "end": 1555.02, "word": " من", "probability": 0.99267578125}, {"start": 1555.02, "end": 1555.58, "word": " السفر", "probability": 0.9326171875}, {"start": 1555.58, "end": 1555.86, "word": " be", "probability": 0.880859375}, {"start": 1555.86, "end": 1556.42, "word": " given", "probability": 0.869140625}, {"start": 1556.42, "end": 1563.12, "word": " فهنا", "probability": 0.9021809895833334}, {"start": 1563.12, "end": 1563.78, "word": " لأي", "probability": 0.9309895833333334}, {"start": 1563.78, "end": 1564.32, "word": " delta", "probability": 0.7978515625}, {"start": 1564.32, "end": 1567.12, "word": " ال", "probability": 0.2059326171875}, {"start": 1567.12, "end": 1571.38, "word": " delta", "probability": 0.7978515625}, {"start": 1571.38, "end": 1571.96, "word": " neighborhood", "probability": 0.438720703125}, {"start": 1571.96, "end": 1574.94, "word": " للسفر", "probability": 0.95166015625}, {"start": 1574.94, "end": 1575.26, "word": " اللي", "probability": 0.87744140625}, {"start": 1575.26, "end": 1575.54, "word": " هو", "probability": 0.94482421875}, {"start": 1575.54, "end": 1576.14, "word": " هيطلع", "probability": 0.9376220703125}], "temperature": 1.0}, {"id": 58, "seek": 159515, "start": 1578.57, "end": 1595.15, "text": "سالب دلتا زاد سفر وموجب دلتا زاد سفر فتقاطع هذا مع الفترة المفتوحة من سفر إلى واحد بساوي", "tokens": [3794, 6027, 3555, 11778, 1211, 2655, 995, 30767, 18513, 8608, 5172, 2288, 4032, 2304, 29245, 3555, 11778, 1211, 2655, 995, 30767, 18513, 8608, 5172, 2288, 6156, 2655, 4587, 41193, 3615, 23758, 20449, 27188, 2655, 25720, 9673, 5172, 2655, 2407, 5016, 3660, 9154, 8608, 5172, 2288, 30731, 36764, 24401, 4724, 3794, 995, 45865], "avg_logprob": -0.14290978098815343, "compression_ratio": 1.5436893203883495, "no_speech_prob": 1.7881393432617188e-07, "words": [{"start": 1578.57, "end": 1579.35, "word": "سالب", "probability": 0.6609700520833334}, {"start": 1579.35, "end": 1579.97, "word": " دلتا", "probability": 0.77044677734375}, {"start": 1579.97, "end": 1581.31, "word": " زاد", "probability": 0.697509765625}, {"start": 1581.31, "end": 1581.81, "word": " سفر", "probability": 0.89208984375}, {"start": 1581.81, "end": 1583.97, "word": " وموجب", "probability": 0.8310546875}, {"start": 1583.97, "end": 1584.45, "word": " دلتا", "probability": 0.9593505859375}, {"start": 1584.45, "end": 1584.81, "word": " زاد", "probability": 0.95947265625}, {"start": 1584.81, "end": 1585.39, "word": " سفر", "probability": 0.9892578125}, {"start": 1585.39, "end": 1587.75, "word": " فتقاطع", "probability": 0.9724609375}, {"start": 1587.75, "end": 1588.23, "word": " هذا", "probability": 0.81982421875}, {"start": 1588.23, "end": 1588.63, "word": " مع", "probability": 0.98291015625}, {"start": 1588.63, "end": 1589.63, "word": " الفترة", "probability": 0.9903971354166666}, {"start": 1589.63, "end": 1590.57, "word": " المفتوحة", "probability": 0.9717610677083334}, {"start": 1590.57, "end": 1590.75, "word": " من", "probability": 0.986328125}, {"start": 1590.75, "end": 1591.57, "word": " سفر", "probability": 0.95654296875}, {"start": 1591.57, "end": 1592.27, "word": " إلى", "probability": 0.75830078125}, {"start": 1592.27, "end": 1593.03, "word": " واحد", "probability": 0.987060546875}, {"start": 1593.03, "end": 1595.15, "word": " بساوي", "probability": 0.7645263671875}], "temperature": 1.0}, {"id": 59, "seek": 162502, "start": 1596.9, "end": 1625.02, "text": "في خيارين إما الفترة المفتوحة من سفر إلى دلتا إذا كانت ال delta أصغر من واحد طبعا أكبر من سفر وبساوي الفترة المفتوحة من سفر إلى واحد إذا كان ال delta أكبر من أو ساوي الواحد زي ما شوفنا في برهان الكلام الأولاني", "tokens": [41185, 16490, 1829, 9640, 9957, 11933, 15042, 27188, 2655, 25720, 9673, 5172, 2655, 2407, 5016, 3660, 9154, 8608, 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هذا البرهان مشابه لبرهان المثال الأول ناخد كمان مثال آخر مثال تالت every every finite", "tokens": [8297, 46811, 10943, 22807, 11933, 15730, 23758, 2423, 26890, 3224, 7649, 37893, 16758, 3224, 5296, 26890, 3224, 7649, 9673, 12984, 6027, 16247, 12610, 8717, 47283, 3215, 9122, 2304, 7649, 50113, 6027, 19753, 34740, 50113, 6027, 6055, 6027, 2655, 633, 633, 19362], "avg_logprob": -0.1517857149952934, "compression_ratio": 1.4298245614035088, "no_speech_prob": 0.0, "words": [{"start": 1938.41, "end": 1938.91, "word": "Okay", "probability": 0.8681640625}, {"start": 1938.91, "end": 1939.71, "word": " تمام؟", "probability": 0.6891276041666666}, {"start": 1939.71, "end": 1939.99, "word": " إذا", "probability": 0.749755859375}, {"start": 1939.99, "end": 1940.77, "word": " هذا", "probability": 0.62060546875}, {"start": 1940.77, "end": 1941.27, "word": " البرهان", "probability": 0.840087890625}, {"start": 1941.27, "end": 1941.87, "word": " مشابه", "probability": 0.982421875}, {"start": 1941.87, 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1.0}, {"id": 72, "seek": 199577, "start": 1966.97, "end": 1995.77, "text": "set A contained in R has no cluster points كل finite set مالهاش ولا cluster point", "tokens": [3854, 316, 16212, 294, 497, 575, 572, 13630, 2793, 28242, 19362, 992, 3714, 6027, 3224, 33599, 49429, 13630, 935], "avg_logprob": -0.3740234404802322, "compression_ratio": 1.0823529411764705, "no_speech_prob": 0.0, "words": [{"start": 1966.97, "end": 1967.95, "word": "set", "probability": 0.26904296875}, {"start": 1967.95, "end": 1969.61, "word": " A", "probability": 0.52734375}, {"start": 1969.61, "end": 1970.37, "word": " contained", "probability": 0.239013671875}, {"start": 1970.37, "end": 1970.73, "word": " in", "probability": 0.89892578125}, {"start": 1970.73, "end": 1971.23, "word": " R", "probability": 0.98681640625}, {"start": 1971.23, "end": 1973.37, "word": " has", "probability": 0.5966796875}, {"start": 1973.37, "end": 1979.35, "word": " no", "probability": 0.94775390625}, {"start": 1979.35, "end": 1983.07, "word": " cluster", "probability": 0.93505859375}, {"start": 1983.07, "end": 1989.73, "word": " points", "probability": 0.58056640625}, {"start": 1989.73, "end": 1990.87, "word": " كل", "probability": 0.52392578125}, {"start": 1990.87, "end": 1991.71, "word": " finite", "probability": 0.701171875}, {"start": 1991.71, "end": 1992.37, "word": " set", "probability": 0.96435546875}, {"start": 1992.37, "end": 1993.23, "word": " مالهاش", "probability": 0.73150634765625}, {"start": 1993.23, "end": 1994.29, "word": " ولا", "probability": 0.84716796875}, {"start": 1994.29, "end": 1994.93, "word": " cluster", "probability": 0.95703125}, {"start": 1994.93, "end": 1995.77, "word": " point", "probability": 0.9306640625}], "temperature": 1.0}, {"id": 73, "seek": 202095, "start": 1997.69, "end": 2020.95, "text": "والبرهان سهل proof say دعنا ال set a نسمي عناصرها a1, a2 الى an هدف مش هدف finite set اذا عناصرهم ممكن اعملهم list a1, a2 الى an", "tokens": [2407, 6027, 26890, 3224, 7649, 8608, 3224, 1211, 8177, 584, 11778, 3615, 8315, 2423, 992, 257, 8717, 38251, 1829, 18871, 33546, 2288, 11296, 257, 16, 11, 257, 17, 2423, 7578, 364, 8032, 3215, 5172, 37893, 8032, 3215, 5172, 19362, 992, 1975, 15730, 18871, 33546, 2288, 16095, 3714, 43020, 1975, 25957, 1211, 16095, 1329, 257, 16, 11, 257, 17, 2423, 7578, 364], "avg_logprob": -0.31350805730588976, "compression_ratio": 1.3868613138686132, "no_speech_prob": 0.0, "words": [{"start": 1997.6899999999998, "end": 1998.83, "word": "والبرهان", "probability": 0.747607421875}, {"start": 1998.83, "end": 1999.97, "word": " سهل", "probability": 0.87646484375}, {"start": 1999.97, "end": 2000.65, "word": " proof", "probability": 0.26025390625}, {"start": 2000.65, "end": 2004.95, "word": " say", "probability": 0.7021484375}, {"start": 2004.95, "end": 2005.93, "word": " دعنا", "probability": 0.7333984375}, {"start": 2005.93, "end": 2007.27, "word": " ال", "probability": 0.7724609375}, {"start": 2007.27, "end": 2007.57, "word": " set", "probability": 0.91552734375}, {"start": 2007.57, "end": 2007.99, "word": " a", "probability": 0.441650390625}, {"start": 2007.99, "end": 2008.57, "word": " نسمي", "probability": 0.6505126953125}, {"start": 2008.57, "end": 2009.29, "word": " عناصرها", "probability": 0.9312744140625}, {"start": 2009.29, "end": 2009.99, "word": " a1,", "probability": 0.783447265625}, {"start": 2010.57, "end": 2012.41, "word": " a2", "probability": 0.82958984375}, {"start": 2012.41, "end": 2014.05, "word": " الى", "probability": 0.5859375}, {"start": 2014.05, "end": 2014.43, "word": " an", "probability": 0.51806640625}, {"start": 2014.43, "end": 2014.91, "word": " هدف", "probability": 0.6925455729166666}, {"start": 2014.91, "end": 2015.03, "word": " مش", "probability": 0.85693359375}, {"start": 2015.03, "end": 2015.27, "word": " هدف", "probability": 0.6387532552083334}, {"start": 2015.27, "end": 2015.55, "word": " finite", "probability": 0.92578125}, {"start": 2015.55, "end": 2016.05, "word": " set", "probability": 0.9306640625}, {"start": 2016.05, "end": 2017.15, "word": " اذا", "probability": 0.5828857421875}, {"start": 2017.15, "end": 2017.85, "word": " عناصرهم", "probability": 0.8934326171875}, {"start": 2017.85, "end": 2018.07, "word": " ممكن", "probability": 0.789306640625}, {"start": 2018.07, "end": 2018.63, "word": " اعملهم", "probability": 0.8433837890625}, {"start": 2018.63, "end": 2018.97, "word": " list", "probability": 0.92822265625}, {"start": 2018.97, "end": 2019.47, "word": " a1,", "probability": 0.9404296875}, {"start": 2019.57, "end": 2019.95, "word": " a2", "probability": 0.97509765625}, {"start": 2019.95, "end": 2020.37, "word": " الى", "probability": 0.847412109375}, {"start": 2020.37, "end": 2020.95, "word": " an", "probability": 0.67578125}], "temperature": 1.0}, {"id": 74, "seek": 204437, "start": 2022.21, "end": 2044.37, "text": "و ممكن اعملهم order ارتبهم حسب المؤشر تبعهم يعني a1 اصغر من a2 اصغر من a3 اصغر من اصغر من am ممكن نعمل كلمة من هذا ولا لأ ممكن by the ordering principle", "tokens": [2407, 3714, 43020, 1975, 25957, 1211, 16095, 1668, 1975, 43500, 3555, 16095, 11331, 35457, 9673, 33604, 46309, 6055, 3555, 3615, 16095, 37495, 22653, 257, 16, 1975, 9381, 17082, 2288, 9154, 257, 17, 1975, 9381, 17082, 2288, 9154, 257, 18, 1975, 9381, 17082, 2288, 9154, 1975, 9381, 17082, 2288, 9154, 669, 3714, 43020, 8717, 25957, 1211, 9122, 19528, 3660, 9154, 23758, 49429, 5296, 10721, 3714, 43020, 538, 264, 21739, 8665], "avg_logprob": -0.20558035586561477, "compression_ratio": 1.5695364238410596, "no_speech_prob": 0.0, "words": [{"start": 2022.21, "end": 2022.43, "word": "و", "probability": 0.482421875}, {"start": 2022.43, "end": 2022.87, "word": " ممكن", "probability": 0.823486328125}, {"start": 2022.87, "end": 2023.65, "word": " اعملهم", "probability": 0.852783203125}, {"start": 2023.65, "end": 2024.19, "word": " order", "probability": 0.89453125}, {"start": 2024.19, "end": 2025.25, "word": " ارتبهم", "probability": 0.8848876953125}, {"start": 2025.25, "end": 2026.17, "word": " حسب", "probability": 0.937255859375}, {"start": 2026.17, "end": 2026.83, "word": " المؤشر", "probability": 0.955078125}, {"start": 2026.83, "end": 2027.53, "word": " تبعهم", "probability": 0.80108642578125}, {"start": 2027.53, "end": 2028.81, "word": " يعني", "probability": 0.685302734375}, {"start": 2028.81, "end": 2029.73, "word": " a1", "probability": 0.530029296875}, {"start": 2029.73, "end": 2031.49, "word": " اصغر", "probability": 0.885986328125}, {"start": 2031.49, "end": 2031.71, "word": " من", "probability": 0.990234375}, {"start": 2031.71, "end": 2032.45, "word": " a2", "probability": 0.988037109375}, {"start": 2032.45, "end": 2033.13, "word": " اصغر", "probability": 0.984375}, {"start": 2033.13, "end": 2033.37, "word": " من", "probability": 0.9931640625}, {"start": 2033.37, "end": 2034.33, "word": " a3", "probability": 0.994140625}, {"start": 2034.33, "end": 2035.69, "word": " اصغر", "probability": 0.978515625}, {"start": 2035.69, "end": 2036.03, "word": " من", "probability": 0.9951171875}, {"start": 2036.03, "end": 2036.69, "word": " اصغر", "probability": 0.91552734375}, {"start": 2036.69, "end": 2036.93, "word": " من", "probability": 0.9892578125}, {"start": 2036.93, "end": 2037.45, "word": " am", "probability": 0.317626953125}, {"start": 2037.45, "end": 2038.79, "word": " ممكن", "probability": 0.943115234375}, {"start": 2038.79, "end": 2039.19, "word": " نعمل", "probability": 0.9625651041666666}, {"start": 2039.19, "end": 2039.41, "word": " كلمة", "probability": 0.6325276692708334}, {"start": 2039.41, "end": 2039.53, "word": " من", "probability": 0.30078125}, {"start": 2039.53, "end": 2039.69, "word": " هذا", "probability": 0.705078125}, {"start": 2039.69, "end": 2039.85, "word": " ولا", "probability": 0.419677734375}, {"start": 2039.85, "end": 2040.15, "word": " لأ", "probability": 0.86376953125}, {"start": 2040.15, "end": 2040.71, "word": " ممكن", "probability": 0.831298828125}, {"start": 2040.71, "end": 2042.89, "word": " by", "probability": 0.6552734375}, {"start": 2042.89, "end": 2043.15, "word": " the", "probability": 0.7880859375}, {"start": 2043.15, "end": 2043.57, "word": " ordering", "probability": 0.92822265625}, {"start": 2043.57, "end": 2044.37, "word": " principle", "probability": 0.94677734375}], "temperature": 1.0}, {"id": 75, "seek": 207908, "start": 2051.36, "end": 2079.08, "text": "أو باستخدام ال ordering تبع ال real numbers إذا هي عندي ال 6A تبعتي هي خط العداد وهي A1 وهي A2 وهي A3 مش شرط المسافة بين كل أنصار والتاني تكون متساوية وهكذا إلى", "tokens": [10721, 2407, 4724, 995, 14851, 9778, 3215, 10943, 2423, 21739, 6055, 3555, 3615, 2423, 957, 3547, 11933, 15730, 39896, 18871, 16254, 2423, 1386, 32, 6055, 3555, 3615, 31371, 39896, 16490, 9566, 18863, 3215, 18513, 37037, 1829, 316, 16, 37037, 1829, 316, 17, 37037, 1829, 316, 18, 37893, 13412, 2288, 9566, 9673, 3794, 31845, 3660, 49374, 28242, 14739, 9381, 9640, 16070, 2655, 7649, 1829, 6055, 30544, 44650, 3794, 995, 2407, 10632, 37037, 4117, 15730, 30731], "avg_logprob": -0.22729166626930236, "compression_ratio": 1.4857142857142858, "no_speech_prob": 5.960464477539063e-08, "words": [{"start": 2051.36, "end": 2051.68, "word": "أو", "probability": 0.704345703125}, {"start": 2051.68, "end": 2052.3, "word": " باستخدام", "probability": 0.97412109375}, {"start": 2052.3, "end": 2052.46, "word": " ال", "probability": 0.66259765625}, {"start": 2052.46, "end": 2052.88, "word": " ordering", "probability": 0.61572265625}, {"start": 2052.88, "end": 2053.36, "word": " تبع", "probability": 0.7110188802083334}, {"start": 2053.36, "end": 2053.46, "word": " ال", "probability": 0.90185546875}, {"start": 2053.46, "end": 2053.64, "word": " real", "probability": 0.76806640625}, {"start": 2053.64, "end": 2054.22, "word": " numbers", "probability": 0.63525390625}, {"start": 2054.22, "end": 2060.78, "word": " إذا", "probability": 0.4100341796875}, {"start": 2060.78, "end": 2061.18, "word": " هي", "probability": 0.3125}, {"start": 2061.18, 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"word": " A3", "probability": 0.989013671875}, {"start": 2071.86, "end": 2073.02, "word": " مش", "probability": 0.67041015625}, {"start": 2073.02, "end": 2073.32, "word": " شرط", "probability": 0.9820963541666666}, {"start": 2073.32, "end": 2073.86, "word": " المسافة", "probability": 0.9857177734375}, {"start": 2073.86, "end": 2074.04, "word": " بين", "probability": 0.94140625}, {"start": 2074.04, "end": 2074.28, "word": " كل", "probability": 0.95654296875}, {"start": 2074.28, "end": 2074.66, "word": " أنصار", "probability": 0.8072916666666666}, {"start": 2074.66, "end": 2075.34, "word": " والتاني", "probability": 0.851318359375}, {"start": 2075.34, "end": 2076.16, "word": " تكون", "probability": 0.880615234375}, {"start": 2076.16, "end": 2077.1, "word": " متساوية", "probability": 0.93798828125}, {"start": 2077.1, "end": 2078.68, "word": " وهكذا", "probability": 0.8267415364583334}, {"start": 2078.68, "end": 2079.08, "word": " إلى", "probability": 0.7998046875}], "temperature": 1.0}, {"id": 76, "seek": 210191, "start": 2080.77, "end": 2101.91, "text": "أخر عنصر AN ف fix X ينتمي إلى R و بد أثبت أن claim X is not a cluster point", "tokens": [10721, 34740, 18871, 9381, 2288, 5252, 6156, 3191, 1783, 7251, 29399, 2304, 1829, 30731, 497, 4032, 47525, 5551, 12984, 3555, 2655, 14739, 3932, 1783, 307, 406, 257, 13630, 935], "avg_logprob": -0.3656249890724818, "compression_ratio": 0.9523809523809523, "no_speech_prob": 0.0, "words": [{"start": 2080.77, "end": 2081.29, "word": "أخر", "probability": 0.580322265625}, {"start": 2081.29, "end": 2081.79, "word": " عنصر", "probability": 0.9596354166666666}, {"start": 2081.79, "end": 2082.35, "word": " AN", "probability": 0.15478515625}, {"start": 2082.35, "end": 2086.23, "word": " ف", "probability": 0.67333984375}, {"start": 2086.23, "end": 2086.89, "word": " fix", "probability": 0.35498046875}, {"start": 2086.89, "end": 2089.91, "word": " X", "probability": 0.5390625}, {"start": 2089.91, "end": 2090.73, "word": " ينتمي", "probability": 0.9429931640625}, {"start": 2090.73, "end": 2090.99, "word": " إلى", "probability": 0.7880859375}, {"start": 2090.99, "end": 2091.43, "word": " R", "probability": 0.982421875}, {"start": 2091.43, "end": 2093.49, "word": " و", "probability": 0.66064453125}, {"start": 2093.49, "end": 2093.79, "word": " بد", "probability": 0.1845703125}, {"start": 2093.79, "end": 2094.33, "word": " أثبت", "probability": 0.88720703125}, {"start": 2094.33, "end": 2094.63, "word": " أن", "probability": 0.37890625}, {"start": 2094.63, "end": 2095.65, "word": " claim", "probability": 0.56201171875}, {"start": 2095.65, "end": 2097.81, "word": " X", "probability": 0.876953125}, {"start": 2097.81, "end": 2098.33, "word": " is", "probability": 0.92822265625}, {"start": 2098.33, "end": 2098.77, "word": " not", "probability": 0.95263671875}, {"start": 2098.77, "end": 2100.75, "word": " a", "probability": 0.71240234375}, {"start": 2100.75, "end": 2101.19, "word": " cluster", "probability": 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ال case 2 case 2 ان ال X لا تنتمي ال ال set A ففي الحالة هذه", "tokens": [8297, 46811, 10943, 1975, 15730, 23758, 8978, 11331, 6027, 3660, 5296, 15042, 2423, 1783, 6055, 30544, 3714, 29245, 23328, 3660, 8978, 316, 21542, 6027, 3660, 16712, 7649, 10632, 16472, 2423, 1389, 568, 1389, 568, 16472, 2423, 1783, 20193, 6055, 29399, 2304, 1829, 2423, 2423, 992, 316, 6156, 41185, 21542, 6027, 3660, 29538], "avg_logprob": -0.20120872641509435, "compression_ratio": 1.5681818181818181, "no_speech_prob": 0.0, "words": [{"start": 2388.44, "end": 2388.86, "word": "Okay", "probability": 0.73974609375}, {"start": 2388.86, "end": 2389.4, "word": " تمام", "probability": 0.7646484375}, {"start": 2389.4, "end": 2390.44, "word": " اذا", "probability": 0.5928955078125}, {"start": 2390.44, "end": 2391.78, "word": " هذا", "probability": 0.44580078125}, {"start": 2391.78, "end": 2392.2, "word": " في", "probability": 0.89990234375}, {"start": 2392.2, "end": 2392.72, "word": " حالة", "probability": 0.9921875}, 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\ No newline at end of file
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/Ejs4dHLsIvo_raw.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/Ejs4dHLsIvo_raw.srt
new file mode 100644
index 0000000000000000000000000000000000000000..0e291faf4092106e4cb94cff1cdc57a366099a50
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/Ejs4dHLsIvo_raw.srt
@@ -0,0 +1,1556 @@
+1
+00:00:20,920 --> 00:00:24,640
+بسم الله الرحمن الرحيم هنبدأ في المحاضرة هذه
+
+2
+00:00:24,640 --> 00:00:29,600
+chapter جديد وهو chapter أربعة في الكتاب المقرر
+
+3
+00:00:29,600 --> 00:00:35,280
+عنوان ال chapter limits of functions و هنبدأ أول
+
+4
+00:00:35,280 --> 00:00:39,540
+section في ال chapter هذا و برضه عنوان ال section
+
+5
+00:00:39,540 --> 00:00:44,180
+الأول هو نفس عنوان ال chapter limits of functions
+
+6
+00:00:44,180 --> 00:00:52,780
+فقبل ما نعرف limit of a functionبدنا نتعرف على
+
+7
+00:00:52,780 --> 00:01:00,060
+مصطلح جديد وهو cluster point of a set نقطة تراكم
+
+8
+00:01:00,060 --> 00:01:04,780
+ال cluster point أو بعض الكتب بيسموها accumulation
+
+9
+00:01:04,780 --> 00:01:12,120
+point و كتب أخرى بيسميها limit point فلو في عندي
+
+10
+00:01:12,120 --> 00:01:18,200
+set A subset من R set of real numbers و C real
+
+11
+00:01:18,200 --> 00:01:23,190
+number فال real number هذابنسميه cluster point
+
+12
+00:01:23,190 --> 00:01:28,030
+للست a if and only if the following condition is
+
+13
+00:01:28,030 --> 00:01:33,770
+satisfied for every delta عدد موجب نقدر نجد x
+
+14
+00:01:33,770 --> 00:01:39,650
+ينتمي إلى المجموعة a و ال x هذه مختلفة عن النقطة c
+
+15
+00:01:39,650 --> 00:01:45,330
+بحيث ان المسافة بين x و c تكون أصغر من delta هذا
+
+16
+00:01:45,330 --> 00:01:49,410
+الشرط هذا
+
+17
+00:01:49,410 --> 00:01:54,020
+الشرطis equivalent to saying بكافئ ان انا اقول
+
+18
+00:01:54,020 --> 00:01:59,160
+every delta neighborhood every delta neighborhood
+
+19
+00:01:59,160 --> 00:02:03,940
+لنقطة c اللى هو الفترة المفتوحة اللى مركزها c ونص
+
+20
+00:02:03,940 --> 00:02:11,500
+قطرة delta every delta neighborhoodof c يتقاطع مع
+
+21
+00:02:11,500 --> 00:02:18,200
+المجموعة a في نقطة واحدة على الأقل x مختلفة عن ال
+
+22
+00:02:18,200 --> 00:02:25,720
+c يعني بمعنى أخر بقدر ألاقي في التقاطع هذا نقطة x
+
+23
+00:02:25,720 --> 00:02:32,440
+يعني التقاطع هذا لا يساوي five okay
+
+24
+00:02:32,440 --> 00:02:37,990
+كمان مرةالنقطة C هذه بتكون cluster point للمجموعة
+
+25
+00:02:37,990 --> 00:02:44,370
+A إذا أي delta neighborhood للنقطة C بيتخاطع مع
+
+26
+00:02:44,370 --> 00:02:52,690
+المجموعة A في نقطة X مختلفة عن الـC بس لازم
+
+27
+00:02:52,690 --> 00:02:58,060
+كل delta neighborhood لـCيتقاطع مع المجموعة A في
+
+28
+00:02:58,060 --> 00:03:02,720
+نقطة X مختلفة عن الـC طب عشان اثبت ان الـC ليست
+
+29
+00:03:02,720 --> 00:03:09,000
+cluster point بنفي الشرط هذا يكفي ان اقول there
+
+30
+00:03:09,000 --> 00:03:14,520
+exist بدل for every delta او every delta
+
+31
+00:03:14,520 --> 00:03:18,860
+neighborhoodيكفى ان اجيب there exists delta
+
+32
+00:03:18,860 --> 00:03:24,720
+neighborhood واحد ل C و التقاطع هذا بساوي فاي يعني
+
+33
+00:03:24,720 --> 00:03:30,540
+بحيث ان ال delta neighborhood لا يتقاطع مع اي
+
+34
+00:03:30,540 --> 00:03:39,180
+مشيول منها C بالمرة ناخد نظرية الأول ال definition
+
+35
+00:03:39,180 --> 00:03:43,180
+هذا بيكافئ النظرية التانية بتقول ان
+
+36
+00:03:46,060 --> 00:03:53,140
+الـ condition هذا تبع التعريف بكافئ شرط تاني اذا
+
+37
+00:03:53,140 --> 00:04:06,140
+هنا let A subset من R و C real number C
+
+38
+00:04:06,140 --> 00:04:16,740
+is a cluster is a cluster pointof the set A if and
+
+39
+00:04:16,740 --> 00:04:22,400
+only if the following condition is satisfied there
+
+40
+00:04:22,400 --> 00:04:26,600
+exist a
+
+41
+00:04:26,600 --> 00:04:38,640
+sequence a n contained in A وكل عناصرها مختلفة
+
+42
+00:04:38,640 --> 00:04:52,650
+عن ال Csuch that limit a n بساوي c اذا هذا الشرط
+
+43
+00:04:52,650 --> 00:04:59,310
+بكافئ الشرط اللي هناك الشرط هذا او اللي بكافه
+
+44
+00:04:59,310 --> 00:05:06,290
+فلبرهان ذلك اذا كمان مرة انا عشان اثبت ان c is a
+
+45
+00:05:06,290 --> 00:05:10,610
+cluster point للمجموع يعني يكفي ان اثبت ان يوجد
+
+46
+00:05:12,310 --> 00:05:17,950
+سيكوانس في المجموعة A وكل على سرها مختلفة لاتساوي
+
+47
+00:05:17,950 --> 00:05:23,370
+C وانهيتها بالساوي لعدد C فتعالى نبرهن النظرية هذه
+
+48
+00:05:23,370 --> 00:05:28,850
+نبرهن ال only if part الأول فال only if part يعني
+
+49
+00:05:28,850 --> 00:05:36,910
+ال assumption assume ان C is a cluster is
+
+50
+00:05:36,910 --> 00:05:39,230
+a cluster point
+
+51
+00:05:40,700 --> 00:05:49,140
+of a then
+
+52
+00:05:49,140 --> 00:06:00,080
+for every n ينتمي إلى n لكل عدد طبيعي n take delta
+
+53
+00:06:00,080 --> 00:06:08,110
+بساوي واحد على n عدد موجببما انه C is a cluster
+
+54
+00:06:08,110 --> 00:06:11,230
+point ل A then by definition of a cluster point
+
+55
+00:06:11,230 --> 00:06:14,410
+then
+
+56
+00:06:14,410 --> 00:06:23,770
+by definition there exist a N ينتمي إلى A مختلف عن
+
+57
+00:06:23,770 --> 00:06:28,210
+ال C such that
+
+58
+00:06:28,210 --> 00:06:31,070
+ال ..
+
+59
+00:06:34,910 --> 00:06:42,950
+الـ AN هذا ينتمي للـ Delta neighborhood للـ
+
+60
+00:06:42,950 --> 00:06:56,090
+C وطبعا ينتمي إلى A negative C هعمل
+
+61
+00:06:56,090 --> 00:07:02,810
+التعريف هذالما انه الـ C is a cluster point لأي
+
+62
+00:07:02,810 --> 00:07:09,410
+Delta أكبر من السفر خد Delta بساوي واحد على N لكل
+
+63
+00:07:09,410 --> 00:07:13,290
+عدد طبيعي N هذا بيطلع عدد موجب لذلك لـ Delta بساوي
+
+64
+00:07:13,290 --> 00:07:18,470
+واحد على N بقدر ألاجني عنصر X اللي هو ساميه An
+
+65
+00:07:18,470 --> 00:07:24,470
+يعتمد على الـ Delta وهذا العنصر موجود في A مختلف
+
+66
+00:07:24,470 --> 00:07:30,570
+عن الـ Cو أيضا موجود في الـ delta neighborhood لـ
+
+67
+00:07:30,570 --> 00:07:38,370
+C طيب
+
+68
+00:07:38,370 --> 00:07:42,610
+إذا و
+
+69
+00:07:42,610 --> 00:07:51,230
+واضح هنا من ال AN ينتمي ل ال
+
+70
+00:07:51,230 --> 00:07:56,150
+AN ينتمي ل DN ال
+
+71
+00:07:56,150 --> 00:08:01,970
+deltaأو الـ 1 على N neighborhood للـ C اللي هو C
+
+72
+00:08:01,970 --> 00:08:09,750
+سالب واحد على N C موجب واحد على N وهذا صحيح لكل N
+
+73
+00:08:09,750 --> 00:08:16,270
+بيقدي انه بيقدي
+
+74
+00:08:16,270 --> 00:08:20,890
+انه C سالم A N أو
+
+75
+00:08:24,630 --> 00:08:31,950
+absolute a n سالب c أصغر من واحد على n وهذا صحيح
+
+76
+00:08:31,950 --> 00:08:36,370
+لكل n هزبوت؟
+
+77
+00:08:36,370 --> 00:08:40,750
+هاي a n أكبر من c سالب واحد على n أصغر من c زاد
+
+78
+00:08:40,750 --> 00:08:45,310
+واحد على n هذا معناه absolute a n minus c أصغر من
+
+79
+00:08:45,310 --> 00:08:48,790
+واحد على n لكل n هذا صحيح لكل n
+
+80
+00:08:52,390 --> 00:08:56,990
+إذا هاي فيها sequence إذا هاي أثبتنا مايوجد
+
+81
+00:08:56,990 --> 00:09:09,350
+sequence إذا am is a sequence in a وكل حدودها
+
+82
+00:09:09,350 --> 00:09:16,610
+مختلفة عن ال c and by theorem اتنين اربعة الشهيرة
+
+83
+00:09:18,560 --> 00:09:21,560
+أنا عندي الـ absolute value هذي أصغر من واحد على N
+
+84
+00:09:21,560 --> 00:09:26,480
+لكل N limit واحد على N بالساوي سفر خد C بالساوي
+
+85
+00:09:26,480 --> 00:09:33,760
+واحد عدن موجب لأن بيطلع limit ال sequence A N as N
+
+86
+00:09:33,760 --> 00:09:39,300
+tends to infinity بساوي S C إذن هين أثبتنا إنه لو
+
+87
+00:09:39,300 --> 00:09:44,920
+كانت C cluster point فأثبتنا إنه يوجد sequence A N
+
+88
+00:09:44,920 --> 00:09:51,380
+في المجموعة Aوكل عناصرها مختلفة عن الـ C ونهايتها
+
+89
+00:09:51,380 --> 00:09:59,080
+بساوي الـ C إذا هذا بثبت جزء ال only if part الآن
+
+90
+00:09:59,080 --> 00:10:04,220
+لثبت العكس لإثبات
+
+91
+00:10:04,220 --> 00:10:04,920
+العكس
+
+92
+00:10:21,910 --> 00:10:28,090
+assume أن الشرط اللي حصل بتتحقق
+
+93
+00:10:28,090 --> 00:10:41,250
+assume ال condition اللي حصل holds يعني بتتحقق to
+
+94
+00:10:41,250 --> 00:10:48,890
+show c is a cluster point of
+
+95
+00:10:48,890 --> 00:11:03,360
+a letdelta أكبر من السفر is given since
+
+96
+00:11:03,360 --> 00:11:15,160
+by star من الشرط star لدي limit لان بساوي c و
+
+97
+00:11:15,160 --> 00:11:19,260
+delta أكبر من السفر is given إذا من تعريف delta
+
+98
+00:11:19,260 --> 00:11:25,560
+capital Nللـ limit لأي delta أو إبسلون عدد موجة
+
+99
+00:11:25,560 --> 00:11:31,320
+there exist n يعتمد على delta natural number دحيث
+
+100
+00:11:31,320 --> 00:11:40,140
+أنه لكل n أكبر من أو سوى capital N هذا بيقدي أنه
+
+101
+00:11:40,140 --> 00:11:47,000
+absolute a n minus c أصغر من delta
+
+102
+00:11:50,570 --> 00:11:57,870
+إذاً هذا بيقدّي إنه هيعندي am
+
+103
+00:12:02,990 --> 00:12:08,910
+طبعا هدف قلبي أن a n أصغر
+
+104
+00:12:08,910 --> 00:12:15,370
+من c زائد delta أكبر من c negative delta يعني a n
+
+105
+00:12:15,370 --> 00:12:22,630
+تنتمي إلى v delta of c وتنتمي طبعا من ال condition
+
+106
+00:12:22,630 --> 00:12:32,780
+star تنتمي إلى a difference cو هذا صحيح لكل n أكبر
+
+107
+00:12:32,780 --> 00:12:38,660
+من أو ساوي capital N إذا هاي كل delta neighborhood
+
+108
+00:12:38,660 --> 00:12:50,240
+ل C بيتقاطع مع A في عدد لانهائي من النقاط المختلفة
+
+109
+00:12:50,240 --> 00:12:57,420
+عن ال C وبالتالي الشرط تبع ال definition هيتحققو
+
+110
+00:12:57,420 --> 00:13:05,660
+then by definition .. by definition C is a cluster
+
+111
+00:13:05,660 --> 00:13:14,000
+point of the set A و هدا بكمل ال F part and
+
+112
+00:13:14,000 --> 00:13:19,140
+therefore completes the proof of the theorem okay
+
+113
+00:13:19,140 --> 00:13:20,000
+تمام؟
+
+114
+00:13:25,140 --> 00:13:27,540
+Fine خلّينا ناخد بعض الأمثلة
+
+115
+00:13:49,070 --> 00:14:05,410
+تشير إلى أن كل X ينتمي إلى مقفل مقفل مقفل
+
+116
+00:14:05,410 --> 00:14:16,770
+مقفل
+
+117
+00:14:23,120 --> 00:14:29,460
+of set A1 بساوي الفترة
+
+118
+00:14:29,460 --> 00:14:40,780
+المفتوحة من سفر إلى واحد من
+
+119
+00:14:40,780 --> 00:14:45,460
+هنا يثبت إن كل X الفترة المغلقة من سفر إلى واحد هي
+
+120
+00:14:45,460 --> 00:14:50,480
+cluster point للفترة المفتوحة من سفر إلى واحد
+
+121
+00:15:01,360 --> 00:15:09,580
+ففي الأول بدي أثبت أنه كل نقطة داخل الفترة المغلقة
+
+122
+00:15:09,580 --> 00:15:14,460
+هي cluster point للمجموع عادي و بعدين في المرحلة
+
+123
+00:15:14,460 --> 00:15:19,840
+التانية حتة من نقاط الأطراف اللي هي 01 تطلع أيضا
+
+124
+00:15:19,840 --> 00:15:26,620
+cluster point لست A1 فنشوف مع بعض ان ال claim 1
+
+125
+00:15:34,010 --> 00:15:45,190
+بنثبت ان كل X ينتمي للفترة المفتوحة is a cluster
+
+126
+00:15:45,190 --> 00:15:50,410
+point of
+
+127
+00:15:50,410 --> 00:15:58,730
+set A1 اللي هي الفترة المفتوحة نفسها لبرهان ذلك to
+
+128
+00:15:58,730 --> 00:16:09,390
+see thisالبرهن ذلك fix ينتمي
+
+129
+00:16:09,390 --> 00:16:17,470
+للفترة المفتوحة ونثبت ان cluster point للمجموعة a1
+
+130
+00:16:17,470 --> 00:16:29,940
+اذا fix x and let delta أكبر من السفر be givenنبدأ
+
+131
+00:16:29,940 --> 00:16:35,900
+بالـ delta أكبر من 0 ونثبت أن كل delta
+
+132
+00:16:35,900 --> 00:16:42,820
+neighborhood للنقطة X بيتقاطع مع المجموعة A1 في
+
+133
+00:16:42,820 --> 00:16:49,400
+نقطة مختلفة عن X وبالتالي الـ X هتطلع cluster
+
+134
+00:16:49,400 --> 00:16:55,960
+point حسب التعريف طيب نحن
+
+135
+00:16:55,960 --> 00:16:56,920
+لدينا اتصالين
+
+136
+00:17:01,270 --> 00:17:07,090
+two cases حالتين الحالة
+
+137
+00:17:07,090 --> 00:17:10,270
+الأولى ان ال delta هذه اللى انا اخترتها العشوائية
+
+138
+00:17:10,270 --> 00:17:17,310
+ال delta اللى انا اخترتها ممكن تكون اصغر من ..
+
+139
+00:17:17,310 --> 00:17:23,590
+طبعا موجبة هي ممكن تكون اصغر من واحد ف in this
+
+140
+00:17:23,590 --> 00:17:27,930
+case in
+
+141
+00:17:27,930 --> 00:17:30,050
+this case في هذه الحالة
+
+142
+00:17:35,000 --> 00:17:49,440
+لدينا in this
+
+143
+00:17:49,440 --> 00:17:58,680
+case الـ delta neighborhood ل X اللي هو X minus
+
+144
+00:17:58,680 --> 00:18:06,650
+Delta X موجة Deltaبنلاحظ انه تقاطع مع المجموعة ايه
+
+145
+00:18:06,650 --> 00:18:14,050
+اللي هي الفترة اللي مفتوحة من سفر لواحد بطلع
+
+146
+00:18:14,050 --> 00:18:23,080
+واحد من الخيارات التاليةاما الفترة المفتوحة x
+
+147
+00:18:23,080 --> 00:18:28,820
+negative delta x positive delta او الفترة المفتوحة
+
+148
+00:18:28,820 --> 00:18:36,420
+من صفر الى اكس positive delta او الفترة
+
+149
+00:18:36,420 --> 00:18:40,340
+المفتوحة من x negative delta الى واحد او الفترة
+
+150
+00:18:40,340 --> 00:18:42,920
+المفتوحة من صفر الى واحد
+
+151
+00:18:48,470 --> 00:18:55,310
+حسب قيمة الـ Delta يعني أنا
+
+152
+00:18:55,310 --> 00:19:03,030
+عندي الفترة المفتوحة من 0 إلى 1 هذه الفترة
+
+153
+00:19:03,030 --> 00:19:12,350
+المفتوحة تبعتي اللي هي A1 هذا ال set A1 و أنا عندي
+
+154
+00:19:12,350 --> 00:19:20,080
+ال Delta عدد موجب أصغر من 1والـ X هذه تنتمي .. الـ
+
+155
+00:19:20,080 --> 00:19:27,300
+X هذه نقطة ما في الفترة
+
+156
+00:19:27,300 --> 00:19:38,320
+fixed number بين 0 و 1 الآن
+
+157
+00:19:38,320 --> 00:19:41,160
+أنا عندي الـ delta neighborhood لـ X هذا هو ممكن
+
+158
+00:19:41,160 --> 00:19:46,510
+يكون زي هيك شكله وبالتالي تقاطعتقاطعه مع الفترة
+
+159
+00:19:46,510 --> 00:19:53,310
+المفتوحة هو نفسه، صح؟ إذا كان زي هيك أو ممكن يكون
+
+160
+00:19:53,310 --> 00:19:57,250
+الـ delta neighborhood لل X يكون شكله زي هيك
+
+161
+00:19:57,250 --> 00:20:01,430
+وبالتالي
+
+162
+00:20:01,430 --> 00:20:06,150
+تقاطعه مع الفترة .. مع الست واحد، هيكون الفترة
+
+163
+00:20:06,150 --> 00:20:10,830
+المفتوحة من صفر إلى X زي الـ delta اللي هي التانية
+
+164
+00:20:10,830 --> 00:20:17,310
+يعنيصح وممكن يكون ال
+
+165
+00:20:17,310 --> 00:20:21,810
+delta neighborhood ال x تكون جريبة من الواحد زي
+
+166
+00:20:21,810 --> 00:20:26,410
+هيك و ال delta neighborhood حوالين ال x يكون زي
+
+167
+00:20:26,410 --> 00:20:34,190
+هيك شكله هاي x negative delta x positive delta
+
+168
+00:20:35,200 --> 00:20:39,440
+وبالتالي تقاطع مع الفترة من صفر إلى واحد هيعطيني
+
+169
+00:20:39,440 --> 00:20:44,700
+الجزء هذا اللي هو فترة مفتوحة من X سالب Delta إلى
+
+170
+00:20:44,700 --> 00:20:48,360
+واحد وممكن
+
+171
+00:20:48,360 --> 00:20:56,170
+ال Delta لبرهود ال X تكون جريبة من المنتصفوالـ
+
+172
+00:20:56,170 --> 00:20:59,250
+Delta تكون قريبة من الواحد قيمتها أصغر من واحد لكن
+
+173
+00:20:59,250 --> 00:21:04,250
+قريبة من واحد وبالتالي الـ Delta neighborhood لل X
+
+174
+00:21:04,250 --> 00:21:09,150
+يكون زي هيك وبالتالي تقاطعه مع المجموعة A واحد
+
+175
+00:21:09,150 --> 00:21:12,970
+بيطلع المجموعة A واحد نفسها، صحيح؟ إن هذه كل
+
+176
+00:21:12,970 --> 00:21:20,590
+احتمالات وفي كل الأحوال التقاطع هذا بيطلع infinite
+
+177
+00:21:20,590 --> 00:21:22,090
+is infinite
+
+178
+00:21:25,670 --> 00:21:29,890
+تقاطع المجمعتين هذول بيطلع فترة و الفترة اي فترة
+
+179
+00:21:29,890 --> 00:21:33,330
+مفتوحة ال cardinal number تبعها بيساوي ال cardinal
+
+180
+00:21:33,330 --> 00:21:36,310
+number تبع ال real numbers اللي هي uncountable set
+
+181
+00:21:36,310 --> 00:21:41,270
+وبالتالي infinite إذا التقاطع هذا infinite وهذا
+
+182
+00:21:41,270 --> 00:21:46,410
+بيقدي ان ال V
+
+183
+00:21:46,410 --> 00:21:57,490
+Delta of X تقاطعالـ a1 منزوعة منها الـ x هيطلع
+
+184
+00:21:57,490 --> 00:22:03,390
+بالتأكيد لا يساوي في لأن التقاطة هذا بيطلع
+
+185
+00:22:03,390 --> 00:22:07,630
+infinite وبالتالي هيك بيكون أثبتنا أن كل delta
+
+186
+00:22:07,630 --> 00:22:13,370
+neighborhood ل x بيتقاطع مع a1 في نقطة مختلفة عن x
+
+187
+00:22:23,000 --> 00:22:31,840
+الحالة التانية case two ان ال delta هذه تكون اكبر
+
+188
+00:22:31,840 --> 00:22:36,670
+من اوسع واحدبرضه فى الحاله دى بنا نثبت انه كل
+
+189
+00:22:36,670 --> 00:22:41,170
+delta neighborhood ال X بتقاطع مع A واحد فى نقطه
+
+190
+00:22:41,170 --> 00:22:49,730
+مختلفه عن ال X نشوف مع بعض in this case in
+
+191
+00:22:49,730 --> 00:22:57,310
+this case ال X negative او ال negative delta X
+
+192
+00:22:57,310 --> 00:23:01,310
+موجب delta هذا اللى هو ال delta neighborhood ل X
+
+193
+00:23:01,310 --> 00:23:06,620
+تقاطعالمجموعة A1 اللي هي الفترة المفتوحة من 0 إلى
+
+194
+00:23:06,620 --> 00:23:17,040
+1 هيطلع بساوي الفترة المفتوحة من 0 إلى 1 لأن الـ
+
+195
+00:23:17,040 --> 00:23:24,780
+Delta هنا أكبر من أو ساوي الواحد يعني هي عندي S 0
+
+196
+00:23:24,780 --> 00:23:36,180
+إلى 1 هذه اللي هي المجموعة A وهي Xنقطة ما داخل
+
+197
+00:23:36,180 --> 00:23:42,300
+الفترة فلما يكون x زايد ال delta لما تكون ال delta
+
+198
+00:23:42,300 --> 00:23:49,160
+تبعتي أكبر من واحد فx زايد ال delta هتكون هنا و x
+
+199
+00:23:49,160 --> 00:23:55,880
+سالب ال delta بالتأكيد هتكون هنا وبالتالي ال delta
+
+200
+00:23:55,880 --> 00:24:01,640
+neighborhood ل x هيحتويالمجموعة a واحد وبالتالي
+
+201
+00:24:01,640 --> 00:24:06,760
+تقاطع معاها تطلع المجموعة a واحد وهذا is infinite
+
+202
+00:24:06,760 --> 00:24:11,680
+وبالتالي
+
+203
+00:24:11,680 --> 00:24:18,060
+اذا ال delta neighborhood هذا تقاطع الفترة
+
+204
+00:24:18,060 --> 00:24:20,460
+المفتوحة minus x
+
+205
+00:24:23,650 --> 00:24:29,910
+لا أكيد بتأكيد لا يساوي five okay تمام اذا في
+
+206
+00:24:29,910 --> 00:24:34,470
+الحالتين ال condition تبع ال cluster point تتحقق
+
+207
+00:24:34,470 --> 00:24:44,730
+therefore by definition x is cluster point is
+
+208
+00:24:44,730 --> 00:24:50,780
+cluster point ofالست ا واحد اللي هي الفترة
+
+209
+00:24:50,780 --> 00:24:56,280
+المفتوحة من صفر الى واحد طبعا إذا هذا بثبت ال
+
+210
+00:24:56,280 --> 00:25:09,140
+claim الأولاني طبعا الآن هثبت claim تاني ال claim
+
+211
+00:25:09,140 --> 00:25:09,880
+التاني
+
+212
+00:25:16,660 --> 00:25:25,180
+النقطة 0 is a cluster point
+
+213
+00:25:25,180 --> 00:25:36,460
+of set A1 الفترة مفتوحة من 0 إلى 1 لبرهان
+
+214
+00:25:36,460 --> 00:25:37,020
+ذلك
+
+215
+00:25:47,440 --> 00:25:55,860
+to see this let نبدأ let delta أكبر من السفر be
+
+216
+00:25:55,860 --> 00:26:03,120
+given فهنا
+
+217
+00:26:03,120 --> 00:26:11,380
+لأي delta ال delta
+
+218
+00:26:11,380 --> 00:26:16,140
+neighborhood للسفر اللي هو هيطلع
+
+219
+00:26:18,570 --> 00:26:28,630
+سالب دلتا زاد سفر وموجب دلتا زاد سفر فتقاطع هذا مع
+
+220
+00:26:28,630 --> 00:26:35,150
+الفترة المفتوحة من سفر إلى واحد بساوي
+
+221
+00:26:36,900 --> 00:26:44,360
+في خيارين إما الفترة المفتوحة من سفر إلى دلتا إذا
+
+222
+00:26:44,360 --> 00:26:52,680
+كانت ال delta أصغر من واحد طبعا أكبر من سفر وبساوي
+
+223
+00:26:52,680 --> 00:26:57,600
+الفترة المفتوحة من سفر إلى واحد إذا كان ال delta
+
+224
+00:26:57,600 --> 00:27:04,220
+أكبر من أو ساوي الواحد زي ما شوفنا في برهان الكلام
+
+225
+00:27:04,220 --> 00:27:12,240
+الأولانيمظبوط هاي الاندي الفترة من سفر إلى واحد
+
+226
+00:27:12,240 --> 00:27:19,520
+هذه المجموعة A1 وهي
+
+227
+00:27:19,520 --> 00:27:27,700
+X نقطة .. لأ هاي السفر بالدفتر أن السفر cluster
+
+228
+00:27:27,700 --> 00:27:34,660
+point للمجموعة A1فأخدت أي delta أكبر من السفر الان
+
+229
+00:27:34,660 --> 00:27:38,260
+ال delta هذه لو كانت ال delta هذه اذا هي سالب
+
+230
+00:27:38,260 --> 00:27:42,720
+delta موجب delta لو كانت ال delta هذه أصغر من واحد
+
+231
+00:27:42,720 --> 00:27:47,120
+فتقاطع ال delta neighborhood مع ال a واحد هيكون
+
+232
+00:27:47,120 --> 00:27:51,140
+الجزء هذا اللي هو الفترة المفتوحة من سفر ل delta
+
+233
+00:27:51,140 --> 00:27:57,100
+صح؟ و لو كانت ال delta هذه أكبر من واحد لو كانت ال
+
+234
+00:27:57,100 --> 00:27:58,940
+delta هذه أكبر من واحد
+
+235
+00:28:01,680 --> 00:28:11,060
+فال .. ف delta هتكون هاي delta أكبر من واحد و سالف
+
+236
+00:28:11,060 --> 00:28:15,120
+delta هتكون هان و بالتالي ال delta لبرهود هذا
+
+237
+00:28:15,120 --> 00:28:24,220
+تقاطع مع a واحد بيساوي a واحد مظبوط صح؟ تمام؟ و في
+
+238
+00:28:24,220 --> 00:28:27,840
+كل الأحوال التقاطع هذا بيطلع infinite is infinite
+
+239
+00:28:27,840 --> 00:28:31,000
+infinite set لأنه open interval
+
+240
+00:28:38,080 --> 00:28:46,040
+تقاطة a-a1 هو
+
+241
+00:28:46,040 --> 00:28:53,940
+نفس تقاطة a1
+
+242
+00:29:00,690 --> 00:29:03,830
+إذن هي اللي أثبتت إن كل delta neighborhood للسفر
+
+243
+00:29:03,830 --> 00:29:09,590
+يتقاطع مع المجموعة A1 في نقطة مختلفة عن السفر في
+
+244
+00:29:09,590 --> 00:29:14,070
+حقيقة الأمر في حقيقة الأمر كل delta neighborhood
+
+245
+00:29:14,070 --> 00:29:19,250
+للسفر بتقاطع مع A1 في عدد لانهائي من النقاط اللي
+
+246
+00:29:19,250 --> 00:29:24,350
+موجودة في A1 ومختلفة عن السفر إذن by definition
+
+247
+00:29:24,350 --> 00:29:27,610
+zero is a cluster
+
+248
+00:29:30,120 --> 00:29:36,540
+point of A1 اللي هي الفترة مفتوحة من صفر لواحد
+
+249
+00:29:36,540 --> 00:29:43,720
+يبقى لإكمال البرهان يمكن أن يظهر الكليل التالت
+
+250
+00:29:43,720 --> 00:29:55,560
+باقي أثبت أن الواحد is a cluster point of set A1
+
+251
+00:29:55,560 --> 00:30:02,300
+اللي هي الفترة مفتوحة من صفر لواحدوبرهان ال claim
+
+252
+00:30:02,300 --> 00:30:07,360
+التالت زي .. similar لبرهان ال claim التالت إذا
+
+253
+00:30:07,360 --> 00:30:19,280
+هنا the proof its proof is similar to
+
+254
+00:30:19,280 --> 00:30:20,260
+claim to
+
+255
+00:30:23,650 --> 00:30:27,990
+فحاسيبكم انتوا تكتبوا وبالتالي هيك بالكون أثبتنا
+
+256
+00:30:27,990 --> 00:30:32,250
+ان كل نقطة في الفترة المغلطة سواء كانت نقطة الطرف
+
+257
+00:30:32,250 --> 00:30:37,570
+اللي هي 0 او 1 او نقطة داخلية interior point نقطة
+
+258
+00:30:37,570 --> 00:30:41,290
+داخل الفترة المغلطة كل النقاط هذه بتطلع cluster
+
+259
+00:30:41,290 --> 00:30:47,970
+points لمجموعة A1 اللي هي الفترة المفتوحة تمام؟
+
+260
+00:30:52,450 --> 00:31:05,790
+بالمثل ممكن إثبات إن
+
+261
+00:31:05,790 --> 00:31:09,250
+كل نقطة في الفترة المغلقة is cluster point
+
+262
+00:31:09,250 --> 00:31:16,990
+للمجموعة A2 اللي هي الفترة المغلقة من 0 إلى 1
+
+263
+00:31:16,990 --> 00:31:20,390
+والبرهان
+
+264
+00:31:20,390 --> 00:31:28,770
+هو نفسهبنعمل three claims وفي كل برهان هيكون الفرق
+
+265
+00:31:28,770 --> 00:31:35,430
+انه عندي انا a بدل a 1 هيكون a 2 فهتكون اللي هو
+
+266
+00:31:35,430 --> 00:31:41,370
+الفترات هذه فترة مغلقة من سفر إلى واحد وبالتالي
+
+267
+00:31:41,370 --> 00:31:47,910
+بيصير هذه الفترة من هنا مغلقة عند السفر ومغلقة عند
+
+268
+00:31:47,910 --> 00:31:54,070
+السفر ومغلقة عند الواحدو هكذا نفس البرهان نسخ لسق
+
+269
+00:31:54,070 --> 00:31:59,790
+مع التعديلات البسيطة ان اي واحد الآن أصبحت بدل ما
+
+270
+00:31:59,790 --> 00:32:03,270
+كانت فترة مفتوحة من صفر لواحد أصبحت فترة مغلقة
+
+271
+00:32:03,270 --> 00:32:06,830
+وبالتالي في التقاطعات الحسابات نغلق اللي هو
+
+272
+00:32:06,830 --> 00:32:13,590
+الفترات and الحاجة المطلوبة okay وبالتالي نفس
+
+273
+00:32:13,590 --> 00:32:18,910
+البرهان will go through هيمشي بالتمام والكمالOkay
+
+274
+00:32:18,910 --> 00:32:24,270
+تمام؟ إذا هذا البرهان مشابه لبرهان المثال الأول
+
+275
+00:32:24,270 --> 00:32:34,010
+ناخد كمان مثال آخر مثال
+
+276
+00:32:34,010 --> 00:32:43,770
+تالت every every
+
+277
+00:32:43,770 --> 00:32:44,310
+finite
+
+278
+00:32:46,970 --> 00:32:59,350
+set A contained in R has no
+
+279
+00:32:59,350 --> 00:33:03,070
+cluster
+
+280
+00:33:03,070 --> 00:33:09,730
+points
+
+281
+00:33:09,730 --> 00:33:19,970
+كل finite set مالهاش ولا cluster pointوالبرهان سهل
+
+282
+00:33:19,970 --> 00:33:24,950
+proof say
+
+283
+00:33:24,950 --> 00:33:35,030
+دعنا ال set a نسمي عناصرها a1, a2 الى an هدف مش
+
+284
+00:33:35,030 --> 00:33:39,950
+هدف finite set اذا عناصرهم ممكن اعملهم list a1, a2
+
+285
+00:33:39,950 --> 00:33:47,530
+الى anو ممكن اعملهم order ارتبهم حسب المؤشر تبعهم
+
+286
+00:33:47,530 --> 00:33:57,450
+يعني a1 اصغر من a2 اصغر من a3 اصغر من اصغر من am
+
+287
+00:33:57,450 --> 00:34:03,570
+ممكن نعمل كلمة من هذا ولا لأ ممكن by the ordering
+
+288
+00:34:03,570 --> 00:34:04,370
+principle
+
+289
+00:34:11,360 --> 00:34:20,780
+أو باستخدام ال ordering تبع ال real numbers إذا
+
+290
+00:34:20,780 --> 00:34:30,880
+هي عندي ال 6A تبعتي هي خط العداد وهي A1 وهي A2 وهي
+
+291
+00:34:30,880 --> 00:34:36,160
+A3 مش شرط المسافة بين كل أنصار والتاني تكون
+
+292
+00:34:36,160 --> 00:34:46,230
+متساوية وهكذا إلىأخر عنصر AN ف
+
+293
+00:34:46,230 --> 00:34:49,910
+fix X
+
+294
+00:34:49,910 --> 00:35:01,190
+ينتمي إلى R و بد أثبت أن claim X is not a cluster
+
+295
+00:35:01,190 --> 00:35:01,910
+point
+
+296
+00:35:09,790 --> 00:35:12,850
+بالتالي المجموع A ليس لديه اي cluster point
+
+297
+00:35:12,850 --> 00:35:16,770
+بالتالي المجموع A ليس لديه اي cluster point
+
+298
+00:35:16,770 --> 00:35:16,790
+بالتالي المجموع A ليس لديه اي cluster point
+
+299
+00:35:16,790 --> 00:35:16,790
+بالتالي المجموع A ليس لديه اي cluster point
+
+300
+00:35:16,790 --> 00:35:17,530
+بالتالي المجموع A ليس لديه اي cluster point
+
+301
+00:35:17,530 --> 00:35:17,750
+بالتالي المجموع A ليس لديه اي cluster point
+
+302
+00:35:17,750 --> 00:35:17,750
+بالتالي المجموع A ليس لديه اي cluster point
+
+303
+00:35:17,750 --> 00:35:18,430
+بالتالي المجموع A ليس لديه اي cluster point
+
+304
+00:35:18,430 --> 00:35:19,070
+بالتالي المجموع A ليس لديه اي cluster point
+
+305
+00:35:19,070 --> 00:35:19,690
+بالتالي المجموع A ليس لديه اي cluster point
+
+306
+00:35:19,690 --> 00:35:25,170
+بالتالي المجموع A ليس لديه اي cluster point
+
+307
+00:35:25,170 --> 00:35:28,870
+بالتالي المجموع A ليس لديه اي cluster point
+
+308
+00:35:28,870 --> 00:35:33,950
+بالتالي المجم
+
+309
+00:35:37,690 --> 00:35:44,410
+إما X تنتمي إلى A أو X تنتمي إلى الـ complement
+
+310
+00:35:44,410 --> 00:35:50,530
+يعني لا تنتمي إلى A صح؟
+
+311
+00:35:50,530 --> 00:35:59,870
+ففي الحالة الأولى case واحد أثر من X تنتمي إلى A
+
+312
+00:35:59,870 --> 00:36:03,970
+وبالدفع تثبت إن X ليست cluster point
+
+313
+00:36:10,180 --> 00:36:18,560
+say x بساوي a m for some m أكبر من أو ساوي واحد
+
+314
+00:36:18,560 --> 00:36:24,180
+أصغر من أو ساوي مش هذا ال x موجود في a و a على
+
+315
+00:36:24,180 --> 00:36:29,460
+سرها a واحد إلى a n إذا هذا ال x هو a m for some m
+
+316
+00:36:29,460 --> 00:36:36,300
+بين واحد و n طيب let
+
+317
+00:36:37,980 --> 00:36:48,860
+delta بساوي نص المسافة ال minimum المسافة
+
+318
+00:36:48,860 --> 00:37:04,760
+بين am minus am minus واحد وam زائد واحد minus am
+
+319
+00:37:11,460 --> 00:37:21,160
+يعني هاي ال X هاي ال M هاي AM وهاي AM زاد واحد
+
+320
+00:37:21,160 --> 00:37:30,960
+والأنصر اللي جاب لها AM minus واحد احنا قلنا ال X
+
+321
+00:37:30,960 --> 00:37:39,850
+سبعتي ال X سبعتي هي ال AM الآن باخدالمسافة هذه
+
+322
+00:37:39,850 --> 00:37:46,030
+اللي هي بين a m زي دول اللي هي المسافة هذه و باخد
+
+323
+00:37:46,030 --> 00:37:51,150
+المسافة هذه بين a m و a m سالب واحد لازم واحدة
+
+324
+00:37:51,150 --> 00:37:55,730
+تكون أصغر من أو يساوي التانية باخدها ال minimum ال
+
+325
+00:37:55,730 --> 00:37:57,730
+minimum .. ال minimum بين المسافتين .. الأصغر بين
+
+326
+00:37:57,730 --> 00:38:02,650
+المسافتين هدول و باخد نصها و باخد نصها بسميها
+
+327
+00:38:02,650 --> 00:38:08,720
+delta فنص .. لو قلنا الأصغرلو قلنا مثلا الأصغر
+
+328
+00:38:08,720 --> 00:38:15,400
+اللي هي هذه فنص الدلتا هذا هي فإذا ال delta هتكون
+
+329
+00:38:15,400 --> 00:38:21,040
+المسافة هذه و بكوّن delta neighborhood حوالين ال X
+
+330
+00:38:21,040 --> 00:38:29,720
+الأن ال delta neighborhood هذا then verify
+
+331
+00:38:29,720 --> 00:38:34,860
+ممكنكم تتحققوا verify that
+
+332
+00:38:37,340 --> 00:38:43,600
+الـ Delta neighborhood V Delta ل A M اللي هي ال X
+
+333
+00:38:43,600 --> 00:38:46,760
+تقاطع
+
+334
+00:38:46,760 --> 00:38:55,280
+ال set A فاي منزوعة منها A M هيطلع بساوي الفاي
+
+335
+00:38:55,280 --> 00:39:01,980
+مافيش تقاطع بينهم وبالتالي therefore by definition
+
+336
+00:39:04,480 --> 00:39:13,200
+ام اكس بساوي ام is not a
+
+337
+00:39:13,200 --> 00:39:22,360
+cluster point of set A لأن عشان تكونما تكونيش
+
+338
+00:39:22,360 --> 00:39:28,040
+cluster point لمجموعة A لازم اثبت انه يوجد there
+
+339
+00:39:28,040 --> 00:39:35,840
+exist delta neighborhood لل X تبعت اللي هي AM بحيث
+
+340
+00:39:35,840 --> 00:39:42,980
+انه ال delta neighborhood هذا مايتقطعش مع ال 6A في
+
+341
+00:39:42,980 --> 00:39:50,440
+اي نقطة مختلفة عن النقطة X وهذا حصلOkay تمام اذا
+
+342
+00:39:50,440 --> 00:39:56,140
+هذا في حالة لما ال X تكون موجودة في A الحالة
+
+343
+00:39:56,140 --> 00:40:03,340
+التانية ان ال case 2 case
+
+344
+00:40:03,340 --> 00:40:13,040
+2 ان ال X لا تنتمي ال ال set A ففي الحالة هذه
+
+345
+00:40:14,910 --> 00:40:20,590
+معناته x مابستويش ولا واحد من العناصر هذه فالحالة
+
+346
+00:40:20,590 --> 00:40:26,690
+هذه ممكن اجزها الى تلت حالات الحالة
+
+347
+00:40:26,690 --> 00:40:32,750
+الأولى ان ال x تبعتي تكون اصغر من a واحدوبالتالي
+
+348
+00:40:32,750 --> 00:40:37,010
+واضح ان المسافة بين X و A واحد كبيرة وباخد نص
+
+349
+00:40:37,010 --> 00:40:43,930
+المسافة ديلتا إذا هيوجد دلتا نبرود ل X ومابتقطعش
+
+350
+00:40:43,930 --> 00:40:47,650
+مع المجموعة A بالمرة وبالتالي X is not cluster
+
+351
+00:40:47,650 --> 00:40:54,250
+point ممكن الحالة التانية أن X تكون أكبر منالـ AM
+
+352
+00:40:54,250 --> 00:40:59,470
+برضه باخد المسافة دي بجيسها و باخد نصها على انه
+
+353
+00:40:59,470 --> 00:41:03,670
+Delta و بكون Delta neighborhood حوالين ال X هذا ال
+
+354
+00:41:03,670 --> 00:41:06,530
+Delta neighborhood واضح انه مابتخطعش مع ال set A
+
+355
+00:41:06,530 --> 00:41:11,910
+بالمرة وبالتالي إذا X في الحالة دي ليست cluster
+
+356
+00:41:11,910 --> 00:41:20,190
+point الحالة التالتة ان X تكون واقعة بين عنصرين من
+
+357
+00:41:20,190 --> 00:41:25,170
+عناصر ال setفباخد اللي هو المسافة الأصغر من
+
+358
+00:41:25,170 --> 00:41:29,310
+المسافتين هدول و هي تكون هادى و باخد نصها delta و
+
+359
+00:41:29,310 --> 00:41:32,890
+بكون delta neighborhood حواليها هذا ال delta
+
+360
+00:41:32,890 --> 00:41:36,510
+neighborhood بتخطعش مع المجموعه هاد المرة وبالتالي
+
+361
+00:41:36,510 --> 00:41:38,990
+حسب التعريف x ليست cluster point
+
+362
+00:41:43,100 --> 00:41:46,360
+ما احنا قلنا اذا كانت X تنتمي لأيه هي برهانة MA لا
+
+363
+00:41:46,360 --> 00:41:52,640
+تنتمي لأ تنتمي اه ليش قدماها بين تنتمي و لا تنتمي
+
+364
+00:41:52,640 --> 00:41:58,420
+ما هي ال X ما تنتميش لأيه فممكن تكون بين عنصرين من
+
+365
+00:41:58,420 --> 00:42:03,360
+عنصرهم هي تنتمي ل R ما تنتميش لأيه فممكن تكون
+
+366
+00:42:03,360 --> 00:42:09,640
+موجودة بين A2 و A3 صح او بين A1 و A2 او بين A3 او
+
+367
+00:42:09,640 --> 00:42:16,530
+A و هكذاأو ممكن تكون ال X على يمين ال AN أو حالة
+
+368
+00:42:16,530 --> 00:42:20,190
+تالتة X تكون على يسار ال A واحد وشوفنا في كل
+
+369
+00:42:20,190 --> 00:42:24,310
+الحالات هذه التلاتة أنه بقدر ألاقي delta
+
+370
+00:42:24,310 --> 00:42:27,990
+neighborhood حوالين ال X لا يتقاطع مع المجموعة A
+
+371
+00:42:27,990 --> 00:42:32,330
+بالمرة وبالتالي X ليست cluster point إذا الكلام
+
+372
+00:42:32,330 --> 00:42:37,110
+هذا واضح حاولوا تكتبوه بطريقة يعني منطقية okay
+
+373
+00:42:37,110 --> 00:42:37,810
+تمام؟
+
+374
+00:42:42,120 --> 00:42:46,900
+أنا متخيلة ال A عبارة عن set دائر مثلا أنا هيك
+
+375
+00:42:46,900 --> 00:42:50,340
+متخيلة و أنه مثلا ال cluster point هي عبارة عن
+
+376
+00:42:50,340 --> 00:42:54,820
+نقطة .. لأ ال A ماتتخيليش ال A عند ال set ال A هي
+
+377
+00:42:54,820 --> 00:43:01,140
+جزء من الأعداد الحقيقية subset من R و R خطلازم
+
+378
+00:43:01,140 --> 00:43:04,200
+يعني تتخيل الحاجات زمان على ال interior point و ال
+
+379
+00:43:04,200 --> 00:43:08,260
+boundary هذا في ال topology حاجة تانية هي زي ات
+
+380
+00:43:08,260 --> 00:43:11,140
+شبهها يعني ممكن نفهمها بهذا الطريقة بس ممكن اه
+
+381
+00:43:11,140 --> 00:43:13,860
+ممكن بس احنا هنا على ال real line يعني خليني احنا
+
+382
+00:43:13,860 --> 00:43:18,820
+نتقيد بال sets اللي موجودة على ال real line اما هو
+
+383
+00:43:18,820 --> 00:43:24,080
+طبعا في تعني من الكلام هذا في حاجات اعم و فرغات
+
+384
+00:43:24,080 --> 00:43:28,550
+اعم من ال ..ال real number اللي هو ال topological
+
+385
+00:43:28,550 --> 00:43:36,030
+spaces و هذا موضوع طبعا متشعب و بده يعني تدرس ال
+
+386
+00:43:36,030 --> 00:43:40,530
+topology عشان تفهم كل شيء okay فى اي اسئلة تانى؟
+
+387
+00:43:40,530 --> 00:43:44,430
+okay لنكتفي بهذا القدر و ان شاء الله بنكمل
+
+388
+00:43:44,430 --> 00:43:52,670
+المحاضرة الجاية الموضوع و بنخش بتعريف ال limit لل
+
+389
+00:43:52,670 --> 00:43:53,110
+functions
+
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/Gx7j9GpXuiI_postprocess.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/Gx7j9GpXuiI_postprocess.srt
new file mode 100644
index 0000000000000000000000000000000000000000..47dcfb670cadcd35075a00427c5e166a7488f467
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/Gx7j9GpXuiI_postprocess.srt
@@ -0,0 +1,1780 @@
+1
+00:00:21,580 --> 00:00:26,880
+بسم الله الرحمن الرحيم اليوم ان شاء الله هنبدأ
+
+2
+00:00:26,880 --> 00:00:34,000
+chapter خمسة و هذا اخر chapter هناخده في ال course
+
+3
+00:00:34,000 --> 00:00:50,080
+فانواع ال chapter هذا continuous
+
+4
+00:00:53,880 --> 00:01:01,820
+functions الدوالة المتصلة و
+
+5
+00:01:01,820 --> 00:01:08,460
+أول section برضه section خمسة واحد في هذا ال
+
+6
+00:01:08,460 --> 00:01:16,320
+chapter برضه عنوانه continuous functions
+
+7
+00:01:24,100 --> 00:01:29,280
+الدولة المتصلة فنعرف شو معنى الدولة تكون متصلة عن
+
+8
+00:01:29,280 --> 00:01:35,160
+نقطة definition let
+
+9
+00:01:35,160 --> 00:01:49,280
+f be function from a to r and c be element of a we
+
+10
+00:01:49,280 --> 00:02:00,630
+sayإنه ال function if is continuous if
+
+11
+00:02:00,630 --> 00:02:05,770
+is continuous at
+
+12
+00:02:05,770 --> 00:02:18,950
+x بساوي c if إذا تحقق الشرط التالي for
+
+13
+00:02:18,950 --> 00:02:20,470
+every
+
+14
+00:02:22,680 --> 00:02:29,400
+إبسلون أكبر من السفر نقدر نرد عليها delta تعتمد
+
+15
+00:02:29,400 --> 00:02:37,840
+على إبسلون positive number بحيث أنه لكل X لكل
+
+16
+00:02:37,840 --> 00:02:44,090
+X في Aو ال absolute value ل x minus c أصغر من
+
+17
+00:02:44,090 --> 00:02:52,170
+delta فهذا بتضمن ان absolute f of x minus f of c
+
+18
+00:02:52,170 --> 00:03:01,630
+أصغر من ال epsilon فهذا
+
+19
+00:03:01,630 --> 00:03:13,010
+بنسميه this is calledthis is called epsilon delta
+
+20
+00:03:13,010 --> 00:03:18,770
+definition of
+
+21
+00:03:18,770 --> 00:03:31,170
+continuity لأن
+
+22
+00:03:31,170 --> 00:03:36,790
+هذا تعريف epsilon delta للاتصال لحظو هذا التعريف
+
+23
+00:03:36,790 --> 00:03:44,530
+تقريبا هوهو تعريف انه limit ال function f of x لما
+
+24
+00:03:44,530 --> 00:03:52,310
+x تقوله c بساوي f of c هدد
+
+25
+00:03:52,310 --> 00:04:07,210
+كانت c is a cluster point طب
+
+26
+00:04:07,210 --> 00:04:13,930
+لحظة انتوالما عرفنا احنا ما معناه انه ال limit ل
+
+27
+00:04:13,930 --> 00:04:18,710
+function and x بيساوي C و C cluster point للمجموع
+
+28
+00:04:18,710 --> 00:04:24,570
+A بيساوي عدد L بدلنا L هنا ب F و C صح؟ معناه كان
+
+29
+00:04:24,570 --> 00:04:30,290
+لكل إبسلون فيه Delta بحيث لكل X في A و ال X هذه
+
+30
+00:04:30,290 --> 00:04:37,540
+كانت مختلفة لا تساوي C فكنا نحط هنا أكبر من 0فإذا
+
+31
+00:04:37,540 --> 00:04:41,480
+كانت المسافة هذه أصغر من دلتا تطلع المسافة من f of
+
+32
+00:04:41,480 --> 00:04:46,040
+x وال L اللي هي ال limit هنا طبعا احنا بدلنا ال L
+
+33
+00:04:46,040 --> 00:04:50,940
+ب F of C فبين هذا يطلع أصغر من X هنا تقريبا نفس
+
+34
+00:04:50,940 --> 00:04:56,480
+التعريف if
+
+35
+00:04:56,480 --> 00:05:00,460
+if
+
+36
+00:05:00,460 --> 00:05:09,090
+is not continuousلو كانت ال F ليست متصلة عند
+
+37
+00:05:09,090 --> 00:05:14,910
+النقطة C فبنقول if
+
+38
+00:05:14,910 --> 00:05:31,810
+F fails to be continuous at C we say ان F is
+
+39
+00:05:31,810 --> 00:05:32,990
+discontinuous
+
+40
+00:05:38,310 --> 00:05:46,350
+discontinuous at c إذا لو كانت الدالة مش متصلة عن
+
+41
+00:05:46,350 --> 00:05:52,710
+c يعني شرط الاتصال هذا مش متحقق فبنقول أن الدالة
+
+42
+00:05:52,710 --> 00:05:57,610
+discontinuous منفصلة عند النقطة c okay تمام
+
+43
+00:06:09,660 --> 00:06:17,360
+بنلاحظ انه ال .. زي ما شوفنا في section 4-1 تعريف
+
+44
+00:06:17,360 --> 00:06:21,840
+epsilon delta لل limits of functions في بكافة
+
+45
+00:06:21,840 --> 00:06:26,600
+neighborhood definition وهنا برضه تعريف ال epsilon
+
+46
+00:06:26,600 --> 00:06:31,760
+delta definition للاتصال عن النقطة في بكافة
+
+47
+00:06:31,760 --> 00:06:36,400
+neighborhood definition فنكتب ال neighborhood
+
+48
+00:06:36,400 --> 00:06:37,340
+definition هذا
+
+49
+00:06:46,200 --> 00:06:53,400
+لت if دي function from a to r و c belong to a then
+
+50
+00:06:53,400 --> 00:07:02,480
+the following statements are equivalent واحد
+
+51
+00:07:02,480 --> 00:07:11,180
+ال function if is continuous is continuous at x
+
+52
+00:07:11,180 --> 00:07:12,540
+بساوي z
+
+53
+00:07:20,900 --> 00:07:26,360
+إتنين هذا طبعا إتنين نسميه in labor hood
+
+54
+00:07:26,360 --> 00:07:31,940
+definition of continuity
+
+55
+00:07:45,120 --> 00:07:48,580
+الـ neighborhood definition للـ continuity ايش
+
+56
+00:07:48,580 --> 00:07:57,920
+بيقول لكل for every epsilon neighborhood v epsilon
+
+57
+00:07:57,920 --> 00:08:05,700
+لنقطة f of c there
+
+58
+00:08:05,700 --> 00:08:18,440
+exist delta neighborhood v deltaof C لنقطة C طبعا
+
+59
+00:08:18,440 --> 00:08:26,200
+هذا epsilon neighborhood ل F of C يوجد delta
+
+60
+00:08:26,200 --> 00:08:38,660
+neighborhood V Delta of C بحيث انه لكل X تنتمي إلى
+
+61
+00:08:38,660 --> 00:08:47,830
+A تقاطع الـ Deltaneighborhood ل C لازم هذا يضمن ان
+
+62
+00:08:47,830 --> 00:08:53,050
+صورة ال X تنتمي
+
+63
+00:08:53,050 --> 00:09:04,590
+الى D epsilon ل F of C that
+
+64
+00:09:04,590 --> 00:09:08,630
+is that
+
+65
+00:09:08,630 --> 00:09:11,910
+is هذا يعني ان ال
+
+66
+00:09:14,980 --> 00:09:23,060
+الـ image للست A تقاطع V Delta of C is contained
+
+67
+00:09:23,060 --> 00:09:34,140
+in الـ Epsilon neighbourhood لـ F of C
+
+68
+00:09:34,140 --> 00:09:40,100
+هاي
+
+69
+00:09:40,100 --> 00:09:47,330
+كان في عنديزي هيك مثلا يكون في اندي فانكشن زي هذه
+
+70
+00:09:47,330 --> 00:09:57,210
+y بساوي f of x وقلنا
+
+71
+00:09:57,210 --> 00:10:03,810
+انه لو كانت x او c c
+
+72
+00:10:03,810 --> 00:10:07,670
+نقطة ال dial عندها متصلة هي f of c
+
+73
+00:10:11,410 --> 00:10:17,830
+ما معناه ان الدالة متصلة عند X بساوي C معناه لو
+
+74
+00:10:17,830 --> 00:10:23,770
+أخدت لأي
+
+75
+00:10:23,770 --> 00:10:30,850
+إبسلون أكبر من سفر فيه Delta أو لو أخدت أي إبسلون
+
+76
+00:10:30,850 --> 00:10:31,290
+neighborhood
+
+77
+00:10:34,530 --> 00:10:38,270
+يعني النقطة هذه F of C زاد Epsilon النقطة هذه
+
+78
+00:10:38,270 --> 00:10:48,430
+المسافة هذه Epsilon فهذه F of C سالب Epsilon فهذه
+
+79
+00:10:48,430 --> 00:10:53,610
+الفترة المفتوحة عبارة عن Epsilon neighborhood ل F
+
+80
+00:10:53,610 --> 00:10:54,150
+of C
+
+81
+00:10:57,200 --> 00:11:01,620
+فلأي إبسلون أكبر من السفر ممكن أكوّن إبسلون
+
+82
+00:11:01,620 --> 00:11:06,420
+neighborhood ل F of C وبالتالي بقدر أرد على الـ
+
+83
+00:11:06,420 --> 00:11:14,580
+Epsilon neighborhood هذا بـ Delta يعني
+
+84
+00:11:14,580 --> 00:11:20,980
+أكوّن Delta neighborhood هنا C minus Delta C موجة
+
+85
+00:11:20,980 --> 00:11:21,460
+بDelta
+
+86
+00:11:26,200 --> 00:11:37,060
+إذاً هذا عبارة عن V Delta V Delta ل C إذاً
+
+87
+00:11:37,060 --> 00:11:43,200
+لأي إبسلون لأي إبسلون neighborhood ل F of C بقدر
+
+88
+00:11:43,200 --> 00:11:52,720
+ألاقي Delta neighborhood للمقطة C بحيث أنه لكل Xلو
+
+89
+00:11:52,720 --> 00:12:01,620
+أخدت x نقطة في الـ delta neighborhood فصورتها f of
+
+90
+00:12:01,620 --> 00:12:09,060
+x هتطلع تنتمي لل epsilon neighborhood لل F of C
+
+91
+00:12:09,060 --> 00:12:17,140
+okay تمام فهذا هو نفسه هذا بكافي التعريف هذا بكافي
+
+92
+00:12:17,140 --> 00:12:20,660
+التعريف ال epsilon delta definition لل continuity
+
+93
+00:12:24,390 --> 00:12:29,850
+هي لكل إبسلون لكل إبسلون أكبر من الصفر يعني كأني
+
+94
+00:12:29,850 --> 00:12:36,450
+بقول لكل إبسلون نبرهود ل F و C يوجد Delta عدد موجب
+
+95
+00:12:36,450 --> 00:12:44,290
+فهذا معناه يوجد Delta نبرهود لل C بحيث أنه لكل X
+
+96
+00:12:44,290 --> 00:12:50,560
+المسافر لكل X تنتمي لكل X في Aو X بالتحقق
+
+97
+00:12:50,560 --> 00:12:55,980
+المتباينة دي معناته X سنتمي المسافة بين X و C أصغر
+
+98
+00:12:55,980 --> 00:12:56,380
+من Delta
+
+99
+00:13:02,120 --> 00:13:07,000
+فهذا بيقدي ان المسافة بين F of X و F of C هي F of
+
+100
+00:13:07,000 --> 00:13:12,160
+X و F of C أصغر من Epsilon يعني ال F of X هذه
+
+101
+00:13:12,160 --> 00:13:17,900
+تنتمي لل Epsilon برهود ل F of C إذن التعريفين هذول
+
+102
+00:13:17,900 --> 00:13:24,800
+متكافئين وهذا واضح من الرسم وبالتالي البرهان جاهز
+
+103
+00:13:24,800 --> 00:13:32,000
+من .. بس ترجمتهالحاجات هذه الى لغة ال neighborhood
+
+104
+00:13:32,000 --> 00:13:39,600
+اذا في لان تعريفين للاتصال على النقطة واحد epsilon
+
+105
+00:13:39,600 --> 00:13:45,400
+delta definition والتاني اللي بكافه neighborhood
+
+106
+00:13:45,400 --> 00:13:50,360
+definition طيب
+
+107
+00:13:50,360 --> 00:13:55,260
+ناخد بعض الملاحظات على تعريف الاتصال
+
+108
+00:14:16,000 --> 00:14:22,640
+إذا C هو مقاومة مقاومة
+
+109
+00:14:22,640 --> 00:14:30,180
+A ثم
+
+110
+00:14:30,180 --> 00:14:38,200
+F مستمر في X بساوي
+
+111
+00:14:42,830 --> 00:14:47,530
+لو كانت الـ C هذه cluster point فالاتصال ان C
+
+112
+00:14:47,530 --> 00:14:55,730
+بكافئ بكافئ ان ال limit ل F of X من تعريف ال
+
+113
+00:14:55,730 --> 00:15:03,570
+limits ان C بساوي F of C وهذا
+
+114
+00:15:03,570 --> 00:15:06,790
+طبعاً
+
+115
+00:15:06,790 --> 00:15:09,090
+this condition
+
+116
+00:15:12,780 --> 00:15:19,680
+is three in
+
+117
+00:15:19,680 --> 00:15:24,800
+one ال
+
+118
+00:15:24,800 --> 00:15:30,480
+definition هذا بكافئ تلت او الشرط هذا بكافئ تلت
+
+119
+00:15:30,480 --> 00:15:37,600
+شروط او هو تلت شروط في واحد اول شرط ان ال function
+
+120
+00:15:37,600 --> 00:15:39,540
+f and c is defined
+
+121
+00:15:43,900 --> 00:15:49,540
+يعني هذا عبارة عن عدد حقيقي name ال limit ل f of x
+
+122
+00:15:49,540 --> 00:15:56,180
+لما x تقول إلى c exist يعني عدد حقيقي والشرط
+
+123
+00:15:56,180 --> 00:16:04,880
+التالت أنه لازم ال limit لل function f and c بساوة
+
+124
+00:16:04,880 --> 00:16:09,980
+قيمة الدالة and c يعني عشان الدالة تكون متصلة عند
+
+125
+00:16:09,980 --> 00:16:16,020
+النقطة c في مجالهاو لو كانت الـ C هي cluster point
+
+126
+00:16:16,020 --> 00:16:21,790
+طبعاأو حتى لو ماكنتش cluster point فلازم التلاتة
+
+127
+00:16:21,790 --> 00:16:25,250
+صوروطها تتحقق الدالة معرفة عن C طبعا هذا لأن C
+
+128
+00:16:25,250 --> 00:16:30,450
+نقطة في مجال الدالة فلازم تكون معرفة عن ده لازم ال
+
+129
+00:16:30,450 --> 00:16:34,830
+limit ل F عن C تكون موجودة وقيمة ال limit بساوي
+
+130
+00:16:34,830 --> 00:16:39,290
+قيمة الدالة عند النقطة C لو أي واحد ماليش صوروط
+
+131
+00:16:39,290 --> 00:16:43,830
+التلاتة هدول اختل فبنقول ان ال function مش متصلة
+
+132
+00:16:43,830 --> 00:16:49,410
+عند النقطة COkay تمام واضح اذا لو كانت ال C هي دي
+
+133
+00:16:49,410 --> 00:16:53,510
+cluster point فتعريف الاتصال النقطة هو بالظبط
+
+134
+00:16:53,510 --> 00:16:58,470
+تعريف ان limited دالة ان C تكون موجودة و بتساوي
+
+135
+00:16:58,470 --> 00:17:02,570
+قيمتها ان C وهذا الشرط هو تلات شروط و ال C في ال A
+
+136
+00:17:02,570 --> 00:17:09,510
+نعم ال C تنتمي ل A اه طبعا ال C تنتمي ل A ال C
+
+137
+00:17:09,510 --> 00:17:11,130
+دائما تنتمي ل A
+
+138
+00:17:17,100 --> 00:17:22,120
+طب لو ماكناش ال c cluster point الملاحظة التانية
+
+139
+00:17:22,120 --> 00:17:29,440
+if c is not يعني لو كان ال c تنتمي طبعا دايما ال c
+
+140
+00:17:29,440 --> 00:17:40,980
+تنتمي ل a is not a cluster point is
+
+141
+00:17:40,980 --> 00:17:44,100
+not cluster point of a
+
+142
+00:17:48,950 --> 00:17:54,070
+then من تعريف ال cluster point لازم نلاقي delta
+
+143
+00:17:54,070 --> 00:18:05,430
+أكبر من السفر such that a تقاطع v delta of c بساوي
+
+144
+00:18:05,430 --> 00:18:06,850
+singleton c
+
+145
+00:18:11,300 --> 00:18:14,580
+ما معناه ان النقطة C الموجودة في A مايعنياش
+
+146
+00:18:14,580 --> 00:18:18,460
+cluster point او ما معناه ان C تنتمي ل A cluster
+
+147
+00:18:18,460 --> 00:18:24,380
+point معناها ان كل delta neighborhood لل C بيتقاطع
+
+148
+00:18:24,380 --> 00:18:30,400
+مع A في نقطة مختلفة عن ال C على الأقلمعناه ان الـ
+
+149
+00:18:30,400 --> 00:18:34,040
+C ما تكونش cluster point معناه ان يوجد delta
+
+150
+00:18:34,040 --> 00:18:37,040
+neighborhood واحد يعني يوجد delta عدد موجب
+
+151
+00:18:37,040 --> 00:18:40,780
+وبالتالي يوجد على الأقل delta neighborhood للـ C
+
+152
+00:18:40,780 --> 00:18:46,620
+وهذا ال delta neighborhoodمابتقاطعش مع a في أي
+
+153
+00:18:46,620 --> 00:18:50,660
+نقطة مختلفة عن ال c يعني التقاطع هذا بس في نقطة
+
+154
+00:18:50,660 --> 00:18:55,300
+واحدة c لأن ال c هي مركز ال neighborhood و c تنتمي
+
+155
+00:18:55,300 --> 00:19:03,320
+ل a فالتقاطع هذا مافيش فيه أي x مختلفة عن ال c في
+
+156
+00:19:03,320 --> 00:19:09,740
+الحالة هذه in this case in
+
+157
+00:19:09,740 --> 00:19:10,580
+this case
+
+158
+00:19:14,230 --> 00:19:23,970
+if is automatically continuous
+
+159
+00:19:23,970 --> 00:19:34,940
+at cالدولة في الحالة هذه بتكون متصلة تلقائيًا عند
+
+160
+00:19:34,940 --> 00:19:39,360
+النقطة C أو التعريف متحقق تلقائيًا ليه؟ لأنه
+
+161
+00:19:39,360 --> 00:19:44,060
+تعالوا نرجع للتعريف ما معناه أن F تكون متصلة عند
+
+162
+00:19:44,060 --> 00:19:49,660
+النقطة C معناه لأي epsilon neighborhood ل F و C
+
+163
+00:19:49,660 --> 00:19:53,680
+نقدر
+
+164
+00:19:53,680 --> 00:19:57,020
+نلاقي يوجد delta neighborhood ل C فخد ال delta
+
+165
+00:19:57,020 --> 00:20:00,040
+neighborhoodفي التعريف هذا خد الـ delta
+
+166
+00:20:00,040 --> 00:20:07,900
+neighborhood هو هذا ففي الحالة هذه لكل x تنتمي إلى
+
+167
+00:20:07,900 --> 00:20:12,340
+a تقاطع v delta و c ما التقاطع هذا مافيش فيه إلا
+
+168
+00:20:12,340 --> 00:20:17,100
+نقطة واحدة اللي هي c صح فلكل x موجود في التقاطع
+
+169
+00:20:17,100 --> 00:20:21,950
+هذا مافيش إلا x بالساوية cفصورة ال X هذه هي صورة
+
+170
+00:20:21,950 --> 00:20:28,210
+ال C وبالتالي صورة ال X هذه هي صورة ال C فهذه أكيد
+
+171
+00:20:28,210 --> 00:20:33,310
+تنتمي لإمسلون برهود ل F of C لأن ال F of C هي
+
+172
+00:20:33,310 --> 00:20:38,850
+المركز تبع الفترة هذه صح فهذا شرط متحقق trivially
+
+173
+00:20:38,850 --> 00:20:44,870
+تلقايا وبالتالي إذا السواء
+
+174
+00:20:46,570 --> 00:20:49,730
+سواء الـ C هنا كانت cluster point أو ماكنتش
+
+175
+00:20:49,730 --> 00:20:55,630
+cluster point فممكن نعتبر أن التعريف لاتصال النقطة
+
+176
+00:20:55,630 --> 00:21:00,450
+هو التعريف هذا لأن لو كانت ال C cluster point
+
+177
+00:21:00,450 --> 00:21:04,190
+فتعريف لاتصال النقطة هو هذا التعريف لو كانت ال C
+
+178
+00:21:04,190 --> 00:21:07,750
+ماهياش cluster point فهذا التعريف متحقق ال trivia
+
+179
+00:21:07,750 --> 00:21:12,380
+اللي بدهيوبالتالي مافيش داعي ان احنا نقول .. لما
+
+180
+00:21:12,380 --> 00:21:14,840
+نيجي نفحص الاتصال على النقطة C نقول هل ال C
+
+181
+00:21:14,840 --> 00:21:18,840
+cluster point او مش cluster point سواء كانت
+
+182
+00:21:18,840 --> 00:21:24,380
+cluster point او ماكانتش cluster point فالاتصال
+
+183
+00:21:24,380 --> 00:21:33,020
+and ال C بيصير هو .. يعني هل هذا شرط بتحقق او لا
+
+184
+00:21:41,130 --> 00:21:44,890
+طبعا زي ما اخدنا احنا ايام ما خدنا دراسنا ال
+
+185
+00:21:44,890 --> 00:21:54,950
+limits لل functions فكان
+
+186
+00:21:54,950 --> 00:21:57,590
+في عندي sequential criterion for limits
+
+187
+00:22:02,270 --> 00:22:06,810
+بنفس الطريقة في هنا sequential criterion for
+
+188
+00:22:06,810 --> 00:22:15,990
+continuity للاتصال إذا في عندي هنا sequential
+
+189
+00:22:15,990 --> 00:22:21,130
+criterion
+
+190
+00:22:21,130 --> 00:22:24,150
+for
+
+191
+00:22:24,150 --> 00:22:25,110
+continuity
+
+192
+00:22:35,670 --> 00:22:44,430
+let f be function from a to r و c نقطة في a then
+
+193
+00:22:44,430 --> 00:22:56,170
+the following statements are equivalent واحد
+
+194
+00:22:56,170 --> 00:23:08,010
+if is continuousif is continuous at c for
+
+195
+00:23:08,010 --> 00:23:11,910
+every for
+
+196
+00:23:11,910 --> 00:23:22,050
+every sequence x in contained in a with
+
+197
+00:23:22,050 --> 00:23:25,370
+limit
+
+198
+00:23:25,370 --> 00:23:41,270
+x in بساوي cنحن لدينا ان ال limit ل f of x n as n
+
+199
+00:23:41,270 --> 00:23:45,790
+tensor infinity بسوي f of c
+
+200
+00:23:51,740 --> 00:23:54,940
+الان الـ sequential criterion for continuity بتقول
+
+201
+00:23:54,940 --> 00:24:00,380
+عشان اثبت ان الدالة F continuous عند نقطة يكفي ان
+
+202
+00:24:00,380 --> 00:24:04,900
+انا اثبت ان لو اخدت اي sequence نهايتها اي
+
+203
+00:24:04,900 --> 00:24:07,660
+sequence في مجال الدالة طبعا كنا في ال limits
+
+204
+00:24:07,660 --> 00:24:13,020
+نشترط ان X in كل انصر في ال sequence مختلف عن ال C
+
+205
+00:24:13,020 --> 00:24:17,200
+هنا لأ ممكن يساوي ال C مش مشكلة هاي الاختلاف بس
+
+206
+00:24:17,200 --> 00:24:21,430
+بين ال sequential criterion for limitsو Sequential
+
+207
+00:24:21,430 --> 00:24:26,030
+criterion for continuity إنه لكل sequence x in في
+
+208
+00:24:26,030 --> 00:24:32,550
+مجال الدالة و نهايتها بتساوي c لازم اطلع عندي
+
+209
+00:24:32,550 --> 00:24:37,990
+نهاية ال image تبعت ال sequence x in بتساوي العدد
+
+210
+00:24:37,990 --> 00:24:42,860
+f و cوبرهان النظرية هذه زي برهان sequential
+
+211
+00:24:42,860 --> 00:24:49,120
+criterion for limits مع تعديلات طفيفة مع التعديلات
+
+212
+00:24:49,120 --> 00:24:58,580
+الطفيفة في التعريفين او في التعريف تبع الاتصال اذا
+
+213
+00:24:58,580 --> 00:25:11,090
+ال proof similar to proof ofsequential criterion
+
+214
+00:25:11,090 --> 00:25:19,570
+for limits for limits sequential criterion for
+
+215
+00:25:19,570 --> 00:25:34,190
+limits of functions in section أربعة واحد with
+
+216
+00:25:34,190 --> 00:25:38,030
+slight modification
+
+217
+00:25:45,120 --> 00:25:51,780
+مع تعديل بسيط مع تعديل بسيط التعديل هنا انه ال هنا
+
+218
+00:25:51,780 --> 00:25:58,180
+كنا نطلب ال X لا تساوي C وكمان كنا هناك نطلب انه C
+
+219
+00:25:58,180 --> 00:26:02,740
+تكون cluster point لكن شوفنا حتى لو C ماكنتش
+
+220
+00:26:02,740 --> 00:26:10,940
+cluster point فهذا برضه متحقق تلقائيا برضه
+
+221
+00:26:10,940 --> 00:26:11,700
+أخدنا
+
+222
+00:26:14,550 --> 00:26:18,230
+بعد ما أخدنا الـ sequential criterion for limits
+
+223
+00:26:18,230 --> 00:26:22,410
+of functions في section 4-1 أخدنا بعدها على طول
+
+224
+00:26:22,410 --> 00:26:29,850
+مباشرة divergence criterion for limits فهنا بقابل
+
+225
+00:26:29,850 --> 00:26:38,190
+ال divergence criterion اللي هو
+
+226
+00:26:38,190 --> 00:26:39,910
+discontinuity criterion
+
+227
+00:26:46,180 --> 00:26:48,980
+discontinuity criterion
+
+228
+00:27:00,500 --> 00:27:10,940
+لت if بي function from a to r و c نقطة في a و d
+
+229
+00:27:10,940 --> 00:27:15,440
+then the
+
+230
+00:27:15,440 --> 00:27:23,000
+following statements are equivalent واحد if is
+
+231
+00:27:23,000 --> 00:27:24,160
+discontinuous
+
+232
+00:27:26,370 --> 00:27:36,730
+إذا كان الـ discontinuous at x بساوي c ثم يوجد
+
+233
+00:27:36,730 --> 00:27:49,070
+سيكوينس x in contained in a with limit x in بساوي
+
+234
+00:27:49,070 --> 00:27:49,610
+c
+
+235
+00:27:53,460 --> 00:28:01,180
+but limit الـ image للـ sequence x in لا يساوي f
+
+236
+00:28:01,180 --> 00:28:08,260
+of z وبرهان
+
+237
+00:28:08,260 --> 00:28:13,100
+النظرية هذه بيجي من النظرية الـ sequential
+
+238
+00:28:13,100 --> 00:28:16,980
+criterion أنا
+
+239
+00:28:16,980 --> 00:28:21,400
+عندي واحد one بكافي اتنين one if and only if two
+
+240
+00:28:24,600 --> 00:28:29,660
+وبالتالي not one نفي one بكافئ نفي two طيب تعالى
+
+241
+00:28:29,660 --> 00:28:35,400
+نشوف نفي one if is discontinuous at c نفي two for
+
+242
+00:28:35,400 --> 00:28:40,420
+every sequence بتحقق الشرط هذا نهايت صورتها بساوي
+
+243
+00:28:40,420 --> 00:28:45,160
+f of c ان في الشرط العبارة هذه فبصير there exist a
+
+244
+00:28:45,160 --> 00:28:50,380
+sequence x in contained in a ونهايتها c لكن نهايت
+
+245
+00:28:50,380 --> 00:28:56,550
+صورتها لا تساوي f of cOkay تمام إذا البرهان نظرية
+
+246
+00:28:56,550 --> 00:29:05,170
+هذه جاي من نفي أو ينتج من النظرية السابقة طب
+
+247
+00:29:05,170 --> 00:29:15,350
+نرجع ناخد قبل ما ناخد أمثلة بدنا ناخد بس تعريف
+
+248
+00:29:15,350 --> 00:29:20,170
+الاتصال على مجموعة definition
+
+249
+00:29:24,990 --> 00:29:32,690
+استخدم الفرصة let f be a function from a to r and
+
+250
+00:29:32,690 --> 00:29:38,050
+let
+
+251
+00:29:38,050 --> 00:29:47,090
+b be a subset of a نقول
+
+252
+00:29:47,090 --> 00:29:50,890
+ان الفرصة is continuous
+
+253
+00:29:54,760 --> 00:30:05,060
+if is continuous on الـ set B on the
+
+254
+00:30:05,060 --> 00:30:16,640
+set B if is continuous on the set B if if is
+
+255
+00:30:16,640 --> 00:30:32,720
+continuous at every at everyما ينتمي إلى دي إذا
+
+256
+00:30:32,720 --> 00:30:38,880
+الإتصال على مجموعة معناه إن الدالة تكون متصلة عند
+
+257
+00:30:38,880 --> 00:30:47,520
+كل نقطة في المجموعة، عند كل نقطة في المجموعة طيب
+
+258
+00:30:47,520 --> 00:30:49,080
+ناخد بعض الأمثلة
+
+259
+00:31:06,780 --> 00:31:17,520
+الـ function f of x بتساوي k و
+
+260
+00:31:17,520 --> 00:31:30,460
+x belong to R is continuous on R الدالة
+
+261
+00:31:30,460 --> 00:31:43,300
+ثابت k continuous على كل الـ Rاحنا شفنا proof fix
+
+262
+00:31:43,300 --> 00:31:46,240
+c أنتمي الار
+
+263
+00:31:51,650 --> 00:32:02,150
+Since limit ل F of X as X Sin C بساوي K احنا
+
+264
+00:32:02,150 --> 00:32:07,850
+اثمتنا قبلين ان limit اي ده لثابته بساوي ثابت K
+
+265
+00:32:07,850 --> 00:32:15,690
+وهذا بساوي F and C فال
+
+266
+00:32:15,690 --> 00:32:29,850
+F is continuousat every c ينتمي إلى r فاكرين
+
+267
+00:32:29,850 --> 00:32:34,430
+احنا هدف بقناه باستخدام تعريف epsilon delta قولنا
+
+268
+00:32:34,430 --> 00:32:39,930
+لأي epsilon أكبر من السفر choose أي delta أكبر من
+
+269
+00:32:39,930 --> 00:32:43,690
+السفر فتعريف
+
+270
+00:32:43,690 --> 00:32:47,670
+ال limit بتحقق
+
+271
+00:32:47,670 --> 00:32:48,790
+وهنا نفس الحاجة
+
+272
+00:33:16,050 --> 00:33:25,330
+طيب المثال تاني لو أخدت f of x بساوي x لكل x ينتمي
+
+273
+00:33:25,330 --> 00:33:31,570
+إلى R ال identity function فبرضه
+
+274
+00:33:31,570 --> 00:33:39,350
+أثبتنا احنا ان ال function هذه is continuous if is
+
+275
+00:33:39,350 --> 00:33:44,290
+continuousعلى مجموعة الأعداد الحقيقية
+
+276
+00:34:07,950 --> 00:34:17,850
+فممكن أن نثبت C ينتمي إلى R و أثبتنا احنا في
+
+277
+00:34:17,850 --> 00:34:24,390
+section أربعة واحد ان limit F of X لما X تقول C
+
+278
+00:34:24,390 --> 00:34:32,530
+طلعت بساوي C صح؟ وهذا عبارة عن F of C فالـ F is
+
+279
+00:34:32,530 --> 00:34:35,610
+continuous at C
+
+280
+00:34:39,860 --> 00:34:48,180
+و بما انه c arbitrary element اذا
+
+281
+00:34:48,180 --> 00:34:55,720
+ال F يكون continuous at every c ينتمي ال R
+
+282
+00:34:55,720 --> 00:35:03,220
+وبالتالي continuous على كل ال R ممكن
+
+283
+00:35:03,220 --> 00:35:08,760
+برضهنستخدم تعريف epsilon دلتا مباشرة بلاش نقول ان
+
+284
+00:35:08,760 --> 00:35:13,440
+احنا اثبتنا ان ال limit ل ال function f and c
+
+285
+00:35:13,440 --> 00:35:17,020
+بالساوية c في section اربعة واحدة انا ممكن اثبت
+
+286
+00:35:17,020 --> 00:35:22,520
+يعني استخدم تعريف epsilon دلتا مباشرة و اقول let
+
+287
+00:35:22,520 --> 00:35:32,180
+if fix اول حاجة fix c تنتمي ل R to showif is
+
+288
+00:35:32,180 --> 00:35:39,820
+continuous at c let epsilon أكبر من السفر be given
+
+289
+00:35:39,820 --> 00:35:44,720
+it
+
+290
+00:35:44,720 --> 00:35:49,540
+shows .. زي ما عملنا في ال limits it shows delta
+
+291
+00:35:49,540 --> 00:35:54,640
+بساوي epsilon لذن
+
+292
+00:35:54,640 --> 00:36:00,160
+هي يوجد delta تعتمد على epsilonThen لهذه الـ Delta
+
+293
+00:36:00,160 --> 00:36:06,600
+لو كان X ينتمي إلى A A هنا اللي هي R و Absolute X
+
+294
+00:36:06,600 --> 00:36:12,360
+minus C أصغر من Delta فهذا بتضمن أنه Absolute F of
+
+295
+00:36:12,360 --> 00:36:20,080
+X Absolute F of X minus F of C هذا بيطلع بساوي
+
+296
+00:36:20,080 --> 00:36:28,590
+Absolute X minus F of X بساويX و F of C بساوي C
+
+297
+00:36:28,590 --> 00:36:33,010
+وهذا أصغر من Delta ماخدين المسافة هذه أصغر من
+
+298
+00:36:33,010 --> 00:36:38,250
+Delta وأنا اختارت Delta بساوي Epsilon إذن هذه
+
+299
+00:36:38,250 --> 00:36:42,110
+أثبتت لكل Epsilon يوجد Delta تعتمد على Epsilon
+
+300
+00:36:42,110 --> 00:36:46,150
+بحيث لكل X في مجال الدالة المسافة بينها و بين C
+
+301
+00:36:46,150 --> 00:36:50,650
+أصغر من Delta طلع المسافة بين F of X و F of C أصغر
+
+302
+00:36:50,650 --> 00:36:58,390
+من Epsilon إذن هذا معناه أن F is continuousat C
+
+303
+00:36:58,390 --> 00:37:06,010
+since C تنتمي ل R was arbitrary اذا F is
+
+304
+00:37:06,010 --> 00:37:12,750
+continuous على كل الأعداد الحقيقية تمام؟ اذا هذا
+
+305
+00:37:12,750 --> 00:37:15,890
+ممكن استخدم تعريف Epsilon Delta مباشرة
+
+306
+00:37:19,990 --> 00:37:23,390
+دون الاعتماد على النتائج اللي عملناها تابعة
+
+307
+00:37:23,390 --> 00:37:28,390
+النهاية في section أربعة واحد بالمثل ممكن مثال زي
+
+308
+00:37:28,390 --> 00:37:35,290
+هذا برضه ال function f
+
+309
+00:37:35,290 --> 00:37:43,790
+of x بساوي x سربية is continuous على كل الأعداد
+
+310
+00:37:43,790 --> 00:37:44,570
+الحقيقية
+
+311
+00:38:05,350 --> 00:38:08,350
+الدالة متصلة عند النقطة C
+
+312
+00:38:13,110 --> 00:38:18,330
+نفس تعريف epsilon دلتا زي ما عملنا في اثبات ان ال
+
+313
+00:38:18,330 --> 00:38:24,490
+limit لل function f of x and x بساوي c بساوي c
+
+314
+00:38:24,490 --> 00:38:30,110
+تربيه اللي هو f of c وذلك
+
+315
+00:38:30,110 --> 00:38:35,710
+بياخد اي epsilon اكبر من صفر و بنجيب دلتا زي ما
+
+316
+00:38:35,710 --> 00:38:38,510
+عملنا في section اربعة واحد دلتا بساوي ال minimum
+
+317
+00:38:38,510 --> 00:38:45,380
+لقمتينو نثبت أنه لكل x المسافة بينها و بين الـC
+
+318
+00:38:45,380 --> 00:38:47,960
+أصغر من الـDelta بيطلع المسافة هذه أصغر من الـC
+
+319
+00:38:47,960 --> 00:38:53,120
+نعيد يعني إيش نفس البرمجة، إذن هذا لو طلب منكم
+
+320
+00:38:53,120 --> 00:38:56,460
+استخدام تعريف epsilon delta لإثبات أن الدالة هذه
+
+321
+00:38:56,460 --> 00:39:00,420
+مقتصرة على R فبتقول لأي epsilon أكبر من السفر
+
+322
+00:39:00,420 --> 00:39:05,060
+choose delta زي ما عملنا في section 4-1 في إثبات
+
+323
+00:39:05,060 --> 00:39:08,900
+أن limit للدالة هذه عن C بساوي C تربية
+
+324
+00:39:12,370 --> 00:39:18,470
+أو ممكن تقولي we should اذا ما طلبش منك استخدم
+
+325
+00:39:18,470 --> 00:39:23,590
+التعريف epsilon دلتا فبتقولي we should أثبتنا in
+
+326
+00:39:23,590 --> 00:39:33,970
+section أربع واحد that limit ل F of X لما X تقول
+
+327
+00:39:33,970 --> 00:39:42,230
+إلى C بساوي C تربية اللي هي F of Cحسب تعريف
+
+328
+00:39:42,230 --> 00:39:45,470
+الاتصال على النقطة بيطلع أي شرط تلاتة في واحد
+
+329
+00:39:45,470 --> 00:39:54,190
+متحقق وبالتاني if is continuous at c okay تمام
+
+330
+00:39:57,190 --> 00:40:00,230
+وطبعاً بما أن الـ C تنتمي الـ R was arbitrary إذن
+
+331
+00:40:00,230 --> 00:40:03,970
+الدالة F continuous على كل الـ R okay إذا دامت
+
+332
+00:40:03,970 --> 00:40:11,050
+ياندي إما نستخدم نتائج section 4-1 أو نعيد البرهان
+
+333
+00:40:11,050 --> 00:40:15,250
+باستخدام تعريف epsilon Delta زي ما عملنا في المثال
+
+334
+00:40:15,250 --> 00:40:22,770
+الأخير أو زي ما عملنا في section 4-1 الدالة كمان
+
+335
+00:40:22,770 --> 00:40:23,930
+عندي الدالة
+
+336
+00:40:32,140 --> 00:40:41,000
+لو أخدت five X بيساوي واحد على X فهذه الدالة is
+
+337
+00:40:41,000 --> 00:40:46,280
+continuous on ال set A
+
+338
+00:40:58,940 --> 00:41:04,860
+اللي هي كل ال X ينتمي إلى R حقيته X أكبر من السفر
+
+339
+00:41:04,860 --> 00:41:11,380
+فاحنا
+
+340
+00:41:11,380 --> 00:41:20,880
+أثبتنا في X C تنتمي إلى A هذا بقدر انه C أكبر من
+
+341
+00:41:20,880 --> 00:41:23,280
+سفر و أثبتنا
+
+342
+00:41:28,820 --> 00:41:35,560
+In section أربع
+
+343
+00:41:35,560 --> 00:41:44,320
+واحد ذات limit لـ function phi of x لما x تقول إلى
+
+344
+00:41:44,320 --> 00:41:52,240
+c بسوى واحد على c بسوى phi of c باستخدام تعريف
+
+345
+00:41:52,240 --> 00:41:58,070
+epsilon دلتا اما نعيدالبرهان هداك لأي epsilon في
+
+346
+00:41:58,070 --> 00:42:03,450
+ديلتا بساوي minimum لقمتين او نقول انه احنا اثبتنا
+
+347
+00:42:03,450 --> 00:42:06,890
+ان limit الدالة هدا عند اي عدد c موجد بساوي واحد
+
+348
+00:42:06,890 --> 00:42:12,590
+على c اللي هو قيمة الدالة عن c وبالتالي اذا الدالة
+
+349
+00:42:12,590 --> 00:42:19,830
+في is continuous at c بما ان ال c تنتمي ل a was
+
+350
+00:42:19,830 --> 00:42:26,450
+arbitrary اذا ال في continuousعلى المجموعة A
+
+351
+00:42:26,450 --> 00:42:30,370
+بالمثل
+
+352
+00:42:30,370 --> 00:42:35,050
+ممكن نثبت ان الدالة دي continuous كمان على
+
+353
+00:42:35,050 --> 00:42:44,530
+المجموعة B اللي هي كل ال X ينتمي ل R حيث X أصغر من
+
+354
+00:42:44,530 --> 00:42:48,990
+0 الدالة
+
+355
+00:42:48,990 --> 00:42:54,190
+دي متصلة عند كل الأعداد الحقيقية مع عدد 0فهي متصلة
+
+356
+00:42:54,190 --> 00:42:57,610
+عند الأعداد الحقيقية الموجبة وعند الأعداد الحقيقية
+
+357
+00:42:57,610 --> 00:43:07,950
+السالبة طيب
+
+358
+00:43:07,950 --> 00:43:13,370
+الدالة five
+
+359
+00:43:13,370 --> 00:43:19,950
+x نفسها برضه بساوي واحد على x is not is
+
+360
+00:43:19,950 --> 00:43:33,190
+discontinuousis discontinuous at c بساوي سفر proof
+
+361
+00:43:33,190 --> 00:43:39,090
+one الدالة
+
+362
+00:43:39,090 --> 00:43:44,530
+هذه ليست متصلة عند السفر فالبرهان ذلك ممكن نقول
+
+363
+00:43:44,530 --> 00:43:49,610
+أنه في في
+
+364
+00:43:52,850 --> 00:43:59,250
+is undefined is undefined is undefined is
+
+365
+00:43:59,250 --> 00:44:05,090
+undefined is undefined is undefined is undefined
+
+366
+00:44:05,090 --> 00:44:05,970
+undefined is undefined is undefined is undefined
+
+367
+00:44:05,970 --> 00:44:07,390
+is undefined is undefined is undefined is
+
+368
+00:44:07,390 --> 00:44:07,470
+undefined is undefined is undefined is undefined
+
+369
+00:44:07,470 --> 00:44:07,830
+is undefined is undefined is undefined is
+
+370
+00:44:07,830 --> 00:44:09,950
+is undefined is undefined is undefined is
+
+371
+00:44:09,950 --> 00:44:16,950
+undefined is undefined is undefined is undefined
+
+372
+00:44:18,990 --> 00:44:25,170
+can't be continuous at x بساوي سفر لأن عشان هي
+
+373
+00:44:25,170 --> 00:44:28,550
+تكون متصلة عند سفر لازم تلات شروط يتحققوا أنها
+
+374
+00:44:28,550 --> 00:44:32,790
+تكون أول chart معرفة عند السفر فده هي مش معرفة عند
+
+375
+00:44:32,790 --> 00:44:38,390
+السفر فكيف تلات شروط هيتحققوا هذا برهان تاني برهان
+
+376
+00:44:38,390 --> 00:44:45,850
+آخر ان ما احنا شوفنا we should
+
+377
+00:44:48,290 --> 00:44:52,870
+in section أربع
+
+378
+00:44:52,870 --> 00:44:57,990
+واحد أو أربع اتنين that
+
+379
+00:44:57,990 --> 00:45:08,290
+limit لفاي of x as x tends to zero does not exist
+
+380
+00:45:08,290 --> 00:45:12,850
+أثبتنا إن الـ function هذه ما لهاش limit عند السفر
+
+381
+00:45:15,830 --> 00:45:21,510
+فا استخدمنا ال divergence criterion ا شفنا ان هناك
+
+382
+00:45:21,510 --> 00:45:27,450
+sequence اللى هى واحد عال ان converge للسفر but
+
+383
+00:45:27,450 --> 00:45:34,690
+limit ال image لل sequence واحد على ان as n tends
+
+384
+00:45:34,690 --> 00:45:40,170
+to infinity بساوي limit in بساوي infinity does not
+
+385
+00:45:40,170 --> 00:45:47,950
+exist in Rوبالتالي by divergence criterion ال
+
+386
+00:45:47,950 --> 00:45:51,270
+function هذه مالهاش limit وبالتالي مش ممكن تكون
+
+387
+00:45:51,270 --> 00:46:02,990
+continuous so if I can't be continuous at x بساوي
+
+388
+00:46:02,990 --> 00:46:09,510
+سفر تمام؟ لأن واحد من الشروط التلاتة تبعت الاتصال
+
+389
+00:46:09,510 --> 00:46:12,650
+عن نقطة غير متحققة تمام؟
+
+390
+00:46:22,580 --> 00:46:28,520
+في كمان مثال أخدناه في section
+
+391
+00:46:28,520 --> 00:46:36,020
+4-1 الـ
+
+392
+00:46:36,020 --> 00:46:42,220
+signum function اللي
+
+393
+00:46:42,220 --> 00:46:52,050
+كان تعريفهابتساوي سفر if x بساوي سفر و x على
+
+394
+00:46:52,050 --> 00:47:00,370
+absolute x إذا كان x لا يساوي سفر is discontinuous
+
+395
+00:47:00,370 --> 00:47:09,170
+is discontinuous at x بساوي سفر why
+
+396
+00:47:18,170 --> 00:47:23,550
+لأنه اثبتنا احنا في section أربعة واحد انه limit ل
+
+397
+00:47:23,550 --> 00:47:31,490
+signum x لما x تقول إلى سفر does not exist
+
+398
+00:47:40,560 --> 00:47:43,240
+اللي هي ان ال limit لل signal function عند السفر
+
+399
+00:47:43,240 --> 00:47:46,580
+does not exist شوفنا ان ال limit من اليمين واحد
+
+400
+00:47:46,580 --> 00:47:50,020
+عند السفر و ال limit و ال limit عند السفر مليار
+
+401
+00:47:50,020 --> 00:47:53,340
+ساعة سالف واحد وبالتالي مش متساوي اتين اذا ال
+
+402
+00:47:53,340 --> 00:48:00,000
+limit عند السفر does not exist okay تمام اذا ال ال
+
+403
+00:48:00,000 --> 00:48:04,700
+function هذه ماهياش متصلة عند السفر لعدم نظرا لعدم
+
+404
+00:48:04,700 --> 00:48:10,970
+وجود ال limit عند السفررغم أن الدالة هذه معرفة عند
+
+405
+00:48:10,970 --> 00:48:17,310
+السفر، الـSignum للسفر هي معرفة عند السفر بساوي
+
+406
+00:48:17,310 --> 00:48:24,930
+سفر تمام؟
+
+407
+00:48:24,930 --> 00:48:30,710
+طيب، لكن ممكن اثبات أن الـSignum function متصلة
+
+408
+00:48:30,710 --> 00:48:32,850
+عند كل X لا يساوي سفر
+
+409
+00:48:45,100 --> 00:48:52,440
+However، الـ signum الـ signum function is
+
+410
+00:48:52,440 --> 00:48:59,280
+continuous at
+
+411
+00:48:59,280 --> 00:49:09,460
+every x لا يساوي سفر لأنه
+
+412
+00:49:22,230 --> 00:49:42,610
+proof fix c لا تنتمي لار وc لا يساوي ستة تمام then
+
+413
+00:49:42,610 --> 00:49:53,460
+absolute signum x minus signumالـ C بساوي absolute
+
+414
+00:49:53,460 --> 00:49:57,420
+X
+
+415
+00:49:57,420 --> 00:50:14,640
+على absolute X أو
+
+416
+00:50:14,640 --> 00:50:15,160
+بلاش
+
+417
+00:50:19,850 --> 00:50:26,730
+then ال limit ل sigma x
+
+418
+00:50:26,730 --> 00:50:34,390
+لما x تقول إلى c بساوي
+
+419
+00:50:34,390 --> 00:50:37,990
+لما
+
+420
+00:50:37,990 --> 00:50:43,670
+x تقول إلى c فهذا عبارة عن limit x على absolute x
+
+421
+00:50:43,670 --> 00:50:45,630
+لما x تقول إلى c
+
+422
+00:51:03,050 --> 00:51:08,750
+فده كانت ال X لا تساوي سفر فاما ال X موجة بقى أو
+
+423
+00:51:08,750 --> 00:51:12,890
+سالي بقى
+
+424
+00:51:12,890 --> 00:51:18,010
+then C أكبر من السفر or C أصغر من سفر
+
+425
+00:51:23,040 --> 00:51:27,120
+الـ C هتكون أكبر من السفر الـ C هنا لأ تساوي سفر
+
+426
+00:51:27,120 --> 00:51:33,240
+إذا أما C أكبر من السفر أو أصغر من السفر case one
+
+427
+00:51:33,240 --> 00:51:41,000
+لو كانت C أكبر من سفر فهذا بقد أنه limit signum X
+
+428
+00:51:41,000 --> 00:51:50,980
+as X tends to C بساوي limit X على absolute X
+
+429
+00:51:59,940 --> 00:52:05,660
+و طبعا ال X أكبر من ال
+
+430
+00:52:05,660 --> 00:52:11,860
+C أكبر من السفر ف absolute .. فهذا بيساوي واحد
+
+431
+00:52:11,860 --> 00:52:21,440
+بيساوي limit واحد as X tends to C بيساوي واحد
+
+432
+00:52:21,440 --> 00:52:32,490
+بيساوي F and Cأو signum C لأن
+
+433
+00:52:32,490 --> 00:52:40,050
+ال C موجبة فلما ال C تكون موجبة ف absolute ال C
+
+434
+00:52:40,050 --> 00:52:47,250
+بساوي ال C بطلع المخضر هذا بطلع واحدو بالتالي إذا
+
+435
+00:52:47,250 --> 00:52:57,970
+ال signal x is continuous at c case 2 إذا كانت ال
+
+436
+00:52:57,970 --> 00:53:11,210
+c أصغر من سفر ف similar to case 1 في
+
+437
+00:53:11,210 --> 00:53:17,600
+الحالة هذهقيمة ال function هتطلع سالب واحد عند c و
+
+438
+00:53:17,600 --> 00:53:22,820
+limit عند c هتطلع سالب واحد وبالتالي في اتصال عند
+
+439
+00:53:22,820 --> 00:53:26,320
+ال c إذا ال sign and function مش متصلة عند الصفر
+
+440
+00:53:26,320 --> 00:53:30,800
+لكنها متصلة عن كل الأعداد الحقيقية المختلفة عن
+
+441
+00:53:30,800 --> 00:53:37,910
+الصفرOkay بنكتفي بهذا القدر و بنكمل طبعا إن شاء
+
+442
+00:53:37,910 --> 00:53:44,390
+الله في المحاضرة القادمة هنعطيكم إن شاء الله break
+
+443
+00:53:44,390 --> 00:53:49,350
+خمس دقائق و بعدين نواصل المحاضرة التانية اللي
+
+444
+00:53:49,350 --> 00:53:56,090
+هناخد فيها discussion أو مناقشة لل chapter أربعة
+
+445
+00:53:56,090 --> 00:53:58,350
+section أربعة واحد و أربعة اتنين
+
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/JXFFuyzuuqA_raw.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/JXFFuyzuuqA_raw.srt
new file mode 100644
index 0000000000000000000000000000000000000000..8e38a6c21f8833b05ff713f5081ef57692a64917
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/JXFFuyzuuqA_raw.srt
@@ -0,0 +1,1864 @@
+1
+00:00:20,220 --> 00:00:25,360
+بسم الله الرحمن الرحيم هندرس اليوم ان شاء الله مع
+
+2
+00:00:25,360 --> 00:00:32,000
+بعض ال section خمسة أربعةاللي بيتحدث عن موضوع ال
+
+3
+00:00:32,000 --> 00:00:36,720
+uniform continuity أو الاتصال المنظم للدوال
+
+4
+00:00:36,720 --> 00:00:40,600
+هحنحاول ناخد أكبر جزء ممكن من ال section الجزء
+
+5
+00:00:40,600 --> 00:00:44,860
+المتبقي ممكن نكمله في المحاضرة الجاية يوم الأتنين
+
+6
+00:00:44,860 --> 00:00:49,820
+فال
+
+7
+00:00:49,820 --> 00:00:54,540
+.. خلّينا الأول نراجع .. نراجع تعريف الاتصال
+
+8
+00:00:54,540 --> 00:00:59,270
+العاديال continuity على مجموعة فلو كان في handy
+
+9
+00:00:59,270 --> 00:01:04,170
+function f من a ل r فالعبارات التالية بتكون
+
+10
+00:01:04,170 --> 00:01:13,410
+متكافئة if is continuous at at
+
+11
+00:01:13,410 --> 00:01:20,810
+every at every u ينتمي إلى a اللي هو مجال الدالة
+
+12
+00:01:20,810 --> 00:01:24,370
+العبارة التانية given
+
+13
+00:01:27,500 --> 00:01:36,300
+epsilon أكبر من السفر and given u ينتمي إلى a يوجد
+
+14
+00:01:36,300 --> 00:01:41,160
+.. بيقدر نلاقي delta و ال delta هذه تعتمد على ال
+
+15
+00:01:41,160 --> 00:01:51,590
+epsilon و على ال u عدد موجببحيث أنه لكل x ينتمي
+
+16
+00:01:51,590 --> 00:01:59,250
+إلى a و absolute x minus u أصغر من delta فهذا
+
+17
+00:01:59,250 --> 00:02:07,830
+بتضمن إلى absolute f of x minus f of u أصغر من
+
+18
+00:02:07,830 --> 00:02:08,310
+epsilon
+
+19
+00:02:19,690 --> 00:02:30,650
+خلّينا بس ناخد المثال التالي consider
+
+20
+00:02:30,650 --> 00:02:41,910
+ال function f of xبتساوي واحد على X و X ينتبه لايه
+
+21
+00:02:41,910 --> 00:02:45,890
+اللي هي الفترة
+
+22
+00:02:45,890 --> 00:02:56,270
+كل ال X في R حيث X أكبر من الصفر إذا ال function F
+
+23
+00:02:56,270 --> 00:03:02,770
+معرفة على كل الأعداد الموجبة احنا
+
+24
+00:03:02,770 --> 00:03:05,770
+أثبتنا قبل هيك و proved
+
+25
+00:03:10,640 --> 00:03:14,920
+earlier فيما سبق في دراساتنا السابقة في section
+
+26
+00:03:14,920 --> 00:03:21,540
+اربعة خمسة ثلاثة او خمسة اتنين اثبتنا ان ال
+
+27
+00:03:21,540 --> 00:03:30,700
+function f is continuous على المجموعة a وخلنا
+
+28
+00:03:30,700 --> 00:03:36,580
+نراجع مع بعض ان مع بعض نراجع البرهان fix
+
+29
+00:03:39,080 --> 00:03:46,920
+fix u ينتمي إلى a given إبصر
+
+30
+00:03:46,920 --> 00:03:49,760
+أكبر من صفر أكبر من صفر أكبر من صفر أكبر من صفر
+
+31
+00:03:49,760 --> 00:03:50,560
+أكبر من صفر أكبر من صفر أكبر من صفر أكبر من صفر
+
+32
+00:03:50,560 --> 00:03:53,060
+أكبر من صفر أكبر من صفر أكبر من صفر أكبر من صفر
+
+33
+00:03:53,060 --> 00:03:56,600
+أكبر من صفر أكبر من صفر أكبر من صفر أكبر من صفر
+
+34
+00:03:56,600 --> 00:03:57,260
+أكبر من صفر أكبر من صفر أكبر من صفر أكبر من صفر
+
+35
+00:03:57,260 --> 00:03:57,360
+أكبر من صفر أكبر من صفر أكبر من صفر أكبر من صفر
+
+36
+00:03:57,360 --> 00:04:00,020
+أكبر من صفر أكبر من صفر أكبر من صفر أكبر من صفر
+
+37
+00:04:00,020 --> 00:04:06,790
+أكبر من صفر أكبر من صفربتطبيق تعريف epsilon delta
+
+38
+00:04:06,790 --> 00:04:12,110
+للاتصال ان نقطة given epsilon اذا بيطلع ارجعه we
+
+39
+00:04:12,110 --> 00:04:19,350
+found delta و ال delta هذه كانت ال minimum لقنتين
+
+40
+00:04:19,350 --> 00:04:24,470
+u ع اتنين او كانت هناك c ع اتنين بدل u كانت النقطة
+
+41
+00:04:24,470 --> 00:04:33,350
+بيسميها c فعندي u ع اتنين و u تربيه على اتنين في
+
+42
+00:04:33,350 --> 00:04:40,450
+epsilonطبعا هذا عدد موجب واضح ان ال delta هذه عدد
+
+43
+00:04:40,450 --> 00:04:44,530
+موجب لان هذا عدد موجب و هذا عدد موجب و بعدين ال
+
+44
+00:04:44,530 --> 00:04:50,530
+delta لاحظوا انها بتعتمد على ال epsilon و على ال U
+
+45
+00:04:52,480 --> 00:04:55,840
+الـ Delta بتعتمد على الـ Epsilon وعلى الـ U مش بس
+
+46
+00:04:55,840 --> 00:04:58,280
+على الـ Epsilon وعلى النقطة U اللى احنا بدنا نفحص
+
+47
+00:04:58,280 --> 00:05:05,020
+عندها الاتصال فشوفنا بعد هيك انه .. اذا for this
+
+48
+00:05:05,020 --> 00:05:11,880
+Delta اذا لو أخدنا X ينتمي إلى A و Absolute X
+
+49
+00:05:11,880 --> 00:05:19,560
+minus U أصغر من Delta فطبعا هذا قدهذا أدى أن الـ
+
+50
+00:05:19,560 --> 00:05:26,240
+delta هنا أصغر من أو يساوي U ع 2 وبالتالي هذا
+
+51
+00:05:26,240 --> 00:05:35,600
+بيقدر أن X أصغر من 3U ع 2 أكبر من U ع 2 لما نحل
+
+52
+00:05:35,600 --> 00:05:42,720
+المعادلة المتبينة هذه في U وهذا
+
+53
+00:05:42,720 --> 00:05:44,520
+بيقدر بدوره
+
+54
+00:05:46,640 --> 00:05:59,580
+أبسلوت f of x minus f of u طالع بيساوي أبسلوت واحد
+
+55
+00:05:59,580 --> 00:06:06,580
+على x minus واحد على u هذا بيساوي أبسلوت u minus x
+
+56
+00:06:06,580 --> 00:06:13,390
+على x في u المفروض أحط هنا أبسلوتأكس في U لكن ال X
+
+57
+00:06:13,390 --> 00:06:17,290
+و ال U عناصر في A و A عناصرها كل أعداد موجبة فلا
+
+58
+00:06:17,290 --> 00:06:21,950
+داعي ال absolute value الأن absolute أنا عندي هنا
+
+59
+00:06:21,950 --> 00:06:31,390
+من المتباينة هذه بيطلع عندي المفروض أنه أنا عندي
+
+60
+00:06:31,390 --> 00:06:43,100
+بيطلع U على 2 أصغر من X صح فهذا بيقدي أنه X فيأضرب
+
+61
+00:06:43,100 --> 00:06:47,420
+في U، U عدد موجب فبطلع U تربيع اتنين اصغر من X
+
+62
+00:06:47,420 --> 00:06:55,520
+وبالتالي واحد مقلوب XU بطلع اصغر من اتنين على U
+
+63
+00:06:55,520 --> 00:07:02,200
+تربيع اذا مقلوب XU اصغر من اتنين على U تربيع في
+
+64
+00:07:02,200 --> 00:07:08,790
+absolute U minus Xو هذي أصغر من دلتا إذاً هذي أصغر
+
+65
+00:07:08,790 --> 00:07:13,830
+من اتنين على U تربية في دلتا طيب الدلتا أنا
+
+66
+00:07:13,830 --> 00:07:18,390
+اختارها ال minimum للقيمة هذه وهذه فبالتالي الدلتا
+
+67
+00:07:18,390 --> 00:07:22,890
+هذه تطلع أصغر من أو ساوي القيمة التانيةإذن اتنين
+
+68
+00:07:22,890 --> 00:07:28,850
+على U تربية ضرب U تربية على اتنين في Epsilon و
+
+69
+00:07:28,850 --> 00:07:33,490
+طبعا هذولا بيروحوا مع بعض و بيظل Epsilon وبالتالي
+
+70
+00:07:33,490 --> 00:07:38,290
+بما أن Epsilon was arbitrary إذا ال F is
+
+71
+00:07:38,290 --> 00:07:48,110
+continuous at U ولمّا كانت U arbitrary since U
+
+72
+00:07:48,110 --> 00:07:49,770
+belonged to A was
+
+73
+00:07:52,720 --> 00:08:00,980
+arbitrary if is continuous على كل المجموعة ايه هذا
+
+74
+00:08:00,980 --> 00:08:05,740
+كان برهانة خلناها قبل هيك طيب ما الغرض مش ايش
+
+75
+00:08:05,740 --> 00:08:10,200
+النقطة ان احنا نعيد البرهان النقطة هي عايزين نفكز
+
+76
+00:08:10,200 --> 00:08:16,160
+او نأكد انه في اثبات الاتصال عند النقطة U لاحظنا
+
+77
+00:08:16,160 --> 00:08:20,330
+ان ال delta بتعتمد على ال epsilon و على ال Uهذا
+
+78
+00:08:20,330 --> 00:08:24,510
+معناه ان الـ delta بتتغير قيمتها مع تغير ال U
+
+79
+00:08:24,510 --> 00:08:28,070
+فمثلا
+
+80
+00:08:28,070 --> 00:08:40,890
+لو جينا نعمل هاي الدالة دي لو جينا رسمناها هاي
+
+81
+00:08:40,890 --> 00:08:47,730
+الدالة واحد على X لو جيت اخدت انا X لو كان هذا
+
+82
+00:08:47,730 --> 00:08:59,250
+واحد هذا اتنينفو هذا نص لو كانت ال U تبعتي لو كانت
+
+83
+00:08:59,250 --> 00:09:07,750
+ال U بساوي نص ف
+
+84
+00:09:07,750 --> 00:09:17,810
+F لنص بساوي هيطلع اتنين هذا بساوي F لنص طب لو جيت
+
+85
+00:09:17,810 --> 00:09:25,470
+أخدتأبسلون نيبرهود لاتنين اذا هذا عبارة عن بي
+
+86
+00:09:25,470 --> 00:09:32,310
+ابسلون لاتنين اللي هو صورة النص فهذا الابسلون
+
+87
+00:09:32,310 --> 00:09:38,130
+نيبرهود هيقابله delta
+
+88
+00:09:38,130 --> 00:09:43,350
+neighborhood هيقابله
+
+89
+00:09:43,350 --> 00:09:44,150
+delta
+
+90
+00:09:50,400 --> 00:09:59,440
+هذا عبارة عن delta neighborhood للنص باللاحظ هنا
+
+91
+00:09:59,440 --> 00:10:02,680
+ان ال delta هي قيمتها
+
+92
+00:10:20,550 --> 00:10:25,830
+هذه اتنين لو اخدت U بساوة اتنين لو اخدت U بساوة
+
+93
+00:10:25,830 --> 00:10:30,230
+اتنين احنا اثبتنا ان الدالة متصلة على الاتنين وهذه
+
+94
+00:10:30,230 --> 00:10:37,730
+ال function شكلها هيكون زي هيك يعني
+
+95
+00:10:37,730 --> 00:10:41,770
+هون ف F لتنين
+
+96
+00:10:44,810 --> 00:10:49,990
+بساوي نص او صورة اتنين بطلع نص اللي هي صورة
+
+97
+00:10:49,990 --> 00:10:54,470
+الاتنين الان لو انا اخدت كوانة epsilon
+
+98
+00:10:54,470 --> 00:11:01,750
+neighborhood لنقطة نص هذه ال epsilon هنا نفس قيمة
+
+99
+00:11:01,750 --> 00:11:06,890
+ال epsilon اللي هنا نفس القيمة وبالتالي الان اذا
+
+100
+00:11:06,890 --> 00:11:13,680
+في عندي انا دي epsilon لن نصفطبعاً لكل epsilon
+
+101
+00:11:13,680 --> 00:11:16,480
+neighborhood للنص بما أن الدلة متصلة عند اتنين
+
+102
+00:11:16,480 --> 00:11:22,480
+هيوجد V Delta يوجد
+
+103
+00:11:22,480 --> 00:11:28,800
+V Delta okay
+
+104
+00:11:28,800 --> 00:11:32,960
+هذا هيكون V Delta
+
+105
+00:11:39,350 --> 00:11:43,010
+هذا عبارة عن V Delta او Delta neighborhood للإفنين
+
+106
+00:11:43,010 --> 00:11:48,190
+فبلاحظ انه رغم ان ال epsilon هنا نفس قيمة ال
+
+107
+00:11:48,190 --> 00:11:52,890
+epsilon هنا الا ان ال delta هنا شوف جدش صغيرة
+
+108
+00:11:52,890 --> 00:12:00,400
+بينما ال delta هنا شايفين ما اكبرها؟تغيرت مين اللي
+
+109
+00:12:00,400 --> 00:12:05,220
+غير ال delta ال U لما ال U كانت نص ال delta كانت
+
+110
+00:12:05,220 --> 00:12:11,340
+صغيرة لما ال U كانت اتنين ال U كبرت اذا ال delta
+
+111
+00:12:11,340 --> 00:12:15,600
+هنا او ال delta نبرهود بيعتمد على ال epsilon او ال
+
+112
+00:12:15,600 --> 00:12:19,200
+delta بتعتمد على ال مش بس على ال epsilon و على ال
+
+113
+00:12:19,200 --> 00:12:23,840
+U و على النقطة نفسها okay واضح اذا هنا ال delta
+
+114
+00:12:23,840 --> 00:12:31,210
+تغيرت مع تغير ال UOkay تمام وبالتالي ال delta لأي
+
+115
+00:12:31,210 --> 00:12:34,470
+epsilon ال delta ده بتعتمد على ال u على ال epsilon
+
+116
+00:12:34,470 --> 00:12:39,410
+أو على النقطة وعلى ال epsilon تمام واضحة النقطة
+
+117
+00:12:39,410 --> 00:12:45,370
+هذه طيب احنا خلينا نقبل ناشية ده المثال خلينا ناخد
+
+118
+00:12:45,370 --> 00:12:54,770
+مثال تاني example
+
+119
+00:12:54,770 --> 00:12:56,210
+2
+
+120
+00:12:59,420 --> 00:13:09,840
+خلّينا ناخد الـ function f of x بساوي 2x و x ينتمي
+
+121
+00:13:09,840 --> 00:13:13,780
+إلى R Note
+
+122
+00:13:13,780 --> 00:13:20,620
+that .. خلّينا نلاحظ أول أن absolute f of x minus
+
+123
+00:13:20,620 --> 00:13:29,440
+f of uبساوي absolute اتنين X minus اتنين U بساوي
+
+124
+00:13:29,440 --> 00:13:38,420
+اتنين في absolute X minus U لكل X و U ينتمي ال R
+
+125
+00:13:38,420 --> 00:13:44,880
+مظبوط هيك؟ طيب
+
+126
+00:13:44,880 --> 00:13:51,760
+الدالة هذه معروفة انها متصلة على R المجال تبعها
+
+127
+00:13:51,760 --> 00:13:52,200
+صح؟
+
+128
+00:14:03,920 --> 00:14:13,000
+على الـ set R فكيف بنعمل فكس بنثبت U في R بنثبت أن
+
+129
+00:14:13,000 --> 00:14:22,180
+F متصل عند الـ U صح؟ and let أكبر من السفر be
+
+130
+00:14:22,180 --> 00:14:22,780
+given
+
+131
+00:14:28,810 --> 00:14:36,250
+تختار دلتا نختار دلتا بساوي أبسلون ع اتنين أكبر من
+
+132
+00:14:36,250 --> 00:14:45,010
+السفر فلهذه الدلتا then لو كان x ينتمي إلى ال a
+
+133
+00:14:45,010 --> 00:14:51,490
+اللي هي r و absolute x minus u أصغر من الدلتا فهذا
+
+134
+00:14:51,490 --> 00:14:58,840
+هيديني absolute f of x minus f of uبتقول إن هذا
+
+135
+00:14:58,840 --> 00:15:03,440
+بيطلع بساوية أصغر من أو ساوية اتنين في absolute x
+
+136
+00:15:03,440 --> 00:15:09,940
+minus u أو بساوية بالأعلى، صح؟ طيب مانا ال X هذه
+
+137
+00:15:09,940 --> 00:15:14,660
+ماخدها بحيث أن absolute x minus u أصغر من ال
+
+138
+00:15:14,660 --> 00:15:20,160
+delta، صح؟عشان ذلك انا اخترت delta بساوي epsilon ع
+
+139
+00:15:20,160 --> 00:15:24,500
+اتنين اه شوفت ايش خدنا delta بساوي epsilon ع اتنين
+
+140
+00:15:24,500 --> 00:15:30,740
+طيب و هذا بساوي epsilon حسب اختيارنا لل delta
+
+141
+00:15:30,740 --> 00:15:37,460
+وبالتالي هيك اذا ال function بما انه epsilon was
+
+142
+00:15:37,460 --> 00:15:44,800
+arbitrarily اذا f is continuousat الـ U وبما أن U
+
+143
+00:15:44,800 --> 00:15:48,060
+belong to R وزر فبتره إذا if continuous على كل ال
+
+144
+00:15:48,060 --> 00:15:55,240
+R كمان مرة النقطة هنا اللي عايزين أن أكد عليها هو
+
+145
+00:15:55,240 --> 00:16:01,520
+إن ال Delta لأي إبسلون و لأي U و لأي إبسلون ال
+
+146
+00:16:01,520 --> 00:16:06,160
+Delta هنا تعتمد على إبسلون فقط مالهاش دعوة في ال U
+
+147
+00:16:06,160 --> 00:16:11,790
+بمعنى آخر لو أنا ال U هذه غيرتهاأخذت U تانية لو
+
+148
+00:16:11,790 --> 00:16:14,670
+كانت تانية مثلا U بالساعة و سفر او واحد او اتنين
+
+149
+00:16:14,670 --> 00:16:19,310
+او تلاتة او اي عدد حقيقي فكل مرة ال delta نفس ال
+
+150
+00:16:19,310 --> 00:16:25,800
+delta F2 will work لل U لكل Uلأي إبسن خدي نفس ال
+
+151
+00:16:25,800 --> 00:16:28,340
+delta إبسن على اتنين هتعطيهم إيه ال implication
+
+152
+00:16:28,340 --> 00:16:33,640
+هذه بغض النظر عن ال U okay؟ وبالتالي هنا في ال ..
+
+153
+00:16:33,640 --> 00:16:37,320
+في ال .. في الاتصال هذا ال delta هنا تعتمد على
+
+154
+00:16:37,320 --> 00:16:40,540
+إبسن فقط و لا تعتمد على U بينما في المثال السابق
+
+155
+00:16:40,540 --> 00:16:45,240
+شوفنا ال delta بتعتمد على Uهذا النوع من الاتصال
+
+156
+00:16:45,240 --> 00:16:48,860
+بنسميه اتصال منتظم اللي فيه ال delta تعتمد على
+
+157
+00:16:48,860 --> 00:16:52,760
+epsilon فقط اتصال منتظم او uniform continuity
+
+158
+00:16:52,760 --> 00:16:55,540
+اتصال اللي جابله اللي ال delta تعتمد على ال
+
+159
+00:16:55,540 --> 00:17:00,510
+epsilon و على النقطة Uهذا نسميه continuity عادية
+
+160
+00:17:00,510 --> 00:17:04,230
+او نقول continuity اتصال اما هذا uniform
+
+161
+00:17:04,230 --> 00:17:08,770
+continuity هنشوف ال gate من التعريف ان ال uniform
+
+162
+00:17:08,770 --> 00:17:13,990
+continuity اقوى و اشمل من ال continuity العادية
+
+163
+00:17:13,990 --> 00:17:22,670
+okay تمام اذا خليني اضع تعريف ال uniform
+
+164
+00:17:22,670 --> 00:17:25,590
+continuity definition
+
+165
+00:17:28,670 --> 00:17:40,970
+فنشطة f من a الى r هي عامة عامة
+
+166
+00:17:40,970 --> 00:17:49,130
+مستمرة عامة مستمرة عامة عامة عامة عامة عامة عامة
+
+167
+00:17:49,130 --> 00:17:49,170
+عامة عامة عامة عامة عامة عامة عامة عامة عامة عامة
+
+168
+00:17:49,170 --> 00:17:49,170
+عامة عامة عامة عامة عامة عامة عامة عامة عامة عامة
+
+169
+00:17:49,170 --> 00:17:54,610
+عامة عامة عامة عامة عامة عامة عامة عامة عامة
+
+170
+00:17:54,610 --> 00:17:55,930
+عامة عامة عامة عامة عامة عامة عامة عامة عامة عامة
+
+171
+00:17:55,930 --> 00:17:55,950
+عامة عامة عامة عامة عامة عامة عامة عامة عامة عامة
+
+172
+00:17:55,950 --> 00:17:56,070
+عامة عامة عامة عامة عامة عامة عامة عامة عامة عامة
+
+173
+00:18:00,400 --> 00:18:06,760
+إبسلون أكبر من السفر يوجد Delta تعتمد على إبسلون
+
+174
+00:18:06,760 --> 00:18:13,920
+فقط، عدد موجب بحيث أنه لكل X و U تنتمي إلى A
+
+175
+00:18:13,920 --> 00:18:20,620
+وأبسليوت X minus U أصغر من Delta هذا بتضمن أن
+
+176
+00:18:20,620 --> 00:18:29,420
+أبسليوت F of X minus F of U أصغر من الإبسلون
+
+177
+00:18:31,760 --> 00:18:35,660
+إذا هنا لأي أبسلون أكبر من السفر في دلتة واحدة
+
+178
+00:18:35,660 --> 00:18:40,100
+تعتمد على أبسلون فقط والدلتة هذه تلف على كل ال X
+
+179
+00:18:40,100 --> 00:18:44,620
+وكل ال U أو لكل ال U مرة واحدة فهنا لكل X و لأي U
+
+180
+00:18:44,620 --> 00:18:48,300
+إذا المسافة بينهم أصغر من الدلتة فالمسافة بين
+
+181
+00:18:48,300 --> 00:18:54,140
+أصغرهم أصغر من X تمام؟ الآن واضح من التعريفات
+
+182
+00:18:58,880 --> 00:19:05,820
+remarks المراحبة الأولى uniform
+
+183
+00:19:05,820 --> 00:19:13,760
+continuity
+
+184
+00:19:13,760 --> 00:19:23,440
+uniform continuity implies continuity
+
+185
+00:19:27,240 --> 00:19:35,720
+الاتصال المنتظر بيؤدي للاتصال العادى و البرهان
+
+186
+00:19:35,720 --> 00:19:39,960
+واضح يعني بمعنى اخر لو في عندي function f from a
+
+187
+00:19:39,960 --> 00:19:46,460
+to r و ال function كانت uniformly continuous فهذا
+
+188
+00:19:46,460 --> 00:19:54,350
+بيؤدي ان f continuous فالبرهان ذلكأفرضي أن F
+
+189
+00:19:54,350 --> 00:20:00,770
+uniformly continuous إذا اشترتها تتحقق تبع الـ
+
+190
+00:20:00,770 --> 00:20:05,750
+Uniform Continuous الآن لإثبات أن F continuous على
+
+191
+00:20:05,750 --> 00:20:11,210
+A بتثبت أن F continuous at every U ينتمي لـ A يعني
+
+192
+00:20:11,210 --> 00:20:17,090
+بتثبت أنه لأي epsilon و لأي U فـ let epsilon be
+
+193
+00:20:17,090 --> 00:20:20,030
+given و let U be fixed element في A
+
+194
+00:20:22,930 --> 00:20:26,390
+من هنا لهذه الـ Epsilon من هنا بما أن هذا الشرط
+
+195
+00:20:26,390 --> 00:20:32,670
+متحقق لأن خد ال Delta لأي ال Epsilon هادي given خد
+
+196
+00:20:32,670 --> 00:20:34,950
+ال Delta اللي هي هذه موجودة في ال uniform
+
+197
+00:20:34,950 --> 00:20:38,650
+continuous اللي بتعتمد على Epsilon فقط خديها هي ال
+
+198
+00:20:38,650 --> 00:20:43,730
+Delta هذه فطبعا هذه ال Delta بتخلي ال implication
+
+199
+00:20:43,730 --> 00:20:51,850
+هذه تتحقق نظرت؟ لو هذه متحققة فهذه متحققةإذن هيك
+
+200
+00:20:51,850 --> 00:20:55,270
+واضح إن ال uniform continuous إذا الواحدة بيقدر أن
+
+201
+00:20:55,270 --> 00:20:58,930
+f continuous and u لما أن u واظهر بترة إذا f
+
+202
+00:20:58,930 --> 00:21:05,310
+continuous and كل على كل المجموعية لكن
+
+203
+00:21:05,310 --> 00:21:11,690
+العكس مش صحيح إذا العكس المواحدة التانية العكس مش
+
+204
+00:21:11,690 --> 00:21:17,570
+صحيح but not conversely
+
+205
+00:21:22,160 --> 00:21:26,380
+العكس مش صحيح، يعني ال continuity لا تؤدي إلى ال
+
+206
+00:21:26,380 --> 00:21:37,360
+uniform continuity و على سبيل المثال for
+
+207
+00:21:37,360 --> 00:21:39,220
+example على سبيل المثال
+
+208
+00:21:46,610 --> 00:21:51,610
+أحنا شوفنا قبل شوية في بداية المحاضرة الـ function
+
+209
+00:21:51,610 --> 00:21:56,870
+f of x بساوي واحد على x و x ينتمي إلى a اللي هي
+
+210
+00:21:56,870 --> 00:22:03,870
+الفترة مفتوحة من سفر لماء لنهاية is continuous on
+
+211
+00:22:03,870 --> 00:22:11,550
+a أثبتت أنها continuous على المجموعة a but
+
+212
+00:22:15,440 --> 00:22:28,480
+but if is not uniformly continuous on a as we
+
+213
+00:22:28,480 --> 00:22:34,140
+shall see in
+
+214
+00:22:34,140 --> 00:22:39,100
+a few minutes
+
+215
+00:22:39,100 --> 00:22:46,160
+كما سنرى بعد لحظات الدالة هذه ليست متصلةإتصالا
+
+216
+00:22:46,160 --> 00:22:52,940
+منتظم هنأخر المرحلة ده شوية و هنبرهنه فلكن في
+
+217
+00:22:52,940 --> 00:22:59,560
+الأول خلينا من التعريف تبع ال uniform continuity
+
+218
+00:22:59,560 --> 00:23:09,720
+نستنتج non uniform continuity criterion من
+
+219
+00:23:09,720 --> 00:23:13,120
+هنا non uniform
+
+220
+00:23:15,380 --> 00:23:22,560
+non uniform continuity criteria
+
+221
+00:23:22,560 --> 00:23:33,940
+let
+
+222
+00:23:33,940 --> 00:23:41,240
+f from a to r be a function then
+
+223
+00:23:44,150 --> 00:23:53,730
+the following statements are equivalent واحد if is
+
+224
+00:23:53,730 --> 00:23:58,810
+not uniformly
+
+225
+00:23:58,810 --> 00:24:09,510
+continuous على المجال تبعها نين there exists
+
+226
+00:24:09,510 --> 00:24:17,380
+epsilon zero أكبر من السفرsuch that for every
+
+227
+00:24:17,380 --> 00:24:26,620
+delta أكبر من السفر يوجد x delta و u delta أناصر
+
+228
+00:24:26,620 --> 00:24:36,220
+في a such that absolute x delta minus u delta أصغر
+
+229
+00:24:36,220 --> 00:24:45,160
+من delta and absolute f of x delta-f of u دلتا
+
+230
+00:24:45,160 --> 00:24:53,160
+أكبر من أو يساوي epsilon zero الأبارع
+
+231
+00:24:53,160 --> 00:25:00,020
+التالتة there exist epsilon zero أكبر من الصفر and
+
+232
+00:25:00,020 --> 00:25:06,200
+two sequences متتاليتين xn
+
+233
+00:25:07,630 --> 00:25:14,930
+و un موجودين في مجال الدالة a such that بحيث ان
+
+234
+00:25:14,930 --> 00:25:23,910
+limit xn minus un بساوي سفر as n tends to infinity
+
+235
+00:25:23,910 --> 00:25:25,690
+and
+
+236
+00:25:27,050 --> 00:25:35,910
+absolute f of xn minus f of un أكبر من أو ساوي
+
+237
+00:25:35,910 --> 00:25:42,350
+epsilon zero هذا صحيح لكل n ينتبه للأعداد الطبيعية
+
+238
+00:25:42,350 --> 00:25:51,070
+okay تمام طيب نشوف البرهان تبع النظرية هذه البرهان
+
+239
+00:25:51,070 --> 00:25:55,440
+تبع النظرية هذه ينتج مباشرة منتعريف ال uniform
+
+240
+00:25:55,440 --> 00:26:01,980
+continuity تعالى نشوف واحد بكافئ اتنين طيب ما
+
+241
+00:26:01,980 --> 00:26:07,300
+معناه if uniform continuous على المجموعة ايه؟
+
+242
+00:26:07,300 --> 00:26:12,920
+معناه الشرط هذا بتحقق طيب ما معناه ان if not
+
+243
+00:26:12,920 --> 00:26:16,540
+uniform continuous على ايه؟ معناه ال negation تبع
+
+244
+00:26:16,540 --> 00:26:19,720
+العبارة دي بتحقق تعالى ننفذ العبارة انفذ العبارة
+
+245
+00:26:20,730 --> 00:26:25,250
+بدل لكل epsilon يوجد epsilon zero بدل يوجد delta
+
+246
+00:26:25,250 --> 00:26:31,550
+لكل delta موجبة بدل لكل x و u يوجد x و u يعتمد كل
+
+247
+00:26:31,550 --> 00:26:36,730
+واحد منهم يعتمد على ال delta بحيث لو كان هذا أصغر
+
+248
+00:26:36,730 --> 00:26:41,950
+من delta فلازم هذا يقدر انه الأصغر هذا أكبر من أو
+
+249
+00:26:41,950 --> 00:26:45,910
+ساوى ال epsilon zeroلأن واضح أن العبارة الأولى
+
+250
+00:26:45,910 --> 00:26:50,330
+بتكافئ التانية لأنه نفي التعريف بكافئ العبارة
+
+251
+00:26:50,330 --> 00:26:55,570
+التانية طيب التانية بتكافئ التالتة وهذا برضه صح
+
+252
+00:26:55,570 --> 00:27:01,650
+افرض أن التانية صحيحة تعني نثبت أن التالتة صحيحة
+
+253
+00:27:01,650 --> 00:27:05,170
+طيب هي التانية يوجد epsilon zero يوجد و هكذا
+
+254
+00:27:12,050 --> 00:27:16,770
+بحيث لكل delta خدي delta بالساوية واحد على n يعني
+
+255
+00:27:16,770 --> 00:27:21,370
+معنى أخر لكل n يوجد delta بالساوية واحد على n عدد
+
+256
+00:27:21,370 --> 00:27:26,730
+موجةوبالتالي يوجد X يعتمد على الـ Delta اللي هي
+
+257
+00:27:26,730 --> 00:27:31,670
+واحد على N اللي بتعتمد على N إذا لكل N لكل N يوجد
+
+258
+00:27:31,670 --> 00:27:37,310
+XN و UN صح؟ وبالتالي يوجد two sequences و ال two
+
+259
+00:27:37,310 --> 00:27:41,470
+sequences هدول بيحققوا أن absolute X واحدة XN
+
+260
+00:27:41,470 --> 00:27:46,310
+minus UN أصغر من واحد على N اللي هي ال Delta و هذا
+
+261
+00:27:46,310 --> 00:27:52,660
+صحيح لكل N إذا ال limitإذا كان هذا أصغر من واحد
+
+262
+00:27:52,660 --> 00:27:55,900
+على xn minus un على absolute أصغر من واحد على n
+
+263
+00:27:55,900 --> 00:28:00,020
+حصّم نظرية اتنين أربعة هذا معناه limit xn minus un
+
+264
+00:28:00,020 --> 00:28:06,420
+بساوة سفر وهذا هي absolute f of xn minus f of un
+
+265
+00:28:06,420 --> 00:28:12,180
+أكبر من أوسع okay فهو واضح وطبعا العكس نفس الحاجة
+
+266
+00:28:12,180 --> 00:28:16,020
+إذن البرهانة النظرية هذه ينتج مباشرة من ال
+
+267
+00:28:16,020 --> 00:28:20,340
+definition تبع ال uniform continuity
+
+268
+00:28:22,600 --> 00:28:27,400
+الان دعونا نرجع للمثال
+
+269
+00:28:27,400 --> 00:28:38,560
+هذا اذا هنا example to
+
+270
+00:28:38,560 --> 00:28:46,710
+show ان ال functionf of x بالساوي واحد على x is
+
+271
+00:28:46,710 --> 00:28:51,190
+not uniformly
+
+272
+00:28:51,190 --> 00:28:58,750
+continuous على المجموعة a اللي هي الفافرة مفتوحة
+
+273
+00:28:58,750 --> 00:29:07,010
+من صفر لما لا نهاية we use non
+
+274
+00:29:07,010 --> 00:29:09,270
+uniform
+
+275
+00:29:11,050 --> 00:29:16,390
+Non-uniform continuity
+
+276
+00:29:16,390 --> 00:29:21,890
+criteria
+
+277
+00:29:37,150 --> 00:29:47,310
+يوجد أبسلون زيرو يوجد
+
+278
+00:29:47,310 --> 00:29:49,870
+عدد أبسلون زيرو موجد
+
+279
+00:30:07,550 --> 00:30:16,570
+تختار خيار Xm بساوي واحد على ان اكيد هذه ال
+
+280
+00:30:16,570 --> 00:30:19,630
+sequence contain في الفترة المفتوحة من سفر للملا
+
+281
+00:30:19,630 --> 00:30:28,210
+نهاية صح؟ and كمان تختار خيار ثاني UN بساوي واحد
+
+282
+00:30:28,210 --> 00:30:33,370
+على ان زايد واحدبرضه هذه ال sequence حدودها كلها
+
+283
+00:30:33,370 --> 00:30:37,730
+موزبة وبالتالي مجموعة جزئية من الفترة المفتوحة من
+
+284
+00:30:37,730 --> 00:30:41,830
+سفر لملنغا Clearly
+
+285
+00:30:41,830 --> 00:30:45,290
+واضح
+
+286
+00:30:45,290 --> 00:30:54,330
+ان ال limit ل xn minus un as n times infinity
+
+287
+00:30:54,330 --> 00:31:04,720
+بساوي limit1 على n minus 1 على n زاد 1 as n equals
+
+288
+00:31:04,720 --> 00:31:11,660
+infinity ف limit الأولى ساوي سفر limit ال sequence
+
+289
+00:31:11,660 --> 00:31:18,200
+التانية سفر وبالتالي بيطلع سفر لأن هنا حققت كل
+
+290
+00:31:18,200 --> 00:31:24,020
+شروط ضايل بس المتبينة هادية also
+
+291
+00:31:28,610 --> 00:31:38,510
+أنا عندي absolute f of x in minus f of u in هذا
+
+292
+00:31:38,510 --> 00:31:46,990
+المفروض بيطلع بيساوي absolute in minus in زد واحد،
+
+293
+00:31:46,990 --> 00:31:53,430
+أزبوتك؟ وهذا بيساوي واحد، واحد أصغر من أوي، بيساوي
+
+294
+00:31:53,430 --> 00:32:00,000
+واحد اللي هو epsilon zeroو هذا صحيح لكل n في n
+
+295
+00:32:00,000 --> 00:32:08,940
+أصبوت هنا هاني انا ايش عملت ال criterion رقم تلاتة
+
+296
+00:32:08,940 --> 00:32:15,660
+اتحققتها اتحققت انها متحققة ها يوجد epsilon zero
+
+297
+00:32:15,660 --> 00:32:21,600
+واحد لاحظوا الواحد علشان انا اختارت واحد ممكن اخد
+
+298
+00:32:21,600 --> 00:32:25,040
+برضه epsilon zero بساوة اتنين لان الواحد اصغر من
+
+299
+00:32:25,040 --> 00:32:29,380
+الاتنينمافي مشكلة بس مش أقل من واحد يعني نص من
+
+300
+00:32:29,380 --> 00:32:32,840
+فعشان لأن هي أثبتت يوجد epsilon zero عدد موجب
+
+301
+00:32:32,840 --> 00:32:36,280
+ويوجد two sequences انا اختارتهم انا اوجدتهم بنفسي
+
+302
+00:32:36,280 --> 00:32:39,780
+واحد على n واحد على n زيادة واحد كلهم موجودين في
+
+303
+00:32:39,780 --> 00:32:45,380
+مجال الدالة ايه و limit الفرق بينهم سفر لكن
+
+304
+00:32:45,380 --> 00:32:52,960
+absolute الفرق بين صورهممش أقوى هذا هيكون بساوي
+
+305
+00:32:52,960 --> 00:32:59,860
+واحد أكبر من أو ساوي .. مش أصغر من أو ساوي بدي
+
+306
+00:32:59,860 --> 00:33:06,140
+أكبر من أو ساوي واحد اللي هو epsilon خليني
+
+307
+00:33:06,140 --> 00:33:09,760
+أنا أسحب الكلام اللي حكيته سابقا و أقول هنا ممكن
+
+308
+00:33:09,760 --> 00:33:13,140
+أخد ال epsilon zero بساوي واحد أو أي حاجة أصغر
+
+309
+00:33:13,140 --> 00:33:19,750
+يعني نص بنفعيعني أبسلون زيرو بساوي نص منفع لكن أي
+
+310
+00:33:19,750 --> 00:33:23,630
+شيء أكبر من واحد منفعش لأن أنا بدي واحد يكون أكبر
+
+311
+00:33:23,630 --> 00:33:28,270
+من أو ساوي أبسلون زيرو تمام؟ إذا هذه أثبتنا
+
+312
+00:33:28,270 --> 00:33:31,750
+وبالتالي حسب ال non-uniform continuity criterion
+
+313
+00:33:31,750 --> 00:33:35,710
+ال .. ال function هذه is not uniform ل continuous
+
+314
+00:33:35,710 --> 00:33:42,250
+تمام؟ لكن أثبتنا سابق جابليك أنها is continuous
+
+315
+00:33:42,250 --> 00:33:48,150
+على المجال تبعهاإذا لو قلنا لكم prove or disprove
+
+316
+00:33:48,150 --> 00:33:51,330
+continuity
+
+317
+00:33:51,330 --> 00:33:55,010
+implies continuity .. ال uniform .. continuity
+
+318
+00:33:55,010 --> 00:33:58,970
+implies uniform continuity هتقولي هذا ال statement
+
+319
+00:33:58,970 --> 00:34:04,150
+false و ال counter example هو هذا هذا مثال على
+
+320
+00:34:04,150 --> 00:34:07,570
+function continuous لكن ليست uniformly continuous
+
+321
+00:34:07,570 --> 00:34:17,820
+تمام؟ طيب، كويسخلّينا الآن نثبت بعض النظريات
+
+322
+00:34:17,820 --> 00:34:24,300
+المهمة اللي بتخص uniform continuity ومن أهم
+
+323
+00:34:24,300 --> 00:34:32,680
+النظريات التي هي النظرية التالية theorem اسمها
+
+324
+00:34:32,680 --> 00:34:36,660
+uniform continuity
+
+325
+00:34:36,660 --> 00:34:40,160
+continuity theorem
+
+326
+00:34:49,430 --> 00:34:56,770
+let I بساوي be
+
+327
+00:34:56,770 --> 00:35:05,570
+a closed and bounded interval
+
+328
+00:35:05,570 --> 00:35:09,350
+اذا
+
+329
+00:35:09,350 --> 00:35:17,360
+I عبارة عن closed and bounded interval لو كانلو
+
+330
+00:35:17,360 --> 00:35:22,980
+كانت الـ function f continuous، if f from I to R
+
+331
+00:35:22,980 --> 00:35:34,040
+is continuous on I، then f is uniformly ..
+
+332
+00:35:34,040 --> 00:35:43,060
+uniformly continuous on
+
+333
+00:35:43,060 --> 00:35:43,620
+I
+
+334
+00:35:46,190 --> 00:35:51,870
+والبرهان السهل prove by contradiction اذا ان بكل
+
+335
+00:35:51,870 --> 00:35:57,070
+بساطة نظرية هذه رغم بساطة بساطة ال statement تبعها
+
+336
+00:35:57,070 --> 00:36:01,710
+اللي انا من اهم النظريات بكل بساطة النظرية اللي
+
+337
+00:36:01,710 --> 00:36:04,850
+بيقول لو كان في عندك function متصل على المجال
+
+338
+00:36:04,850 --> 00:36:08,550
+تبعها والمجال تبعها closed bounded interval اذا
+
+339
+00:36:08,550 --> 00:36:13,970
+الاتصال العادي يصبح اتصال منتظمإن ان هذه الحالة
+
+340
+00:36:13,970 --> 00:36:18,030
+الوحيدة اللي او يعني احد الحالات اللي فيها بيكون
+
+341
+00:36:18,030 --> 00:36:22,650
+الاتصال العادى بقدر الاتصال المنظم ان احنا اضافنا
+
+342
+00:36:22,650 --> 00:36:26,630
+شرط ان مجال تبع الدالة مايكونش اي set لازم يكون
+
+343
+00:36:26,630 --> 00:36:31,090
+closed bounded interval لبرهان ذلك بال
+
+344
+00:36:31,090 --> 00:36:39,670
+contradiction assume on contrary that
+
+345
+00:36:41,290 --> 00:36:55,010
+if is not uniformly continuous on I then by non
+
+346
+00:36:55,010 --> 00:37:03,550
+uniform continuity criteria النظرية
+
+347
+00:37:03,550 --> 00:37:10,620
+اللي فوقيوجد إبسلون زيرو أكبر من السفر و two
+
+348
+00:37:10,620 --> 00:37:15,620
+sequences and
+
+349
+00:37:15,620 --> 00:37:25,040
+two sequences واحدة نسميها x in والتانية un
+
+350
+00:37:25,040 --> 00:37:37,510
+contained in I بحيث أنهabsolute xn minus un أصغر
+
+351
+00:37:37,510 --> 00:37:46,390
+من واحد على n لكل n and absolute f of xn minus f
+
+352
+00:37:46,390 --> 00:37:56,420
+of unأكبر من أو ساوي epsilon zero لكل n في n كل
+
+353
+00:37:56,420 --> 00:38:01,300
+هذا ناخده من ال non uniform continuity criterion
+
+354
+00:38:01,300 --> 00:38:11,500
+الآن بدنا نصل لتناقض طيب
+
+355
+00:38:11,500 --> 00:38:15,980
+عشان نصل لتناقض since
+
+356
+00:38:18,370 --> 00:38:25,750
+I is bounded الفترة دي احنا فرضين انها bounded و
+
+357
+00:38:25,750 --> 00:38:32,550
+ال sequence x in contained in I then ال sequence x
+
+358
+00:38:32,550 --> 00:38:35,450
+in is bounded
+
+359
+00:38:41,210 --> 00:38:57,810
+هنا باستخدام حسب bolzano
+
+360
+00:38:57,810 --> 00:39:01,890
+weierstrass
+
+361
+00:39:01,890 --> 00:39:02,350
+firm
+
+362
+00:39:11,180 --> 00:39:23,360
+السيكوينس هناك سبسيكوينس سميها xnk of xn such that
+
+363
+00:39:23,360 --> 00:39:28,740
+السيكوينس had a convergence limit xnk as k tends
+
+364
+00:39:28,740 --> 00:39:33,840
+to infinity as
+
+365
+00:39:33,840 --> 00:39:40,030
+k tends to infinity بساوي z ينتمي إلى rبالنسبة لـ
+
+366
+00:39:40,030 --> 00:39:45,090
+some z and some r بلزانو فيروس عسكرية كل sequence
+
+367
+00:39:45,090 --> 00:39:48,570
+لها convergence subsequence سم السبسيكوينس هكذا
+
+368
+00:39:48,570 --> 00:39:50,330
+وسم ال limit تبعتها هكذا
+
+369
+00:39:54,530 --> 00:40:00,450
+الـ sub-sequence X in K contained in I التي هي
+
+370
+00:40:00,450 --> 00:40:05,450
+الفترة المغلقة من A إلى B فأحنا أخدنا نظرية تقول
+
+371
+00:40:05,450 --> 00:40:08,290
+أن لو كان هناك sequence حدودها محصورة بين A وB
+
+372
+00:40:08,290 --> 00:40:13,230
+ومتقاربة فنهايتها أيضًا محصورة بين A وB فهذا سيؤدي
+
+373
+00:40:13,230 --> 00:40:18,230
+إلى أن Z تنتمي إلى الفترة المغلقة من A إلى B التي
+
+374
+00:40:18,230 --> 00:40:18,890
+هي I
+
+375
+00:40:24,140 --> 00:40:28,340
+الذي يدفع الاتصال
+
+376
+00:40:28,340 --> 00:40:31,620
+الاتصال
+
+377
+00:40:31,620 --> 00:40:34,420
+الاتصال الاتصال الاتصال الاتصال الاتصال الاتصال
+
+378
+00:40:34,420 --> 00:40:44,240
+الاتصال الاتصال
+
+379
+00:40:51,890 --> 00:41:02,550
+موجودين في I موجودين في I موجودين
+
+380
+00:41:02,550 --> 00:41:12,990
+في I موجودين في Iالـ subsequence UN برضه لها
+
+381
+00:41:12,990 --> 00:41:16,970
+subsequence مشابهة وconvergent لنفس الـ Z هذا مش
+
+382
+00:41:16,970 --> 00:41:25,390
+واضح لثبته لثباته to see this to see this note
+
+383
+00:41:25,390 --> 00:41:28,250
+that
+
+384
+00:41:31,680 --> 00:41:37,040
+بنقدر اخل الفرق بين
+
+385
+00:41:37,040 --> 00:41:47,580
+unk و z أصغر من أي epsilon فهذا
+
+386
+00:41:47,580 --> 00:41:58,020
+أصغر من أو ساوي unk minus xnk زاد absolute xnk
+
+387
+00:41:58,020 --> 00:42:03,380
+minus zهو في الأصل أن أنا المفروض أكتب انا اشعر
+
+388
+00:42:03,380 --> 00:42:07,840
+بالإضطراحة x in k و رجعتها و استخدمت ال triangle
+
+389
+00:42:07,840 --> 00:42:18,420
+inequality طيب هذا صحيح لكل k ينتمي إلى n طيب أنا
+
+390
+00:42:18,420 --> 00:42:22,220
+عندي limit
+
+391
+00:42:22,220 --> 00:42:30,290
+x in minus u in بالساوية سفرلأن هذا صحيح لكل n ف
+
+392
+00:42:30,290 --> 00:42:36,130
+limit u in k minus x in k برضه بيساوي سفر فهذا
+
+393
+00:42:36,130 --> 00:42:43,750
+بيروح لسفر as k tends to infinity وعندي أنا برضه u
+
+394
+00:42:43,750 --> 00:42:50,290
+ال x in k جلنا تقول إلى z فبالتالي ال absolute
+
+395
+00:42:50,290 --> 00:42:56,630
+value هذه بتروح لسفر as k tends to infinityوهذا
+
+396
+00:42:56,630 --> 00:43:08,170
+أكبر من سفر، إذن by squeeze theorem ال sequence
+
+397
+00:43:08,170 --> 00:43:13,030
+هذه محصورة بين ال sequence هذه بالسفر ومجموعة two
+
+398
+00:43:13,030 --> 00:43:18,270
+sequences بيقولوا للسفرإذا من ال limit ل absolute
+
+399
+00:43:18,270 --> 00:43:25,570
+u in k minus z as k tends to infinity بساوي سفر و
+
+400
+00:43:25,570 --> 00:43:31,270
+منها بطلع ال limit u in k as k tends to infinity
+
+401
+00:43:31,270 --> 00:43:38,230
+بساوي z وبالتالي هذا بثبت ال claim تمام؟ إذا هنا
+
+402
+00:43:38,230 --> 00:43:43,830
+أثبتنا ال claim الآن بعد ما أثبتنا ال claim
+
+403
+00:43:57,160 --> 00:44:04,320
+طيب طيب now انا
+
+404
+00:44:04,320 --> 00:44:12,300
+اندي قولنا اثبتنا انه النقطة z تنتمي .. z تنتمي ل
+
+405
+00:44:12,300 --> 00:44:16,880
+I ال limit تبعت ال subsequence تنتمي ل I وال F
+
+406
+00:44:16,880 --> 00:44:17,460
+continuous
+
+407
+00:44:21,950 --> 00:44:25,850
+إن الهدف بقدم if is continuous لأن if continuous
+
+408
+00:44:25,850 --> 00:44:32,210
+على I وبالتالي continuous عند أي نقطة في ال I ولا
+
+409
+00:44:32,210 --> 00:44:36,990
+تكن ال Z hence
+
+410
+00:44:36,990 --> 00:44:40,730
+by
+
+411
+00:44:40,730 --> 00:44:46,510
+sequential criterion by sequential criterion for
+
+412
+00:44:46,510 --> 00:44:50,500
+continuous functionالـ function continuous عند
+
+413
+00:44:50,500 --> 00:44:54,640
+النقطة z وفي عندي sequence x in k converged ل z
+
+414
+00:44:54,640 --> 00:45:01,260
+اذا ال limit لصورة ال sequence او ال subsequence
+
+415
+00:45:01,260 --> 00:45:10,180
+لما كتره ل infinity بساوي f of z و كذلك ايضاAnd
+
+416
+00:45:10,180 --> 00:45:13,760
+برضه ال limit أنا عندي برضه ال sequence هذي
+
+417
+00:45:13,760 --> 00:45:20,220
+converge ل z فنهاية صورة ال subsequence u in k as
+
+418
+00:45:20,220 --> 00:45:27,260
+k tends to infinity برضه بيساوي f of z تمام
+
+419
+00:45:27,260 --> 00:45:31,520
+طيب
+
+420
+00:45:31,520 --> 00:45:35,000
+لكن
+
+421
+00:45:35,000 --> 00:45:43,090
+أنا عنديأنا عندي المتباينة الـ but أنا عندي
+
+422
+00:45:43,090 --> 00:45:47,170
+absolute f of x in
+
+423
+00:45:55,850 --> 00:46:01,570
+من الفرض هيها من الفرض ان ال function not
+
+424
+00:46:01,570 --> 00:46:07,070
+uniformly continuous انا عندي هذا اكبر من او يساوي
+
+425
+00:46:07,070 --> 00:46:10,050
+epsilon zero لكل n لكل حدود ال sequences
+
+426
+00:46:14,300 --> 00:46:20,060
+فهذا بيقدّي .. هذا بدوره بيقدّي انه epsilon zero
+
+427
+00:46:20,060 --> 00:46:27,180
+هي epsilon zero أصغر من أو ساوي absolute f of x in
+
+428
+00:46:27,180 --> 00:46:36,220
+k minus f of u in k تمام؟
+
+429
+00:46:37,840 --> 00:46:41,720
+هذا صحيح لـ sequence x in و لـ sequence u in إذا
+
+430
+00:46:41,720 --> 00:46:46,200
+صحيح للـ subsequence للـ subsequences إذا هذه جاية
+
+431
+00:46:46,200 --> 00:46:50,620
+من هنا طيب و by triangle inequality by triangle
+
+432
+00:46:50,620 --> 00:46:56,760
+inequality ممكن أخلي هذا أصغر لو ساوي f of x nk
+
+433
+00:46:56,760 --> 00:47:10,090
+minus f of z زاد absolute f of z-F of U in K انا
+
+434
+00:47:10,090 --> 00:47:14,610
+شو انا عاملة اتراحت من هنا F of Z و رجعتها اه و
+
+435
+00:47:14,610 --> 00:47:17,810
+استخدمت ال triangle equality فصار اندي اصلا مجموعة
+
+436
+00:47:17,810 --> 00:47:24,070
+two absolute values طيب
+
+437
+00:47:24,070 --> 00:47:30,660
+ما انا ممكن اخليأنا عندي limit ال sequence هذه
+
+438
+00:47:30,660 --> 00:47:36,800
+بساوي f of z فلأي given epsilon أكبر من الصفر ممكن
+
+439
+00:47:36,800 --> 00:47:42,300
+أخلي absolute الفرخ هذا أصغر من epsilon على 2 ونفس
+
+440
+00:47:42,300 --> 00:47:47,260
+الحاجة أنا عندي ال sequence f of u and k converged
+
+441
+00:47:47,260 --> 00:47:51,840
+ل f of z إذا ممكن أخلي ال absolute value للفرخ هذه
+
+442
+00:47:51,840 --> 00:47:59,540
+أصغر من epsilon على 2وبالتالي بيطلع المجموعة
+
+443
+00:47:59,540 --> 00:48:06,420
+epsilon هذا صحيح لكل K أكبر من أو ساوي كابتل K أو
+
+444
+00:48:06,420 --> 00:48:12,360
+كابتل N واضح تمام؟ يعني لا أي epsilon أكبر من صفر
+
+445
+00:48:12,360 --> 00:48:15,820
+أو لا عفو ان ال epsilon نفس ال epsilon zero هذه ال
+
+446
+00:48:15,820 --> 00:48:19,380
+epsilon هي نفس ال epsilon zeroهي عندي الـ Epsilon
+
+447
+00:48:19,380 --> 00:48:23,780
+Zero given لما ان ال sequence هي ال converge إذا
+
+448
+00:48:23,780 --> 00:48:28,400
+يوجد capital N واحد يعتمد على Epsilon Zero بحيث أن
+
+449
+00:48:28,400 --> 00:48:33,580
+أبسلوت الفرق هذا أصغر من أو ساوي Epsilon على اتنين
+
+450
+00:48:33,580 --> 00:48:37,140
+لكل K أكبر من أو ساوي capital K واحد او capital N
+
+451
+00:48:37,140 --> 00:48:42,560
+واحد ونفس الحاجة لنفس ال Epsilon Zero يوجد N اتنين
+
+452
+00:48:43,830 --> 00:48:47,730
+بحيث انه بما انه هذه ال sequence converge اذا
+
+453
+00:48:47,730 --> 00:48:52,310
+الفرخة ده بقدر اخليه لكل n اكبر من او لكل k اكبر
+
+454
+00:48:52,310 --> 00:48:56,630
+من او ساوي n اتنين اصغر من يبسلون اتنين الان خدي n
+
+455
+00:48:56,630 --> 00:49:05,410
+بساوي ال maximum ل n واحد و n اتنين فبقدر
+
+456
+00:49:05,410 --> 00:49:11,930
+اخلي هذا اصغر من يبسلون زيرو لكل k اكبر من او ساوي
+
+457
+00:49:11,930 --> 00:49:16,590
+nففي النهاية بيطلع عندى epsilon zero أقل من
+
+458
+00:49:16,590 --> 00:49:19,650
+epsilon أصغر من epsilon zero هذا مديني
+
+459
+00:49:19,650 --> 00:49:23,590
+contradiction لأن هذا التناقض بيقول لي أن ال
+
+460
+00:49:23,590 --> 00:49:28,150
+assumption تبعنا أن ال function not uniformly
+
+461
+00:49:28,150 --> 00:49:32,050
+continuous كان assumption خطأ لأن الصح أن ال F
+
+462
+00:49:32,050 --> 00:49:37,810
+تكون uniformly continuous okay تمام واضح؟Okay إذا
+
+463
+00:49:37,810 --> 00:49:44,230
+بنوقف ان شاء الله هنا عند نهاية البرهان هذا و
+
+464
+00:49:44,230 --> 00:49:51,690
+بيكون هيك احنا يعني خلصنا جزء مش بسيط في section
+
+465
+00:49:51,690 --> 00:49:56,730
+خمسة أربعة و نكتفي بهذا القدر و يعطيكم ألف عافية و
+
+466
+00:49:56,730 --> 00:49:58,590
+شكرا لحصن أصغائكم
+
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/Kfhi1a_WpFk_postprocess.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/Kfhi1a_WpFk_postprocess.srt
new file mode 100644
index 0000000000000000000000000000000000000000..e98332f8e4973dbefe907d239f4476f51c3a9750
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/Kfhi1a_WpFk_postprocess.srt
@@ -0,0 +1,1616 @@
+1
+00:00:20,650 --> 00:00:26,490
+طبعا احنا زي ما اتفجنا معاكم هنعمل
+
+2
+00:00:26,490 --> 00:00:36,030
+مناقشة لل course و هنبدأ طبعا ب chapter اتنين اللي
+
+3
+00:00:36,030 --> 00:00:42,270
+هو اول chapter درسناه و هنبدأ ب section اتنين واحد
+
+4
+00:00:42,270 --> 00:00:46,270
+و اذا في وجب طبعا هنحاول نجاوب على أسئلة في
+
+5
+00:00:46,270 --> 00:00:47,190
+section اتنين اتنين
+
+6
+00:00:50,880 --> 00:00:55,480
+باطمن عليكم انه يعني طبعا الأسئلة عددها كبير و مش
+
+7
+00:00:55,480 --> 00:01:00,980
+هانلحج نحل كل المسائل لكن باطمن عليكم انكم تسألوا
+
+8
+00:01:00,980 --> 00:01:06,020
+الأسئلة اللي انتوا يعني وجدتوا فيها صعوبة في حالها
+
+9
+00:01:06,020 --> 00:01:10,000
+حتى
+
+10
+00:01:10,000 --> 00:01:17,240
+تكون يعني الفايدة تعمق أكتر فنبدأ ب section 2 1 هل
+
+11
+00:01:17,240 --> 00:01:23,860
+في أي سؤال في section 2 1حاولتوا تحلوه ماعرفتوش
+
+12
+00:01:23,860 --> 00:01:33,540
+تحلوه أو وجدته صعوبة في حاله ففي
+
+13
+00:01:33,540 --> 00:01:36,640
+أي سؤال من الأسئلة اللي احنا حددناها في ال
+
+14
+00:01:36,640 --> 00:01:42,080
+syllabus و قولنا لكم حلوها في أي سؤال في section 2
+
+15
+00:01:42,080 --> 00:01:49,810
+و 1 تحبوا تسألوا عنه؟ستاذ ممكن نسأل .. نشرح ..
+
+16
+00:01:49,810 --> 00:01:54,650
+نشرح .. نظرية ما .. ما نثبتش .. مش عارف كيف نثبت
+
+17
+00:01:54,650 --> 00:02:02,090
+ما هي عامة exercise اه إيش هي دي؟ X على Z زي Y على
+
+18
+00:02:02,090 --> 00:02:07,930
+W يوم ساوي X Z زي Z Y على Z W كيف نستخدم .. كيف
+
+19
+00:02:07,930 --> 00:02:09,430
+نبدأ فيها؟ مش عارف
+
+20
+00:02:17,430 --> 00:02:21,510
+يعني الخواص اللى .. مش هادى خاصيا من الخواص اللى
+
+21
+00:02:21,510 --> 00:02:27,210
+خدناها حاول تشوف يعني كيف احنا برهنا الخواص الأخرى
+
+22
+00:02:27,210 --> 00:02:37,050
+و تستفيدي منها و .. و تحاول تبرهنيها يعني ممكن
+
+23
+00:02:37,050 --> 00:02:41,310
+كمان تبص في الكتاب المقرر و تشوف يعني هل هو حلها
+
+24
+00:02:41,310 --> 00:02:47,940
+او محلاش لكن انا يعنيالخواص اللي احنا مابرهنهاش
+
+25
+00:02:47,940 --> 00:02:53,700
+يعني .. يعني كان برهانها سهل و ممكن تتبرهنيها بنفس
+
+26
+00:02:53,700 --> 00:03:00,280
+الأسلوب اللي احنا برهننا فيه الأزاية الأخرى فخليني
+
+27
+00:03:00,280 --> 00:03:03,900
+أترك الإجابة على السؤال هذا إليك تحاولي فيه مرة
+
+28
+00:03:03,900 --> 00:03:08,360
+تانية و إذا ماعرفتيش ممكن تجيلي على المكتب و ممكن
+
+29
+00:03:08,360 --> 00:03:09,280
+نتناقش
+
+30
+00:03:11,940 --> 00:03:16,180
+ياريت تسألوني أسئلة من التمرين من ال exercises إذا
+
+31
+00:03:16,180 --> 00:03:26,140
+سمحتوا، تفضلي السؤال الرابع في section السابع في
+
+32
+00:03:26,140 --> 00:03:30,360
+section اتنين واحد، حاضر
+
+33
+00:03:39,140 --> 00:03:50,140
+إذا حل السؤال سبعة section اتنين واحد modify
+
+34
+00:03:50,140 --> 00:03:53,360
+the proof of theorem اتنين واحد اربعة في الكتاب
+
+35
+00:03:53,360 --> 00:03:59,200
+المقرر to show that there
+
+36
+00:03:59,200 --> 00:04:05,640
+does not exist لا يوجد there does not exist
+
+37
+00:04:08,790 --> 00:04:19,170
+T ينتمي للـ Q بحيث أن T تربية بساوية تلاتة بمعنى
+
+38
+00:04:19,170 --> 00:04:23,770
+آخر يعني الجذر التلاتة بنا نثبت أنه جذر التلاتة
+
+39
+00:04:23,770 --> 00:04:28,730
+ليس عدد نسبي فالمفروض
+
+40
+00:04:28,730 --> 00:04:33,010
+أنكم يعني تفهموا و تحاولوا تفهموا البرهان تبع
+
+41
+00:04:33,010 --> 00:04:37,810
+اثبات أنه جذر اتنين ليس عدد نسبي وتحاولوا تعملوا
+
+42
+00:04:37,810 --> 00:04:45,910
+برهان مشابهفي أدة براهين للسؤال هذا فأحد البراهين
+
+43
+00:04:45,910 --> 00:04:52,590
+شبه البرهان اللي أخدنا بتابع جداد الأتنين ليس عدد
+
+44
+00:04:52,590 --> 00:04:59,330
+نسبي فخلينا نشوفه مع بعض خلينا نشوف إذن البرهان
+
+45
+00:04:59,330 --> 00:05:04,810
+prove طبعا البرهان بالتناقض assume
+
+46
+00:05:07,480 --> 00:05:18,180
+on contrary نفرض على النقيد there exist T بساوي A
+
+47
+00:05:18,180 --> 00:05:29,420
+على B عدد نسبي بحيث انه ال greatest common divisor
+
+48
+00:05:29,420 --> 00:05:34,760
+لل A والB بساوي واحد and
+
+49
+00:05:37,230 --> 00:05:43,970
+T تربيع اللي هو بساوي A على B لكل تربيع بساوي
+
+50
+00:05:43,970 --> 00:05:52,030
+تلاتة اذا هذا النقيض او النفي تبع يوجد عدد نسبي
+
+51
+00:05:52,030 --> 00:06:00,050
+مربعه بساوي تلاتة النفي تبعه يوجد عدد نسبي مربعه
+
+52
+00:06:00,050 --> 00:06:05,840
+بساوي تلاتة وطبعا ممكن نفرض انه العدد النسبيالـ
+
+53
+00:06:05,840 --> 00:06:10,160
+greatest common divisor للـ bus والمقام تبعه بساوي
+
+54
+00:06:10,160 --> 00:06:17,300
+واحد زي ما عملنا في حالة ال square root of two طيب
+
+55
+00:06:17,300 --> 00:06:22,740
+then في الحالة هذه لو ربع .. لو هنا من المعادلة
+
+56
+00:06:22,740 --> 00:06:32,820
+هذه بنحصل على a تربية بساوي تلاتة b تربية وهذا
+
+57
+00:06:32,820 --> 00:06:42,870
+بيقدّيإن ال B تقسم A تربية ال
+
+58
+00:06:42,870 --> 00:06:48,170
+B تقسم A تربية او A تربية اللي هي تلاتة B تربية
+
+59
+00:06:48,170 --> 00:06:54,810
+تقبل القسم على B بدون باطل طيب
+
+60
+00:06:54,810 --> 00:07:03,250
+ال و في الحالة هذه بقدر
+
+61
+00:07:04,790 --> 00:07:10,850
+أفصل حالتين العدد بي هذا ممكن .. هذا طبعا عدد ..
+
+62
+00:07:10,850 --> 00:07:20,430
+عدد صحيح ممكن يكون أكبر من الواحد و ممكن يكون
+
+63
+00:07:20,430 --> 00:07:28,990
+بساوي واحد فنفرض أن ال بي أكبر من واحد then في
+
+64
+00:07:28,990 --> 00:07:32,770
+الحالة هذه بي can be written
+
+65
+00:07:37,720 --> 00:07:45,000
+as product of
+
+66
+00:07:45,000 --> 00:07:49,640
+primes ال
+
+67
+00:07:49,640 --> 00:07:54,820
+بيه ده عدد صحيح أكبر من واحد فممكن نكتبه على صورة
+
+68
+00:07:55,540 --> 00:08:01,280
+حاصل ضرب أعداد أولية أي عدد صحيح أكبر من واحد ممكن
+
+69
+00:08:01,280 --> 00:08:06,180
+كتابته على صورة حاصل ضرب أعداد أولية product of
+
+70
+00:08:06,180 --> 00:08:12,100
+primes prime عدد أول هذا حقيقة معروفة في نظرية
+
+71
+00:08:12,100 --> 00:08:17,020
+الأعداد وبالتالي
+
+72
+00:08:17,020 --> 00:08:20,240
+hence وبالتالي
+
+73
+00:08:22,840 --> 00:08:33,180
+يوجد a prime يوجد عدد أولي a prime P بحيث أن هذا
+
+74
+00:08:33,180 --> 00:08:39,120
+ال P يقسم ال B يعني
+
+75
+00:08:39,120 --> 00:08:44,880
+أنا ال B هذا هي product of primes ممكن يكون بساوي
+
+76
+00:08:44,880 --> 00:08:50,840
+P1 ضرب P2 ضرب PN
+
+77
+00:08:52,290 --> 00:08:59,710
+حيث و P1 و P2 و PN كلهم Primes أعداد أولية فأكيد
+
+78
+00:08:59,710 --> 00:09:07,390
+لو أخدت أي واحد منهم فهذا بيقسم بي أو بيقبل قسم
+
+79
+00:09:07,390 --> 00:09:16,530
+عليه بس إذا يوجد يوجد Prime سمي P يقسم بي أو بيقبل
+
+80
+00:09:16,530 --> 00:09:20,710
+قسم عليه فهذا
+
+81
+00:09:20,710 --> 00:09:33,050
+بيؤديهذا بيقدي ان P يقسم ال A تربية لان انا عندي P
+
+82
+00:09:33,050 --> 00:09:40,570
+يقسم A تربية و P يقسم B اذا
+
+83
+00:09:40,570 --> 00:09:51,030
+ال P هذا يقسم A تربية okay تمام طيب و منهاهذا
+
+84
+00:09:51,030 --> 00:09:57,970
+بيدّي أن P يقسم A إذا P يقسم A تربية فممكن اثبات
+
+85
+00:09:57,970 --> 00:10:08,830
+أن P يقسم العدد الصحيح A وهكذا أثبتنا الـ
+
+86
+00:10:08,830 --> 00:10:20,170
+greatest common divisor لـ A وB أكبر من أو ساوي Pو
+
+87
+00:10:20,170 --> 00:10:27,090
+P هذا طبعا أكبر من واحد لأن
+
+88
+00:10:27,090 --> 00:10:39,430
+ال P يقسم ال A و P يقسم ال B فمعناته
+
+89
+00:10:39,430 --> 00:10:46,350
+في عامل مشترك في common divisor اللي هو Pبين a و b
+
+90
+00:10:46,350 --> 00:10:50,050
+لأن ال greatest common divisor سيكون على الأقل p
+
+91
+00:10:50,050 --> 00:10:53,770
+ويمكن أن يكون أكبر وبالتالي greatest common
+
+92
+00:10:53,770 --> 00:10:59,070
+divisor ل a و b أكبر من واحد وهذا بتناقض مع فرضنا
+
+93
+00:10:59,070 --> 00:11:02,650
+أن greatest common divisor ل a و b بساوي واحد
+
+94
+00:11:02,650 --> 00:11:07,850
+which is
+
+95
+00:11:07,850 --> 00:11:08,870
+a contradiction
+
+96
+00:11:15,680 --> 00:11:23,960
+وهذا التناقض بكمل البرهان يعني
+
+97
+00:11:23,960 --> 00:11:30,740
+فرضنا هذا أنه في عدد نسبي مربع بساوي تلاتة كان فرض
+
+98
+00:11:30,740 --> 00:11:36,780
+خطأ الصح أنه لا يوجد عدد نسبي مربع بساوي تلاتةOkay
+
+99
+00:11:36,780 --> 00:11:39,740
+تمام إذا هذا برهان وفيه طبعا براهين أخرى ممكن
+
+100
+00:11:39,740 --> 00:11:43,460
+تلاقوها تجدوها في كتب ال real analysis لكن هذا أحد
+
+101
+00:11:43,460 --> 00:11:52,580
+البراهين تمام؟ مين عنده سؤال تاني؟ في أي سؤال
+
+102
+00:11:52,580 --> 00:12:03,220
+تاني؟ في section 2.1 أو 2.2؟
+
+103
+00:12:03,220 --> 00:12:04,680
+نعم
+
+104
+00:12:06,770 --> 00:12:10,590
+لا نفس السؤال مش في اكتر من حالة ولا بس باخد ال D
+
+105
+00:12:10,590 --> 00:12:15,390
+أكتر من حالة؟ اه في كمان حالة صحيح ال case التانية
+
+106
+00:12:15,390 --> 00:12:23,290
+مظبوط ال case التانية خليني اكتبها هناك صحيح في
+
+107
+00:12:23,290 --> 00:12:24,030
+حالة تانية
+
+108
+00:12:42,990 --> 00:12:48,270
+كاس اتنين ال بي بالساوية واحد لو كانت ال بي
+
+109
+00:12:48,270 --> 00:12:55,330
+بالساوية واحد فهذا بيقدّي انا عندي ا تربية من هنا
+
+110
+00:12:55,330 --> 00:13:00,270
+في عندي ا تربية بالساوية تلاتة بي تربية فلو ال بي
+
+111
+00:13:00,270 --> 00:13:08,730
+بالساوية واحد معناه ا تربية تطلع بالساوية تلاتةو
+
+112
+00:13:08,730 --> 00:13:20,170
+هذا يعني مستحيل which is impossible هذا
+
+113
+00:13:20,170 --> 00:13:28,690
+مستحيل لأنه لأنه مافيش عدد صحيح عدد صحيح مربع
+
+114
+00:13:28,690 --> 00:13:34,130
+بساوي تلاتة since there does not exist integer
+
+115
+00:13:34,130 --> 00:13:36,950
+integer
+
+116
+00:13:38,790 --> 00:13:46,690
+a such that a تربية بساوي تلاتة اذا في الحالة هذه
+
+117
+00:13:46,690 --> 00:13:54,510
+حصلنا على حاجة impossible يعني تناقض وهنا
+
+118
+00:13:54,510 --> 00:14:01,510
+كمان حصلنا على تناقض ان ال assumption تبعنا انه
+
+119
+00:14:01,510 --> 00:14:06,550
+يوجد عدد نسبي مربع بساوي تلاتة كان فرض خاطئ وهذا
+
+120
+00:14:06,550 --> 00:14:08,090
+يكمل البرهان في الحالتين
+
+121
+00:14:10,890 --> 00:14:17,630
+في اي سؤال تانى؟ في
+
+122
+00:14:17,630 --> 00:14:25,890
+اسئلة تانية في ال section 2 1 او 2 2 انا
+
+123
+00:14:25,890 --> 00:14:30,190
+كنت متوقع ان يكون عندكم اسئلة كتيرة واضح جدا انكم
+
+124
+00:14:30,190 --> 00:14:35,150
+انتوا لا محاولين تحلوا الاسئلة وبالتالي ما عندكمش
+
+125
+00:14:35,150 --> 00:14:38,550
+يعني استفسارات
+
+126
+00:14:40,490 --> 00:14:42,690
+طبعا ست و عشرين
+
+127
+00:15:13,320 --> 00:15:20,080
+سؤال ستة وعشرين سكتشن اتنين واحد show
+
+128
+00:15:20,080 --> 00:15:28,560
+by induction that
+
+129
+00:15:28,560 --> 00:15:36,860
+لو
+
+130
+00:15:36,860 --> 00:15:46,900
+كان A ينتمي ل Rو M و N أعداد طبيعية فهذا بيقدي أن
+
+131
+00:15:46,900 --> 00:15:52,480
+A to M
+
+132
+00:15:52,480 --> 00:16:04,080
+plus N بيساوي A to M في A to N نريد
+
+133
+00:16:04,080 --> 00:16:10,300
+أن نثبت أن استخدام ال inductionإنه لأي عدد حقيقي A
+
+134
+00:16:10,300 --> 00:16:15,640
+و لأي عدد طبيعي M و N A to M زاد N بساوي A to M
+
+135
+00:16:15,640 --> 00:16:24,800
+ضرب A to N هذا أحد قوانين الأساس فطبعا
+
+136
+00:16:24,800 --> 00:16:29,420
+هنعمل induction أو بيسموه double induction على M و
+
+137
+00:16:29,420 --> 00:16:32,860
+N في نفس الواجت فالحالة الأولى
+
+138
+00:16:36,700 --> 00:16:40,780
+فم بيساوي ان بيساوي واحد لو كان الام والان كلها
+
+139
+00:16:40,780 --> 00:16:46,040
+بيساوي واحد then
+
+140
+00:16:46,040 --> 00:16:56,900
+a to m زي الان بيساوي a to واحد زي الواحد بيساوي a
+
+141
+00:16:56,900 --> 00:17:06,340
+ترمية andA to M ضرب A to N بيساوي A to 1 ضرب A to
+
+142
+00:17:06,340 --> 00:17:13,780
+1 بيساوي A تلبيه وبالتالي الطرفين المعادلة هذه
+
+143
+00:17:13,780 --> 00:17:21,200
+متحققة لأن الطرفين بيساوي نفس المقدار A تلبيه
+
+144
+00:17:21,200 --> 00:17:26,300
+اذا اذا
+
+145
+00:17:26,300 --> 00:17:27,680
+star holds
+
+146
+00:17:30,970 --> 00:17:39,310
+in case M بساوي M بساوي 1 المعادلة star متحققه في
+
+147
+00:17:39,310 --> 00:17:43,610
+حالة M بساوي M بساوي 1 الان ال induction
+
+148
+00:17:43,610 --> 00:17:48,370
+hypothesis ان هذا induction عادي بس يعني بدل من
+
+149
+00:17:48,370 --> 00:17:53,530
+قيان على M يكون على M و M ال induction hypothesis
+
+150
+00:17:53,530 --> 00:17:59,270
+الفرض تبع ال induction بنفرض assume
+
+151
+00:18:03,340 --> 00:18:09,860
+assume star holds المعادلة
+
+152
+00:18:09,860 --> 00:18:18,760
+star صحيحة for m بساوي k أكبر من واحد and n بساوي
+
+153
+00:18:18,760 --> 00:18:28,940
+j أكبر من واحد this
+
+154
+00:18:28,940 --> 00:18:42,170
+meansهذا معناه ان a to k plus j بساوي a to k ضرب a
+
+155
+00:18:42,170 --> 00:18:48,710
+to j اذا احنا فرضين صحة المعادلة هذه عندما m بساوي
+
+156
+00:18:48,710 --> 00:18:57,560
+k و عندما n بساوي jالان نريد اثبات صحتها عندما M
+
+157
+00:18:57,560 --> 00:19:01,440
+بيساوي K زايد واحد وعندما N بيساوي G زايد واحد
+
+158
+00:19:01,440 --> 00:19:08,840
+تعالى نثبت صحتها في الحالة يعني اذا now A ناخد A
+
+159
+00:19:08,840 --> 00:19:16,320
+to K زايد واحد زايد G زايد واحد
+
+160
+00:19:20,610 --> 00:19:25,150
+ان نتبع صحة المعادلة star عندما M بساوي K زايد
+
+161
+00:19:25,150 --> 00:19:29,370
+واحد و N بساوي G زايد واحد في الحالة هذه الطرف
+
+162
+00:19:29,370 --> 00:19:37,310
+اليسار لstar بساوي كلام هذا وهذا ممكن نجزئه إلى A
+
+163
+00:19:37,310 --> 00:19:49,290
+to K زايد G زايد واحد زايد واحدالأُس هذا ممكن نقعد
+
+164
+00:19:49,290 --> 00:19:53,450
+الكتابة على صورة ك زائد جي زائد واحد مع بعض زائد
+
+165
+00:19:53,450 --> 00:20:01,550
+واحد وهذا بيساوي a to ك زائد جي زائد واحد ضرب a
+
+166
+00:20:01,550 --> 00:20:06,090
+إذن
+
+167
+00:20:06,090 --> 00:20:13,630
+a أُس الكلام هذا ضرب a أُس واحد صح؟ وهذا بيساوي
+
+168
+00:20:15,990 --> 00:20:20,710
+A أُس K زائد J زائد واحد عبارة عن A أُس K زائد J
+
+169
+00:20:20,710 --> 00:20:30,370
+ضرب A إذن الجزء هذا A أُس K زائد J زائد واحد عبارة
+
+170
+00:20:30,370 --> 00:20:34,850
+عن A أُس K زائد J ضرب A وفي أندم الأول ضرب A
+
+171
+00:20:34,850 --> 00:20:39,410
+باستخدام
+
+172
+00:20:39,410 --> 00:20:44,690
+ال induction hypothesisالفرض طبع ال induction احنا
+
+173
+00:20:44,690 --> 00:20:52,950
+فرضين ان a to k زي j بساوي a to k ضرب a ضرب a to j
+
+174
+00:20:52,950 --> 00:21:03,390
+وفي عندي من الأول a في a اذا
+
+175
+00:21:03,390 --> 00:21:06,910
+هذا
+
+176
+00:21:06,910 --> 00:21:15,050
+بساوي a to kضرب a مع بعض ممكن أخدهم مع بعض ضرب a
+
+177
+00:21:15,050 --> 00:21:22,670
+to j في a مع بعض لأن هنا استخدمنا ال fact ان عملية
+
+178
+00:21:22,670 --> 00:21:30,430
+ضرب الأعداد الحقيقية associative الآن a to k ضرب a
+
+179
+00:21:30,430 --> 00:21:38,650
+بساوي a to k زائد واحد و a to j ضرب a بساوي a to j
+
+180
+00:21:38,650 --> 00:21:48,390
+زائد واحدوهذا هو المطلوب إذاً هذا بثبت إذاً
+
+181
+00:21:48,390 --> 00:21:54,090
+هنا أثبتنا صحة ال star هاي الطرف الشمال ل star
+
+182
+00:21:54,090 --> 00:22:00,110
+عندما M بساوي K زايد واحد و M بساوي K زايد واحد و
+
+183
+00:22:00,110 --> 00:22:04,010
+هذا هو الطرف اليمين ل star عندما M بساوي K زايد
+
+184
+00:22:04,010 --> 00:22:09,430
+واحد و M بساوي G زايد واحد إذاً star holds
+
+185
+00:22:12,700 --> 00:22:27,220
+ن بساوي كزايد واحد and ان بساوي جي زايد واحد this
+
+186
+00:22:27,220 --> 00:22:32,620
+completes the
+
+187
+00:22:32,620 --> 00:22:33,140
+induction
+
+188
+00:22:38,010 --> 00:22:43,590
+إن ان هذا بيكمل البرهان by induction تمام واضح؟ في
+
+189
+00:22:43,590 --> 00:22:51,510
+أي سؤال؟ مفهوم؟ في أسئلة تانية؟ أي أسئلة تانية؟
+
+190
+00:23:15,480 --> 00:23:20,400
+سؤال اربعتاش؟ بس نجيب example لنا اسمها بس نجيب
+
+191
+00:23:20,400 --> 00:23:21,980
+example لنا اسمها بس نجيب example لنا اسمها بس
+
+192
+00:23:21,980 --> 00:23:24,560
+نجيب example لنا اسمها بس نجيب example لنا اسمها
+
+193
+00:23:24,560 --> 00:23:25,020
+بس نجيب example لنا اسمها بس نجيب example لنا
+
+194
+00:23:25,020 --> 00:23:28,740
+اسمها بس نجيب example لنا اسمها بس نجيب example
+
+195
+00:23:28,740 --> 00:23:29,260
+لنا اسمها بس نجيب example لنا اسمها بس نجيب
+
+196
+00:23:29,260 --> 00:23:29,440
+example لنا اسمها بس نجيب example لنا اسمها بس
+
+197
+00:23:29,440 --> 00:23:29,680
+نجيب example لنا اسمها بس نجيب example لنا اسمها
+
+198
+00:23:29,680 --> 00:23:29,900
+بس نجيب example لنا اسمها بس نجيب example لنا
+
+199
+00:23:29,900 --> 00:23:33,360
+اسمها بس نجيب example لنا اسمها بس نجيب example
+
+200
+00:23:33,360 --> 00:23:37,520
+لنا
+
+201
+00:23:37,520 --> 00:23:37,620
+اسمها
+
+202
+00:23:42,390 --> 00:23:45,070
+جزء الأول ولا التاني؟ الأول
+
+203
+00:24:16,920 --> 00:24:25,800
+إذا الجزء الأول مسؤال أربعتاش F zero
+
+204
+00:24:25,800 --> 00:24:36,560
+less than or equal a less than b show أثبتي أنه a
+
+205
+00:24:36,560 --> 00:24:43,160
+تربية أصغر من أو ساوي a في b أصغر من b تربية
+
+206
+00:25:01,170 --> 00:25:07,290
+proof case واحد a بساوي واحد انتوا ان ال a أكبر من
+
+207
+00:25:07,290 --> 00:25:12,430
+أو ساوي سفر فناخد الأول a بساوي سفر و بعدين a أكبر
+
+208
+00:25:12,430 --> 00:25:21,050
+من سفر فلو كان a بساوي سفر هذا بيقدي انه
+
+209
+00:25:21,050 --> 00:25:32,740
+a تربية اللي هي بتساوي سفرأصغر من أوي ساوي A في B
+
+210
+00:25:32,740 --> 00:25:39,740
+اللي هو سفر و
+
+211
+00:25:39,740 --> 00:25:44,180
+طبعا ال B في الحالة هذه أكبر من A يعني أكبر من سفر
+
+212
+00:25:44,180 --> 00:25:47,760
+لأن هذا أصغر من B
+
+213
+00:25:56,410 --> 00:26:01,650
+و أصغر من B تربية لأن
+
+214
+00:26:01,650 --> 00:26:12,050
+هذا بيساوي سفر و B تربية و B أكبر من سفر B أكبر من
+
+215
+00:26:12,050 --> 00:26:17,190
+A اللي هو بيساوي سفر يعني B أكبر من سفر يعني B
+
+216
+00:26:17,190 --> 00:26:22,370
+تربية أكبر من سفر إذا المتبايلة متحققة
+
+217
+00:26:27,190 --> 00:26:38,030
+إذا المتباينة اللي احنا عايزين نثبتها نسميها
+
+218
+00:26:38,030 --> 00:26:48,770
+star إذا star holds in this case الحالة
+
+219
+00:26:48,770 --> 00:27:01,890
+التانية أن a أكبر من 7لو كانت a أكبر من صفر by
+
+220
+00:27:01,890 --> 00:27:08,270
+hypothesis من الفرض أنا عندي الآن a أكبر من صفر
+
+221
+00:27:08,270 --> 00:27:13,230
+أصغر من b فنضرب
+
+222
+00:27:13,230 --> 00:27:17,590
+multiply multiply
+
+223
+00:27:17,590 --> 00:27:18,970
+by
+
+224
+00:27:20,970 --> 00:27:28,150
+أو multiply by صحيح by a أكبر من سفر لما أضرب
+
+225
+00:27:28,150 --> 00:27:32,450
+متباينة في عدد موجة بشريفتها طابقة زي ما هي فهذا
+
+226
+00:27:32,450 --> 00:27:40,710
+بيقدّي فهذا بيقدّي أنه السفر في a بيطلع سفر أصغر
+
+227
+00:27:40,710 --> 00:27:47,550
+من a تربية أصغر من a في b وهذا اللي بدنا يعني
+
+228
+00:27:51,030 --> 00:28:02,650
+لأ هذا .. هاد حاجة كمان and then بعد هيك multiply
+
+229
+00:28:02,650 --> 00:28:08,890
+.. multiply by
+
+230
+00:28:08,890 --> 00:28:14,730
+B اللي هو أكبر من سفر أيضا لإن ال B أكبر من A أكبر
+
+231
+00:28:14,730 --> 00:28:20,340
+من سفر صح؟فلو ضربنا المتباينة هذه في بي اللي هو
+
+232
+00:28:20,340 --> 00:28:25,980
+عدد موجب برضه إشارتها هتبقى زي ما هي إذن هذا
+
+233
+00:28:25,980 --> 00:28:31,760
+بيقدّي لما أضرب المتباينة هذه في بي عدد موجب
+
+234
+00:28:31,760 --> 00:28:37,020
+فهيطلع عند سفر أصغر من a في بي أصغر من b تربية
+
+235
+00:28:37,020 --> 00:28:47,280
+نسمي هذه واحد والمتباينة هذه اتنين الأن واحدعندي
+
+236
+00:28:47,280 --> 00:28:51,760
+اتنين بيقدّوا يعني
+
+237
+00:28:51,760 --> 00:29:00,540
+عندي a تربية أكبر من سفر أصغر من a بيه وعندي a بيه
+
+238
+00:29:00,540 --> 00:29:05,120
+أصغر من b تربية وهذه هي ال star
+
+239
+00:29:16,200 --> 00:29:22,040
+إذا in both cases
+
+240
+00:29:22,040 --> 00:29:30,280
+في الحالة الأولى والتانية we have أثبتنا أن a
+
+241
+00:29:30,280 --> 00:29:34,680
+تربية أصغر
+
+242
+00:29:34,680 --> 00:29:40,760
+أو يساوي ab و
+
+243
+00:29:40,760 --> 00:29:47,120
+ab في الحالتين أصغر من b تربيةكمان مرة في الحالة
+
+244
+00:29:47,120 --> 00:29:53,080
+الأولى a تربية أصغر من أو يساوي a b في الحالة
+
+245
+00:29:53,080 --> 00:29:58,020
+التانية a تربية أصغر من a b إذا في الحالتين a
+
+246
+00:29:58,020 --> 00:30:02,420
+تربية أصغر من أو يساوي a b في الحالة الأولى a b
+
+247
+00:30:02,420 --> 00:30:05,680
+أصغر من b تربية وفي الحالة التانية a b أصغر من b
+
+248
+00:30:05,680 --> 00:30:10,300
+تربية إذا في الحالتين a b أصغر من b تربية و هذا
+
+249
+00:30:10,300 --> 00:30:13,820
+اللي أحنا عايزين نثبته تمام إذا هيك بتكون أثبتنا
+
+250
+00:30:13,820 --> 00:30:21,140
+جزءالأول من السؤال الجزء
+
+251
+00:30:21,140 --> 00:30:27,960
+التاني بيقول انه احنا مانقدرش المتباينة هذه star
+
+252
+00:30:27,960 --> 00:30:38,500
+نبدلها ال
+
+253
+00:30:38,500 --> 00:30:43,760
+star can't be replaced
+
+254
+00:30:46,040 --> 00:30:53,660
+replaced by a تربيع أصغر من a بي أصغر من بي تربيع
+
+255
+00:30:53,660 --> 00:31:01,140
+يعني هنا ال inequality هذه أصغر من أو ساوي مانقدرش
+
+256
+00:31:01,140 --> 00:31:08,260
+نبدلها بأصغر من strictly أصغر من فبنوضح ذلك بمثال
+
+257
+00:31:08,260 --> 00:31:15,280
+فاخدي a بساوي سفر و b أي عدد example
+
+258
+00:31:37,110 --> 00:31:39,170
+زي ما عملنا في الحلقة الأولى
+
+259
+00:31:43,550 --> 00:31:50,550
+فهيطلع عندى هذه مساوية و ليست .. و ليست اكبر من ..
+
+260
+00:31:50,550 --> 00:32:01,390
+اه؟ okay فى أسئلة تانية؟ مين عند السؤال التانى؟
+
+261
+00:32:01,390 --> 00:32:05,050
+section اتنين اتنين؟ فى اي أسئلة في section اتنين
+
+262
+00:32:05,050 --> 00:32:08,910
+اتنين؟ section اتنين واحد اتنين اتنين
+
+263
+00:32:12,890 --> 00:32:21,830
+في أسئلة كتيرة حلوة section
+
+264
+00:32:21,830 --> 00:32:37,450
+1122 فينا مجموعة كبيرة من الأسئلة الناس
+
+265
+00:32:37,450 --> 00:32:39,270
+بتدرس الناس بتحل مثال
+
+266
+00:32:42,490 --> 00:32:46,430
+حليتوا كل المسائل؟ مش عندكم أسئلة؟ ال section
+
+267
+00:32:46,430 --> 00:32:56,890
+اتنين واحد و اتنين اتنين؟
+
+268
+00:32:56,890 --> 00:33:03,850
+مان لديها سؤال؟ السؤال اتنين في section اتنين
+
+269
+00:33:03,850 --> 00:33:10,710
+اتنين حاضر اتنين اتنين اتنين نكتب النص تبع السؤال
+
+270
+00:33:14,400 --> 00:33:25,960
+if a و b are real numbers show ifpty أنه
+
+271
+00:33:25,960 --> 00:33:35,160
+absolute a زائد b بساوي absolute a زايد absolute b
+
+272
+00:33:35,160 --> 00:33:47,110
+if and only if a في b أكبر منأو يساوي سفر هيك
+
+273
+00:33:47,110 --> 00:33:59,610
+مكتوب طيب
+
+274
+00:33:59,610 --> 00:34:07,590
+البرهان ال proof راحظوا هذا if and only if
+
+275
+00:34:07,590 --> 00:34:11,730
+statement فخلينا
+
+276
+00:34:11,730 --> 00:34:21,730
+الأولنثبت ال claim الأول claim one للدعاء الأول أن
+
+277
+00:34:21,730 --> 00:34:28,350
+a ضرب b أكبر من أو ساوي سفر if and only if
+
+278
+00:34:28,350 --> 00:34:36,290
+absolute a ضرب b بساوي a في b
+
+279
+00:34:43,590 --> 00:34:47,970
+هل هذا واضح هذا
+
+280
+00:34:47,970 --> 00:35:01,250
+واضح من تعريف that this is clear by definition of
+
+281
+00:35:01,250 --> 00:35:09,370
+absolute value طيب
+
+282
+00:35:09,370 --> 00:35:10,490
+الكلام التاني
+
+283
+00:35:16,440 --> 00:35:24,700
+الكلام التاني absolute a plus absolute b الكل
+
+284
+00:35:24,700 --> 00:35:35,560
+تربية بساوي a زي b الكل تربية if
+
+285
+00:35:35,560 --> 00:35:43,500
+and only if absolute a ضرب b بساوي a في b
+
+286
+00:35:49,120 --> 00:35:56,920
+بروف للـ claim هذا هاي لو أخدت absolute a زاد
+
+287
+00:35:56,920 --> 00:36:04,520
+absolute b وربعت فهذا المفروض يساوي absolute a
+
+288
+00:36:04,520 --> 00:36:12,780
+الكتر بيها زاد absolute b الكتر بيها زاد اتنين
+
+289
+00:36:12,780 --> 00:36:19,600
+absolute a في absolute bو هذا بساوي absolute a
+
+290
+00:36:19,600 --> 00:36:26,080
+الكل
+
+291
+00:36:26,080 --> 00:36:33,480
+تربية زار
+
+292
+00:36:33,480 --> 00:36:44,050
+absolute b الكل تربية زاد 2 absolute a ضارب bلأن
+
+293
+00:36:44,050 --> 00:36:54,670
+absolute a ضرب absolute b بيطلع absolute a في b و
+
+294
+00:36:54,670 --> 00:36:59,210
+absolute absolute
+
+295
+00:36:59,210 --> 00:37:06,850
+a بساوي الجدر التربية a ل a تربية هذا
+
+296
+00:37:06,850 --> 00:37:18,950
+بكافئ أن a أو absoluteكل تربية بيساوي ا تربية لو
+
+297
+00:37:18,950 --> 00:37:25,030
+ربعت الطرفين فبطلع absolute a تربية بيساوي a تربية
+
+298
+00:37:25,030 --> 00:37:30,490
+وبالتالي ممكن استبدل absolute a تربية ممكن استبدل
+
+299
+00:37:30,490 --> 00:37:36,530
+absolute a تربية ب a تربية و
+
+300
+00:37:36,530 --> 00:37:41,450
+استبدل absolute b تربية ب b تربية
+
+301
+00:37:50,020 --> 00:37:57,000
+الان هذا بيساوي هذا بيساوي a تربية زائد b تربية
+
+302
+00:37:57,000 --> 00:38:06,460
+زائد اتنين a b if and only if if and only if
+
+303
+00:38:06,460 --> 00:38:10,940
+absolute a b بيساوي a في b
+
+304
+00:38:13,940 --> 00:38:17,600
+هذا المقدار اللي فوق متى بيساوي a تربية زايد بي
+
+305
+00:38:17,600 --> 00:38:22,500
+تربية زايد اتنين a ب a اذا absolute ب a في b هو
+
+306
+00:38:22,500 --> 00:38:33,140
+عبارة عن a في b صح؟ وهذا
+
+307
+00:38:33,140 --> 00:38:39,720
+الأخير بيساوي يعني
+
+308
+00:38:39,720 --> 00:38:50,460
+هذا السطر هذاهي تعالى نشوف بقدّه
+
+309
+00:38:50,460 --> 00:39:00,000
+ان absolute a زاد absolute b الكل تربية بساوي
+
+310
+00:39:03,190 --> 00:39:11,210
+A زائد B الكل ترمية if and only if absolute A B
+
+311
+00:39:11,210 --> 00:39:27,050
+بساوي A B absolute
+
+312
+00:39:27,050 --> 00:39:32,390
+A B بساوي A B ومن claim واحد
+
+313
+00:39:36,690 --> 00:39:43,890
+عمر ال claim واحد هذا بتحقق if and only if a ضربي
+
+314
+00:39:43,890 --> 00:39:49,610
+أكبر من أو ساوى 0 لأن هذا نتيجة أو بضم ال two
+
+315
+00:39:49,610 --> 00:39:54,070
+claims مع بعض طيب
+
+316
+00:39:54,070 --> 00:39:58,290
+لو أخلنا جدر التربية للطرفين فضي جدر التربية يعني
+
+317
+00:39:58,290 --> 00:40:02,290
+take square root
+
+318
+00:40:05,210 --> 00:40:14,850
+of both sides فهذا
+
+319
+00:40:14,850 --> 00:40:20,410
+بيقدّي أنه absolute a زائد absolute b الجدر
+
+320
+00:40:20,410 --> 00:40:23,030
+التربيعي هنا بعطيني ال absolute value ل a زائد
+
+321
+00:40:23,030 --> 00:40:28,570
+absolute b وهنا بعطيني الجدر التربيعي الجدر
+
+322
+00:40:28,570 --> 00:40:31,370
+التربيعي العدد التربيعي بعطيني القيمة المطلقة
+
+323
+00:40:31,370 --> 00:40:32,010
+تبعته
+
+324
+00:40:34,820 --> 00:40:41,180
+هذا بتحقق if and only if a ضرب b أكبر من أو ساوى 0
+
+325
+00:40:41,180 --> 00:40:48,200
+وهذا هو المطلوب وهذا هو المطلوب okay تمام فهذا هو
+
+326
+00:40:48,200 --> 00:40:53,140
+المطلوب في
+
+327
+00:40:53,140 --> 00:40:58,220
+أي أسئلة تانية في
+
+328
+00:40:58,220 --> 00:41:01,700
+عندكم أي سؤال تاني section 2 أو section 1
+
+329
+00:41:07,070 --> 00:41:12,510
+أن أسئلة كتيرة لو جيت بعدي و أنتوا مابتسألوش و مش
+
+330
+00:41:12,510 --> 00:41:17,650
+هيكون في راجعة بعد هيك لأ أساكاشنا دي المرة الجاية
+
+331
+00:41:17,650 --> 00:41:21,830
+هناخد مناقشة في section اتنين تلاتة و اتنين أربعة
+
+332
+00:41:21,830 --> 00:41:31,370
+في أي أسئلة أفندم في أي section اتنين اتنين
+
+333
+00:41:37,770 --> 00:41:38,370
+حاضر
+
+334
+00:42:02,930 --> 00:42:11,490
+السؤال خمستاش سيكشن اتنين نيلو نشوف ايه هو السؤال
+
+335
+00:42:11,490 --> 00:42:25,670
+show
+
+336
+00:42:25,670 --> 00:42:29,790
+if a و b are real numbers
+
+337
+00:42:32,160 --> 00:42:38,560
+و a لا يساوي b then
+
+338
+00:42:38,560 --> 00:42:49,100
+there exist epsilon neighborhoods u
+
+339
+00:42:49,100 --> 00:43:09,390
+of a and v of b such thatU تقاطع V بساوي five إذن
+
+340
+00:43:09,390 --> 00:43:24,630
+كمان مرة أنا
+
+341
+00:43:24,630 --> 00:43:25,090
+عندي
+
+342
+00:43:28,550 --> 00:43:35,990
+A وB أعداد حقيقية و A لا يساوي B هذا معناه أن
+
+343
+00:43:35,990 --> 00:43:43,970
+either A less than B ف
+
+344
+00:43:43,970 --> 00:43:52,310
+assume نأخد الحالة الأولى case one أن
+
+345
+00:43:52,310 --> 00:44:01,330
+A أصغر من B إذا هي خط الأعدادهذا خط الاعداد وهذه a
+
+346
+00:44:01,330 --> 00:44:12,670
+وهذه b و a أصغر من b بدي أثبت أنه في جوار ل a بعمق
+
+347
+00:44:12,670 --> 00:44:22,450
+epsilon اسمه u وفي جوار ل b بعمق epsilon والجوارين
+
+348
+00:44:22,450 --> 00:44:26,170
+هذول المبروهود سقطوهم بسوء في يعني منفصلين عن بعض
+
+349
+00:44:27,430 --> 00:44:35,570
+فبكل بساطة بجيس المسافة من a و b و باخد نص المسافة
+
+350
+00:44:35,570 --> 00:44:43,670
+يعني لأن هنا take epsilon
+
+351
+00:44:43,670 --> 00:44:48,770
+بساوي
+
+352
+00:44:48,770 --> 00:44:54,270
+نص المسافة نص ال b minus
+
+353
+00:44:57,480 --> 00:45:13,820
+أو نص او نص المسافة بين A وB فهي
+
+354
+00:45:13,820 --> 00:45:15,920
+نص المسافة لو كونت فترة
+
+355
+00:45:23,000 --> 00:45:28,520
+فهي منتصف المسافة هذه المسافة
+
+356
+00:45:28,520 --> 00:45:33,260
+هذه منتصف المسافة سمنها epsilon فهذه النقطة هتكون
+
+357
+00:45:33,260 --> 00:45:40,960
+a زاد epsilon وهي نفس المسافة a سالب epsilon وكون
+
+358
+00:45:40,960 --> 00:45:45,660
+فترة مفتوحة وسم
+
+359
+00:45:45,660 --> 00:45:48,960
+الفترة المفتوحة هذه new
+
+360
+00:46:02,460 --> 00:46:07,900
+فنسمي الفترة المفتوحة هذه U مركزها E و نصف قطرها E
+
+361
+00:46:07,900 --> 00:46:16,200
+و نكوّن فترة تانية برضه مركزها B و نصف قطرها E
+
+362
+00:46:16,200 --> 00:46:22,080
+يعني هذه النقطة هتصير B زاد E وهذه النقطة هتصير
+
+363
+00:46:25,910 --> 00:46:33,530
+بزائد ابسلون و النقطة هذه بسالب ابسلون و نكون فترة
+
+364
+00:46:33,530 --> 00:46:39,970
+مفتوحة تمام
+
+365
+00:46:39,970 --> 00:46:43,310
+اذا
+
+366
+00:46:43,310 --> 00:46:50,890
+انا في اندي و نسمي الفترة المفتوحة هذه نسميها
+
+367
+00:46:50,890 --> 00:46:51,030
+ب
+
+368
+00:46:57,180 --> 00:47:01,040
+أذن هذا عبارة عن الـ U الفترة المفتوحة اذا هنا led
+
+369
+00:47:01,040 --> 00:47:14,300
+take
+
+370
+00:47:14,300 --> 00:47:16,480
+u
+
+371
+00:47:18,050 --> 00:47:26,410
+by definition بتساوي a سالب إبسلون و a مجد إبسلون
+
+372
+00:47:26,410 --> 00:47:40,810
+و V بساوي B سالب إبسلون B plus إبسلون فواضح
+
+373
+00:47:40,810 --> 00:47:41,410
+nearly
+
+374
+00:47:46,910 --> 00:47:58,030
+Clearly U is an epsilon neighborhood of A and V is
+
+375
+00:47:58,030 --> 00:48:05,410
+an epsilon neighborhood of B ومش هيكوا بس and ممكن
+
+376
+00:48:05,410 --> 00:48:09,370
+اثبات ان U تقاطع B بساوي فاي
+
+377
+00:48:12,440 --> 00:48:21,120
+يعني لو أخدنا هاي واضح هاي جوار هذا مافيش ولا نقطة
+
+378
+00:48:21,120 --> 00:48:27,940
+فيه موجودة في الجوار التاني هذه الفترة المفتوحة
+
+379
+00:48:27,940 --> 00:48:35,540
+منفصلة عن الفترة المفتوحة يوم طبعا؟
+
+380
+00:48:39,190 --> 00:48:44,970
+إن هذا الكلام يعني واضح ان هذا عبارة عن epsilon
+
+381
+00:48:44,970 --> 00:48:50,150
+neighborhood ل a فترة مفتوحة مركزها a نصف قطرة
+
+382
+00:48:50,150 --> 00:48:53,590
+epsilon هذا نسميه epsilon neighborhood ل a و V
+
+383
+00:48:53,590 --> 00:48:57,510
+الفترة المفتوحة هذه بي سالب epsilon و V موجب
+
+384
+00:48:57,510 --> 00:49:04,310
+epsilon برضه عبارة عن epsilon neighborhood ل B و
+
+385
+00:49:04,310 --> 00:49:09,900
+اتنين حسب الرسم منفصلين و هذا ممكن اثباتهباستخدام
+
+386
+00:49:09,900 --> 00:49:14,380
+التناقض يعني افرض انه في عنصر يعني التقاطع هذا لا
+
+387
+00:49:14,380 --> 00:49:19,900
+يساوي فيه وبالتالي في عنصر موجود في U وموجود في V
+
+388
+00:49:19,900 --> 00:49:25,540
+في أن و واحد واصلي إليه تناقض okay هذا ممكن اثباته
+
+389
+00:49:25,540 --> 00:49:34,280
+بطريقة تحليلية طبعا في السؤال في الحل في حل السؤال
+
+390
+00:49:34,280 --> 00:49:41,370
+هذا يقول مافي عندنا حلتينالحالة الأولى a أصغر من b
+
+391
+00:49:41,370 --> 00:49:48,110
+والحالة الثانية b أصغر من a وشوفنا هنا أخدنا
+
+392
+00:49:48,110 --> 00:49:51,230
+الحالة اللي فيها a أصغر من b زي اللي هو النضال تحت
+
+393
+00:49:51,230 --> 00:49:57,090
+الاسم هاي a أصغر من b و أثبتنا في الحالة هذه أن u
+
+394
+00:49:57,090 --> 00:50:00,070
+جد epsilon never move u ل a
+
+395
+00:50:17,240 --> 00:50:24,820
+باقي الحالة التانية case 2 نثبت
+
+396
+00:50:24,820 --> 00:50:27,120
+برضه المطلوب
+
+397
+00:50:33,160 --> 00:50:38,320
+القرآن في الحلقة التانية مماثل للقرآن اللي عملناه
+
+398
+00:50:38,320 --> 00:50:38,880
+في الحلقة
+
+399
+00:50:50,040 --> 00:50:54,040
+و هذا طبعا يكمل الفرحة، إذا الحالة التانية اللي
+
+400
+00:50:54,040 --> 00:50:58,120
+فيها P أصفر من A، بس نضع P هنا و A هنا، و نفس
+
+401
+00:50:58,120 --> 00:51:05,440
+العادة، فتصير هذا الـP و هذا الـU، و يعطينا نفس
+
+402
+00:51:05,440 --> 00:51:13,020
+النتيجة، إت من كون كملنا المرهان اللي هو التمريد
+
+403
+00:51:14,530 --> 00:51:17,970
+وطبعا بعد ذلك ان شاء الله هنكمل حل ال تمارين لل
+
+404
+00:51:17,970 --> 00:51:20,870
+sport الخادم في نفس الموضوع
+
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/Kfhi1a_WpFk_raw.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/Kfhi1a_WpFk_raw.srt
new file mode 100644
index 0000000000000000000000000000000000000000..e98332f8e4973dbefe907d239f4476f51c3a9750
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/Kfhi1a_WpFk_raw.srt
@@ -0,0 +1,1616 @@
+1
+00:00:20,650 --> 00:00:26,490
+طبعا احنا زي ما اتفجنا معاكم هنعمل
+
+2
+00:00:26,490 --> 00:00:36,030
+مناقشة لل course و هنبدأ طبعا ب chapter اتنين اللي
+
+3
+00:00:36,030 --> 00:00:42,270
+هو اول chapter درسناه و هنبدأ ب section اتنين واحد
+
+4
+00:00:42,270 --> 00:00:46,270
+و اذا في وجب طبعا هنحاول نجاوب على أسئلة في
+
+5
+00:00:46,270 --> 00:00:47,190
+section اتنين اتنين
+
+6
+00:00:50,880 --> 00:00:55,480
+باطمن عليكم انه يعني طبعا الأسئلة عددها كبير و مش
+
+7
+00:00:55,480 --> 00:01:00,980
+هانلحج نحل كل المسائل لكن باطمن عليكم انكم تسألوا
+
+8
+00:01:00,980 --> 00:01:06,020
+الأسئلة اللي انتوا يعني وجدتوا فيها صعوبة في حالها
+
+9
+00:01:06,020 --> 00:01:10,000
+حتى
+
+10
+00:01:10,000 --> 00:01:17,240
+تكون يعني الفايدة تعمق أكتر فنبدأ ب section 2 1 هل
+
+11
+00:01:17,240 --> 00:01:23,860
+في أي سؤال في section 2 1حاولتوا تحلوه ماعرفتوش
+
+12
+00:01:23,860 --> 00:01:33,540
+تحلوه أو وجدته صعوبة في حاله ففي
+
+13
+00:01:33,540 --> 00:01:36,640
+أي سؤال من الأسئلة اللي احنا حددناها في ال
+
+14
+00:01:36,640 --> 00:01:42,080
+syllabus و قولنا لكم حلوها في أي سؤال في section 2
+
+15
+00:01:42,080 --> 00:01:49,810
+و 1 تحبوا تسألوا عنه؟ستاذ ممكن نسأل .. نشرح ..
+
+16
+00:01:49,810 --> 00:01:54,650
+نشرح .. نظرية ما .. ما نثبتش .. مش عارف كيف نثبت
+
+17
+00:01:54,650 --> 00:02:02,090
+ما هي عامة exercise اه إيش هي دي؟ X على Z زي Y على
+
+18
+00:02:02,090 --> 00:02:07,930
+W يوم ساوي X Z زي Z Y على Z W كيف نستخدم .. كيف
+
+19
+00:02:07,930 --> 00:02:09,430
+نبدأ فيها؟ مش عارف
+
+20
+00:02:17,430 --> 00:02:21,510
+يعني الخواص اللى .. مش هادى خاصيا من الخواص اللى
+
+21
+00:02:21,510 --> 00:02:27,210
+خدناها حاول تشوف يعني كيف احنا برهنا الخواص الأخرى
+
+22
+00:02:27,210 --> 00:02:37,050
+و تستفيدي منها و .. و تحاول تبرهنيها يعني ممكن
+
+23
+00:02:37,050 --> 00:02:41,310
+كمان تبص في الكتاب المقرر و تشوف يعني هل هو حلها
+
+24
+00:02:41,310 --> 00:02:47,940
+او محلاش لكن انا يعنيالخواص اللي احنا مابرهنهاش
+
+25
+00:02:47,940 --> 00:02:53,700
+يعني .. يعني كان برهانها سهل و ممكن تتبرهنيها بنفس
+
+26
+00:02:53,700 --> 00:03:00,280
+الأسلوب اللي احنا برهننا فيه الأزاية الأخرى فخليني
+
+27
+00:03:00,280 --> 00:03:03,900
+أترك الإجابة على السؤال هذا إليك تحاولي فيه مرة
+
+28
+00:03:03,900 --> 00:03:08,360
+تانية و إذا ماعرفتيش ممكن تجيلي على المكتب و ممكن
+
+29
+00:03:08,360 --> 00:03:09,280
+نتناقش
+
+30
+00:03:11,940 --> 00:03:16,180
+ياريت تسألوني أسئلة من التمرين من ال exercises إذا
+
+31
+00:03:16,180 --> 00:03:26,140
+سمحتوا، تفضلي السؤال الرابع في section السابع في
+
+32
+00:03:26,140 --> 00:03:30,360
+section اتنين واحد، حاضر
+
+33
+00:03:39,140 --> 00:03:50,140
+إذا حل السؤال سبعة section اتنين واحد modify
+
+34
+00:03:50,140 --> 00:03:53,360
+the proof of theorem اتنين واحد اربعة في الكتاب
+
+35
+00:03:53,360 --> 00:03:59,200
+المقرر to show that there
+
+36
+00:03:59,200 --> 00:04:05,640
+does not exist لا يوجد there does not exist
+
+37
+00:04:08,790 --> 00:04:19,170
+T ينتمي للـ Q بحيث أن T تربية بساوية تلاتة بمعنى
+
+38
+00:04:19,170 --> 00:04:23,770
+آخر يعني الجذر التلاتة بنا نثبت أنه جذر التلاتة
+
+39
+00:04:23,770 --> 00:04:28,730
+ليس عدد نسبي فالمفروض
+
+40
+00:04:28,730 --> 00:04:33,010
+أنكم يعني تفهموا و تحاولوا تفهموا البرهان تبع
+
+41
+00:04:33,010 --> 00:04:37,810
+اثبات أنه جذر اتنين ليس عدد نسبي وتحاولوا تعملوا
+
+42
+00:04:37,810 --> 00:04:45,910
+برهان مشابهفي أدة براهين للسؤال هذا فأحد البراهين
+
+43
+00:04:45,910 --> 00:04:52,590
+شبه البرهان اللي أخدنا بتابع جداد الأتنين ليس عدد
+
+44
+00:04:52,590 --> 00:04:59,330
+نسبي فخلينا نشوفه مع بعض خلينا نشوف إذن البرهان
+
+45
+00:04:59,330 --> 00:05:04,810
+prove طبعا البرهان بالتناقض assume
+
+46
+00:05:07,480 --> 00:05:18,180
+on contrary نفرض على النقيد there exist T بساوي A
+
+47
+00:05:18,180 --> 00:05:29,420
+على B عدد نسبي بحيث انه ال greatest common divisor
+
+48
+00:05:29,420 --> 00:05:34,760
+لل A والB بساوي واحد and
+
+49
+00:05:37,230 --> 00:05:43,970
+T تربيع اللي هو بساوي A على B لكل تربيع بساوي
+
+50
+00:05:43,970 --> 00:05:52,030
+تلاتة اذا هذا النقيض او النفي تبع يوجد عدد نسبي
+
+51
+00:05:52,030 --> 00:06:00,050
+مربعه بساوي تلاتة النفي تبعه يوجد عدد نسبي مربعه
+
+52
+00:06:00,050 --> 00:06:05,840
+بساوي تلاتة وطبعا ممكن نفرض انه العدد النسبيالـ
+
+53
+00:06:05,840 --> 00:06:10,160
+greatest common divisor للـ bus والمقام تبعه بساوي
+
+54
+00:06:10,160 --> 00:06:17,300
+واحد زي ما عملنا في حالة ال square root of two طيب
+
+55
+00:06:17,300 --> 00:06:22,740
+then في الحالة هذه لو ربع .. لو هنا من المعادلة
+
+56
+00:06:22,740 --> 00:06:32,820
+هذه بنحصل على a تربية بساوي تلاتة b تربية وهذا
+
+57
+00:06:32,820 --> 00:06:42,870
+بيقدّيإن ال B تقسم A تربية ال
+
+58
+00:06:42,870 --> 00:06:48,170
+B تقسم A تربية او A تربية اللي هي تلاتة B تربية
+
+59
+00:06:48,170 --> 00:06:54,810
+تقبل القسم على B بدون باطل طيب
+
+60
+00:06:54,810 --> 00:07:03,250
+ال و في الحالة هذه بقدر
+
+61
+00:07:04,790 --> 00:07:10,850
+أفصل حالتين العدد بي هذا ممكن .. هذا طبعا عدد ..
+
+62
+00:07:10,850 --> 00:07:20,430
+عدد صحيح ممكن يكون أكبر من الواحد و ممكن يكون
+
+63
+00:07:20,430 --> 00:07:28,990
+بساوي واحد فنفرض أن ال بي أكبر من واحد then في
+
+64
+00:07:28,990 --> 00:07:32,770
+الحالة هذه بي can be written
+
+65
+00:07:37,720 --> 00:07:45,000
+as product of
+
+66
+00:07:45,000 --> 00:07:49,640
+primes ال
+
+67
+00:07:49,640 --> 00:07:54,820
+بيه ده عدد صحيح أكبر من واحد فممكن نكتبه على صورة
+
+68
+00:07:55,540 --> 00:08:01,280
+حاصل ضرب أعداد أولية أي عدد صحيح أكبر من واحد ممكن
+
+69
+00:08:01,280 --> 00:08:06,180
+كتابته على صورة حاصل ضرب أعداد أولية product of
+
+70
+00:08:06,180 --> 00:08:12,100
+primes prime عدد أول هذا حقيقة معروفة في نظرية
+
+71
+00:08:12,100 --> 00:08:17,020
+الأعداد وبالتالي
+
+72
+00:08:17,020 --> 00:08:20,240
+hence وبالتالي
+
+73
+00:08:22,840 --> 00:08:33,180
+يوجد a prime يوجد عدد أولي a prime P بحيث أن هذا
+
+74
+00:08:33,180 --> 00:08:39,120
+ال P يقسم ال B يعني
+
+75
+00:08:39,120 --> 00:08:44,880
+أنا ال B هذا هي product of primes ممكن يكون بساوي
+
+76
+00:08:44,880 --> 00:08:50,840
+P1 ضرب P2 ضرب PN
+
+77
+00:08:52,290 --> 00:08:59,710
+حيث و P1 و P2 و PN كلهم Primes أعداد أولية فأكيد
+
+78
+00:08:59,710 --> 00:09:07,390
+لو أخدت أي واحد منهم فهذا بيقسم بي أو بيقبل قسم
+
+79
+00:09:07,390 --> 00:09:16,530
+عليه بس إذا يوجد يوجد Prime سمي P يقسم بي أو بيقبل
+
+80
+00:09:16,530 --> 00:09:20,710
+قسم عليه فهذا
+
+81
+00:09:20,710 --> 00:09:33,050
+بيؤديهذا بيقدي ان P يقسم ال A تربية لان انا عندي P
+
+82
+00:09:33,050 --> 00:09:40,570
+يقسم A تربية و P يقسم B اذا
+
+83
+00:09:40,570 --> 00:09:51,030
+ال P هذا يقسم A تربية okay تمام طيب و منهاهذا
+
+84
+00:09:51,030 --> 00:09:57,970
+بيدّي أن P يقسم A إذا P يقسم A تربية فممكن اثبات
+
+85
+00:09:57,970 --> 00:10:08,830
+أن P يقسم العدد الصحيح A وهكذا أثبتنا الـ
+
+86
+00:10:08,830 --> 00:10:20,170
+greatest common divisor لـ A وB أكبر من أو ساوي Pو
+
+87
+00:10:20,170 --> 00:10:27,090
+P هذا طبعا أكبر من واحد لأن
+
+88
+00:10:27,090 --> 00:10:39,430
+ال P يقسم ال A و P يقسم ال B فمعناته
+
+89
+00:10:39,430 --> 00:10:46,350
+في عامل مشترك في common divisor اللي هو Pبين a و b
+
+90
+00:10:46,350 --> 00:10:50,050
+لأن ال greatest common divisor سيكون على الأقل p
+
+91
+00:10:50,050 --> 00:10:53,770
+ويمكن أن يكون أكبر وبالتالي greatest common
+
+92
+00:10:53,770 --> 00:10:59,070
+divisor ل a و b أكبر من واحد وهذا بتناقض مع فرضنا
+
+93
+00:10:59,070 --> 00:11:02,650
+أن greatest common divisor ل a و b بساوي واحد
+
+94
+00:11:02,650 --> 00:11:07,850
+which is
+
+95
+00:11:07,850 --> 00:11:08,870
+a contradiction
+
+96
+00:11:15,680 --> 00:11:23,960
+وهذا التناقض بكمل البرهان يعني
+
+97
+00:11:23,960 --> 00:11:30,740
+فرضنا هذا أنه في عدد نسبي مربع بساوي تلاتة كان فرض
+
+98
+00:11:30,740 --> 00:11:36,780
+خطأ الصح أنه لا يوجد عدد نسبي مربع بساوي تلاتةOkay
+
+99
+00:11:36,780 --> 00:11:39,740
+تمام إذا هذا برهان وفيه طبعا براهين أخرى ممكن
+
+100
+00:11:39,740 --> 00:11:43,460
+تلاقوها تجدوها في كتب ال real analysis لكن هذا أحد
+
+101
+00:11:43,460 --> 00:11:52,580
+البراهين تمام؟ مين عنده سؤال تاني؟ في أي سؤال
+
+102
+00:11:52,580 --> 00:12:03,220
+تاني؟ في section 2.1 أو 2.2؟
+
+103
+00:12:03,220 --> 00:12:04,680
+نعم
+
+104
+00:12:06,770 --> 00:12:10,590
+لا نفس السؤال مش في اكتر من حالة ولا بس باخد ال D
+
+105
+00:12:10,590 --> 00:12:15,390
+أكتر من حالة؟ اه في كمان حالة صحيح ال case التانية
+
+106
+00:12:15,390 --> 00:12:23,290
+مظبوط ال case التانية خليني اكتبها هناك صحيح في
+
+107
+00:12:23,290 --> 00:12:24,030
+حالة تانية
+
+108
+00:12:42,990 --> 00:12:48,270
+كاس اتنين ال بي بالساوية واحد لو كانت ال بي
+
+109
+00:12:48,270 --> 00:12:55,330
+بالساوية واحد فهذا بيقدّي انا عندي ا تربية من هنا
+
+110
+00:12:55,330 --> 00:13:00,270
+في عندي ا تربية بالساوية تلاتة بي تربية فلو ال بي
+
+111
+00:13:00,270 --> 00:13:08,730
+بالساوية واحد معناه ا تربية تطلع بالساوية تلاتةو
+
+112
+00:13:08,730 --> 00:13:20,170
+هذا يعني مستحيل which is impossible هذا
+
+113
+00:13:20,170 --> 00:13:28,690
+مستحيل لأنه لأنه مافيش عدد صحيح عدد صحيح مربع
+
+114
+00:13:28,690 --> 00:13:34,130
+بساوي تلاتة since there does not exist integer
+
+115
+00:13:34,130 --> 00:13:36,950
+integer
+
+116
+00:13:38,790 --> 00:13:46,690
+a such that a تربية بساوي تلاتة اذا في الحالة هذه
+
+117
+00:13:46,690 --> 00:13:54,510
+حصلنا على حاجة impossible يعني تناقض وهنا
+
+118
+00:13:54,510 --> 00:14:01,510
+كمان حصلنا على تناقض ان ال assumption تبعنا انه
+
+119
+00:14:01,510 --> 00:14:06,550
+يوجد عدد نسبي مربع بساوي تلاتة كان فرض خاطئ وهذا
+
+120
+00:14:06,550 --> 00:14:08,090
+يكمل البرهان في الحالتين
+
+121
+00:14:10,890 --> 00:14:17,630
+في اي سؤال تانى؟ في
+
+122
+00:14:17,630 --> 00:14:25,890
+اسئلة تانية في ال section 2 1 او 2 2 انا
+
+123
+00:14:25,890 --> 00:14:30,190
+كنت متوقع ان يكون عندكم اسئلة كتيرة واضح جدا انكم
+
+124
+00:14:30,190 --> 00:14:35,150
+انتوا لا محاولين تحلوا الاسئلة وبالتالي ما عندكمش
+
+125
+00:14:35,150 --> 00:14:38,550
+يعني استفسارات
+
+126
+00:14:40,490 --> 00:14:42,690
+طبعا ست و عشرين
+
+127
+00:15:13,320 --> 00:15:20,080
+سؤال ستة وعشرين سكتشن اتنين واحد show
+
+128
+00:15:20,080 --> 00:15:28,560
+by induction that
+
+129
+00:15:28,560 --> 00:15:36,860
+لو
+
+130
+00:15:36,860 --> 00:15:46,900
+كان A ينتمي ل Rو M و N أعداد طبيعية فهذا بيقدي أن
+
+131
+00:15:46,900 --> 00:15:52,480
+A to M
+
+132
+00:15:52,480 --> 00:16:04,080
+plus N بيساوي A to M في A to N نريد
+
+133
+00:16:04,080 --> 00:16:10,300
+أن نثبت أن استخدام ال inductionإنه لأي عدد حقيقي A
+
+134
+00:16:10,300 --> 00:16:15,640
+و لأي عدد طبيعي M و N A to M زاد N بساوي A to M
+
+135
+00:16:15,640 --> 00:16:24,800
+ضرب A to N هذا أحد قوانين الأساس فطبعا
+
+136
+00:16:24,800 --> 00:16:29,420
+هنعمل induction أو بيسموه double induction على M و
+
+137
+00:16:29,420 --> 00:16:32,860
+N في نفس الواجت فالحالة الأولى
+
+138
+00:16:36,700 --> 00:16:40,780
+فم بيساوي ان بيساوي واحد لو كان الام والان كلها
+
+139
+00:16:40,780 --> 00:16:46,040
+بيساوي واحد then
+
+140
+00:16:46,040 --> 00:16:56,900
+a to m زي الان بيساوي a to واحد زي الواحد بيساوي a
+
+141
+00:16:56,900 --> 00:17:06,340
+ترمية andA to M ضرب A to N بيساوي A to 1 ضرب A to
+
+142
+00:17:06,340 --> 00:17:13,780
+1 بيساوي A تلبيه وبالتالي الطرفين المعادلة هذه
+
+143
+00:17:13,780 --> 00:17:21,200
+متحققة لأن الطرفين بيساوي نفس المقدار A تلبيه
+
+144
+00:17:21,200 --> 00:17:26,300
+اذا اذا
+
+145
+00:17:26,300 --> 00:17:27,680
+star holds
+
+146
+00:17:30,970 --> 00:17:39,310
+in case M بساوي M بساوي 1 المعادلة star متحققه في
+
+147
+00:17:39,310 --> 00:17:43,610
+حالة M بساوي M بساوي 1 الان ال induction
+
+148
+00:17:43,610 --> 00:17:48,370
+hypothesis ان هذا induction عادي بس يعني بدل من
+
+149
+00:17:48,370 --> 00:17:53,530
+قيان على M يكون على M و M ال induction hypothesis
+
+150
+00:17:53,530 --> 00:17:59,270
+الفرض تبع ال induction بنفرض assume
+
+151
+00:18:03,340 --> 00:18:09,860
+assume star holds المعادلة
+
+152
+00:18:09,860 --> 00:18:18,760
+star صحيحة for m بساوي k أكبر من واحد and n بساوي
+
+153
+00:18:18,760 --> 00:18:28,940
+j أكبر من واحد this
+
+154
+00:18:28,940 --> 00:18:42,170
+meansهذا معناه ان a to k plus j بساوي a to k ضرب a
+
+155
+00:18:42,170 --> 00:18:48,710
+to j اذا احنا فرضين صحة المعادلة هذه عندما m بساوي
+
+156
+00:18:48,710 --> 00:18:57,560
+k و عندما n بساوي jالان نريد اثبات صحتها عندما M
+
+157
+00:18:57,560 --> 00:19:01,440
+بيساوي K زايد واحد وعندما N بيساوي G زايد واحد
+
+158
+00:19:01,440 --> 00:19:08,840
+تعالى نثبت صحتها في الحالة يعني اذا now A ناخد A
+
+159
+00:19:08,840 --> 00:19:16,320
+to K زايد واحد زايد G زايد واحد
+
+160
+00:19:20,610 --> 00:19:25,150
+ان نتبع صحة المعادلة star عندما M بساوي K زايد
+
+161
+00:19:25,150 --> 00:19:29,370
+واحد و N بساوي G زايد واحد في الحالة هذه الطرف
+
+162
+00:19:29,370 --> 00:19:37,310
+اليسار لstar بساوي كلام هذا وهذا ممكن نجزئه إلى A
+
+163
+00:19:37,310 --> 00:19:49,290
+to K زايد G زايد واحد زايد واحدالأُس هذا ممكن نقعد
+
+164
+00:19:49,290 --> 00:19:53,450
+الكتابة على صورة ك زائد جي زائد واحد مع بعض زائد
+
+165
+00:19:53,450 --> 00:20:01,550
+واحد وهذا بيساوي a to ك زائد جي زائد واحد ضرب a
+
+166
+00:20:01,550 --> 00:20:06,090
+إذن
+
+167
+00:20:06,090 --> 00:20:13,630
+a أُس الكلام هذا ضرب a أُس واحد صح؟ وهذا بيساوي
+
+168
+00:20:15,990 --> 00:20:20,710
+A أُس K زائد J زائد واحد عبارة عن A أُس K زائد J
+
+169
+00:20:20,710 --> 00:20:30,370
+ضرب A إذن الجزء هذا A أُس K زائد J زائد واحد عبارة
+
+170
+00:20:30,370 --> 00:20:34,850
+عن A أُس K زائد J ضرب A وفي أندم الأول ضرب A
+
+171
+00:20:34,850 --> 00:20:39,410
+باستخدام
+
+172
+00:20:39,410 --> 00:20:44,690
+ال induction hypothesisالفرض طبع ال induction احنا
+
+173
+00:20:44,690 --> 00:20:52,950
+فرضين ان a to k زي j بساوي a to k ضرب a ضرب a to j
+
+174
+00:20:52,950 --> 00:21:03,390
+وفي عندي من الأول a في a اذا
+
+175
+00:21:03,390 --> 00:21:06,910
+هذا
+
+176
+00:21:06,910 --> 00:21:15,050
+بساوي a to kضرب a مع بعض ممكن أخدهم مع بعض ضرب a
+
+177
+00:21:15,050 --> 00:21:22,670
+to j في a مع بعض لأن هنا استخدمنا ال fact ان عملية
+
+178
+00:21:22,670 --> 00:21:30,430
+ضرب الأعداد الحقيقية associative الآن a to k ضرب a
+
+179
+00:21:30,430 --> 00:21:38,650
+بساوي a to k زائد واحد و a to j ضرب a بساوي a to j
+
+180
+00:21:38,650 --> 00:21:48,390
+زائد واحدوهذا هو المطلوب إذاً هذا بثبت إذاً
+
+181
+00:21:48,390 --> 00:21:54,090
+هنا أثبتنا صحة ال star هاي الطرف الشمال ل star
+
+182
+00:21:54,090 --> 00:22:00,110
+عندما M بساوي K زايد واحد و M بساوي K زايد واحد و
+
+183
+00:22:00,110 --> 00:22:04,010
+هذا هو الطرف اليمين ل star عندما M بساوي K زايد
+
+184
+00:22:04,010 --> 00:22:09,430
+واحد و M بساوي G زايد واحد إذاً star holds
+
+185
+00:22:12,700 --> 00:22:27,220
+ن بساوي كزايد واحد and ان بساوي جي زايد واحد this
+
+186
+00:22:27,220 --> 00:22:32,620
+completes the
+
+187
+00:22:32,620 --> 00:22:33,140
+induction
+
+188
+00:22:38,010 --> 00:22:43,590
+إن ان هذا بيكمل البرهان by induction تمام واضح؟ في
+
+189
+00:22:43,590 --> 00:22:51,510
+أي سؤال؟ مفهوم؟ في أسئلة تانية؟ أي أسئلة تانية؟
+
+190
+00:23:15,480 --> 00:23:20,400
+سؤال اربعتاش؟ بس نجيب example لنا اسمها بس نجيب
+
+191
+00:23:20,400 --> 00:23:21,980
+example لنا اسمها بس نجيب example لنا اسمها بس
+
+192
+00:23:21,980 --> 00:23:24,560
+نجيب example لنا اسمها بس نجيب example لنا اسمها
+
+193
+00:23:24,560 --> 00:23:25,020
+بس نجيب example لنا اسمها بس نجيب example لنا
+
+194
+00:23:25,020 --> 00:23:28,740
+اسمها بس نجيب example لنا اسمها بس نجيب example
+
+195
+00:23:28,740 --> 00:23:29,260
+لنا اسمها بس نجيب example لنا اسمها بس نجيب
+
+196
+00:23:29,260 --> 00:23:29,440
+example لنا اسمها بس نجيب example لنا اسمها بس
+
+197
+00:23:29,440 --> 00:23:29,680
+نجيب example لنا اسمها بس نجيب example لنا اسمها
+
+198
+00:23:29,680 --> 00:23:29,900
+بس نجيب example لنا اسمها بس نجيب example لنا
+
+199
+00:23:29,900 --> 00:23:33,360
+اسمها بس نجيب example لنا اسمها بس نجيب example
+
+200
+00:23:33,360 --> 00:23:37,520
+لنا
+
+201
+00:23:37,520 --> 00:23:37,620
+اسمها
+
+202
+00:23:42,390 --> 00:23:45,070
+جزء الأول ولا التاني؟ الأول
+
+203
+00:24:16,920 --> 00:24:25,800
+إذا الجزء الأول مسؤال أربعتاش F zero
+
+204
+00:24:25,800 --> 00:24:36,560
+less than or equal a less than b show أثبتي أنه a
+
+205
+00:24:36,560 --> 00:24:43,160
+تربية أصغر من أو ساوي a في b أصغر من b تربية
+
+206
+00:25:01,170 --> 00:25:07,290
+proof case واحد a بساوي واحد انتوا ان ال a أكبر من
+
+207
+00:25:07,290 --> 00:25:12,430
+أو ساوي سفر فناخد الأول a بساوي سفر و بعدين a أكبر
+
+208
+00:25:12,430 --> 00:25:21,050
+من سفر فلو كان a بساوي سفر هذا بيقدي انه
+
+209
+00:25:21,050 --> 00:25:32,740
+a تربية اللي هي بتساوي سفرأصغر من أوي ساوي A في B
+
+210
+00:25:32,740 --> 00:25:39,740
+اللي هو سفر و
+
+211
+00:25:39,740 --> 00:25:44,180
+طبعا ال B في الحالة هذه أكبر من A يعني أكبر من سفر
+
+212
+00:25:44,180 --> 00:25:47,760
+لأن هذا أصغر من B
+
+213
+00:25:56,410 --> 00:26:01,650
+و أصغر من B تربية لأن
+
+214
+00:26:01,650 --> 00:26:12,050
+هذا بيساوي سفر و B تربية و B أكبر من سفر B أكبر من
+
+215
+00:26:12,050 --> 00:26:17,190
+A اللي هو بيساوي سفر يعني B أكبر من سفر يعني B
+
+216
+00:26:17,190 --> 00:26:22,370
+تربية أكبر من سفر إذا المتبايلة متحققة
+
+217
+00:26:27,190 --> 00:26:38,030
+إذا المتباينة اللي احنا عايزين نثبتها نسميها
+
+218
+00:26:38,030 --> 00:26:48,770
+star إذا star holds in this case الحالة
+
+219
+00:26:48,770 --> 00:27:01,890
+التانية أن a أكبر من 7لو كانت a أكبر من صفر by
+
+220
+00:27:01,890 --> 00:27:08,270
+hypothesis من الفرض أنا عندي الآن a أكبر من صفر
+
+221
+00:27:08,270 --> 00:27:13,230
+أصغر من b فنضرب
+
+222
+00:27:13,230 --> 00:27:17,590
+multiply multiply
+
+223
+00:27:17,590 --> 00:27:18,970
+by
+
+224
+00:27:20,970 --> 00:27:28,150
+أو multiply by صحيح by a أكبر من سفر لما أضرب
+
+225
+00:27:28,150 --> 00:27:32,450
+متباينة في عدد موجة بشريفتها طابقة زي ما هي فهذا
+
+226
+00:27:32,450 --> 00:27:40,710
+بيقدّي فهذا بيقدّي أنه السفر في a بيطلع سفر أصغر
+
+227
+00:27:40,710 --> 00:27:47,550
+من a تربية أصغر من a في b وهذا اللي بدنا يعني
+
+228
+00:27:51,030 --> 00:28:02,650
+لأ هذا .. هاد حاجة كمان and then بعد هيك multiply
+
+229
+00:28:02,650 --> 00:28:08,890
+.. multiply by
+
+230
+00:28:08,890 --> 00:28:14,730
+B اللي هو أكبر من سفر أيضا لإن ال B أكبر من A أكبر
+
+231
+00:28:14,730 --> 00:28:20,340
+من سفر صح؟فلو ضربنا المتباينة هذه في بي اللي هو
+
+232
+00:28:20,340 --> 00:28:25,980
+عدد موجب برضه إشارتها هتبقى زي ما هي إذن هذا
+
+233
+00:28:25,980 --> 00:28:31,760
+بيقدّي لما أضرب المتباينة هذه في بي عدد موجب
+
+234
+00:28:31,760 --> 00:28:37,020
+فهيطلع عند سفر أصغر من a في بي أصغر من b تربية
+
+235
+00:28:37,020 --> 00:28:47,280
+نسمي هذه واحد والمتباينة هذه اتنين الأن واحدعندي
+
+236
+00:28:47,280 --> 00:28:51,760
+اتنين بيقدّوا يعني
+
+237
+00:28:51,760 --> 00:29:00,540
+عندي a تربية أكبر من سفر أصغر من a بيه وعندي a بيه
+
+238
+00:29:00,540 --> 00:29:05,120
+أصغر من b تربية وهذه هي ال star
+
+239
+00:29:16,200 --> 00:29:22,040
+إذا in both cases
+
+240
+00:29:22,040 --> 00:29:30,280
+في الحالة الأولى والتانية we have أثبتنا أن a
+
+241
+00:29:30,280 --> 00:29:34,680
+تربية أصغر
+
+242
+00:29:34,680 --> 00:29:40,760
+أو يساوي ab و
+
+243
+00:29:40,760 --> 00:29:47,120
+ab في الحالتين أصغر من b تربيةكمان مرة في الحالة
+
+244
+00:29:47,120 --> 00:29:53,080
+الأولى a تربية أصغر من أو يساوي a b في الحالة
+
+245
+00:29:53,080 --> 00:29:58,020
+التانية a تربية أصغر من a b إذا في الحالتين a
+
+246
+00:29:58,020 --> 00:30:02,420
+تربية أصغر من أو يساوي a b في الحالة الأولى a b
+
+247
+00:30:02,420 --> 00:30:05,680
+أصغر من b تربية وفي الحالة التانية a b أصغر من b
+
+248
+00:30:05,680 --> 00:30:10,300
+تربية إذا في الحالتين a b أصغر من b تربية و هذا
+
+249
+00:30:10,300 --> 00:30:13,820
+اللي أحنا عايزين نثبته تمام إذا هيك بتكون أثبتنا
+
+250
+00:30:13,820 --> 00:30:21,140
+جزءالأول من السؤال الجزء
+
+251
+00:30:21,140 --> 00:30:27,960
+التاني بيقول انه احنا مانقدرش المتباينة هذه star
+
+252
+00:30:27,960 --> 00:30:38,500
+نبدلها ال
+
+253
+00:30:38,500 --> 00:30:43,760
+star can't be replaced
+
+254
+00:30:46,040 --> 00:30:53,660
+replaced by a تربيع أصغر من a بي أصغر من بي تربيع
+
+255
+00:30:53,660 --> 00:31:01,140
+يعني هنا ال inequality هذه أصغر من أو ساوي مانقدرش
+
+256
+00:31:01,140 --> 00:31:08,260
+نبدلها بأصغر من strictly أصغر من فبنوضح ذلك بمثال
+
+257
+00:31:08,260 --> 00:31:15,280
+فاخدي a بساوي سفر و b أي عدد example
+
+258
+00:31:37,110 --> 00:31:39,170
+زي ما عملنا في الحلقة الأولى
+
+259
+00:31:43,550 --> 00:31:50,550
+فهيطلع عندى هذه مساوية و ليست .. و ليست اكبر من ..
+
+260
+00:31:50,550 --> 00:32:01,390
+اه؟ okay فى أسئلة تانية؟ مين عند السؤال التانى؟
+
+261
+00:32:01,390 --> 00:32:05,050
+section اتنين اتنين؟ فى اي أسئلة في section اتنين
+
+262
+00:32:05,050 --> 00:32:08,910
+اتنين؟ section اتنين واحد اتنين اتنين
+
+263
+00:32:12,890 --> 00:32:21,830
+في أسئلة كتيرة حلوة section
+
+264
+00:32:21,830 --> 00:32:37,450
+1122 فينا مجموعة كبيرة من الأسئلة الناس
+
+265
+00:32:37,450 --> 00:32:39,270
+بتدرس الناس بتحل مثال
+
+266
+00:32:42,490 --> 00:32:46,430
+حليتوا كل المسائل؟ مش عندكم أسئلة؟ ال section
+
+267
+00:32:46,430 --> 00:32:56,890
+اتنين واحد و اتنين اتنين؟
+
+268
+00:32:56,890 --> 00:33:03,850
+مان لديها سؤال؟ السؤال اتنين في section اتنين
+
+269
+00:33:03,850 --> 00:33:10,710
+اتنين حاضر اتنين اتنين اتنين نكتب النص تبع السؤال
+
+270
+00:33:14,400 --> 00:33:25,960
+if a و b are real numbers show ifpty أنه
+
+271
+00:33:25,960 --> 00:33:35,160
+absolute a زائد b بساوي absolute a زايد absolute b
+
+272
+00:33:35,160 --> 00:33:47,110
+if and only if a في b أكبر منأو يساوي سفر هيك
+
+273
+00:33:47,110 --> 00:33:59,610
+مكتوب طيب
+
+274
+00:33:59,610 --> 00:34:07,590
+البرهان ال proof راحظوا هذا if and only if
+
+275
+00:34:07,590 --> 00:34:11,730
+statement فخلينا
+
+276
+00:34:11,730 --> 00:34:21,730
+الأولنثبت ال claim الأول claim one للدعاء الأول أن
+
+277
+00:34:21,730 --> 00:34:28,350
+a ضرب b أكبر من أو ساوي سفر if and only if
+
+278
+00:34:28,350 --> 00:34:36,290
+absolute a ضرب b بساوي a في b
+
+279
+00:34:43,590 --> 00:34:47,970
+هل هذا واضح هذا
+
+280
+00:34:47,970 --> 00:35:01,250
+واضح من تعريف that this is clear by definition of
+
+281
+00:35:01,250 --> 00:35:09,370
+absolute value طيب
+
+282
+00:35:09,370 --> 00:35:10,490
+الكلام التاني
+
+283
+00:35:16,440 --> 00:35:24,700
+الكلام التاني absolute a plus absolute b الكل
+
+284
+00:35:24,700 --> 00:35:35,560
+تربية بساوي a زي b الكل تربية if
+
+285
+00:35:35,560 --> 00:35:43,500
+and only if absolute a ضرب b بساوي a في b
+
+286
+00:35:49,120 --> 00:35:56,920
+بروف للـ claim هذا هاي لو أخدت absolute a زاد
+
+287
+00:35:56,920 --> 00:36:04,520
+absolute b وربعت فهذا المفروض يساوي absolute a
+
+288
+00:36:04,520 --> 00:36:12,780
+الكتر بيها زاد absolute b الكتر بيها زاد اتنين
+
+289
+00:36:12,780 --> 00:36:19,600
+absolute a في absolute bو هذا بساوي absolute a
+
+290
+00:36:19,600 --> 00:36:26,080
+الكل
+
+291
+00:36:26,080 --> 00:36:33,480
+تربية زار
+
+292
+00:36:33,480 --> 00:36:44,050
+absolute b الكل تربية زاد 2 absolute a ضارب bلأن
+
+293
+00:36:44,050 --> 00:36:54,670
+absolute a ضرب absolute b بيطلع absolute a في b و
+
+294
+00:36:54,670 --> 00:36:59,210
+absolute absolute
+
+295
+00:36:59,210 --> 00:37:06,850
+a بساوي الجدر التربية a ل a تربية هذا
+
+296
+00:37:06,850 --> 00:37:18,950
+بكافئ أن a أو absoluteكل تربية بيساوي ا تربية لو
+
+297
+00:37:18,950 --> 00:37:25,030
+ربعت الطرفين فبطلع absolute a تربية بيساوي a تربية
+
+298
+00:37:25,030 --> 00:37:30,490
+وبالتالي ممكن استبدل absolute a تربية ممكن استبدل
+
+299
+00:37:30,490 --> 00:37:36,530
+absolute a تربية ب a تربية و
+
+300
+00:37:36,530 --> 00:37:41,450
+استبدل absolute b تربية ب b تربية
+
+301
+00:37:50,020 --> 00:37:57,000
+الان هذا بيساوي هذا بيساوي a تربية زائد b تربية
+
+302
+00:37:57,000 --> 00:38:06,460
+زائد اتنين a b if and only if if and only if
+
+303
+00:38:06,460 --> 00:38:10,940
+absolute a b بيساوي a في b
+
+304
+00:38:13,940 --> 00:38:17,600
+هذا المقدار اللي فوق متى بيساوي a تربية زايد بي
+
+305
+00:38:17,600 --> 00:38:22,500
+تربية زايد اتنين a ب a اذا absolute ب a في b هو
+
+306
+00:38:22,500 --> 00:38:33,140
+عبارة عن a في b صح؟ وهذا
+
+307
+00:38:33,140 --> 00:38:39,720
+الأخير بيساوي يعني
+
+308
+00:38:39,720 --> 00:38:50,460
+هذا السطر هذاهي تعالى نشوف بقدّه
+
+309
+00:38:50,460 --> 00:39:00,000
+ان absolute a زاد absolute b الكل تربية بساوي
+
+310
+00:39:03,190 --> 00:39:11,210
+A زائد B الكل ترمية if and only if absolute A B
+
+311
+00:39:11,210 --> 00:39:27,050
+بساوي A B absolute
+
+312
+00:39:27,050 --> 00:39:32,390
+A B بساوي A B ومن claim واحد
+
+313
+00:39:36,690 --> 00:39:43,890
+عمر ال claim واحد هذا بتحقق if and only if a ضربي
+
+314
+00:39:43,890 --> 00:39:49,610
+أكبر من أو ساوى 0 لأن هذا نتيجة أو بضم ال two
+
+315
+00:39:49,610 --> 00:39:54,070
+claims مع بعض طيب
+
+316
+00:39:54,070 --> 00:39:58,290
+لو أخلنا جدر التربية للطرفين فضي جدر التربية يعني
+
+317
+00:39:58,290 --> 00:40:02,290
+take square root
+
+318
+00:40:05,210 --> 00:40:14,850
+of both sides فهذا
+
+319
+00:40:14,850 --> 00:40:20,410
+بيقدّي أنه absolute a زائد absolute b الجدر
+
+320
+00:40:20,410 --> 00:40:23,030
+التربيعي هنا بعطيني ال absolute value ل a زائد
+
+321
+00:40:23,030 --> 00:40:28,570
+absolute b وهنا بعطيني الجدر التربيعي الجدر
+
+322
+00:40:28,570 --> 00:40:31,370
+التربيعي العدد التربيعي بعطيني القيمة المطلقة
+
+323
+00:40:31,370 --> 00:40:32,010
+تبعته
+
+324
+00:40:34,820 --> 00:40:41,180
+هذا بتحقق if and only if a ضرب b أكبر من أو ساوى 0
+
+325
+00:40:41,180 --> 00:40:48,200
+وهذا هو المطلوب وهذا هو المطلوب okay تمام فهذا هو
+
+326
+00:40:48,200 --> 00:40:53,140
+المطلوب في
+
+327
+00:40:53,140 --> 00:40:58,220
+أي أسئلة تانية في
+
+328
+00:40:58,220 --> 00:41:01,700
+عندكم أي سؤال تاني section 2 أو section 1
+
+329
+00:41:07,070 --> 00:41:12,510
+أن أسئلة كتيرة لو جيت بعدي و أنتوا مابتسألوش و مش
+
+330
+00:41:12,510 --> 00:41:17,650
+هيكون في راجعة بعد هيك لأ أساكاشنا دي المرة الجاية
+
+331
+00:41:17,650 --> 00:41:21,830
+هناخد مناقشة في section اتنين تلاتة و اتنين أربعة
+
+332
+00:41:21,830 --> 00:41:31,370
+في أي أسئلة أفندم في أي section اتنين اتنين
+
+333
+00:41:37,770 --> 00:41:38,370
+حاضر
+
+334
+00:42:02,930 --> 00:42:11,490
+السؤال خمستاش سيكشن اتنين نيلو نشوف ايه هو السؤال
+
+335
+00:42:11,490 --> 00:42:25,670
+show
+
+336
+00:42:25,670 --> 00:42:29,790
+if a و b are real numbers
+
+337
+00:42:32,160 --> 00:42:38,560
+و a لا يساوي b then
+
+338
+00:42:38,560 --> 00:42:49,100
+there exist epsilon neighborhoods u
+
+339
+00:42:49,100 --> 00:43:09,390
+of a and v of b such thatU تقاطع V بساوي five إذن
+
+340
+00:43:09,390 --> 00:43:24,630
+كمان مرة أنا
+
+341
+00:43:24,630 --> 00:43:25,090
+عندي
+
+342
+00:43:28,550 --> 00:43:35,990
+A وB أعداد حقيقية و A لا يساوي B هذا معناه أن
+
+343
+00:43:35,990 --> 00:43:43,970
+either A less than B ف
+
+344
+00:43:43,970 --> 00:43:52,310
+assume نأخد الحالة الأولى case one أن
+
+345
+00:43:52,310 --> 00:44:01,330
+A أصغر من B إذا هي خط الأعدادهذا خط الاعداد وهذه a
+
+346
+00:44:01,330 --> 00:44:12,670
+وهذه b و a أصغر من b بدي أثبت أنه في جوار ل a بعمق
+
+347
+00:44:12,670 --> 00:44:22,450
+epsilon اسمه u وفي جوار ل b بعمق epsilon والجوارين
+
+348
+00:44:22,450 --> 00:44:26,170
+هذول المبروهود سقطوهم بسوء في يعني منفصلين عن بعض
+
+349
+00:44:27,430 --> 00:44:35,570
+فبكل بساطة بجيس المسافة من a و b و باخد نص المسافة
+
+350
+00:44:35,570 --> 00:44:43,670
+يعني لأن هنا take epsilon
+
+351
+00:44:43,670 --> 00:44:48,770
+بساوي
+
+352
+00:44:48,770 --> 00:44:54,270
+نص المسافة نص ال b minus
+
+353
+00:44:57,480 --> 00:45:13,820
+أو نص او نص المسافة بين A وB فهي
+
+354
+00:45:13,820 --> 00:45:15,920
+نص المسافة لو كونت فترة
+
+355
+00:45:23,000 --> 00:45:28,520
+فهي منتصف المسافة هذه المسافة
+
+356
+00:45:28,520 --> 00:45:33,260
+هذه منتصف المسافة سمنها epsilon فهذه النقطة هتكون
+
+357
+00:45:33,260 --> 00:45:40,960
+a زاد epsilon وهي نفس المسافة a سالب epsilon وكون
+
+358
+00:45:40,960 --> 00:45:45,660
+فترة مفتوحة وسم
+
+359
+00:45:45,660 --> 00:45:48,960
+الفترة المفتوحة هذه new
+
+360
+00:46:02,460 --> 00:46:07,900
+فنسمي الفترة المفتوحة هذه U مركزها E و نصف قطرها E
+
+361
+00:46:07,900 --> 00:46:16,200
+و نكوّن فترة تانية برضه مركزها B و نصف قطرها E
+
+362
+00:46:16,200 --> 00:46:22,080
+يعني هذه النقطة هتصير B زاد E وهذه النقطة هتصير
+
+363
+00:46:25,910 --> 00:46:33,530
+بزائد ابسلون و النقطة هذه بسالب ابسلون و نكون فترة
+
+364
+00:46:33,530 --> 00:46:39,970
+مفتوحة تمام
+
+365
+00:46:39,970 --> 00:46:43,310
+اذا
+
+366
+00:46:43,310 --> 00:46:50,890
+انا في اندي و نسمي الفترة المفتوحة هذه نسميها
+
+367
+00:46:50,890 --> 00:46:51,030
+ب
+
+368
+00:46:57,180 --> 00:47:01,040
+أذن هذا عبارة عن الـ U الفترة المفتوحة اذا هنا led
+
+369
+00:47:01,040 --> 00:47:14,300
+take
+
+370
+00:47:14,300 --> 00:47:16,480
+u
+
+371
+00:47:18,050 --> 00:47:26,410
+by definition بتساوي a سالب إبسلون و a مجد إبسلون
+
+372
+00:47:26,410 --> 00:47:40,810
+و V بساوي B سالب إبسلون B plus إبسلون فواضح
+
+373
+00:47:40,810 --> 00:47:41,410
+nearly
+
+374
+00:47:46,910 --> 00:47:58,030
+Clearly U is an epsilon neighborhood of A and V is
+
+375
+00:47:58,030 --> 00:48:05,410
+an epsilon neighborhood of B ومش هيكوا بس and ممكن
+
+376
+00:48:05,410 --> 00:48:09,370
+اثبات ان U تقاطع B بساوي فاي
+
+377
+00:48:12,440 --> 00:48:21,120
+يعني لو أخدنا هاي واضح هاي جوار هذا مافيش ولا نقطة
+
+378
+00:48:21,120 --> 00:48:27,940
+فيه موجودة في الجوار التاني هذه الفترة المفتوحة
+
+379
+00:48:27,940 --> 00:48:35,540
+منفصلة عن الفترة المفتوحة يوم طبعا؟
+
+380
+00:48:39,190 --> 00:48:44,970
+إن هذا الكلام يعني واضح ان هذا عبارة عن epsilon
+
+381
+00:48:44,970 --> 00:48:50,150
+neighborhood ل a فترة مفتوحة مركزها a نصف قطرة
+
+382
+00:48:50,150 --> 00:48:53,590
+epsilon هذا نسميه epsilon neighborhood ل a و V
+
+383
+00:48:53,590 --> 00:48:57,510
+الفترة المفتوحة هذه بي سالب epsilon و V موجب
+
+384
+00:48:57,510 --> 00:49:04,310
+epsilon برضه عبارة عن epsilon neighborhood ل B و
+
+385
+00:49:04,310 --> 00:49:09,900
+اتنين حسب الرسم منفصلين و هذا ممكن اثباتهباستخدام
+
+386
+00:49:09,900 --> 00:49:14,380
+التناقض يعني افرض انه في عنصر يعني التقاطع هذا لا
+
+387
+00:49:14,380 --> 00:49:19,900
+يساوي فيه وبالتالي في عنصر موجود في U وموجود في V
+
+388
+00:49:19,900 --> 00:49:25,540
+في أن و واحد واصلي إليه تناقض okay هذا ممكن اثباته
+
+389
+00:49:25,540 --> 00:49:34,280
+بطريقة تحليلية طبعا في السؤال في الحل في حل السؤال
+
+390
+00:49:34,280 --> 00:49:41,370
+هذا يقول مافي عندنا حلتينالحالة الأولى a أصغر من b
+
+391
+00:49:41,370 --> 00:49:48,110
+والحالة الثانية b أصغر من a وشوفنا هنا أخدنا
+
+392
+00:49:48,110 --> 00:49:51,230
+الحالة اللي فيها a أصغر من b زي اللي هو النضال تحت
+
+393
+00:49:51,230 --> 00:49:57,090
+الاسم هاي a أصغر من b و أثبتنا في الحالة هذه أن u
+
+394
+00:49:57,090 --> 00:50:00,070
+جد epsilon never move u ل a
+
+395
+00:50:17,240 --> 00:50:24,820
+باقي الحالة التانية case 2 نثبت
+
+396
+00:50:24,820 --> 00:50:27,120
+برضه المطلوب
+
+397
+00:50:33,160 --> 00:50:38,320
+القرآن في الحلقة التانية مماثل للقرآن اللي عملناه
+
+398
+00:50:38,320 --> 00:50:38,880
+في الحلقة
+
+399
+00:50:50,040 --> 00:50:54,040
+و هذا طبعا يكمل الفرحة، إذا الحالة التانية اللي
+
+400
+00:50:54,040 --> 00:50:58,120
+فيها P أصفر من A، بس نضع P هنا و A هنا، و نفس
+
+401
+00:50:58,120 --> 00:51:05,440
+العادة، فتصير هذا الـP و هذا الـU، و يعطينا نفس
+
+402
+00:51:05,440 --> 00:51:13,020
+النتيجة، إت من كون كملنا المرهان اللي هو التمريد
+
+403
+00:51:14,530 --> 00:51:17,970
+وطبعا بعد ذلك ان شاء الله هنكمل حل ال تمارين لل
+
+404
+00:51:17,970 --> 00:51:20,870
+sport الخادم في نفس الموضوع
+
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/Lc2K-uxXK74.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/Lc2K-uxXK74.srt
new file mode 100644
index 0000000000000000000000000000000000000000..ed268d5424e53947414d0c473c3ffe1e834ebd95
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/Lc2K-uxXK74.srt
@@ -0,0 +1,1779 @@
+1
+00:00:20,750 --> 00:00:26,090
+Okay إذا اليوم إن شاء الله هنكمل موضوع ال limit
+
+2
+00:00:26,090 --> 00:00:32,390
+theorems أو نظريات النهايات ومن النظريات المهمة 
+
+3
+00:00:32,390 --> 00:00:39,710
+هذه هي نظرية 12 بتقول لو في عندي sequence x<sub>n</sub> و
+
+4
+00:00:39,710 --> 00:00:44,570
+ال sequence هذي convergent لـ x فالـ sequence of
+
+5
+00:00:44,570 --> 00:00:49,350
+absolute values بتطلع convergent والـ limit تبعتها
+
+6
+00:00:49,350 --> 00:00:55,490
+تطلع absolute ... absolute limit تبعت الـ sequence
+
+7
+00:00:55,490 --> 00:01:00,750
+x فالبرهان 
+
+8
+00:01:00,750 --> 00:01:04,470
+بيعتمد على ال triangle inequality
+
+9
+00:01:07,360 --> 00:01:13,720
+أحد صور ال triangle inequality كانت المتباينة هذه 
+
+10
+00:01:13,720 --> 00:01:20,740
+|a| - |b| وأخد ال absolute value
+
+11
+00:01:20,740 --> 00:01:28,600
+هذا أصغر من أو يساوي |a - b| فلو أخدت هنا
+
+12
+00:01:28,600 --> 00:01:36,160
+a بساوي x<sub>n</sub> و b بساوي x فبطلع الكلام هذا صحيح لكل
+
+13
+00:01:36,160 --> 00:01:43,760
+الأعداد الطبيعية n الآن أنا عندي x<sub>n</sub> converges to x
+
+14
+00:01:43,760 --> 00:01:51,740
+فلو أخدت أي epsilon أكبر من الصفر given
+
+15
+00:01:55,040 --> 00:02:00,540
+وأنـا x<sub>n</sub> converge لـ x، إذا هذا بيدّي أنه
+
+16
+00:02:00,540 --> 00:02:03,580
+يوجد 
+
+17
+00:02:03,580 --> 00:02:13,660
+N عدد طبيعي يعتمد على epsilon بحيث أنه لو
+
+18
+00:02:13,660 --> 00:02:18,260
+كان n أكبر من أو يساوي N فهذا بيدّي ان
+
+19
+00:02:18,260 --> 00:02:22,080
+|x<sub>n</sub> - x| أصغر من epsilon
+
+20
+00:02:25,260 --> 00:02:30,300
+وبالتالي من هنا إذا الهدف بيطلع أصغر من epsilon
+
+21
+00:02:30,300 --> 00:02:34,260
+لكل
+
+22
+00:02:34,260 --> 00:02:41,180
+n أكبر من أو يساوي N إذا
+
+23
+00:02:41,180 --> 00:02:44,800
+أنا هيك بكون أثبتت إنه لأي epsilon أكبر من صفر
+
+24
+00:02:44,800 --> 00:02:50,760
+يوجد N يعتمد على epsilon عدد طبيعي بحيث
+
+25
+00:02:50,760 --> 00:02:57,040
+لكل n أكبر من أو يساوي N القيمة المطلقة لـ
+
+26
+00:02:57,040 --> 00:03:02,480
+|x<sub>n</sub>| - |x| أصغر من epsilon إذا
+
+27
+00:03:02,480 --> 00:03:07,900
+حسب تعريف epsilon N for limits هذا معناه
+
+28
+00:03:07,900 --> 00:03:14,260
+بالظبط أن limit |x<sub>n</sub>| as n tends to infinity
+
+29
+00:03:14,260 --> 00:03:21,790
+بساوي |x| وهو المطلوب Okay تمام إذا هذا
+
+30
+00:03:21,790 --> 00:03:32,890
+بيكمل برهان نظرية 12 تمام واضح النظرية
+
+31
+00:03:32,890 --> 00:03:39,270
+اللي بعدها نظرية 13 بتقول لو أنا في عندي
+
+32
+00:03:39,270 --> 00:03:45,490
+sequence حدودها كلها غير سالبة حدود الـ sequence x<sub>n</sub>
+
+33
+00:03:45,490 --> 00:03:50,750
+كلها غير سالبة أعداد غير سالبة والـ sequence لو
+
+34
+00:03:50,750 --> 00:03:57,730
+كانت الـ sequence x<sub>n</sub> convergent to some x فالـ
+
+35
+00:03:57,730 --> 00:04:02,730
+limit للـ sequence of square roots لـ x<sub>n</sub> تطلع
+
+36
+00:04:02,730 --> 00:04:08,470
+convergent والـ limit تبعتها بساوي square root للـ
+
+37
+00:04:08,470 --> 00:04:09,890
+limit للـ sequence x<sub>n</sub>
+
+38
+00:04:13,780 --> 00:04:19,760
+والبرهان تبع النظرية دي سهل أنا أول شيء عندي احنا
+
+39
+00:04:19,760 --> 00:04:25,060
+فرضين أن الـ limit لـ x<sub>n</sub> بساوي x في نظرية 8
+
+40
+00:04:25,060 --> 00:04:28,700
+قلنا أن لو كانت حدود الـ sequence x<sub>n</sub> كلها غير سالبة
+
+41
+00:04:28,700 --> 00:04:34,360
+فـ limit لـ sequence x<sub>n</sub> اللي هي x أيضا تطلع غير
+
+42
+00:04:34,360 --> 00:04:40,840
+سالبة إذا x أكبر من أو يساوي 0 الآن في عندي حالتين
+
+43
+00:04:40,840 --> 00:04:46,300
+الـ x هنا أكبر من أو يساوي صفر ففي عندي احتمالين اما
+
+44
+00:04:46,300 --> 00:04:54,260
+x بساوي صفر أو x أكبر من الصفر تمام وفي كل حالة
+
+45
+00:04:54,920 --> 00:04:59,540
+مطلوب مني أن أثبت أن limit الـ square root لـ x<sub>n</sub>
+
+46
+00:04:59,540 --> 00:05:03,960
+بساوي الـ square root of x تمام؟ نشوف في الحالة
+
+47
+00:05:03,960 --> 00:05:08,520
+الأولى لو كانت الـ x بساوي صفر وأنا عندي من الفرض
+
+48
+00:05:08,520 --> 00:05:15,550
+x<sub>n</sub> converges to x اللي هي صفر إذا لو أخدت أي epsilon
+
+49
+00:05:15,550 --> 00:05:20,250
+أكبر من الصفر من كون الـ sequence هذه converge
+
+50
+00:05:20,250 --> 00:05:24,270
+للسفر إذا لأي epsilon يوجد N يعتمد على
+
+51
+00:05:24,270 --> 00:05:30,150
+epsilon بحيث المسافة بين x<sub>n</sub> والصفر أصغر من epsilon
+
+52
+00:05:30,150 --> 00:05:33,470
+تربيع لكل n أكبر من أو يساوي N هذا من
+
+53
+00:05:33,470 --> 00:05:36,690
+تعريف الـ convergence ممكن أحط هنا epsilon أو epsilon
+
+54
+00:05:36,690 --> 00:05:42,940
+تربيع مافي مشكلة طيب أنا عندي x<sub>n</sub> من الفرض الـ x<sub>n</sub>
+
+55
+00:05:42,940 --> 00:05:48,840
+كلهم أكبر من أو يساوي صفر وبالتالي القيمة المطلقة
+
+56
+00:05:48,840 --> 00:05:53,800
+لـ x<sub>n</sub> بساوي نفسها ناخد 
+
+57
+00:05:53,800 --> 00:05:59,120
+الجذر التربيعي للحدود المتباينة هذه هي الـ square
+
+58
+00:05:59,120 --> 00:06:04,790
+root of x<sub>n</sub> بساوي الـ absolute value لـ square root لـ
+
+59
+00:06:04,790 --> 00:06:10,190
+x<sub>n</sub> - صفر وهذا أصغر من epsilon square root
+
+60
+00:06:10,190 --> 00:06:13,870
+لـ epsilon تربيع بيطلع epsilon هذا الكلام صحيح for
+
+61
+00:06:13,870 --> 00:06:18,830
+every n bigger than or equal N طب هذا
+
+62
+00:06:18,830 --> 00:06:23,050
+معناه بما أن epsilon was arbitrarily بما أن احنا
+
+63
+00:06:23,050 --> 00:06:29,850
+أثبتنا هذا الكلام لكل epsilon عدد موجب هذا من تعريف
+
+64
+00:06:29,850 --> 00:06:34,350
+epsilon N for limits للنهايات هذا معناه
+
+65
+00:06:34,350 --> 00:06:40,970
+limit الـ square root لـ x<sub>n</sub> بساوي الصفر لما n تؤول لـ
+
+66
+00:06:40,970 --> 00:06:47,140
+N وهذا ايه هذا اللي هو المطلوب طيب الصفر هنا احنا 
+
+67
+00:06:47,140 --> 00:06:50,780
+ماخدين x بالساوي صفر فالصفر هذا هو square root لـ
+
+68
+00:06:50,780 --> 00:06:54,360
+x إذا هيك اثبتت أن limit square root لـ x<sub>n</sub>
+
+69
+00:06:54,360 --> 00:06:58,820
+بالساوي square root لـ x في حالة لما x بالساوي
+
+70
+00:06:58,820 --> 00:07:07,280
+صفر تمام باقي نثبت النتيجة نفسها في حالة لما x أكبر
+
+71
+00:07:07,280 --> 00:07:11,740
+من 0 تفضلي قالت جيت حكيت أنه ممكن أخد epsilon مش 
+
+72
+00:07:11,740 --> 00:07:15,740
+epsilon تربيع لما أكمل خطوة بعد تطلع جذر الـ epsilon
+
+73
+00:07:15,740 --> 00:07:19,660
+يعني أقل من جذر الـ epsilon جذر الـ epsilon قلت أن احنا
+
+74
+00:07:19,660 --> 00:07:23,600
+خلينا epsilon تربيع عشان لما أخد الجذر يطلع epsilon
+
+75
+00:07:23,600 --> 00:07:29,520
+مافي مشكلة يعني اعتبر هذه هي الـ epsilon مش الـ epsilon
+
+76
+00:07:29,520 --> 00:07:34,300
+أكبر عدد أكبر من 0 given إذا epsilon تربيع برضه عدد
+
+77
+00:07:34,300 --> 00:07:39,880
+موجب بقى تقريبا هو الـ given وبالتالي يوجد N تعتمد
+
+78
+00:07:39,880 --> 00:07:44,320
+على epsilon تربيع بدل epsilon طب epsilon تربيع تعتمد
+
+79
+00:07:44,320 --> 00:07:48,420
+على epsilon إذا ليش ما نقول إذا يوجد N تعتمد على
+
+80
+00:07:48,420 --> 00:07:52,240
+epsilon وإعتبر الـ epsilon تربيع بدل epsilon في الـ
+
+81
+00:07:52,240 --> 00:07:55,920
+definition فمافي مشكلة بس خدناها الـ epsilon تربيع
+
+82
+00:07:55,920 --> 00:07:59,660
+عشان لما ناخد جذر التربيع يطلع عندي أصغر من epsilon
+
+83
+00:07:59,660 --> 00:08:03,760
+وبالتالي نقول حسب التعريف إذا limit جذر x<sub>n</sub> بساوي 
+
+84
+00:08:03,760 --> 00:08:11,840
+6 تمام اللي هي جذر x في أي سؤال ثاني؟ طيب، نشوف 
+
+85
+00:08:11,840 --> 00:08:16,800
+الحالة الثانية، لو كانت الـ x هذه أكبر من صفر، إذا 
+
+86
+00:08:16,800 --> 00:08:20,640
+جذر الـ x بالتأكيد أكبر من الصفر، وبالتالي جذر x<sub>n</sub>
+
+87
+00:08:20,640 --> 00:08:26,120
+زي جذر x أكبر من أو يساوي جذر الـ x، لأن هذا أكبر من
+
+88
+00:08:26,120 --> 00:08:35,430
+أو يساوي صفر، وهذا موجب، لأن الـ x موجبة طيب، الآن 
+
+89
+00:08:35,430 --> 00:08:40,630
+هذا المقدار أكبر من أو يساوي هذا واتنين موجبين، إذا 
+
+90
+00:08:40,630 --> 00:08:47,950
+المقلوب الكبير أصغر من أو يساوي المقلوب الصغير هذه
+
+91
+00:08:47,950 --> 00:08:53,010
+الخاصية أخذناها في chapter one وبناء عليه
+
+92
+00:09:01,430 --> 00:09:06,810
+بناء على ذلك أنا ممكن أحسب جذر x<sub>n</sub> - جذر الـ x
+
+93
+00:09:06,810 --> 00:09:12,870
+بضرب المقدار هذا في المرافق تبعه بسطه مقامه، هاي
+
+94
+00:09:12,870 --> 00:09:16,870
+المرافق تبعه بسطه مقام فكأني ضربت المقدار هذا في 
+
+95
+00:09:16,870 --> 00:09:23,030
+واحد، إذا هذا بساوي نفسه ضرب مرافقه على مرافقه، 
+
+96
+00:09:23,030 --> 00:09:27,870
+تمام؟ الآن الـ numerator تحليل الفرق بين المربعين فبطلع
+
+97
+00:09:27,870 --> 00:09:33,170
+مربع هذا سالب مربع هذا اللي هو x<sub>n</sub> - x و
+
+98
+00:09:33,170 --> 00:09:38,310
+بيبقى الـ denominator في المقام المقدار هذا الآن ناخد
+
+99
+00:09:38,310 --> 00:09:43,370
+القيمة المطلقة للكلام هذا بيساوي القيمة المطلقة
+
+100
+00:09:43,370 --> 00:09:48,230
+للطرف اليمين القيمة المطلقة للـ numerator على القيمة
+
+101
+00:09:48,230 --> 00:09:53,070
+المطلقة للمقام المقام هذا موجب فالقيمة المطلقة له
+
+102
+00:09:53,070 --> 00:09:58,770
+نفسه إذا الآن أنا في عندي sequence اللي هي الحد
+
+103
+00:09:58,770 --> 00:10:02,810
+العام تبعها square root of x<sub>n</sub> وفي عندي عدد square
+
+104
+00:10:02,810 --> 00:10:10,390
+root of x المسافة بينهم أصغر من أو يساوي أصغر من
+
+105
+00:10:10,390 --> 00:10:15,610
+أو يساوي هي المسافة هذه بالساوي 1 على square
+
+106
+00:10:15,610 --> 00:10:21,870
+root of x<sub>n</sub> + square root of x والكسر هذا من
+
+107
+00:10:21,870 --> 00:10:27,950
+المتباينة 9 هذا الكل أصغر من أو يساوي 1 على
+
+108
+00:10:27,950 --> 00:10:32,610
+square root of x ضرب |x<sub>n</sub> - x| الآن
+
+109
+00:10:32,610 --> 00:10:43,830
+ارجعوا لنظرية 2.4 with
+
+110
+00:10:43,830 --> 00:10:52,060
+c عدد موجب يساوي 1 على جذر الـ x هذا عدد موجب و a<sub>n</sub>
+
+111
+00:10:52,060 --> 00:10:59,780
+بساوي x<sub>n</sub> - x إذن
+
+112
+00:10:59,780 --> 00:11:03,940
+هي يوجد c عدد موجب اللي هو 1 على جذر الـ x وهي 
+
+113
+00:11:03,940 --> 00:11:08,820
+في عندي sequence a<sub>n</sub> الحد العام تبعها x<sub>n</sub> - x و
+
+114
+00:11:08,820 --> 00:11:14,680
+الـ sequence هذه تؤول إلى صفر as n tends to
+
+115
+00:11:14,680 --> 00:11:19,870
+infinity لأن أنا من المعطيات عندي x<sub>n</sub> تؤول لـ x أو
+
+116
+00:11:19,870 --> 00:11:24,490
+limit x<sub>n</sub> بساوي x، لذلك limit الفرق بساوي صفر، لذلك
+
+117
+00:11:24,490 --> 00:11:29,890
+حسب نظرية 2.4، كل شروطها متحققة، وبالتالي، لذلك حسب
+
+118
+00:11:29,890 --> 00:11:34,630
+النظرية هذه، by theorem 2.4، بيطلع عندي limit
+
+119
+00:11:34,630 --> 00:11:41,190
+square root لـ x<sub>n</sub> بساوي square root لـ x وهو المطلوب 
+
+120
+00:11:41,190 --> 00:11:46,690
+إثباته إذا هاي اثبتنا أن limit الـ square root لـ x
+
+121
+00:11:46,690 --> 00:11:50,410
+<sub>n</sub> بساوي الـ square root لـ x في حالة لما x تكون
+
+122
+00:11:50,410 --> 00:11:54,750
+موجبة والحالة الأولى في حالة لما x صفر برضه
+
+123
+00:11:54,750 --> 00:11:58,410
+اثبتنا نفس الحاجة لذلك بنكون كملنا برهان نظرية
+
+124
+00:11:58,410 --> 00:12:02,690
+تمام؟ في حد عنده أي سؤال أو استفسار واضح البرهان؟
+
+125
+00:12:05,660 --> 00:12:12,800
+في نظرية هنا ممكن نسميها نعتبرها ratio test اختبار 
+
+126
+00:12:12,800 --> 00:12:21,660
+الكسور أو النسبة أو ايش
+
+127
+00:12:21,660 --> 00:12:27,300
+الـ ratio test ماذا هذا الـ ratio test بيقول هذا الـ
+
+128
+00:12:27,300 --> 00:12:31,120
+ratio test بتعلق بـ sequences of positive numbers
+
+129
+00:12:32,030 --> 00:12:35,090
+يعني عشان أنا أطبق الـ ratio test لازم الـ sequence
+
+130
+00:12:35,090 --> 00:12:39,170
+تبعتي تكون حدودها كلها موجبة بقى فلو في عندي
+
+131
+00:12:39,170 --> 00:12:44,310
+sequence of positive real numbers such that limit
+
+132
+00:12:44,310 --> 00:12:49,050
+الـ ratio لـ x<sub>n+1</sub> على x<sub>n</sub> exists موجود أو 
+
+133
+00:12:49,050 --> 00:12:54,370
+بتساوي عدد حقيقي L ولو كان هذا العدد L أصغر من
+
+134
+00:12:54,370 --> 00:13:01,300
+واحد فـ limit الـ sequence x<sub>n</sub> بتساوي صفر هذا هو الـ
+
+135
+00:13:01,300 --> 00:13:07,380
+ratio test برهان الـ test أو النظرية هذه موجود في
+
+136
+00:13:07,380 --> 00:13:11,680
+الكتاب نظرية 3.2.11 فحاسبكم تقرأوا
+
+137
+00:13:11,680 --> 00:13:15,780
+البرهان برهان سهل مش صعب بيعتمد على الحاجات اللي
+
+138
+00:13:15,780 --> 00:13:20,340
+أخذناها فعايزينكم 
+
+139
+00:13:20,340 --> 00:13:23,660
+تفتحوا الكتاب وتقرأوا برهان وتفهموا لحالكم بعد
+
+140
+00:13:23,660 --> 00:13:28,800
+ما أخذنا كل هالبرهين بدنا إياكم تعتمدوا عن أنفسكم
+
+141
+00:13:28,800 --> 00:13:33,440
+شوية تمام؟ واللي عنده أي صعوبة في فهم البرهان
+
+142
+00:13:33,440 --> 00:13:38,840
+يرجع له إذا هسا كم تخرق البرهان من الكتاب طيب نهار 
+
+143
+00:13:38,840 --> 00:13:42,540
+.. الآن الكتاب للأسف مش فيه أمثلة في الـ section هذا
+
+144
+00:13:42,540 --> 00:13:49,000
+تلاتة اتنين فهعطيلكم أس .. examples أو أمثلة بحال
+
+145
+00:13:49,000 --> 00:13:52,100
+من التمرين بحال بعض التمرين فأول مثل
+
+146
+00:13:58,060 --> 00:14:02,820
+فأول مثال هو exercise ثمانية عشر الفرع c في section
+
+147
+00:14:02,820 --> 00:14:06,700
+تلاتة اتنين أو صفحة ثمانية وستين في الكتاب المقرر
+
+148
+00:14:06,700 --> 00:14:10,300
+السؤال هذا بيقول discuss the convergence of the
+
+149
+00:14:10,300 --> 00:14:15,820
+sequence xn اللي لحد العام الـ nth term تبعها b to 
+
+150
+00:14:15,820 --> 00:14:20,600
+n على n factorial حيث بيه عدد حقيقي أكبر من واحد
+
+151
+00:14:21,470 --> 00:14:24,070
+Discuss the Convergence يعني بين هل الـ sequence
+
+152
+00:14:24,070 --> 00:14:27,850
+هذي Convergent ولا Divergent وده كانت Convergent
+
+153
+00:14:27,850 --> 00:14:35,790
+عايزين نجيب الـ limit تبعتها طيب تعالوا أول شي احنا 
+
+154
+00:14:35,790 --> 00:14:41,150
+طبعا هنطبق الـ ratio test نظرية 2.14 اللي هو الرسم
+
+155
+00:14:41,150 --> 00:14:45,490
+منها الـ ratio test لتطبيق الـ ratio test بلزمني
+
+156
+00:14:45,490 --> 00:14:50,690
+أتأكد ان الـ sequence xn حدودها موجبة وهذا صحيح لأن
+
+157
+00:14:50,690 --> 00:14:54,970
+الـ b اكبر من واحد و b أكبر من واحد و n 
+
+158
+00:14:54,970 --> 00:14:57,830
+factorial عدد موجب لأن هذه sequence of positive
+
+159
+00:14:57,830 --> 00:15:07,550
+real numbers الآن الـ ratio لـ xn زيادة واحد و xn هي
+
+160
+00:15:07,550 --> 00:15:12,230
+عندي xn زيادة واحد عوض عنها بدل n بـ n زيادة واحد
+
+161
+00:15:13,160 --> 00:15:18,740
+وضربها في مقلوب xn هي مقلوب xn وطبعا احنا عارفين
+
+162
+00:15:18,740 --> 00:15:25,460
+ان n plus one factorial بتساوي n plus one في n 
+
+163
+00:15:25,460 --> 00:15:31,900
+factorial هذا بنفك حاصل ضرب زي هذا n factorial
+
+164
+00:15:31,900 --> 00:15:37,640
+بتروح مع n factorial وb to n بتروح مع b to n بضل b
+
+165
+00:15:38,750 --> 00:15:43,210
+بعد الاختصارات والتبسيط الكاسر هذا بيطلع ب على n 
+
+166
+00:15:43,210 --> 00:15:47,870
+زيادة واحد الآن لما انت تقول لـ infinity ان زيادة واحد
+
+167
+00:15:47,870 --> 00:15:54,050
+بتقول لـ infinity مقلوبة بتروح لصفر ضرب ب عدد موجب
+
+168
+00:15:54,050 --> 00:15:58,990
+بتروح لصفر إذا limit ب على ان زيادة واحد بساوي ب
+
+169
+00:15:58,990 --> 00:16:03,290
+في limit واحد على ان زيادة واحد اللي هي صفر ب في
+
+170
+00:16:03,290 --> 00:16:10,590
+صفر بساوي صفر تمام؟ إذا أنا عندي L اللي هو بمثل
+
+171
+00:16:10,590 --> 00:16:17,570
+limit الـ ratio هذا طلع بساوي صفر عدد حقيقي أصغر من
+
+172
+00:16:17,570 --> 00:16:23,910
+واحد إذا حسب الـ ratio test limit للـ sequence xn 
+
+173
+00:16:23,910 --> 00:16:28,030
+بساوي صفر إذا هنا أثبتنا إن الـ sequence convergent
+
+174
+00:16:28,030 --> 00:16:34,010
+ونهيتها بتطلع بالساوي صفر تمام؟ واضح؟ إذا تطبيق
+
+175
+00:16:34,010 --> 00:16:35,510
+مباشر على الـ ratio test
+
+176
+00:16:38,490 --> 00:16:42,730
+مثال تاني مثال
+
+177
+00:16:42,730 --> 00:16:46,330
+تاني عبارة عن exercise اتنين فرع a section تلاتة 
+
+178
+00:16:46,330 --> 00:16:54,610
+اتنين بنشوف ايه الـ exercise هذا بيقول give
+
+179
+00:16:54,610 --> 00:17:01,930
+an example of two divergent sequences two
+
+180
+00:17:01,930 --> 00:17:04,090
+divergent sequences
+
+181
+00:17:06,940 --> 00:17:12,840
+such that there are some مجموعهم there
+
+182
+00:17:12,840 --> 00:17:19,020
+are some converges نعطي
+
+183
+00:17:19,020 --> 00:17:24,060
+مثال لـ two divergent sequences تنتهي from two
+
+184
+00:17:24,060 --> 00:17:29,140
+divergent لكن مجموعهم convergent فأسهل مثال هو مثل
+
+185
+00:17:29,140 --> 00:17:36,270
+هذا الحلناخد الـ sequence xn للحد العام تبعها سالب 
+
+186
+00:17:36,270 --> 00:17:42,430
+واحد to n و n بتبدأ من واحد إلى ما نهاية طبعا الـ
+
+187
+00:17:42,430 --> 00:17:48,210
+sequence هذه لو بينا انفرفتها فحدودها هتكون هكذا
+
+188
+00:17:48,210 --> 00:17:53,670
+أول حد سالب واحد، تاني واحد، تالت سالب واحد، 
+
+189
+00:17:53,670 --> 00:18:00,040
+الرابع واحد، وهكذا وناخد الـ sequence yn الحد العام
+
+190
+00:18:00,040 --> 00:18:04,760
+تبعها سالب واحد قص ان زيادة واحد وان طبعا تبدأ من 
+
+191
+00:18:04,760 --> 00:18:12,080
+واحد فهذه الـ sequence حدودها هتكون أول حد واحد،
+
+192
+00:18:12,080 --> 00:18:17,160
+التاني سالب واحد، التالت واحد، الرابع سالب واحد و
+
+193
+00:18:17,160 --> 00:18:17,620
+هكذا
+
+194
+00:18:20,300 --> 00:18:25,720
+تمام احنا اثبتنا بالتفصيل ان الـ sequence xn هذي
+
+195
+00:18:25,720 --> 00:18:29,660
+divergent by contradiction فرضنا انها convergent
+
+196
+00:18:29,660 --> 00:18:35,960
+وصلنا الى تناقض صح؟ طب ما هذي هي هذي هي الـ
+
+197
+00:18:35,960 --> 00:18:43,750
+sequence الـ sequence yn هي سالب الـ sequence xn و Xn
+
+198
+00:18:43,750 --> 00:18:47,710
+is divergent و Yn is divergent أو بنفس البرهان
+
+199
+00:18:47,710 --> 00:18:51,530
+ممكن نعمل نفس البرهان إذا هي عندي مثال على two
+
+200
+00:18:51,530 --> 00:18:57,670
+sequences كلاهما both are divergent لكن لما نيجي
+
+201
+00:18:57,670 --> 00:19:04,750
+نجمعهم لو أخدت الـ sequence جديدة الـ nth term تبعها
+
+202
+00:19:04,750 --> 00:19:09,070
+أو الحد العام تبعها هو مجموع الـ nth term زي Xn
+
+203
+00:19:09,070 --> 00:19:15,280
+وYn هذه sequence تالتة جديدة ما هو الحد العام لهذه
+
+204
+00:19:15,280 --> 00:19:21,360
+الـ sequence؟ اجمع الحد الأول على الأول بيطلع صفر، 
+
+205
+00:19:21,360 --> 00:19:25,740
+التاني على التاني صفر، إذا هذه عبارة عن الـ
+
+206
+00:19:25,740 --> 00:19:30,300
+sequence constant zero ثابت صفر أو الـ sequence
+
+207
+00:19:30,300 --> 00:19:35,480
+الحد العام تبعها ثابت صفر وطبعا أي sequence ثابتة
+
+208
+00:19:35,480 --> 00:19:39,880
+بتكون convergent و limit تبعتها هي الحد الثابت
+
+209
+00:19:39,880 --> 00:19:45,000
+نفسه، لذلك limit لهذه الـ sequence ثابت صفر إذا هذا 
+
+210
+00:19:45,000 --> 00:19:50,700
+مثال على two divergent sequences their sum is
+
+211
+00:19:50,700 --> 00:19:55,900
+convergent okay تمام؟ في برضه حاجات زي هذه ممكن 
+
+212
+00:19:55,900 --> 00:20:00,200
+ينقلب منكم جيبي مثال على two sequences contain
+
+213
+00:20:00,200 --> 00:20:05,820
+مثلا convergent لكن حصل ضربهم divergent يعني حاجات
+
+214
+00:20:05,820 --> 00:20:11,880
+زي هيك وهكذا في الكتاب في تمارين على هذا السياق
+
+215
+00:20:11,880 --> 00:20:22,020
+هتشوفوها تمام؟ مفهوم؟ واضح المثال هذا؟ طيب مثال 
+
+216
+00:20:22,020 --> 00:20:29,440
+رقم تلاتة هذا
+
+217
+00:20:29,440 --> 00:20:32,900
+عبارة عن exercise أربعة عشر في section تلاتة اتنين
+
+218
+00:20:35,100 --> 00:20:41,360
+بيقول خد zn بساوي a to n plus b to n to the power
+
+219
+00:20:41,360 --> 00:20:47,240
+one over n where a و b are positive numbers and a
+
+220
+00:20:47,240 --> 00:20:56,260
+less than b prove أن limit zn بساوي العدد b تمام؟
+
+221
+00:20:56,260 --> 00:21:02,420
+لبرهان ذلك أنا عندي من الفرض a positive إذا a to n 
+
+222
+00:21:02,420 --> 00:21:09,520
+positive وكذلك وبالتالي b to n أصغر من a to n plus
+
+223
+00:21:09,520 --> 00:21:16,460
+b to n الآن ناخد الـ nth root لطرفي المتباينة هذه
+
+224
+00:21:16,460 --> 00:21:22,700
+فبيطلع b أصغر من الـ nth root للمجموعة ده اللي احنا 
+
+225
+00:21:22,700 --> 00:21:33,440
+سمناه zn إذا الآن أنا عندي zn بساوي a n زائد b n to
+
+226
+00:21:33,440 --> 00:21:39,580
+the power one over n والآن أنا عندي بما انه a أصغر 
+
+227
+00:21:39,580 --> 00:21:45,740
+من b a أصغر من b من الفرض هي فهذا بالتأكيد بيقودى
+
+228
+00:21:45,740 --> 00:21:52,200
+انه a to n أصغر من b to n إذا 
+
+229
+00:21:52,200 --> 00:21:59,680
+هشيل الـ a to n هذه و أضع خليها أصغر من b to n زائد
+
+230
+00:21:59,680 --> 00:22:07,730
+b to n الكل to one over n طب هذا بيطلع two ضرب b to
+
+231
+00:22:07,730 --> 00:22:14,450
+n الكل to power one over n وزع الـ power فبيطلع two
+
+232
+00:22:14,450 --> 00:22:22,290
+to one over n ضرب b صح؟ الآن الـ sequence إذا 
+
+233
+00:22:22,290 --> 00:22:28,470
+أنا أصبح عندي لو دمجت المتباينتين عشرة و أحد عشر مع
+
+234
+00:22:28,470 --> 00:22:35,870
+بعض فبيطلع عندي b من المتباينة عشرة الـ B هذا هي
+
+235
+00:22:35,870 --> 00:22:42,590
+أصغر من الـ ZN ومن المتباينة أحد عشر الـ ZN أصغر من
+
+236
+00:22:42,590 --> 00:22:47,610
+two to one over N times B for every N natural
+
+237
+00:22:47,610 --> 00:22:56,780
+number احنا اتوصلنا لالمتباينة هذه صحيحة لكل N أنا 
+
+238
+00:22:56,780 --> 00:23:01,660
+لأن عندي الـ sequence ZN هذه اللي أنا عايز أثبت ان
+
+239
+00:23:01,660 --> 00:23:07,120
+الـ limit تبعتها بالساوي بيه is squeezed between
+
+240
+00:23:07,120 --> 00:23:13,680
+two sequences محصورة من متتاليتين تنتين هاي
+
+241
+00:23:13,680 --> 00:23:20,620
+متتالية وهاي متتالية المتتالية هذه الحد العام
+
+242
+00:23:20,620 --> 00:23:27,340
+تبعها ثابت بيه وبالتالي الـ limit تبعت ب لما ب
+
+243
+00:23:27,340 --> 00:23:35,340
+تقول لـ infinity بتساوي ب و limit الـ sequence هذي 
+
+244
+00:23:35,340 --> 00:23:39,380
+two to واحد على n limit two to واحد على n بتساوي 
+
+245
+00:23:39,380 --> 00:23:44,760
+واحد اثبتنا احنا قبل هيك ان لو n دي c عدد موجب ف
+
+246
+00:23:44,760 --> 00:23:52,170
+limit c to 1 على n as n tends to infinity بساوي
+
+247
+00:23:52,170 --> 00:23:59,230
+واحد صح فان دي c هنا بساوي اتنين لان الـ limit لـ two
+
+248
+00:23:59,230 --> 00:24:02,450
+to one over n as n tends to infinity بساوي واحد 
+
+249
+00:24:02,450 --> 00:24:07,290
+وبالتالي limit two to one over n times constant b 
+
+250
+00:24:07,290 --> 00:24:12,170
+بساوي واحد في b أو b في واحد ف limit الـ sequence
+
+251
+00:24:12,170 --> 00:24:18,000
+هذه ايضا تطلع b لما تنتقل لـ infinity، إذا by
+
+252
+00:24:18,000 --> 00:24:23,000
+squeeze theorem بيطلع عندي limit الـ sequence zm 
+
+253
+00:24:23,000 --> 00:24:28,240
+المحصورة في النص بساوي بيه، okay؟ إذا هاي هنا
+
+254
+00:24:28,240 --> 00:24:34,120
+استخدامنا الـ sandwich أو الـ squeeze، تمام؟ واضح؟
+
+255
+00:24:36,340 --> 00:24:40,080
+Okay إذا هذه يعني بعض الأسئلة هي اللي حلناها،
+
+256
+00:24:40,080 --> 00:24:43,480
+حالها مش صعب إما تطبيق على الـ sandwich theorem أو 
+
+257
+00:24:43,480 --> 00:24:48,680
+على نظرية 2.4 أو الحاجات اللي أخذناها في الـ
+
+258
+00:24:48,680 --> 00:24:52,740
+section هذا أو في الـ succession السابق أو بالتالي
+
+259
+00:24:52,740 --> 00:24:58,760
+مافيش حاجة يعني غريبة أو تستدعي ان احنا نستخدم حاجة
+
+260
+00:24:58,760 --> 00:25:05,270
+مش موجودة في المناهج إذا ما يكون إلا من شطارتكم 
+
+261
+00:25:05,270 --> 00:25:10,210
+تحاولوا تحلوا باقي التمرين اللي في الـ section هذا
+
+262
+00:25:10,210 --> 00:25:15,550
+طبعا هنا لهنا الامتحان .. الامتحان داخل لحد
+
+263
+00:25:15,550 --> 00:25:21,590
+التمرين هذه الجزء اللي بعد هيك مش داخل في الامتحان
+
+264
+00:25:21,590 --> 00:25:22,310
+النصف الأول
+
+265
+00:25:26,220 --> 00:25:32,640
+تمام فإذا هنا الـ section جديد أو عنوان جديد الـ
+
+266
+00:25:32,640 --> 00:25:38,160
+monotone sequences المتتاليات اللي بيسموها
+
+267
+00:25:38,160 --> 00:25:42,380
+الواتيرية المتتاليات الواتيرية الـ monotone
+
+268
+00:25:42,380 --> 00:25:46,960
+sequence يعني متتالية واتيرية يعني إما متزايدة أو
+
+269
+00:25:46,960 --> 00:25:55,200
+متناقصة فناخد تعريف let x in be a sequence of real
+
+270
+00:25:55,200 --> 00:26:02,880
+numbers سنقول إن سيكوينس Xn increasing متزايدة إذا
+
+271
+00:26:02,880 --> 00:26:07,400
+كان Xn less
+
+272
+00:26:07,400 --> 00:26:11,800
+than or equal to Xn plus one for every n لو كان كل 
+
+273
+00:26:11,800 --> 00:26:17,260
+حد أصغر من أو يساوي اللي بعده فالسيكوينس في الحالة دي 
+
+274
+00:26:17,260 --> 00:26:23,860
+بنسميها increasing و بنسميها decreasing إذا كان كل 
+
+275
+00:26:23,860 --> 00:26:32,760
+حد أكبر من أو يساوي اللي بعده تمام؟ 
+
+276
+00:26:32,760 --> 00:26:40,000
+طيب بنسمي الـ sequence monotone الـ sequence بنسميها
+
+277
+00:26:40,000 --> 00:26:45,460
+monotone أو واتيرية if it is either increasing or
+
+278
+00:26:45,460 --> 00:26:46,040
+decreasing
+
+279
+00:26:48,950 --> 00:26:53,170
+إن المتتالية الوطرية هي متتالية إما increasing أو
+
+280
+00:26:53,170 --> 00:26:58,250
+decreasing معنى 
+
+281
+00:26:58,250 --> 00:27:01,490
+Every increasing sequence is a monotone sequence
+
+282
+00:27:01,490 --> 00:27:06,090
+and every decreasing sequence is a monotone sequence
+
+283
+00:27:06,090 --> 00:27:14,370
+طب هاي أمثلة على monotone sequences فندّي هنا
+
+284
+00:27:14,370 --> 00:27:21,540
+The sequence of natural numbers is increasing واضح أن
+
+285
+00:27:21,540 --> 00:27:26,440
+xn = n  أصغر من أو يساوي xn+1 اللي هو n+1
+
+286
+00:27:26,440 --> 00:27:31,440
+زاد واحد لأن هذا increasing وهذا increasing ال
+
+287
+00:27:31,440 --> 00:27:36,040
+sequence اللي ال nth term تبعها 2 to the power n اللي هي
+
+288
+00:27:36,040 --> 00:27:41,440
+هذه is increasing بينما
+
+289
+00:27:41,440 --> 00:27:46,720
+ال sequence 1 over n decreasing هي كل حد أكبر من
+
+290
+00:27:46,720 --> 00:27:52,140
+أو يساوي للبعده وكذلك ال sequence 1 over 2 to the power n
+
+291
+00:27:52,140 --> 00:27:57,580
+طيب، في سؤال هنا بطرح نفسه، هل كل sequence لازم
+
+292
+00:27:57,580 --> 00:28:01,720
+تكون monotone sequence؟ لا، مو لا، مش شرط، مش شرط،
+
+293
+00:28:01,720 --> 00:28:03,620
+مش شرط، مش شرط، مش شرط، مش شرط، مش شرط، مش شرط، مش
+
+294
+00:28:03,620 --> 00:28:03,840
+شرط، مش شرط، مش شرط، مش شرط، مش شرط، مش شرط، مش
+
+295
+00:28:03,840 --> 00:28:07,680
+شرط، مش شرط، مش شرط، مش شرط، مش شرط، مش شرط، مش
+
+296
+00:28:07,680 --> 00:28:12,580
+شرط، مش شرط، مش شرط، مش شرط، مش شرط، مش شرط، مش
+
+297
+00:28:12,580 --> 00:28:17,150
+شرط، مش شرط، مش شرط The following sequence is
+
+298
+00:28:17,150 --> 00:28:20,010
+a sequence the nth term of which is (-1) to the power n
+
+299
+00:28:20,010 --> 00:28:24,830
+or n+1  which is the alternating sequence
+
+300
+00:28:24,830 --> 00:28:29,750
+ال sequence هذه المتدبدبة alternating يعني
+
+301
+00:28:29,750 --> 00:28:34,930
+المتدبدبة في الإشارة 1، -1، 1، -1 
+
+302
+00:28:34,930 --> 00:28:40,450
+هذه ليست convergent ليست monotone is not
+
+303
+00:28:40,450 --> 00:28:46,980
+increasing and it is not decreasing نفس الشيء ال
+
+304
+00:28:46,980 --> 00:28:51,180
+sequence اللي حدها nth  تبعها (-1) to the power n اللي
+
+305
+00:28:51,180 --> 00:28:56,040
+هي -1، 2، -3، وهكذا ال
+
+306
+00:28:56,040 --> 00:29:00,560
+sequence هذه is not monotone لا increasing ولا
+
+307
+00:29:00,560 --> 00:29:05,590
+decreasing تمام واضح إذا ال sequence .. أي .. لو
+
+308
+00:29:05,590 --> 00:29:09,310
+أخذنا أي sequence عشوائية فممكن تكون increasing،
+
+309
+00:29:09,310 --> 00:29:14,070
+ممكن تكون decreasing، ممكن تكون neither، neither
+
+310
+00:29:14,070 --> 00:29:16,810
+increasing nor decreasing زي ال function ممكن تكون
+
+311
+00:29:16,810 --> 00:29:22,950
+odd أو even أو neither، لا odd ولا even، أه؟ تمام؟
+
+312
+00:29:22,950 --> 00:29:26,870
+طيب ال .. في نظرية مهمة هنا في هذا السياق
+
+313
+00:29:29,970 --> 00:29:34,290
+بتخص الـ monotone sequences وبالتالي بنسميها
+
+314
+00:29:34,290 --> 00:29:39,170
+monotone convergence theorem وبنستخدم اختصارات
+
+315
+00:29:39,170 --> 00:29:46,690
+MCT (monotone convergence theorem) النظرية
+
+316
+00:29:46,690 --> 00:29:52,550
+هذه بتقول خذي let x be a monotone
+
+317
+00:29:52,550 --> 00:29:57,650
+sequence خلينا نأخذ monotone sequence الآن هذه الـ
+
+318
+00:29:57,650 --> 00:30:01,130
+monotone sequence بتكون convergent if and only if
+
+319
+00:30:01,130 --> 00:30:05,830
+it is bounded تمام؟
+
+320
+00:30:05,830 --> 00:30:10,470
+Moreover إضافة إلى ذلك لو كانت ال sequence x 
+
+321
+00:30:10,470 --> 00:30:16,970
+هذه bounded and increasing فأكيد طبعًا convergent و
+
+322
+00:30:16,970 --> 00:30:22,370
+ال limit تبعتها يساوي ال supremum لها ك set كذلك
+
+323
+00:30:22,370 --> 00:30:25,170
+لو كانت ال sequence x bounded و decreasing
+
+324
+00:30:27,590 --> 00:30:31,190
+فبتكون طبعًا convergent و ال limit بتبعتها يساوي ال
+
+325
+00:30:31,190 --> 00:30:36,270
+infimum لها ك set طيب
+
+326
+00:30:36,270 --> 00:30:39,430
+أنا عندي هنا two statements أو ثلاثة
+
+327
+00:30:39,430 --> 00:30:47,010
+statements أنا عندي العبارة هذه أنا
+
+328
+00:30:47,010 --> 00:30:53,490
+عندي بتثبت العبارة هذه و العبارتين هدول فكيف
+
+329
+00:30:53,490 --> 00:31:00,150
+البرهان بيتم؟ أوّل شيء العبارة الأولى اللي في البرواز
+
+330
+00:31:00,150 --> 00:31:08,610
+هذه if and only if statement صح ففي two parts واحد
+
+331
+00:31:08,610 --> 00:31:15,750
+هذا ال part only if part و ال if part نشوف ال
+
+332
+00:31:15,750 --> 00:31:21,260
+only if part يعني لو كانت x convergent بينا نثبت
+
+333
+00:31:21,260 --> 00:31:25,680
+إنها it is bounded وهذا أثبتناه في نظرية سابقة
+
+334
+00:31:25,680 --> 00:31:31,120
+أثبتنا إن كل sequence convergent is bounded اختبار
+
+335
+00:31:31,120 --> 00:31:41,320
+الدم فاكرين؟ إذا هذا was proved
+
+336
+00:31:41,320 --> 00:31:49,530
+earlier تم إثباته سابقا في نظرية سابقة لو كانت
+
+337
+00:31:49,530 --> 00:31:54,830
+السيكونس تبقى convergent ضروري تكون bounded سواء
+
+338
+00:31:54,830 --> 00:31:58,690
+كانت السيكونس monotone ولا حتى مش monotone okay؟
+
+339
+00:31:58,690 --> 00:32:02,950
+تمام؟ إن هاي برهان الجزء لهذا موجود في نظرية سابقة
+
+340
+00:32:02,950 --> 00:32:08,970
+باقي نثبت الجزء هذا يعني بنا نثبت أنه لو كانت
+
+341
+00:32:08,970 --> 00:32:17,730
+السيكونس bounded السيكونس لو كانت bounded و
+
+342
+00:32:17,730 --> 00:32:18,510
+monotone
+
+343
+00:32:21,420 --> 00:32:25,800
+طبعًا احنا فرضنا انها monotone اه من البداية x
+
+344
+00:32:25,800 --> 00:32:32,020
+is monotone فالآن عشان نكمل برهان العبارة هذه ال
+
+345
+00:32:32,020 --> 00:32:35,060
+if and only if او ال bi-conditional statement هذا
+
+346
+00:32:35,060 --> 00:32:40,920
+فبدنا نثبت أن لو كانت ال sequence bounded و
+
+347
+00:32:40,920 --> 00:32:49,520
+monotone فبتطلع convergent طيب
+
+348
+00:32:49,520 --> 00:32:54,920
+monotone مونوتون لما ال sequence تكون مونوتون
+
+349
+00:32:54,920 --> 00:33:04,060
+معناها إما increasing أو decreasing أو decreasing
+
+350
+00:33:04,060 --> 00:33:08,260
+إذا
+
+351
+00:33:08,260 --> 00:33:16,500
+عشان اثبت الجزء هذا بده اثبت a و b هذا الجزء هذا
+
+352
+00:33:16,500 --> 00:33:25,750
+لبرهانه بده برهين a و b لأن جزء A بيقول لو كانت ال
+
+353
+00:33:25,750 --> 00:33:29,330
+sequence bounded و increasing فبتثبت أنها
+
+354
+00:33:29,330 --> 00:33:33,510
+convergent صح؟ فهي لو كانت ال sequence bounded و
+
+355
+00:33:33,510 --> 00:33:37,930
+increasing فبتثبت أنها convergent و ال limit
+
+356
+00:33:37,930 --> 00:33:43,530
+تبعتها هي ال supremum لها كمجموعة و الجزء B
+
+357
+00:33:43,530 --> 00:33:47,690
+بيثبت أن لو كانت ال sequence bounded و decreasing
+
+358
+00:33:47,690 --> 00:33:54,510
+فبتطلع convergent وإضافة لذلك إن ال limit تبعتها هي
+
+359
+00:33:54,510 --> 00:34:00,390
+ال infimum لها كسب إذا إكمال برهان الاتجاه هذا و
+
+360
+00:34:00,390 --> 00:34:05,690
+برهان a و b وبالتالي نكمل برهان النظرية يكفي إن
+
+361
+00:34:05,690 --> 00:34:11,290
+احنا نثبت a و b  يكفي إن اثبتنا العبارة من 
+
+362
+00:34:11,290 --> 00:34:16,750
+بروزة هذه و a و b يعني برهاننا للنظرية كاملة تمام؟
+
+363
+00:34:17,990 --> 00:34:39,030
+نثبت الآن باقي إثبات a و b نثبت الجزء a فخلينا
+
+364
+00:34:39,030 --> 00:34:43,130
+نفرض أن ال sequence x is bounded قلنا bounded
+
+365
+00:34:43,130 --> 00:34:48,700
+و increasing طيب من تعريف الـ bounded sequence
+
+366
+00:34:48,700 --> 00:34:54,840
+مدام ال sequence bounded إذا يوجد عدد حقيقي موجب M
+
+367
+00:34:54,840 --> 00:35:03,840
+بحيث أن |Xn| أصغر من أو يساوي M لكل n طيب
+
+368
+00:35:03,840 --> 00:35:07,540
+معروف أن أي عدد حقيقي Xn أصغر من أو يساوي القيمة
+
+369
+00:35:07,540 --> 00:35:14,200
+المطلقة له، مظبوط؟ إذا من ال boundedness من فرض أن
+
+370
+00:35:14,200 --> 00:35:18,260
+ال sequence bounded في عدد موجود بحيث أن xn أصغر
+
+371
+00:35:18,260 --> 00:35:23,640
+من أو يساوي M لكل n تمام واضح طيب الآن إذا ال
+
+372
+00:35:23,640 --> 00:35:27,800
+sequence xn bounded above وبالتالي by supremum ال
+
+373
+00:35:27,800 --> 00:35:33,120
+property ال supremum تبعها exist سميّه x*
+
+374
+00:35:35,800 --> 00:35:40,000
+الآن بيدّثبت الادعاء هذا ال claim الادعاء بيدّثبت
+
+375
+00:35:40,000 --> 00:35:45,260
+أن limit ال sequence xn يساوي ال x* اللي هو
+
+376
+00:35:45,260 --> 00:35:51,580
+ال supremum لـ {xn} فلو أثبتت هذا الادعاء معناته
+
+377
+00:35:51,580 --> 00:35:55,600
+أثبتت أنا أن ال sequence xn is convergent و ال
+
+378
+00:35:55,600 --> 00:36:00,650
+limit تبعتها يساوي ال supremum لها كست تعالوا نشوف
+
+379
+00:36:00,650 --> 00:36:04,930
+كيف نثبت ال claim to see this لبرهان ال claim أنا
+
+380
+00:36:04,930 --> 00:36:09,430
+ايش بتثبت؟ بتثبت أن ال sequence xn convergent و
+
+381
+00:36:09,430 --> 00:36:13,630
+ال limit تبعتها يساوي العدد x* فهستخدم تعريف
+
+382
+00:36:13,630 --> 00:36:17,830
+epsilon N لل limit فلازم ابدأ let epsilon
+
+383
+00:36:17,830 --> 00:36:25,090
+أكبر من الصفر be given الآن ال x* هذاهو ال
+
+384
+00:36:25,090 --> 00:36:28,430
+supremum لل set هذه لما نطرح من ال supremum عدد
+
+385
+00:36:28,430 --> 00:36:33,830
+موجب بيصبح ليس upper bound  بيصبح ليس upper bound لأن ال x
+
+386
+00:36:33,830 --> 00:36:37,690
+* هو أصغر upper bound اطرح منه عدد موجب بيصبح ليس
+
+387
+00:36:37,690 --> 00:36:41,590
+upper bound إذا هذا العدد x* - y is not an
+
+388
+00:36:41,590 --> 00:36:46,710
+upper bound معناته في عنصر في ال set هذه اللي هو xn
+
+389
+00:36:46,710 --> 00:36:51,450
+برقم N أكبر من العدد هذا اللي هو ما هو
+
+390
+00:36:51,450 --> 00:36:55,860
+upper bound وطبعًا العدد هذا المؤشر أو ال index
+
+391
+00:36:55,860 --> 00:37:00,040
+N ده يعتمد على ال epsilon مرتبط بال
+
+392
+00:37:00,040 --> 00:37:05,500
+epsilon اللي بنيت فيه طبعًا أنا فرضت أن ال sequence
+
+393
+00:37:05,500 --> 00:37:10,860
+xn increasing وبالتالي xn أصغر من أو يساوي xn
+
+394
+00:37:10,860 --> 00:37:14,880
+لكل n أكبر من أو يساوي N من تعريف ال
+
+395
+00:37:14,880 --> 00:37:20,500
+increasing sequence إذا أنا في عندي هنا هي عندي x
+
+396
+00:37:20,500 --> 00:37:28,280
+N هي xn أصغر من أو يساوي xn لكل n أكبر من أو
+
+397
+00:37:28,280 --> 00:37:36,360
+يساوي N طيب و x* هو ال supremum of ال sequence xn و
+
+398
+00:37:36,360 --> 00:37:42,440
+xn هذا عنصر في ال sequence و x* upper bound لل
+
+399
+00:37:42,440 --> 00:37:49,540
+sequence إذن xn أصغر من أو يساوي x* طيب و x* أصغر
+
+400
+00:37:49,540 --> 00:37:57,820
+من x* + y هذا مافي شك من هنا .. أيوه
+
+401
+00:37:57,820 --> 00:38:03,460
+.. من المتباينة هذه هي عندي xn أكبر من x
+
+402
+00:38:03,460 --> 00:38:11,420
+* - y إذا أنا طلع عندي الآن x* أكبر من
+
+403
+00:38:11,420 --> 00:38:13,160
+.. أو xn
+
+404
+00:38:15,810 --> 00:38:25,070
+أكبر من x* - y أصغر من x* + y لكل n
+
+405
+00:38:25,070 --> 00:38:30,910
+أكبر من أو يساوي N فظبطت صح؟ طيب مهاد
+
+406
+00:38:30,910 --> 00:38:37,890
+المتباينة هي نفسها xn - x* أصغر من y أكبر
+
+407
+00:38:37,890 --> 00:38:44,610
+من -y لكل n أكبر من أو يساوي N طب
+
+408
+00:38:44,610 --> 00:38:49,930
+المتباينة هذه هي .. صح؟ أظبط؟ إذن |xn
+
+409
+00:38:49,930 --> 00:38:53,210
+- x*| أصغر من epsilon لكل n أكبر من أو
+
+410
+00:38:53,210 --> 00:38:58,370
+يساوي N الآن since epsilon was arbitrary هذا
+
+411
+00:38:58,370 --> 00:39:03,810
+بالضبط تعريف epsilon N لل limit أه؟ بأن هذا
+
+412
+00:39:03,810 --> 00:39:08,470
+الكلام صحيح لكل epsilon أكبر من صفر إذن هذا معناه
+
+413
+00:39:08,470 --> 00:39:13,190
+حسب التعريف أن limit xn يساوي x*
+
+414
+00:39:18,780 --> 00:39:23,660
+إذا هذا بيثبت ال claim وبالتالي هكذا نكون أثبتنا
+
+415
+00:39:23,660 --> 00:39:30,560
+الجزء A من النظرية فالجزء
+
+416
+00:39:30,560 --> 00:39:35,300
+الثاني B ممكن نستخدم A في برهان ال B
+
+417
+00:39:38,510 --> 00:39:42,310
+ففي الجزء B الآن أنا عندي ال sequence تبعتي
+
+418
+00:39:42,310 --> 00:39:46,570
+bounded و decreasing إذا I assume xn is bounded
+
+419
+00:39:46,570 --> 00:39:50,770
+and decreasing فأيش 
+
+420
+00:39:50,770 --> 00:39:55,690
+عمل هعرف sequence جديدة yn اللي هي negative الحد
+
+421
+00:39:55,690 --> 00:40:01,530
+العام تبعها negative x in تمام؟ الآن بما أن x in
+
+422
+00:40:01,530 --> 00:40:05,170
+decreasing إذا الـ sequence سالب x in تطلع 
+
+423
+00:40:05,170 --> 00:40:10,610
+increasing وطبعا بما أن الـ sequence x in bounded
+
+424
+00:40:10,610 --> 00:40:15,670
+إذا الـ sequence سالب x in أيضا bounded إذا الآن
+
+425
+00:40:15,670 --> 00:40:18,790
+أنا في عندي sequence جديد اللي هي sequence yn
+
+426
+00:40:18,790 --> 00:40:26,310
+bounded و in crazy إذا حسب الجزء a by part a limit
+
+427
+00:40:26,310 --> 00:40:32,790
+الـ sequence yn تطلع exist و بتساوي الـ supremum لكل
+
+428
+00:40:32,790 --> 00:40:37,870
+الـ y in الـ supremum لعناصر الـ sequence اللي هي y
+
+429
+00:40:37,870 --> 00:40:41,510
+in تمام؟
+
+430
+00:40:41,510 --> 00:40:47,370
+إنها ده من إيه؟ من الجزء إيه من النظرية؟ طيب الـ
+
+431
+00:40:47,370 --> 00:40:51,450
+supremum لـ سالب xn هيفوق  العدد الطبيعي احنا خدنا قبل
+
+432
+00:40:51,450 --> 00:40:56,490
+هيك exercise بيقول supremum أو infimum سالب حاجة 
+
+433
+00:40:56,490 --> 00:41:02,190
+بساوي سالب الـ infimum فهنا بصير هذا سالب الـ
+
+434
+00:41:02,190 --> 00:41:07,530
+infimum تمام؟ إذا أنا عندي بيطلع عندي limit xn
+
+435
+00:41:07,530 --> 00:41:15,180
+بساوي سالب limit سالب xn تمام؟ أضربوا هنا هيندي
+
+436
+00:41:15,180 --> 00:41:18,940
+limit سالب xn أضربوا المعادلة هذه بالسالب واحد
+
+437
+00:41:18,940 --> 00:41:24,700
+فبطلع سالب limit سالب xn بيساوي سالب سالب موجب اللي
+
+438
+00:41:24,700 --> 00:41:29,000
+هو الـ infimum لـ xn وهذا اللي بدنا يعني لأن هي
+
+439
+00:41:29,000 --> 00:41:33,280
+أثبتنا أن limit xn موجودة exist يعني الـ sequence
+
+440
+00:41:33,280 --> 00:41:37,640
+xn convergent والـ limit تبعتها بتساوي الـ infimum
+
+441
+00:41:40,760 --> 00:41:44,680
+بنكمل برهان الـ monotone convergence theorem طبعا
+
+442
+00:41:44,680 --> 00:41:49,280
+الأمثلة هذه اللي هنا كلها أمثلة تطبيق على الـ
+
+443
+00:41:49,280 --> 00:41:53,180
+monotone convergence theorem فأرجو أنكم تحاولوا
+
+444
+00:41:53,180 --> 00:41:56,080
+تخرجوا الأمثلة هذه و تشوفوا كيف نستخدم الـ
+
+445
+00:41:56,080 --> 00:41:58,440
+monotone convergence theorem في 
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/Mk2487qcZF8_raw.json b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/Mk2487qcZF8_raw.json
new file mode 100644
index 0000000000000000000000000000000000000000..a6a99ca74c19567d388df6c9fc66723a0992c00e
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/Mk2487qcZF8_raw.json
@@ -0,0 +1 @@
+{"segments": [{"id": 1, "seek": 4923, "start": 21.45, "end": 49.23, "text": "Okay إذا احنا في المحاضرة الأخيرة أخدنا الـ monotone convergence theorem وشوفنا أنه في النظرية هذه لو في عندي sequence و ال sequence هذه monotone يعني increasing أو decreasing فعشان فبتكون convergent if and only if it is bounded", "tokens": [8297, 11933, 15730, 1975, 5016, 8315, 8978, 9673, 5016, 46958, 25720, 16247, 9778, 48923, 5551, 9778, 3215, 8315, 2423, 39184, 1108, 310, 546, 32181, 20904, 4032, 8592, 38688, 8315, 14739, 3224, 8978, 28239, 19913, 2288, 10632, 29538, 45164, 8978, 18871, 16254, 8310, 4032, 2423, 8310, 29538, 1108, 310, 546, 37495, 22653, 5662, 34051, 23223, 6156, 3615, 8592, 7649, 6156, 3555, 2655, 30544, 9652, 6930, 498, 293, 787, 498, 309, 307, 37498], "avg_logprob": -0.2000868007954624, "compression_ratio": 1.56, "no_speech_prob": 5.960464477539063e-08, "words": [{"start": 21.45, "end": 21.89, "word": "Okay", "probability": 0.06500244140625}, {"start": 21.89, "end": 22.23, "word": " إذا", "probability": 0.691162109375}, {"start": 22.23, "end": 22.87, "word": " احنا", "probability": 0.7845052083333334}, {"start": 22.87, "end": 23.17, "word": " في", "probability": 0.9482421875}, {"start": 23.17, "end": 24.23, "word": " المحاضرة", "probability": 0.97412109375}, {"start": 24.23, "end": 25.05, "word": " الأخيرة", "probability": 0.97314453125}, {"start": 25.05, "end": 27.55, "word": " أخدنا", "probability": 0.91162109375}, {"start": 27.55, "end": 27.95, "word": " الـ", "probability": 0.848876953125}, {"start": 27.95, "end": 28.39, "word": " monotone", "probability": 0.7667643229166666}, {"start": 28.39, "end": 29.15, "word": " convergence", "probability": 0.9697265625}, {"start": 29.15, "end": 29.75, "word": " theorem", "probability": 0.89990234375}, {"start": 29.75, "end": 33.67, "word": " وشوفنا", "probability": 0.8851318359375}, {"start": 33.67, "end": 34.57, "word": " أنه", "probability": 0.668701171875}, {"start": 34.57, "end": 35.15, "word": " في", "probability": 0.75}, {"start": 35.15, "end": 35.67, "word": " النظرية", "probability": 0.897705078125}, {"start": 35.67, "end": 36.15, "word": " هذه", "probability": 0.91455078125}, {"start": 36.15, "end": 37.27, "word": " لو", "probability": 0.72265625}, {"start": 37.27, "end": 37.49, "word": " في", "probability": 0.8447265625}, {"start": 37.49, "end": 37.91, "word": " عندي", "probability": 0.77587890625}, {"start": 37.91, "end": 38.67, "word": " sequence", "probability": 0.88330078125}, {"start": 38.67, "end": 38.87, "word": " و", "probability": 0.81640625}, {"start": 38.87, "end": 38.97, "word": " ال", "probability": 0.469482421875}, {"start": 38.97, "end": 39.29, "word": " sequence", "probability": 0.84521484375}, {"start": 39.29, "end": 39.61, "word": " هذه", "probability": 0.68017578125}, {"start": 39.61, "end": 40.29, "word": " monotone", "probability": 0.88818359375}, {"start": 40.29, "end": 40.69, "word": " يعني", "probability": 0.91455078125}, {"start": 40.69, "end": 41.35, "word": " increasing", "probability": 0.87890625}, {"start": 41.35, "end": 42.71, "word": " أو", "probability": 0.8310546875}, {"start": 42.71, "end": 43.31, "word": " decreasing", "probability": 0.98486328125}, {"start": 43.31, "end": 45.81, "word": " فعشان", "probability": 0.89111328125}, {"start": 45.81, "end": 46.63, "word": " فبتكون", "probability": 0.841796875}, {"start": 46.63, "end": 47.47, "word": " convergent", "probability": 0.843994140625}, {"start": 47.47, "end": 47.75, "word": " if", "probability": 0.94384765625}, {"start": 47.75, "end": 48.01, "word": " and", "probability": 0.94140625}, {"start": 48.01, "end": 48.23, "word": " only", "probability": 0.92578125}, {"start": 48.23, "end": 48.51, "word": " if", "probability": 0.97265625}, {"start": 48.51, "end": 48.69, "word": " it", "probability": 0.9697265625}, {"start": 48.69, "end": 48.89, "word": " is", "probability": 0.94091796875}, {"start": 48.89, "end": 49.23, "word": " bounded", "probability": 0.912109375}], "temperature": 1.0}, {"id": 2, "seek": 7931, "start": 50.42, "end": 79.32, "text": "إذا الـ monotone sequence converges if and only if it is bounded طيب ال monotone sequence نوعين إما increasing أو decreasing فلو كانت ال sequence increasing و طبعا bounded فشوفنا إنها بتكون convergent حسب ال statement الأول و ال limit تبعتها بساوي ال supremum اللي لها ك set و لو كانت ال sequence decreasing و بالطبع bounded", "tokens": [28814, 15730, 2423, 39184, 1108, 310, 546, 8310, 9652, 2880, 498, 293, 787, 498, 309, 307, 37498, 23032, 1829, 3555, 2423, 1108, 310, 546, 8310, 8717, 45367, 9957, 11933, 15042, 5662, 34051, 23223, 6156, 1211, 2407, 25961, 2655, 2423, 8310, 5662, 4032, 23032, 3555, 3615, 995, 37498, 6156, 8592, 38688, 8315, 36145, 11296, 39894, 30544, 9652, 6930, 11331, 35457, 2423, 5629, 16247, 12610, 4032, 2423, 4948, 6055, 3555, 34268, 11296, 4724, 3794, 995, 45865, 2423, 23710, 449, 13672, 1829, 5296, 11296, 9122, 992, 4032, 45164, 25961, 2655, 2423, 8310, 23223, 4032, 20666, 9566, 3555, 3615, 37498], "avg_logprob": -0.18814433481275422, "compression_ratio": 1.781512605042017, "no_speech_prob": 0.0, "words": [{"start": 50.42, "end": 50.84, "word": "إذا", "probability": 0.75390625}, {"start": 50.84, "end": 51.06, "word": " الـ", "probability": 0.4573974609375}, {"start": 51.06, "end": 51.6, "word": " monotone", "probability": 0.8772786458333334}, {"start": 51.6, "end": 52.04, "word": " sequence", "probability": 0.96875}, {"start": 52.04, "end": 53.06, "word": " converges", "probability": 0.863037109375}, {"start": 53.06, "end": 53.32, "word": " if", "probability": 0.888671875}, {"start": 53.32, "end": 53.56, "word": " and", "probability": 0.91357421875}, {"start": 53.56, "end": 53.78, "word": " only", "probability": 0.92236328125}, {"start": 53.78, "end": 54.06, "word": " if", "probability": 0.978515625}, {"start": 54.06, "end": 54.24, "word": " it", "probability": 0.93359375}, {"start": 54.24, "end": 54.42, "word": " is", "probability": 0.896484375}, {"start": 54.42, "end": 54.76, "word": " bounded", "probability": 0.927734375}, {"start": 54.76, "end": 61.36, "word": " طيب", "probability": 0.826171875}, {"start": 61.36, "end": 61.64, "word": " ال", "probability": 0.286865234375}, {"start": 61.64, "end": 62.16, "word": " monotone", "probability": 0.833984375}, {"start": 62.16, "end": 62.66, "word": " sequence", "probability": 0.94384765625}, {"start": 62.66, "end": 63.22, "word": " نوعين", "probability": 0.7548828125}, {"start": 63.22, "end": 63.6, "word": " إما", "probability": 0.81982421875}, {"start": 63.6, "end": 64.12, "word": " increasing", "probability": 0.95166015625}, {"start": 64.12, "end": 64.42, "word": " أو", "probability": 0.89501953125}, {"start": 64.42, "end": 64.92, "word": " decreasing", "probability": 0.98193359375}, {"start": 64.92, "end": 65.3, "word": " فلو", "probability": 0.9601236979166666}, {"start": 65.3, "end": 65.62, "word": " كانت", "probability": 0.991455078125}, {"start": 65.62, "end": 65.74, "word": " ال", "probability": 0.93505859375}, {"start": 65.74, "end": 66.04, "word": " sequence", "probability": 0.95458984375}, {"start": 66.04, "end": 66.78, "word": " increasing", "probability": 0.97412109375}, {"start": 66.78, "end": 67.02, "word": " و", "probability": 0.93994140625}, {"start": 67.02, "end": 67.36, "word": " طبعا", "probability": 0.82879638671875}, {"start": 67.36, "end": 67.86, "word": " bounded", "probability": 0.85791015625}, {"start": 67.86, "end": 69.26, "word": " فشوفنا", "probability": 0.9527587890625}, {"start": 69.26, "end": 69.48, "word": " إنها", "probability": 0.7327880859375}, {"start": 69.48, "end": 69.76, "word": " بتكون", "probability": 0.903564453125}, {"start": 69.76, "end": 70.42, "word": " convergent", "probability": 0.951171875}, {"start": 70.42, "end": 70.78, "word": " حسب", "probability": 0.976318359375}, {"start": 70.78, "end": 70.94, "word": " ال", "probability": 0.923828125}, {"start": 70.94, "end": 71.28, "word": " statement", "probability": 0.9453125}, {"start": 71.28, "end": 71.94, "word": " الأول", "probability": 0.99462890625}, {"start": 71.94, "end": 72.7, "word": " و", "probability": 0.7880859375}, {"start": 72.7, "end": 72.8, "word": " ال", "probability": 0.64892578125}, {"start": 72.8, "end": 72.98, "word": " limit", "probability": 0.9423828125}, {"start": 72.98, "end": 73.5, "word": " تبعتها", "probability": 0.745849609375}, {"start": 73.5, "end": 73.88, "word": " بساوي", "probability": 0.7847900390625}, {"start": 73.88, "end": 74.04, "word": " ال", "probability": 0.74462890625}, {"start": 74.04, "end": 74.5, "word": " supremum", "probability": 0.72998046875}, {"start": 74.5, "end": 74.7, "word": " اللي", "probability": 0.59552001953125}, {"start": 74.7, "end": 74.78, "word": " لها", "probability": 0.7166748046875}, {"start": 74.78, "end": 75.0, "word": " ك", "probability": 0.8662109375}, {"start": 75.0, "end": 75.28, "word": " set", "probability": 0.71533203125}, {"start": 75.28, "end": 76.3, "word": " و", "probability": 0.79541015625}, {"start": 76.3, "end": 76.48, "word": " لو", "probability": 0.9677734375}, {"start": 76.48, "end": 76.8, "word": " كانت", "probability": 0.996337890625}, {"start": 76.8, "end": 76.92, "word": " ال", "probability": 0.97509765625}, {"start": 76.92, "end": 77.28, "word": " sequence", "probability": 0.99462890625}, {"start": 77.28, "end": 77.8, "word": " decreasing", "probability": 0.96484375}, {"start": 77.8, "end": 78.08, "word": " و", "probability": 0.9189453125}, {"start": 78.08, "end": 78.58, "word": " بالطبع", "probability": 0.9739990234375}, {"start": 78.58, "end": 79.32, "word": " bounded", "probability": 0.9462890625}], "temperature": 1.0}, {"id": 3, "seek": 9556, "start": 80.76, "end": 95.56, "text": "فحسب ال statement الأول تطلع convergence ونهايتها هي ال infimum تبعها ك Z وشوفنا برهانة مغرية في المحاضرة السابقة", "tokens": [5172, 5016, 35457, 2423, 5629, 16247, 12610, 6055, 9566, 1211, 3615, 32181, 4032, 1863, 11296, 36081, 11296, 39896, 2423, 1536, 332, 449, 6055, 3555, 3615, 11296, 9122, 1176, 4032, 8592, 38688, 8315, 4724, 2288, 3224, 7649, 3660, 3714, 17082, 2288, 10632, 8978, 9673, 5016, 46958, 25720, 21136, 16758, 28671], "avg_logprob": -0.2507812362909317, "compression_ratio": 1.2765957446808511, "no_speech_prob": 0.0, "words": [{"start": 80.76, "end": 81.34, "word": "فحسب", "probability": 0.6822916666666666}, {"start": 81.34, "end": 81.54, "word": " ال", "probability": 0.7001953125}, {"start": 81.54, "end": 81.86, "word": " statement", "probability": 0.53857421875}, {"start": 81.86, "end": 82.42, "word": " الأول", "probability": 0.96826171875}, {"start": 82.42, "end": 82.84, "word": " تطلع", "probability": 0.83935546875}, {"start": 82.84, "end": 83.34, "word": " convergence", "probability": 0.1868896484375}, {"start": 83.34, "end": 84.5, "word": " ونهايتها", "probability": 0.8267578125}, {"start": 84.5, "end": 86.34, "word": " هي", "probability": 0.6064453125}, {"start": 86.34, "end": 86.52, "word": " ال", "probability": 0.96044921875}, {"start": 86.52, "end": 87.04, "word": " infimum", "probability": 0.6484375}, {"start": 87.04, "end": 88.22, "word": " تبعها", "probability": 0.9390869140625}, {"start": 88.22, "end": 88.46, "word": " ك", "probability": 0.8310546875}, {"start": 88.46, "end": 88.72, "word": " Z", "probability": 0.25830078125}, {"start": 88.72, "end": 93.04, "word": " وشوفنا", "probability": 0.8980712890625}, {"start": 93.04, "end": 93.46, "word": " برهانة", "probability": 0.92578125}, {"start": 93.46, "end": 93.94, "word": " مغرية", "probability": 0.7799072265625}, {"start": 93.94, "end": 94.26, "word": " في", "probability": 0.96630859375}, {"start": 94.26, "end": 94.94, "word": " المحاضرة", "probability": 0.97900390625}, {"start": 94.94, "end": 95.56, "word": " السابقة", "probability": 0.9900716145833334}], "temperature": 1.0}, {"id": 4, "seek": 12772, "start": 98.06, "end": 127.72, "text": "الان بنشوف كيف نطبق النظرية في إثبات إن certain sequences are convergent أو divergent النظرية هذه بالمناسبة ممكن نستخدمها في إثبات إنه monotone sequence معينة إما convergent أو divergent عشان أثبت إن ال monotone sequence إذا عندي أنا monotone sequence عشان أثبت إنها convergent لازم أثبت إنها bounded العكس لو في عنده monotone sequence", "tokens": [6027, 7649, 4724, 1863, 8592, 38688, 9122, 33911, 8717, 9566, 3555, 4587, 28239, 19913, 2288, 10632, 8978, 11933, 12984, 3555, 9307, 36145, 1629, 22978, 366, 9652, 6930, 34051, 18558, 6930, 28239, 19913, 2288, 10632, 29538, 20666, 2304, 8315, 35457, 3660, 3714, 43020, 8717, 14851, 9778, 40448, 11296, 8978, 11933, 12984, 3555, 9307, 36145, 3224, 1108, 310, 546, 8310, 20449, 9957, 3660, 11933, 15042, 9652, 6930, 34051, 18558, 6930, 6225, 8592, 7649, 5551, 12984, 3555, 2655, 36145, 2423, 1108, 310, 546, 8310, 11933, 15730, 18871, 16254, 41850, 1108, 310, 546, 8310, 6225, 8592, 7649, 5551, 12984, 3555, 2655, 36145, 11296, 9652, 6930, 5296, 31377, 2304, 5551, 12984, 3555, 2655, 36145, 11296, 37498, 18863, 4117, 3794, 45164, 8978, 43242, 3224, 1108, 310, 546, 8310], "avg_logprob": -0.16628557225553, "compression_ratio": 2.047008547008547, "no_speech_prob": 0.0, "words": [{"start": 98.06, "end": 98.44, "word": "الان", "probability": 0.694091796875}, {"start": 98.44, "end": 99.06, "word": " بنشوف", "probability": 0.68597412109375}, {"start": 99.06, "end": 99.3, "word": " كيف", "probability": 0.9677734375}, {"start": 99.3, "end": 99.78, "word": " نطبق", "probability": 0.963623046875}, {"start": 99.78, "end": 100.4, "word": " النظرية", "probability": 0.9268798828125}, {"start": 100.4, "end": 100.78, "word": " في", "probability": 0.8623046875}, {"start": 100.78, "end": 102.38, "word": " إثبات", "probability": 0.858154296875}, {"start": 102.38, "end": 102.56, "word": " إن", "probability": 0.2415771484375}, {"start": 102.56, "end": 103.06, "word": " certain", "probability": 0.44921875}, {"start": 103.06, "end": 104.04, "word": " sequences", "probability": 0.916015625}, {"start": 104.04, "end": 104.38, "word": " are", "probability": 0.912109375}, {"start": 104.38, "end": 105.1, "word": " convergent", "probability": 0.962890625}, {"start": 105.1, "end": 105.36, "word": " أو", "probability": 0.9150390625}, {"start": 105.36, "end": 106.06, "word": " divergent", "probability": 0.6607666015625}, {"start": 106.06, "end": 107.02, "word": " النظرية", "probability": 0.8824462890625}, {"start": 107.02, "end": 107.2, "word": " هذه", "probability": 0.857421875}, {"start": 107.2, "end": 107.72, "word": " بالمناسبة", "probability": 0.95390625}, {"start": 107.72, "end": 107.9, "word": " ممكن", "probability": 0.91845703125}, {"start": 107.9, "end": 108.56, "word": " نستخدمها", "probability": 0.961328125}, {"start": 108.56, "end": 109.14, "word": " في", "probability": 0.84326171875}, {"start": 109.14, "end": 109.56, "word": " إثبات", "probability": 0.9832763671875}, {"start": 109.56, "end": 110.74, "word": " إنه", "probability": 0.626708984375}, {"start": 110.74, "end": 111.18, "word": " monotone", "probability": 0.6890055338541666}, {"start": 111.18, "end": 111.68, "word": " sequence", "probability": 0.96484375}, {"start": 111.68, "end": 112.38, "word": " معينة", "probability": 0.9724934895833334}, {"start": 112.38, "end": 112.96, "word": " إما", "probability": 0.854248046875}, {"start": 112.96, "end": 113.86, "word": " convergent", "probability": 0.905517578125}, {"start": 113.86, "end": 114.12, "word": " أو", "probability": 0.9609375}, {"start": 114.12, "end": 114.9, "word": " divergent", "probability": 0.882080078125}, {"start": 114.9, "end": 115.96, "word": " عشان", "probability": 0.9669596354166666}, {"start": 115.96, "end": 116.3, "word": " أثبت", "probability": 0.9798583984375}, {"start": 116.3, "end": 116.44, "word": " إن", "probability": 0.8359375}, {"start": 116.44, "end": 116.58, "word": " ال", "probability": 0.51904296875}, {"start": 116.58, "end": 117.08, "word": " monotone", "probability": 0.8369140625}, {"start": 117.08, "end": 117.7, "word": " sequence", "probability": 0.96875}, {"start": 117.7, "end": 118.22, "word": " إذا", "probability": 0.6368408203125}, {"start": 118.22, "end": 118.52, "word": " عندي", "probability": 0.907958984375}, {"start": 118.52, "end": 118.68, "word": " أنا", "probability": 0.578125}, {"start": 118.68, "end": 119.12, "word": " monotone", "probability": 0.9168294270833334}, {"start": 119.12, "end": 119.72, "word": " sequence", "probability": 0.97021484375}, {"start": 119.72, "end": 120.74, "word": " عشان", "probability": 0.9931640625}, {"start": 120.74, "end": 121.04, "word": " أثبت", "probability": 0.99560546875}, {"start": 121.04, "end": 121.2, "word": " إنها", "probability": 0.764404296875}, {"start": 121.2, "end": 121.82, "word": " convergent", "probability": 0.92724609375}, {"start": 121.82, "end": 122.22, "word": " لازم", "probability": 0.9796549479166666}, {"start": 122.22, "end": 122.56, "word": " أثبت", "probability": 0.9927978515625}, {"start": 122.56, "end": 122.76, "word": " إنها", "probability": 0.885498046875}, {"start": 122.76, "end": 123.14, "word": " bounded", "probability": 0.931640625}, {"start": 123.14, "end": 125.72, "word": " العكس", "probability": 0.9435221354166666}, {"start": 125.72, "end": 126.24, "word": " لو", "probability": 0.63916015625}, {"start": 126.24, "end": 126.42, "word": " في", "probability": 0.3935546875}, {"start": 126.42, "end": 126.66, "word": " عنده", "probability": 0.777099609375}, {"start": 126.66, "end": 127.1, "word": " monotone", "probability": 0.9860026041666666}, {"start": 127.1, "end": 127.72, "word": " sequence", "probability": 0.9208984375}], "temperature": 1.0}, {"id": 5, "seek": 15308, "start": 128.52, "end": 153.08, "text": "وبدي اثبت انها divergent يكفي ان اثبت انها unbounded not bounded فهي ان ال sequence xn بساوي واحد على n هاد ال sequence معروف انه ال limit انها convergent وits limit is zero زيها زي ال sequence واحد على n", "tokens": [37746, 16254, 1975, 12984, 3555, 2655, 16472, 11296, 18558, 6930, 7251, 4117, 41185, 16472, 1975, 12984, 3555, 2655, 16472, 11296, 517, 18767, 292, 406, 37498, 6156, 3224, 1829, 16472, 2423, 8310, 2031, 77, 4724, 3794, 995, 45865, 36764, 24401, 15844, 297, 8032, 18513, 2423, 8310, 20449, 32887, 5172, 16472, 3224, 2423, 4948, 16472, 11296, 9652, 6930, 4032, 1208, 4948, 307, 4018, 30767, 1829, 11296, 30767, 1829, 2423, 8310, 36764, 24401, 15844, 297], "avg_logprob": -0.2735445205479452, "compression_ratio": 1.763975155279503, "no_speech_prob": 0.0, "words": [{"start": 128.52, "end": 128.98, "word": "وبدي", "probability": 0.21173095703125}, {"start": 128.98, "end": 129.32, "word": " اثبت", "probability": 0.85076904296875}, {"start": 129.32, "end": 129.5, "word": " انها", "probability": 0.841796875}, {"start": 129.5, "end": 130.4, "word": " divergent", "probability": 0.61590576171875}, {"start": 130.4, "end": 132.8, "word": " يكفي", "probability": 0.8284505208333334}, {"start": 132.8, "end": 133.04, "word": " ان", "probability": 0.82958984375}, {"start": 133.04, "end": 133.52, "word": " اثبت", "probability": 0.977783203125}, {"start": 133.52, "end": 133.82, "word": " انها", "probability": 0.96240234375}, {"start": 133.82, "end": 134.64, "word": " unbounded", "probability": 0.826171875}, {"start": 134.64, "end": 134.94, "word": " not", "probability": 0.311767578125}, {"start": 134.94, "end": 135.46, "word": " bounded", "probability": 0.92041015625}, {"start": 135.46, "end": 138.3, "word": " فهي", "probability": 0.8038736979166666}, {"start": 138.3, "end": 138.5, "word": " ان", "probability": 0.440185546875}, {"start": 138.5, "end": 138.64, "word": " ال", "probability": 0.59423828125}, {"start": 138.64, "end": 139.26, "word": " sequence", "probability": 0.91845703125}, {"start": 139.26, "end": 140.18, "word": " xn", "probability": 0.513427734375}, {"start": 140.18, "end": 140.84, "word": " بساوي", "probability": 0.6241455078125}, {"start": 140.84, "end": 141.34, "word": " واحد", "probability": 0.863525390625}, {"start": 141.34, "end": 141.54, "word": " على", "probability": 0.74609375}, {"start": 141.54, "end": 141.84, "word": " n", "probability": 0.662109375}, {"start": 141.84, "end": 142.6, "word": " هاد", "probability": 0.439453125}, {"start": 142.6, "end": 142.72, "word": " ال", "probability": 0.97314453125}, {"start": 142.72, "end": 143.14, "word": " sequence", "probability": 0.96337890625}, {"start": 143.14, "end": 143.82, "word": " معروف", "probability": 0.9894205729166666}, {"start": 143.82, "end": 145.16, "word": " انه", "probability": 0.75537109375}, {"start": 145.16, "end": 145.38, "word": " ال", "probability": 0.8447265625}, {"start": 145.38, "end": 145.76, "word": " limit", "probability": 0.95654296875}, {"start": 145.76, "end": 146.5, "word": " انها", "probability": 0.908203125}, {"start": 146.5, "end": 147.24, "word": " convergent", "probability": 0.78515625}, {"start": 147.24, "end": 148.32, "word": " وits", "probability": 0.5908203125}, {"start": 148.32, "end": 148.7, "word": " limit", "probability": 0.96240234375}, {"start": 148.7, "end": 148.94, "word": " is", "probability": 0.5830078125}, {"start": 148.94, "end": 149.32, "word": " zero", "probability": 0.8076171875}, {"start": 149.32, "end": 151.18, "word": " زيها", "probability": 0.8816731770833334}, {"start": 151.18, "end": 151.54, "word": " زي", "probability": 0.984619140625}, {"start": 151.54, "end": 151.82, "word": " ال", "probability": 0.91162109375}, {"start": 151.82, "end": 152.26, "word": " sequence", "probability": 0.9873046875}, {"start": 152.26, "end": 152.66, "word": " واحد", "probability": 0.95654296875}, {"start": 152.66, "end": 152.82, "word": " على", "probability": 0.86767578125}, {"start": 152.82, "end": 153.08, "word": " n", "probability": 0.7958984375}], "temperature": 1.0}, {"id": 6, "seek": 18107, "start": 154.35, "end": 181.07, "text": "و ممكن تبرهن أن الـ sequence هذه convergence و نهايتها بالساعة و سفر باستخدام تعريف epsilon capital N لل limit بالظبط زي ما عملنا في برهان ال limit أن ال limit لل sequence واحد علي N 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"text": "اللي هو باستخدام تعريف epsilon capital N زائد الـarchimedean property بالظبط زي ما أثبتنا limit 1 على N بساوية 0 اليوم هنشوف برهان تاني باستخدام الـ monotone convergence theorem", "tokens": [6027, 20292, 31439, 4724, 995, 14851, 9778, 3215, 10943, 37279, 16572, 5172, 17889, 4238, 426, 30767, 16373, 3215, 2423, 39184, 1178, 332, 4858, 282, 4707, 20666, 19913, 3555, 9566, 30767, 1829, 19446, 5551, 12984, 3555, 2655, 8315, 4948, 502, 15844, 426, 4724, 3794, 995, 2407, 10632, 1958, 45595, 20498, 8032, 1863, 8592, 38688, 4724, 2288, 3224, 7649, 6055, 7649, 1829, 4724, 995, 14851, 9778, 3215, 10943, 2423, 39184, 1108, 310, 546, 32181, 20904], "avg_logprob": -0.2125211126498274, "compression_ratio": 1.3367875647668395, "no_speech_prob": 0.0, "words": [{"start": 181.71, "end": 182.05, "word": "اللي", "probability": 0.60791015625}, {"start": 182.05, "end": 182.29, "word": " هو", "probability": 0.953125}, {"start": 182.29, "end": 183.09, "word": " باستخدام", "probability": 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10, "seek": 26774, "start": 247.0, "end": 267.74, "text": "decreasing و bounded اذا by monotone convergence theorem ال sequence هذه هتكون convergent و ال limit تبعتها بساوي ال infimum طيب ال infimum للمجموعة هذه بتساوي سفر", "tokens": [1479, 14066, 3349, 4032, 37498, 1975, 15730, 538, 1108, 310, 546, 32181, 20904, 2423, 8310, 29538, 8032, 2655, 30544, 9652, 6930, 4032, 2423, 4948, 6055, 3555, 34268, 11296, 4724, 3794, 995, 45865, 2423, 1536, 332, 449, 23032, 1829, 3555, 2423, 1536, 332, 449, 5296, 19528, 7435, 2304, 2407, 27884, 29538, 39894, 3794, 995, 45865, 8608, 5172, 2288], "avg_logprob": -0.27774783968925476, "compression_ratio": 1.3885350318471337, "no_speech_prob": 0.0, "words": [{"start": 247.0, "end": 247.76, "word": "decreasing", "probability": 0.6033528645833334}, {"start": 247.76, "end": 249.66, "word": " و", "probability": 0.31103515625}, {"start": 249.66, "end": 250.26, "word": " bounded", "probability": 0.5849609375}, {"start": 250.26, "end": 251.16, "word": " اذا", "probability": 0.622802734375}, {"start": 251.16, "end": 251.46, "word": " by", "probability": 0.705078125}, {"start": 251.46, "end": 252.0, "word": " monotone", "probability": 0.715576171875}, {"start": 252.0, "end": 252.68, "word": " convergence", "probability": 0.93408203125}, {"start": 252.68, "end": 253.26, "word": " theorem", "probability": 0.89990234375}, {"start": 253.26, "end": 254.66, "word": " ال", "probability": 0.50439453125}, {"start": 254.66, "end": 255.26, "word": " sequence", "probability": 0.7255859375}, {"start": 255.26, "end": 255.82, "word": " هذه", "probability": 0.399169921875}, {"start": 255.82, "end": 256.62, "word": " هتكون", "probability": 0.96533203125}, {"start": 256.62, "end": 257.6, "word": " convergent", "probability": 0.784912109375}, {"start": 257.6, "end": 258.38, "word": " و", "probability": 0.61572265625}, {"start": 258.38, "end": 258.48, "word": " ال", "probability": 0.74755859375}, {"start": 258.48, "end": 258.72, "word": " limit", "probability": 0.96533203125}, {"start": 258.72, "end": 259.46, "word": " تبعتها", "probability": 0.904296875}, {"start": 259.46, "end": 260.42, "word": " بساوي", "probability": 0.71148681640625}, {"start": 260.42, "end": 260.58, "word": " ال", "probability": 0.85205078125}, {"start": 260.58, "end": 261.44, "word": " infimum", "probability": 0.7430013020833334}, {"start": 261.44, "end": 263.22, "word": " طيب", "probability": 0.8663736979166666}, {"start": 263.22, "end": 263.4, "word": " ال", "probability": 0.5068359375}, {"start": 263.4, "end": 263.92, "word": " infimum", "probability": 0.91748046875}, {"start": 263.92, "end": 264.66, "word": " للمجموعة", "probability": 0.8916829427083334}, {"start": 264.66, "end": 265.2, "word": " هذه", "probability": 0.92041015625}, {"start": 265.2, "end": 267.22, "word": " بتساوي", "probability": 0.963623046875}, {"start": 267.22, "end": 267.74, "word": " سفر", "probability": 0.819091796875}], "temperature": 1.0}, {"id": 11, "seek": 29039, "start": 270.57, "end": 290.39, "text": "وبرهان ذلك شبيه ببرهان الـ infimum للـ sequence 1 على n بالساوي 0 باستخدام الـ Archimedean property راجعوا برهان أن الـ infimum للـ sequence 1 على n وهذا ثمتناه في المحاضرات السابقة راجعوا البرهان وكتبوا برهان مشابه له", "tokens": [2407, 26890, 3224, 7649, 29910, 23275, 13412, 21292, 3224, 4724, 26890, 3224, 7649, 2423, 39184, 1536, 332, 449, 24976, 39184, 8310, 502, 15844, 297, 20666, 3794, 995, 45865, 1958, 4724, 995, 14851, 9778, 3215, 10943, 2423, 39184, 10984, 332, 4858, 282, 4707, 12602, 26108, 3615, 14407, 4724, 2288, 3224, 7649, 14739, 2423, 39184, 1536, 332, 449, 24976, 39184, 8310, 502, 15844, 297, 37037, 15730, 38637, 2304, 2655, 8315, 3224, 8978, 9673, 5016, 46958, 2288, 9307, 21136, 16758, 28671, 12602, 26108, 3615, 14407, 2423, 26890, 3224, 7649, 4032, 4117, 2655, 3555, 14407, 4724, 2288, 3224, 7649, 37893, 16758, 3224, 46740], "avg_logprob": -0.18640624433755876, "compression_ratio": 1.7692307692307692, "no_speech_prob": 0.0, "words": [{"start": 270.57, "end": 271.43, "word": "وبرهان", "probability": 0.822021484375}, {"start": 271.43, "end": 272.03, "word": " ذلك", "probability": 0.97607421875}, {"start": 272.03, "end": 272.81, "word": " شبيه", "probability": 0.8512369791666666}, {"start": 272.81, "end": 273.39, "word": " ببرهان", "probability": 0.950439453125}, {"start": 273.39, "end": 273.59, "word": " الـ", "probability": 0.52789306640625}, {"start": 273.59, "end": 274.19, "word": " infimum", "probability": 0.9007161458333334}, {"start": 274.19, "end": 274.91, "word": " للـ", "probability": 0.5147705078125}, {"start": 274.91, "end": 275.31, "word": " sequence", "probability": 0.8818359375}, {"start": 275.31, "end": 275.71, "word": " 1", "probability": 0.5888671875}, {"start": 275.71, "end": 275.89, "word": " على", "probability": 0.7470703125}, {"start": 275.89, "end": 276.11, "word": " n", "probability": 0.62939453125}, {"start": 276.11, "end": 276.73, "word": " بالساوي", "probability": 0.8131103515625}, {"start": 276.73, "end": 277.17, "word": " 0", "probability": 0.53173828125}, {"start": 277.17, "end": 278.87, "word": " باستخدام", "probability": 0.9698079427083334}, {"start": 278.87, "end": 279.09, "word": " الـ", "probability": 0.636474609375}, {"start": 279.09, "end": 279.73, "word": " Archimedean", "probability": 0.8460693359375}, {"start": 279.73, "end": 280.31, "word": " property", "probability": 0.7080078125}, {"start": 280.31, "end": 280.87, "word": " راجعوا", "probability": 0.9375}, {"start": 280.87, "end": 281.37, "word": " برهان", "probability": 0.9468994140625}, {"start": 281.37, "end": 282.51, "word": " أن", "probability": 0.305419921875}, {"start": 282.51, "end": 282.77, "word": " الـ", "probability": 0.637451171875}, {"start": 282.77, "end": 283.11, "word": " infimum", "probability": 0.9783528645833334}, {"start": 283.11, "end": 283.39, "word": " للـ", "probability": 0.642333984375}, {"start": 283.39, "end": 283.65, "word": " sequence", "probability": 0.97998046875}, {"start": 283.65, "end": 284.01, "word": " 1", "probability": 0.93505859375}, {"start": 284.01, "end": 284.21, "word": " على", "probability": 0.9140625}, {"start": 284.21, "end": 284.35, "word": " n", "probability": 0.904296875}, {"start": 284.35, "end": 284.61, "word": " وهذا", "probability": 0.596435546875}, {"start": 284.61, "end": 285.11, "word": " ثمتناه", "probability": 0.8171875}, {"start": 285.11, "end": 285.33, "word": " في", "probability": 0.97314453125}, {"start": 285.33, "end": 286.07, "word": " المحاضرات", "probability": 0.95341796875}, {"start": 286.07, "end": 286.65, "word": " السابقة", "probability": 0.9951171875}, {"start": 286.65, "end": 288.17, "word": " راجعوا", "probability": 0.90966796875}, {"start": 288.17, "end": 288.61, "word": " البرهان", "probability": 0.9346923828125}, {"start": 288.61, "end": 289.09, "word": " وكتبوا", "probability": 0.84765625}, {"start": 289.09, "end": 289.49, "word": " برهان", "probability": 0.991943359375}, {"start": 289.49, "end": 290.09, "word": " مشابه", "probability": 0.9876302083333334}, {"start": 290.09, "end": 290.39, "word": " له", "probability": 0.896484375}], "temperature": 1.0}, {"id": 12, "seek": 31256, "start": 292.46, "end": 312.56, "text": "بنفس الطريقة نثبت ان الانثرام لسيكوانس هادى او الست هادى سفر اذا حسب الـ monotone convergence theorem ال sequence واحد على جذر M is convergent و ال limit تبعتها بساوي الانثرام تبعها اللى هو سفر اذا هى مثال على تطبيق الـ monotone convergence theorem", "tokens": [3555, 1863, 36178, 41950, 16572, 28671, 8717, 12984, 3555, 2655, 16472, 2423, 7649, 12984, 2288, 10943, 5296, 3794, 1829, 4117, 2407, 7649, 3794, 8032, 18513, 7578, 1975, 2407, 2423, 14851, 8032, 18513, 7578, 8608, 5172, 2288, 1975, 15730, 11331, 35457, 2423, 39184, 1108, 310, 546, 32181, 20904, 2423, 8310, 36764, 24401, 15844, 10874, 8848, 2288, 376, 307, 9652, 6930, 4032, 2423, 4948, 6055, 3555, 34268, 11296, 4724, 3794, 995, 45865, 2423, 7649, 12984, 2288, 10943, 6055, 3555, 3615, 11296, 13672, 7578, 31439, 8608, 5172, 2288, 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"word": " هادى", "probability": 0.9498697916666666}, {"start": 296.9, "end": 297.22, "word": " سفر", "probability": 0.7529296875}, {"start": 297.22, "end": 298.02, "word": " اذا", "probability": 0.5670166015625}, {"start": 298.02, "end": 298.44, "word": " حسب", "probability": 0.920166015625}, {"start": 298.44, "end": 298.66, "word": " الـ", "probability": 0.469879150390625}, {"start": 298.66, "end": 298.9, "word": " monotone", "probability": 0.6710611979166666}, {"start": 298.9, "end": 299.48, "word": " convergence", "probability": 0.962890625}, {"start": 299.48, "end": 299.88, "word": " theorem", "probability": 0.77880859375}, {"start": 299.88, "end": 300.22, "word": " ال", "probability": 0.55810546875}, {"start": 300.22, "end": 300.72, "word": " sequence", "probability": 0.5419921875}, {"start": 300.72, "end": 301.24, "word": " واحد", "probability": 0.86376953125}, {"start": 301.24, "end": 301.4, "word": " على", "probability": 0.60009765625}, {"start": 301.4, "end": 301.78, "word": " جذر", "probability": 0.7099609375}, {"start": 301.78, "end": 302.02, "word": " M", "probability": 0.36083984375}, {"start": 302.02, "end": 302.66, "word": " is", "probability": 0.06561279296875}, {"start": 302.66, "end": 303.4, "word": " convergent", "probability": 0.95068359375}, {"start": 303.4, "end": 304.16, "word": " و", "probability": 0.640625}, {"start": 304.16, "end": 304.52, "word": " ال", "probability": 0.2069091796875}, {"start": 304.52, "end": 304.76, "word": " limit", "probability": 0.8720703125}, {"start": 304.76, "end": 305.22, "word": " تبعتها", "probability": 0.7550048828125}, {"start": 305.22, "end": 305.7, "word": " بساوي", "probability": 0.675048828125}, {"start": 305.7, "end": 306.22, "word": " الانثرام", "probability": 0.736669921875}, {"start": 306.22, "end": 306.68, "word": " تبعها", "probability": 0.70684814453125}, {"start": 306.68, "end": 306.84, "word": " اللى", "probability": 0.708984375}, {"start": 306.84, "end": 307.06, "word": " هو", "probability": 0.94091796875}, {"start": 307.06, "end": 307.56, "word": " سفر", "probability": 0.9596354166666666}, {"start": 307.56, "end": 308.76, "word": " اذا", "probability": 0.6424560546875}, {"start": 308.76, "end": 308.94, "word": " هى", "probability": 0.434326171875}, {"start": 308.94, "end": 309.38, "word": " مثال", "probability": 0.926025390625}, {"start": 309.38, "end": 309.58, "word": " على", "probability": 0.84521484375}, {"start": 309.58, "end": 310.24, "word": " تطبيق", "probability": 0.9871826171875}, {"start": 310.24, "end": 311.22, "word": " الـ", "probability": 0.706787109375}, {"start": 311.22, "end": 311.52, "word": " monotone", "probability": 0.9449869791666666}, {"start": 311.52, "end": 312.1, "word": " convergence", "probability": 0.9619140625}, {"start": 312.1, "end": 312.56, "word": " theorem", "probability": 0.91796875}], "temperature": 1.0}, {"id": 13, "seek": 34081, "start": 314.19, "end": 340.81, "text": "كذلك ممكن برضه زي ما قلتلكم نستخدم الـ monotone convergence theorem في أثبات أن سيكوانس زي هذه is divergent نشوف مع بعض كيف .. إذا هنا هاي سيكوانس الحد العام تبعها xn هذا الint partial sum بالمناسبة هذا الint partial sum في ال harmonic series", "tokens": [4117, 8848, 23275, 3714, 43020, 4724, 43042, 3224, 30767, 1829, 19446, 12174, 1211, 2655, 23275, 2304, 8717, 14851, 9778, 40448, 2423, 39184, 1108, 310, 546, 32181, 20904, 8978, 5551, 12984, 3555, 9307, 14739, 8608, 1829, 4117, 2407, 7649, 3794, 30767, 1829, 29538, 307, 18558, 6930, 8717, 8592, 38688, 20449, 45030, 11242, 9122, 33911, 4386, 11933, 15730, 34105, 8032, 47302, 8608, 1829, 4117, 2407, 7649, 3794, 21542, 3215, 18863, 10943, 6055, 3555, 3615, 11296, 2031, 77, 23758, 2423, 686, 14641, 2408, 20666, 2304, 8315, 35457, 3660, 23758, 2423, 686, 14641, 2408, 8978, 2423, 32270, 2638], "avg_logprob": -0.2294407888462669, "compression_ratio": 1.5541125541125542, "no_speech_prob": 0.0, "words": [{"start": 314.19, "end": 314.73, "word": "كذلك", "probability": 0.87158203125}, {"start": 314.73, 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{"start": 518.78, "end": 519.84, "word": " الحدود", "probability": 0.9560546875}, {"start": 519.84, "end": 520.1, "word": " هذه", "probability": 0.468994140625}, {"start": 520.1, "end": 520.36, "word": " كلها", "probability": 0.975830078125}, {"start": 520.36, "end": 520.84, "word": " موجودة", "probability": 0.9814453125}, {"start": 520.84, "end": 521.1, "word": " هنا", "probability": 0.79345703125}, {"start": 521.1, "end": 522.86, "word": " صح؟", "probability": 0.8885091145833334}, {"start": 522.86, "end": 523.2, "word": " فهذه", "probability": 0.9485677083333334}, {"start": 523.2, "end": 523.8, "word": " بنسميها", "probability": 0.8287353515625}, {"start": 523.8, "end": 524.64, "word": " subsequence", "probability": 0.81396484375}, {"start": 524.64, "end": 524.84, "word": " من", "probability": 0.98388671875}, {"start": 524.84, "end": 524.98, "word": " ال", "probability": 0.90966796875}, {"start": 524.98, "end": 525.34, "word": " sequence", "probability": 0.98291015625}, {"start": 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546.29, "end": 546.47, "word": " عند", "probability": 0.94970703125}, {"start": 546.47, "end": 546.95, "word": " المجموعة", "probability": 0.96748046875}, {"start": 546.95, "end": 547.41, "word": " واحد", "probability": 0.956787109375}, {"start": 547.41, "end": 547.65, "word": " زاد", "probability": 0.974365234375}, {"start": 547.65, "end": 547.99, "word": " نص", "probability": 0.9931640625}, {"start": 547.99, "end": 548.29, "word": " زاد", "probability": 0.9716796875}, {"start": 548.29, "end": 548.75, "word": " تلت", "probability": 0.9933268229166666}, {"start": 548.75, "end": 549.43, "word": " الى", "probability": 0.70849609375}, {"start": 549.43, "end": 549.95, "word": " اخر", "probability": 0.962158203125}, {"start": 549.95, "end": 550.25, "word": " حد", "probability": 0.99609375}, {"start": 550.25, "end": 550.65, "word": " واحد", "probability": 0.97509765625}, {"start": 550.65, "end": 550.81, "word": " على", "probability": 0.78369140625}, {"start": 550.81, "end": 551.13, "word": " 2", "probability": 0.8779296875}, {"start": 551.13, "end": 551.41, "word": " أس", "probability": 0.9677734375}, {"start": 551.41, "end": 551.63, "word": " n", "probability": 0.767578125}, {"start": 551.63, "end": 552.43, "word": " صح؟", "probability": 0.8004557291666666}, {"start": 552.43, "end": 552.67, "word": " هذا", "probability": 0.75}, {"start": 552.67, "end": 552.85, "word": " من", "probability": 0.99267578125}, {"start": 552.85, "end": 553.25, "word": " تعريف", "probability": 0.98828125}, {"start": 553.25, "end": 553.39, "word": " ال", "probability": 0.978515625}, {"start": 553.39, "end": 553.71, "word": " sequence", "probability": 0.9833984375}], "temperature": 1.0}, {"id": 22, "seek": 58154, "start": 555.68, "end": 581.54, "text": "الان الحدود هذه تبع المجموع هعملها blocks، بلوكات فواحد لوحده في block، نص لوحده، هاخد التلت والربع مع بعض، بعدين ال block الرابع هتكون خمس و سدس و سبعة و تمان، أربع حدود مع بعض، جمهم مع بعض", "tokens": [6027, 7649, 21542, 3215, 23328, 29538, 6055, 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{"start": 577.62, "end": 577.94, "word": " سبعة", "probability": 0.75927734375}, {"start": 577.94, "end": 578.08, "word": " و", "probability": 0.97119140625}, {"start": 578.08, "end": 578.46, "word": " تمان،", "probability": 0.7298177083333334}, {"start": 578.46, "end": 578.76, "word": " أربع", "probability": 0.7057291666666666}, {"start": 578.76, "end": 579.16, "word": " حدود", "probability": 0.98486328125}, {"start": 579.16, "end": 579.38, "word": " مع", "probability": 0.77294921875}, {"start": 579.38, "end": 579.98, "word": " بعض،", "probability": 0.8727213541666666}, {"start": 579.98, "end": 580.94, "word": " جمهم", "probability": 0.5633951822916666}, {"start": 580.94, "end": 581.18, "word": " مع", "probability": 0.98388671875}, {"start": 581.18, "end": 581.54, "word": " بعض", "probability": 0.99658203125}], "temperature": 1.0}, {"id": 23, "seek": 60958, "start": 581.98, "end": 609.58, "text": "و هكذا إلى ال block الأخيرة اللي هي 1 على 2 أس N سالب 1 زاد 1 إلى 1 على 2 أس N طيب هاي 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"probability": 0.95703125}, {"start": 604.72, "end": 604.94, "word": " هذه", "probability": 0.31494140625}, {"start": 604.94, "end": 605.34, "word": " صارت", "probability": 0.89794921875}, {"start": 605.34, "end": 606.04, "word": " أكبر،", "probability": 0.9229736328125}, {"start": 606.04, "end": 606.24, "word": " يعني", "probability": 0.960693359375}, {"start": 606.24, "end": 606.82, "word": " بدلت", "probability": 0.8938802083333334}, {"start": 606.82, "end": 607.24, "word": " التلت", "probability": 0.9427083333333334}, {"start": 607.24, "end": 608.24, "word": " بربع،", "probability": 0.84814453125}, {"start": 608.24, "end": 608.74, "word": " والتلت", "probability": 0.8902587890625}, {"start": 608.74, "end": 609.1, "word": " أكبر", "probability": 0.9855143229166666}, {"start": 609.1, "end": 609.28, "word": " من", "probability": 0.99755859375}, {"start": 609.28, "end": 609.58, "word": " ربع", "probability": 0.9757486979166666}], "temperature": 1.0}, {"id": 24, "seek": 63621, "start": 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13924, 655], "avg_logprob": -0.20861816068645567, "compression_ratio": 1.9588477366255144, "no_speech_prob": 0.0, "words": [{"start": 686.79, "end": 687.13, "word": "in", "probability": 0.364013671875}, {"start": 687.13, "end": 687.31, "word": " من", "probability": 0.76708984375}, {"start": 687.31, "end": 687.89, "word": " الحدود", "probability": 0.9539388020833334}, {"start": 687.89, "end": 688.33, "word": " وهذا", "probability": 0.80029296875}, {"start": 688.33, "end": 688.69, "word": " وعد", "probability": 0.30950927734375}, {"start": 688.69, "end": 688.97, "word": " هاي", "probability": 0.275390625}, {"start": 688.97, "end": 689.13, "word": " in", "probability": 0.5986328125}, {"start": 689.13, "end": 689.47, "word": " زاد", "probability": 0.65673828125}, {"start": 689.47, "end": 689.95, "word": " واحد", "probability": 0.894287109375}, {"start": 689.95, "end": 690.65, "word": " من", "probability": 0.89111328125}, {"start": 690.65, "end": 691.11, "word": " الحدود", "probability": 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"end": 696.57, "word": " بيطلع", "probability": 0.777392578125}, {"start": 696.57, "end": 696.89, "word": " in", "probability": 0.83837890625}, {"start": 696.89, "end": 697.13, "word": " في", "probability": 0.56103515625}, {"start": 697.13, "end": 697.43, "word": " نص", "probability": 0.9599609375}, {"start": 697.43, "end": 697.67, "word": " اللي", "probability": 0.950927734375}, {"start": 697.67, "end": 697.81, "word": " هو", "probability": 0.94189453125}, {"start": 697.81, "end": 698.01, "word": " in", "probability": 0.6708984375}, {"start": 698.01, "end": 698.41, "word": " عتنين", "probability": 0.68914794921875}, {"start": 698.41, "end": 698.81, "word": " زاد", "probability": 0.90576171875}, {"start": 698.81, "end": 699.15, "word": " واحد", "probability": 0.966796875}, {"start": 699.15, "end": 700.13, "word": " طيب", "probability": 0.8974609375}, {"start": 700.13, "end": 700.41, "word": " لما", "probability": 0.9404296875}, {"start": 700.41, "end": 700.65, "word": " in", "probability": 0.467529296875}, {"start": 700.65, "end": 700.93, "word": " تقول", "probability": 0.907958984375}, {"start": 700.93, "end": 701.07, "word": " ل", "probability": 0.705078125}, {"start": 701.07, "end": 701.59, "word": " infinity", "probability": 0.50732421875}, {"start": 701.59, "end": 701.91, "word": " in", "probability": 0.6240234375}, {"start": 701.91, "end": 702.35, "word": " عتنين", "probability": 0.9537353515625}, {"start": 702.35, "end": 702.65, "word": " يقول", "probability": 0.964599609375}, {"start": 702.65, "end": 702.77, "word": " ل", "probability": 0.72509765625}, {"start": 702.77, "end": 703.23, "word": " infinity", "probability": 0.83349609375}, {"start": 703.23, "end": 703.91, "word": " وبالتالي", "probability": 0.91455078125}, {"start": 703.91, "end": 704.41, "word": " واحد", "probability": 0.8623046875}, {"start": 704.41, "end": 705.37, "word": " زاد", "probability": 0.972900390625}, {"start": 705.37, "end": 705.55, "word": " in", "probability": 0.9580078125}, {"start": 705.55, "end": 706.01, "word": " عتنين", "probability": 0.953125}, {"start": 706.01, "end": 706.43, "word": " بيروح", "probability": 0.9300537109375}, {"start": 706.43, "end": 706.57, "word": " ل", "probability": 0.91552734375}, {"start": 706.57, "end": 707.07, "word": " infinity", "probability": 0.79296875}, {"start": 707.07, "end": 710.41, "word": " تمام؟", "probability": 0.7335611979166666}, {"start": 710.41, "end": 710.61, "word": " إذا", "probability": 0.7236328125}, {"start": 710.61, "end": 710.83, "word": " أنا", "probability": 0.69189453125}, {"start": 710.83, "end": 711.25, "word": " عندي", "probability": 0.85595703125}, {"start": 711.25, "end": 711.87, "word": " الحد", "probability": 0.9921875}, {"start": 711.87, "end": 712.29, "word": " العام", "probability": 0.986328125}, {"start": 712.29, "end": 712.51, "word": " لل", "probability": 0.7998046875}, {"start": 712.51, "end": 712.97, "word": " sequence", "probability": 0.94384765625}, {"start": 712.97, "end": 713.47, "word": " هذه", "probability": 0.537109375}, {"start": 713.47, "end": 714.55, "word": " أو", "probability": 0.8955078125}, {"start": 714.55, "end": 714.69, "word": " ال", "probability": 0.85595703125}, {"start": 714.69, "end": 715.51, "word": " subsequence", "probability": 0.867919921875}], "temperature": 1.0}, {"id": 28, "seek": 74253, "start": 716.65, "end": 742.53, "text": "طولة أكبر من مقدار يقول لـ infinity هو بالتالي إذا أنا عندي x to two to n tends to infinity as n tends to infinity", "tokens": [9566, 12610, 3660, 5551, 4117, 26890, 9154, 3714, 28543, 9640, 7251, 39648, 5296, 39184, 13202, 31439, 20666, 2655, 6027, 1829, 11933, 15730, 41850, 18871, 16254, 2031, 281, 732, 281, 297, 12258, 281, 13202, 382, 297, 12258, 281, 13202], "avg_logprob": -0.34014423993917614, "compression_ratio": 1.3275862068965518, "no_speech_prob": 0.0, "words": [{"start": 716.65, "end": 717.23, "word": "طولة", "probability": 0.5087890625}, {"start": 717.23, "end": 718.49, "word": " أكبر", "probability": 0.8896484375}, {"start": 718.49, "end": 718.95, "word": " من", "probability": 0.9794921875}, {"start": 718.95, "end": 721.33, "word": " مقدار", "probability": 0.8907877604166666}, {"start": 721.33, "end": 721.75, "word": " يقول", "probability": 0.864990234375}, {"start": 721.75, "end": 722.01, "word": " لـ", "probability": 0.3397216796875}, {"start": 722.01, "end": 722.33, "word": " infinity", "probability": 0.39599609375}, {"start": 722.33, "end": 722.73, "word": " هو", "probability": 0.400634765625}, {"start": 722.73, "end": 725.39, "word": " بالتالي", "probability": 0.974365234375}, {"start": 725.39, "end": 734.51, "word": " إذا", "probability": 0.6229248046875}, {"start": 734.51, "end": 734.85, "word": " أنا", "probability": 0.429443359375}, {"start": 734.85, "end": 735.37, "word": " عندي", "probability": 0.88427734375}, {"start": 735.37, "end": 736.41, "word": " x", "probability": 0.58544921875}, {"start": 736.41, "end": 737.07, "word": " to", "probability": 0.236328125}, {"start": 737.07, "end": 737.35, "word": " two", "probability": 0.61572265625}, {"start": 737.35, "end": 737.57, "word": " to", "probability": 0.92919921875}, {"start": 737.57, "end": 737.99, "word": " n", "probability": 0.68505859375}, {"start": 737.99, "end": 739.27, "word": " tends", "probability": 0.7978515625}, {"start": 739.27, "end": 739.53, "word": " to", "probability": 0.96142578125}, {"start": 739.53, "end": 740.15, "word": " infinity", "probability": 0.90380859375}, {"start": 740.15, "end": 741.07, "word": " as", "probability": 0.85400390625}, {"start": 741.07, "end": 741.51, "word": " n", "probability": 0.86279296875}, {"start": 741.51, "end": 741.89, "word": " tends", "probability": 0.80322265625}, {"start": 741.89, "end": 742.03, "word": " to", "probability": 0.95654296875}, {"start": 742.03, "end": 742.53, "word": " infinity", "probability": 0.89111328125}], "temperature": 1.0}, {"id": 29, "seek": 77315, "start": 744.05, "end": 773.15, "text": "وبالتالي هذا معناه أن X على 2 نص M أكبر من M لكل M عدد موجب ما معناه أن ال limit لحد هذا أو ال sequence X المؤشرات تبقية 2 نص M تقول infinity معناته هذا الحد العام لها أكبر من أي عدد موجب", "tokens": [37746, 6027, 2655, 6027, 1829, 23758, 20449, 8315, 3224, 14739, 1783, 15844, 568, 8717, 9381, 376, 5551, 4117, 26890, 9154, 376, 5296, 28820, 376, 6225, 3215, 3215, 3714, 29245, 3555, 19446, 20449, 8315, 3224, 14739, 2423, 4948, 5296, 24401, 23758, 34051, 2423, 8310, 1783, 9673, 33604, 46309, 9307, 6055, 3555, 4587, 10632, 568, 8717, 9381, 376, 6055, 4587, 12610, 13202, 20449, 8315, 47395, 23758, 21542, 3215, 18863, 10943, 5296, 11296, 5551, 4117, 26890, 9154, 36632, 6225, 3215, 3215, 3714, 29245, 3555], "avg_logprob": -0.26543444940229743, "compression_ratio": 1.6777777777777778, "no_speech_prob": 0.0, "words": [{"start": 744.05, "end": 744.87, "word": "وبالتالي", "probability": 0.90966796875}, {"start": 744.87, "end": 745.19, "word": " هذا", "probability": 0.693359375}, {"start": 745.19, "end": 745.95, "word": " معناه", "probability": 0.9064127604166666}, {"start": 745.95, "end": 746.23, "word": " أن", "probability": 0.3642578125}, {"start": 746.23, "end": 746.83, "word": " X", "probability": 0.40283203125}, {"start": 746.83, "end": 748.57, "word": " على", "probability": 0.556640625}, {"start": 748.57, "end": 748.99, "word": " 2", "probability": 0.435791015625}, {"start": 748.99, "end": 749.27, "word": " نص", "probability": 0.747802734375}, {"start": 749.27, "end": 749.57, "word": " M", "probability": 0.7431640625}, {"start": 749.57, "end": 750.27, "word": " أكبر", "probability": 0.9617513020833334}, {"start": 750.27, "end": 750.61, "word": " من", "probability": 0.98779296875}, {"start": 750.61, "end": 751.07, "word": " M", "probability": 0.966796875}, {"start": 751.07, "end": 752.53, "word": " لكل", "probability": 0.96923828125}, {"start": 752.53, "end": 752.97, "word": " M", "probability": 0.93359375}, {"start": 752.97, "end": 753.37, "word": " عدد", "probability": 0.9710286458333334}, {"start": 753.37, "end": 753.83, "word": " موجب", "probability": 0.9596354166666666}, {"start": 753.83, "end": 756.51, "word": " ما", "probability": 0.751953125}, {"start": 756.51, "end": 757.07, "word": " معناه", "probability": 0.7976888020833334}, {"start": 757.07, "end": 757.89, "word": " أن", "probability": 0.79296875}, {"start": 757.89, "end": 757.99, "word": " ال", "probability": 0.277099609375}, {"start": 757.99, "end": 758.33, "word": " limit", "probability": 0.505859375}, {"start": 758.33, "end": 762.73, "word": " لحد", "probability": 0.50634765625}, {"start": 762.73, "end": 763.09, "word": " هذا", "probability": 0.434814453125}, {"start": 763.09, "end": 763.33, "word": " أو", "probability": 0.5576171875}, {"start": 763.33, "end": 763.63, "word": " ال", "probability": 0.7548828125}, {"start": 763.63, "end": 764.13, "word": " sequence", "probability": 0.95166015625}, {"start": 764.13, "end": 764.61, "word": " X", "probability": 0.9296875}, {"start": 764.61, "end": 766.11, "word": " المؤشرات", "probability": 0.7890625}, {"start": 766.11, "end": 766.43, "word": " تبقية", "probability": 0.75299072265625}, {"start": 766.43, "end": 766.75, "word": " 2", "probability": 0.396240234375}, {"start": 766.75, "end": 767.07, "word": " نص", "probability": 0.97314453125}, {"start": 767.07, "end": 767.33, "word": " M", "probability": 0.93408203125}, {"start": 767.33, "end": 767.65, "word": " تقول", "probability": 0.606689453125}, {"start": 767.65, "end": 768.09, "word": " infinity", "probability": 0.1641845703125}, {"start": 768.09, "end": 768.73, "word": " معناته", "probability": 0.8287760416666666}, {"start": 768.73, "end": 769.05, "word": " هذا", "probability": 0.60546875}, {"start": 769.05, "end": 769.73, "word": " الحد", "probability": 0.98291015625}, {"start": 769.73, "end": 770.13, "word": " العام", "probability": 0.98095703125}, {"start": 770.13, "end": 771.21, "word": " لها", "probability": 0.58099365234375}, {"start": 771.21, "end": 771.99, "word": " أكبر", "probability": 0.9791666666666666}, {"start": 771.99, "end": 772.15, "word": " من", "probability": 0.9892578125}, {"start": 772.15, "end": 772.37, "word": " أي", "probability": 0.80859375}, {"start": 772.37, "end": 772.75, "word": " عدد", "probability": 0.99267578125}, {"start": 772.75, "end": 773.15, "word": " موجب", "probability": 0.9889322916666666}], "temperature": 1.0}, {"id": 30, "seek": 79884, "start": 774.58, "end": 798.84, "text": "وبالتالي إذا الـ subsequence هذه اللي هي الحد العام تبعها X plus 2N is unbounded وبالتالي فهذا بيقدي إن ال sequence XN نفسها is unbounded لأنه لو كانت ال sequence bounded", "tokens": [37746, 6027, 2655, 6027, 1829, 11933, 15730, 2423, 39184, 13924, 655, 29538, 13672, 1829, 39896, 21542, 3215, 18863, 10943, 6055, 3555, 3615, 11296, 1783, 1804, 568, 45, 307, 517, 18767, 292, 46599, 6027, 2655, 6027, 1829, 6156, 3224, 15730, 4724, 1829, 4587, 16254, 36145, 2423, 8310, 1783, 45, 8717, 36178, 11296, 307, 517, 18767, 292, 5296, 33456, 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"text": "المثال التالت برضه هنستخدم .. هنطبق عليه اللي هو الـ monotone convergence theorem", "tokens": [45340, 12984, 6027, 16712, 6027, 2655, 4724, 43042, 3224, 8032, 1863, 14851, 9778, 40448, 4386, 8032, 1863, 9566, 3555, 4587, 47356, 13672, 1829, 31439, 2423, 39184, 1108, 310, 546, 32181, 20904], "avg_logprob": -0.13330078404396772, "compression_ratio": 1.196078431372549, "no_speech_prob": 0.0, "words": [{"start": 856.61, "end": 857.23, "word": "المثال", "probability": 0.9609375}, {"start": 857.23, "end": 857.61, "word": " التالت", "probability": 0.9498697916666666}, {"start": 857.61, "end": 857.95, "word": " برضه", "probability": 0.8797200520833334}, {"start": 857.95, "end": 858.79, "word": " هنستخدم", "probability": 0.888671875}, {"start": 858.79, "end": 859.19, "word": " ..", "probability": 0.309814453125}, {"start": 859.19, "end": 860.27, "word": " هنطبق", "probability": 0.94296875}, {"start": 860.27, "end": 860.59, "word": " عليه", "probability": 0.98486328125}, {"start": 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انا", "probability": 0.818115234375}, {"start": 880.52, "end": 880.74, "word": " بدي", "probability": 0.4932861328125}, {"start": 880.74, "end": 881.02, "word": " اعرف", "probability": 0.90771484375}, {"start": 881.02, "end": 881.16, "word": " ال", "probability": 0.63525390625}, {"start": 881.16, "end": 881.44, "word": " sequence", "probability": 0.7236328125}, {"start": 881.44, "end": 881.84, "word": " x", "probability": 0.46728515625}, {"start": 881.84, "end": 882.34, "word": " in", "probability": 0.54052734375}, {"start": 882.34, "end": 884.02, "word": " inductively", "probability": 0.6593017578125}, {"start": 884.02, "end": 884.82, "word": " بطريقة", "probability": 0.9578857421875}, {"start": 884.82, "end": 886.94, "word": " استقرائية", "probability": 0.96796875}, {"start": 886.94, "end": 892.86, "word": " شوفنا", "probability": 0.7679036458333334}, {"start": 892.86, "end": 893.08, "word": " احنا", "probability": 0.9202473958333334}, {"start": 893.08, "end": 893.34, "word": " لما", 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أثبت", "probability": 0.910888671875}, {"start": 1196.39, "end": 1197.33, "word": " المتباينة", "probability": 0.9241536458333334}, {"start": 1197.33, "end": 1197.93, "word": " هذه", "probability": 0.417724609375}, {"start": 1197.93, "end": 1199.17, "word": " طبعا", "probability": 0.77032470703125}, {"start": 1199.17, "end": 1200.99, "word": " فتعالى", "probability": 0.88271484375}, {"start": 1200.99, "end": 1201.53, "word": " نشوف", "probability": 0.9864908854166666}, {"start": 1201.53, "end": 1201.93, "word": " كيف", "probability": 0.95556640625}, {"start": 1201.93, "end": 1202.59, "word": " نثبت", "probability": 0.9853515625}, {"start": 1202.59, "end": 1204.07, "word": " المتباينة", "probability": 0.9861653645833334}, {"start": 1204.07, "end": 1205.11, "word": " هذه", "probability": 0.93701171875}, {"start": 1205.11, "end": 1221.69, "word": " طيب", "probability": 0.8712565104166666}, {"start": 1221.69, "end": 1221.91, "word": " أنا", "probability": 0.59423828125}, {"start": 1221.91, "end": 1222.49, "word": " عندي", "probability": 0.8154296875}], "temperature": 1.0}, {"id": 47, "seek": 125328, "start": 1223.84, "end": 1253.28, "text": "هي عندي المتباينة هذه احنا فرضين ان المتباينة هذه صحيحة احنا فرضين من induction hypothesis ان هذه المتباينة صحيحة", "tokens": [3224, 1829, 18871, 16254, 9673, 2655, 3555, 995, 9957, 3660, 29538, 1975, 5016, 8315, 6156, 43042, 9957, 16472, 9673, 2655, 3555, 995, 9957, 3660, 29538, 20328, 5016, 1829, 5016, 3660, 1975, 5016, 8315, 6156, 43042, 9957, 9154, 33371, 17291, 16472, 29538, 9673, 2655, 3555, 995, 9957, 3660, 20328, 5016, 1829, 5016, 3660], "avg_logprob": -0.11497641509433962, "compression_ratio": 1.7339449541284404, "no_speech_prob": 0.0, "words": [{"start": 1223.84, "end": 1224.14, "word": "هي", "probability": 0.5521240234375}, {"start": 1224.14, "end": 1224.46, "word": " عندي", "probability": 0.768310546875}, {"start": 1224.46, "end": 1225.3, "word": " المتباينة", "probability": 0.955322265625}, {"start": 1225.3, "end": 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"end": 1251.76, "word": " ان", "probability": 0.7451171875}, {"start": 1251.76, "end": 1252.04, "word": " هذه", "probability": 0.56884765625}, {"start": 1252.04, "end": 1252.64, "word": " المتباينة", "probability": 0.9903157552083334}, {"start": 1252.64, "end": 1253.28, "word": " صحيحة", "probability": 0.9892578125}], "temperature": 1.0}, {"id": 48, "seek": 127157, "start": 1254.45, "end": 1271.57, "text": "أضرب المتباينة هذه في اتنين هي اضرب كل الأطراف في اتنين فبصير اتنين اصغر من اتنين XK اصغر من اتنين XK زائد واحد اصغر من اربع وهذا بيقدي ان واحد اصغر من جدر اتنين", "tokens": [10721, 11242, 25513, 9673, 2655, 3555, 995, 9957, 3660, 29538, 8978, 1975, 2655, 1863, 9957, 39896, 1975, 11242, 25513, 28242, 16247, 9566, 23557, 5172, 8978, 1975, 2655, 1863, 9957, 6156, 3555, 9381, 13546, 1975, 2655, 1863, 9957, 1975, 9381, 17082, 2288, 9154, 1975, 2655, 1863, 9957, 1783, 42, 1975, 9381, 17082, 2288, 9154, 1975, 2655, 1863, 9957, 1783, 42, 30767, 16373, 3215, 36764, 24401, 1975, 9381, 17082, 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الأطراف", "probability": 0.850830078125}, {"start": 1259.39, "end": 1259.53, "word": " في", "probability": 0.7958984375}, {"start": 1259.53, "end": 1259.87, "word": " اتنين", "probability": 0.9781494140625}, {"start": 1259.87, "end": 1260.51, "word": " فبصير", "probability": 0.81329345703125}, {"start": 1260.51, "end": 1260.93, "word": " اتنين", "probability": 0.9471435546875}, {"start": 1260.93, "end": 1261.35, "word": " اصغر", "probability": 0.874755859375}, {"start": 1261.35, "end": 1261.49, "word": " من", "probability": 0.98291015625}, {"start": 1261.49, "end": 1261.85, "word": " اتنين", "probability": 0.9820556640625}, {"start": 1261.85, "end": 1262.31, "word": " XK", "probability": 0.5582275390625}, {"start": 1262.31, "end": 1262.71, "word": " اصغر", "probability": 0.948974609375}, {"start": 1262.71, "end": 1262.83, "word": " من", "probability": 0.98828125}, {"start": 1262.83, "end": 1263.13, "word": " اتنين", "probability": 0.98779296875}, {"start": 1263.13, "end": 1263.59, 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"word": " جدر", "probability": 0.9573567708333334}, {"start": 1271.05, "end": 1271.57, "word": " اتنين", "probability": 0.9854736328125}], "temperature": 1.0}, {"id": 49, "seek": 130473, "start": 1275.73, "end": 1304.73, "text": "و اذا انا الان باخد الجذر التربيعى لكل الأطراف هذه اخد الجذر التربيعى فهي جذر اتنين طبعا اكبر من واحد اصغر منه يساوي جذر اتنين XK اللى هو XK زاد واحد هذا طبعا من التعريف تبع ال sequence من ال inductive formula جذر اتنين XK حسب التعريف بساوي XK زاد واحد", "tokens": [2407, 1975, 15730, 1975, 8315, 2423, 7649, 4724, 47283, 3215, 25724, 8848, 2288, 16712, 2288, 21292, 3615, 7578, 5296, 28820, 16247, 9566, 23557, 5172, 29538, 1975, 9778, 3215, 25724, 8848, 2288, 16712, 2288, 21292, 3615, 7578, 6156, 3224, 1829, 10874, 8848, 2288, 1975, 2655, 1863, 9957, 23032, 3555, 3615, 995, 1975, 4117, 26890, 9154, 36764, 24401, 1975, 9381, 17082, 2288, 9154, 3224, 7251, 3794, 995, 45865, 10874, 8848, 2288, 1975, 2655, 1863, 9957, 1783, 42, 13672, 7578, 31439, 1783, 42, 30767, 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جذر", "probability": 0.9855143229166666}, {"start": 1299.71, "end": 1300.03, "word": " اتنين", "probability": 0.9871826171875}, {"start": 1300.03, "end": 1300.51, "word": " XK", "probability": 0.9775390625}, {"start": 1300.51, "end": 1302.01, "word": " حسب", "probability": 0.77490234375}, {"start": 1302.01, "end": 1302.51, "word": " التعريف", "probability": 0.99462890625}, {"start": 1302.51, "end": 1303.13, "word": " بساوي", "probability": 0.7833251953125}, {"start": 1303.13, "end": 1304.01, "word": " XK", "probability": 0.97265625}, {"start": 1304.01, "end": 1304.37, "word": " زاد", "probability": 0.950927734375}, {"start": 1304.37, "end": 1304.73, "word": " واحد", "probability": 0.989013671875}], "temperature": 1.0}, {"id": 50, "seek": 133420, "start": 1306.26, "end": 1334.2, "text": "وهذا أصغر من هنا جدر اتنين xk أصغر من جدر اتنين xk زائد واحد وهذا أصغر من جدر الأربع اللي هو الأتنين إذا هاي بيطلع عندي واحد أصغر من أو يساوي xk زائد واحد وهذا برضه من ال inductive formula الجدر التربيع هذا بيساوي xk زائد اتنين", "tokens": [2407, 3224, 15730, 5551, 9381, 17082, 2288, 9154, 34105, 10874, 3215, 2288, 1975, 2655, 1863, 9957, 2031, 74, 5551, 9381, 17082, 2288, 9154, 10874, 3215, 2288, 1975, 2655, 1863, 9957, 2031, 74, 30767, 16373, 3215, 36764, 24401, 37037, 15730, 5551, 9381, 17082, 2288, 9154, 10874, 3215, 2288, 16247, 25513, 3615, 13672, 1829, 31439, 16247, 2655, 1863, 9957, 11933, 15730, 8032, 47302, 4724, 1829, 9566, 1211, 3615, 18871, 16254, 36764, 24401, 5551, 9381, 17082, 2288, 9154, 34051, 7251, 3794, 995, 45865, 2031, 74, 30767, 16373, 3215, 36764, 24401, 37037, 15730, 4724, 43042, 3224, 9154, 2423, 31612, 488, 8513, 25724, 3215, 2288, 16712, 2288, 21292, 3615, 23758, 4724, 1829, 3794, 995, 45865, 2031, 74, 30767, 16373, 3215, 1975, 2655, 1863, 9957], "avg_logprob": -0.16289061990876993, "compression_ratio": 2.169491525423729, "no_speech_prob": 0.0, "words": [{"start": 1306.26, "end": 1306.94, "word": "وهذا", "probability": 0.749755859375}, {"start": 1306.94, 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{"start": 1535.04, "end": 1535.26, "word": " اذا", "probability": 0.562744140625}, {"start": 1535.26, "end": 1535.48, "word": " هي", "probability": 0.8408203125}, {"start": 1535.48, "end": 1536.02, "word": " x", "probability": 0.91162109375}, {"start": 1536.02, "end": 1536.72, "word": " و", "probability": 0.53466796875}, {"start": 1536.72, "end": 1536.98, "word": " جذر", "probability": 0.8416341145833334}, {"start": 1536.98, "end": 1537.38, "word": " اتنين", "probability": 0.9542236328125}, {"start": 1537.38, "end": 1537.54, "word": " في", "probability": 0.7431640625}, {"start": 1537.54, "end": 1537.66, "word": " ال", "probability": 0.74853515625}, {"start": 1537.66, "end": 1537.92, "word": " limit", "probability": 0.97119140625}, {"start": 1537.92, "end": 1538.18, "word": " هذه", "probability": 0.7685546875}, {"start": 1538.18, "end": 1538.6, "word": " بتطلع", "probability": 0.894775390625}, {"start": 1538.6, "end": 1538.92, "word": " جذر", "probability": 0.9949544270833334}, {"start": 1538.92, "end": 1539.08, "word": " ال", "probability": 0.923828125}, {"start": 1539.08, "end": 1539.28, "word": " x", "probability": 0.8369140625}], "temperature": 1.0}, {"id": 58, "seek": 156840, "start": 1539.74, "end": 1568.4, "text": "إذا أصبح أثيان دي معادلة في مجهول واحد x ممكن أحلها و ذلك بتربيع الطرفين فالمعادلة هذه بعد مربعها تصير هيك وهذه في إلها حالين إما x بيطلع بساوي سفر أو x بساوي اتنين احنا عايزين ال x ناخد x بساوي اتنين و نرفض x بساوي سفر طب ليه نرفض x بساوي سفر؟", "tokens": [28814, 15730, 5551, 9381, 49628, 5551, 12984, 1829, 7649, 11778, 1829, 20449, 18513, 37977, 8978, 3714, 7435, 3224, 12610, 36764, 24401, 2031, 3714, 43020, 5551, 5016, 1211, 11296, 4032, 29910, 23275, 39894, 2288, 21292, 3615, 41950, 28480, 9957, 6156, 45340, 3615, 18513, 37977, 29538, 39182, 3714, 25513, 3615, 11296, 6055, 9381, 13546, 39896, 4117, 37037, 24192, 8978, 11933, 1211, 11296, 11331, 6027, 9957, 11933, 15042, 2031, 4724, 1829, 9566, 1211, 3615, 4724, 3794, 995, 45865, 8608, 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induction أن xn أكبر من أوسع واحد أصغر من الأتنين و أثبتنا أن ال sequence هذه convergent، إذا حسب نظرية سابقة", "tokens": [1211, 33456, 1975, 12984, 3555, 2655, 8315, 34105, 538, 33371, 14739, 2031, 77, 5551, 4117, 26890, 9154, 34051, 3794, 3615, 36764, 24401, 5551, 9381, 17082, 2288, 9154, 16247, 2655, 1863, 9957, 4032, 5551, 12984, 3555, 2655, 8315, 14739, 2423, 8310, 29538, 9652, 6930, 12399, 11933, 15730, 11331, 35457, 8717, 19913, 2288, 10632, 8608, 16758, 28671], "avg_logprob": -0.29045759035008295, "compression_ratio": 1.3445945945945945, "no_speech_prob": 0.0, "words": [{"start": 1568.68, "end": 1569.2, "word": "لأن", "probability": 0.6163330078125}, {"start": 1569.2, "end": 1570.12, "word": " اثبتنا", "probability": 0.82412109375}, {"start": 1570.12, "end": 1570.4, "word": " هنا", "probability": 0.34130859375}, {"start": 1570.4, "end": 1570.7, "word": " by", "probability": 0.630859375}, {"start": 1570.7, "end": 1572.9, "word": " induction", "probability": 0.97607421875}, 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"word": " عندي", "probability": 0.893310546875}, {"start": 1638.3, "end": 1638.8, "word": " سفر", "probability": 0.97314453125}, {"start": 1638.8, "end": 1638.94, "word": " او", "probability": 0.955810546875}, {"start": 1638.94, "end": 1639.3, "word": " اتنين", "probability": 0.99365234375}, {"start": 1639.3, "end": 1639.5, "word": " اذا", "probability": 0.784912109375}, {"start": 1639.5, "end": 1639.82, "word": " لازم", "probability": 0.9908854166666666}, {"start": 1639.82, "end": 1640.16, "word": " الساوي", "probability": 0.7537434895833334}, {"start": 1640.16, "end": 1640.46, "word": " اتنين", "probability": 0.98828125}], "temperature": 1.0}, {"id": 62, "seek": 166989, "start": 1641.55, "end": 1669.89, "text": "Okay إذا هين هيك استخدمنا الـ monotone convergence بالمثل في أد التمرين زي هذه sequences معرفة inductively و هتثبتوا أنها convergent و تجيبوا قيمة ال limit بنفس الطريقة و بنفس الأسلوب فحاولوا تستفيدوا من حل المثال الأخير هذا في حل مثل هذه التمرين Okay تمام واضح", "tokens": 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" ممنعش", "probability": 0.81298828125}, {"start": 1764.48, "end": 1764.64, "word": " ان", "probability": 0.75634765625}, {"start": 1764.64, "end": 1765.0, "word": " ممكن", "probability": 0.60296630859375}, {"start": 1765.0, "end": 1765.7, "word": " حد", "probability": 0.87841796875}, {"start": 1765.7, "end": 1766.2, "word": " معين", "probability": 0.99072265625}, {"start": 1766.2, "end": 1766.64, "word": " يثبت", "probability": 0.9952392578125}, {"start": 1766.64, "end": 1766.78, "word": " ان", "probability": 0.8125}, {"start": 1766.78, "end": 1766.94, "word": " ال", "probability": 0.82275390625}, {"start": 1766.94, "end": 1767.28, "word": " suprem", "probability": 0.7626953125}, {"start": 1767.28, "end": 1768.1, "word": " هو", "probability": 0.98828125}, {"start": 1768.1, "end": 1768.64, "word": " اتنين", "probability": 0.9505615234375}, {"start": 1768.64, "end": 1769.68, "word": " اذا", "probability": 0.758544921875}, {"start": 1769.68, "end": 1769.96, "word": " كان", "probability": 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theorem، ال sub-sequences شوفنا قبل شوية sub-sequence", "tokens": [6027, 39184, 1422, 12, 11834, 2667, 293, 14331, 89, 3730, 12, 4360, 811, 9733, 640, 20904, 12399, 2423, 1422, 12, 11834, 2667, 13412, 38688, 8315, 12174, 36150, 13412, 2407, 10632, 1422, 12, 11834, 655], "avg_logprob": -0.3723214217594692, "compression_ratio": 1.202127659574468, "no_speech_prob": 0.0, "words": [{"start": 1789.63, "end": 1789.95, "word": "الـ", "probability": 0.685302734375}, {"start": 1789.95, "end": 1790.17, "word": " sub", "probability": 0.2213134765625}, {"start": 1790.17, "end": 1791.01, "word": "-sequences", "probability": 0.7611490885416666}, {"start": 1791.01, "end": 1793.03, "word": " and", "probability": 0.274169921875}, {"start": 1793.03, "end": 1793.87, "word": " Bolzano", "probability": 0.7632649739583334}, {"start": 1793.87, "end": 1794.61, "word": "-Weierstrass", "probability": 0.7654296875}, {"start": 1794.61, "end": 1796.07, "word": " theorem،", "probability": 0.596435546875}, {"start": 1796.07, "end": 1796.27, "word": " ال", "probability": 0.546875}, {"start": 1796.27, "end": 1796.75, "word": " sub", "probability": 0.2919921875}, {"start": 1796.75, "end": 1797.43, "word": "-sequences", "probability": 0.9635416666666666}, {"start": 1797.43, "end": 1797.85, "word": " شوفنا", "probability": 0.6277669270833334}, {"start": 1797.85, "end": 1798.07, "word": " قبل", "probability": 0.97021484375}, {"start": 1798.07, "end": 1798.45, "word": " شوية", "probability": 0.96142578125}, {"start": 1798.45, "end": 1798.75, "word": " sub", "probability": 0.9052734375}, {"start": 1798.75, "end": 1799.35, "word": "-sequence", "probability": 0.9182942708333334}], "temperature": 1.0}, {"id": 68, "seek": 183494, "start": 1811.18, "end": 1834.94, "text": "شوفنا قبل لحظات في المثال التاني انه في عنده sequence هي عنده sequence xn حدودها x1, x2, x3, x4 و هكذا و في كانت sequence تانية لحد الآن تبعها 2 أُس n", "tokens": [8592, 38688, 8315, 12174, 36150, 5296, 5016, 19913, 9307, 8978, 9673, 12984, 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انه", "probability": 0.3836669921875}, {"start": 1814.92, "end": 1815.02, "word": " في", "probability": 0.7451171875}, {"start": 1815.02, "end": 1815.4, "word": " عنده", "probability": 0.576416015625}, {"start": 1815.4, "end": 1815.96, "word": " sequence", "probability": 0.84716796875}, {"start": 1815.96, "end": 1816.24, "word": " هي", "probability": 0.53271484375}, {"start": 1816.24, "end": 1816.6, "word": " عنده", "probability": 0.91357421875}, {"start": 1816.6, "end": 1817.04, "word": " sequence", "probability": 0.97314453125}, {"start": 1817.04, "end": 1817.84, "word": " xn", "probability": 0.591796875}, {"start": 1817.84, "end": 1819.7, "word": " حدودها", "probability": 0.988037109375}, {"start": 1819.7, "end": 1820.64, "word": " x1,", "probability": 0.885498046875}, {"start": 1820.94, "end": 1821.8, "word": " x2,", "probability": 0.90966796875}, {"start": 1822.42, "end": 1823.82, "word": " x3,", "probability": 0.974609375}, {"start": 1824.64, "end": 1826.54, "word": " x4", "probability": 0.985107421875}, {"start": 1826.54, "end": 1828.9, "word": " و", "probability": 0.79052734375}, {"start": 1828.9, "end": 1829.54, "word": " هكذا", "probability": 0.800537109375}, {"start": 1829.54, "end": 1830.78, "word": " و", "probability": 0.88671875}, {"start": 1830.78, "end": 1831.0, "word": " في", "probability": 0.75830078125}, {"start": 1831.0, "end": 1831.54, "word": " كانت", "probability": 0.75830078125}, {"start": 1831.54, "end": 1831.94, "word": " sequence", "probability": 0.78173828125}, {"start": 1831.94, "end": 1832.44, "word": " تانية", "probability": 0.9378255208333334}, {"start": 1832.44, "end": 1832.84, "word": " لحد", "probability": 0.67333984375}, {"start": 1832.84, "end": 1833.06, "word": " الآن", "probability": 0.69580078125}, {"start": 1833.06, "end": 1833.58, "word": " تبعها", "probability": 0.93359375}, {"start": 1833.58, "end": 1834.16, "word": " 2", "probability": 0.66650390625}, {"start": 1834.16, "end": 1834.6, "word": " أُس", "probability": 0.5687662760416666}, {"start": 1834.6, "end": 1834.94, "word": " n", "probability": 0.56005859375}], "temperature": 1.0}, {"id": 69, "seek": 186134, "start": 1836.18, "end": 1861.34, "text": "الحدود هذي هتكون X2 X4 X8 و هكذا صح؟ لو سمنا الاتنين هذي R1 والاربعة هذي سمنها R2 والتمانية R3", "tokens": [6027, 24401, 23328, 8032, 8848, 1829, 8032, 2655, 30544, 1783, 17, 1783, 19, 1783, 23, 4032, 8032, 4117, 15730, 20328, 5016, 22807, 45164, 8608, 2304, 8315, 2423, 9307, 1863, 9957, 8032, 8848, 1829, 497, 16, 16070, 9640, 3555, 27884, 8032, 8848, 1829, 8608, 27842, 11296, 497, 17, 16070, 39237, 7649, 10632, 497, 18], "avg_logprob": -0.19994213100936678, "compression_ratio": 1.3771929824561404, "no_speech_prob": 0.0, "words": [{"start": 1836.18, "end": 1836.96, "word": "الحدود", "probability": 0.8243815104166666}, {"start": 1836.96, "end": 1837.54, "word": " هذي", "probability": 0.4276529947916667}, {"start": 1837.54, "end": 1838.26, "word": " هتكون", "probability": 0.9488932291666666}, {"start": 1838.26, "end": 1840.76, "word": " X2", "probability": 0.6240234375}, {"start": 1840.76, "end": 1843.42, "word": " X4", "probability": 0.878173828125}, {"start": 1843.42, "end": 1846.02, "word": " X8", "probability": 0.980224609375}, {"start": 1846.02, "end": 1847.74, "word": " و", "probability": 0.64404296875}, {"start": 1847.74, "end": 1848.34, "word": " هكذا", "probability": 0.8971354166666666}, {"start": 1848.34, "end": 1849.9, "word": " صح؟", "probability": 0.8056640625}, {"start": 1849.9, "end": 1851.92, "word": " لو", "probability": 0.919921875}, {"start": 1851.92, "end": 1852.42, "word": " سمنا", "probability": 0.7664388020833334}, {"start": 1852.42, "end": 1853.02, "word": " الاتنين", "probability": 0.81591796875}, {"start": 1853.02, "end": 1853.52, "word": " هذي", "probability": 0.85546875}, {"start": 1853.52, "end": 1854.32, "word": " R1", "probability": 0.931884765625}, {"start": 1854.32, "end": 1856.34, "word": " والاربعة", "probability": 0.8411865234375}, {"start": 1856.34, "end": 1856.7, "word": " هذي", "probability": 0.93359375}, {"start": 1856.7, "end": 1857.28, "word": " سمنها", "probability": 0.8310546875}, {"start": 1857.28, "end": 1858.1, "word": " R2", "probability": 0.986328125}, {"start": 1858.1, "end": 1860.66, "word": " والتمانية", "probability": 0.8917236328125}, {"start": 1860.66, "end": 1861.34, "word": " R3", "probability": 0.973388671875}], "temperature": 1.0}, {"id": 70, "seek": 188576, "start": 1864.82, "end": 1885.76, "text": "فبنلاحظ أن R1 أكبر من أو ساوء واحد، عدد طبيعي أكبر من أو ساوء واحد وR2 أكبر من R1، اللي هو أربعة أكبر من اتنين وR3 اللي هو تمانية أكبر من R2", "tokens": [5172, 3555, 1863, 15040, 5016, 19913, 14739, 497, 16, 5551, 4117, 26890, 9154, 34051, 8608, 995, 2407, 38207, 36764, 24401, 12399, 6225, 3215, 3215, 23032, 21292, 3615, 1829, 5551, 4117, 26890, 9154, 34051, 8608, 995, 2407, 38207, 36764, 24401, 4032, 49, 17, 5551, 4117, 26890, 9154, 497, 16, 12399, 13672, 1829, 31439, 5551, 25513, 27884, 5551, 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هكذا اذا ال sub sequence المؤشرات تبعتها او ال indices انا بسميه index مجموعة index indices ال indices او المؤشرات لل sub sequence هي عداد طبيعية هذا هي اتنين اربعة تمانية هي عداد طبيعية والعداد الطبيعية هذه بتشكل sequence هذه عبارة عن sequence of natural numbers", "tokens": [2407, 8032, 4117, 15730, 1975, 15730, 2423, 1422, 8310, 9673, 33604, 46309, 9307, 6055, 3555, 34268, 11296, 1975, 2407, 2423, 43840, 1975, 8315, 4724, 38251, 1829, 3224, 8186, 3714, 7435, 2304, 2407, 27884, 8186, 43840, 2423, 43840, 1975, 2407, 9673, 33604, 46309, 9307, 24976, 1422, 8310, 39896, 6225, 3215, 18513, 23032, 21292, 3615, 10632, 23758, 39896, 1975, 2655, 1863, 9957, 1975, 25513, 27884, 46811, 7649, 10632, 39896, 6225, 3215, 18513, 23032, 21292, 3615, 10632, 16070, 22488, 18513, 41950, 21292, 3615, 10632, 29538, 39894, 8592, 28820, 8310, 29538, 6225, 3555, 9640, 3660, 18871, 8310, 295, 3303, 3547], "avg_logprob": -0.16196842291920455, "compression_ratio": 1.9752475247524752, "no_speech_prob": 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"probability": 0.691162109375}, {"start": 1894.81, "end": 1895.39, "word": " بسميه", "probability": 0.9149169921875}, {"start": 1895.39, "end": 1895.93, "word": " index", "probability": 0.79736328125}, {"start": 1895.93, "end": 1897.11, "word": " مجموعة", "probability": 0.882421875}, {"start": 1897.11, "end": 1897.55, "word": " index", "probability": 0.79541015625}, {"start": 1897.55, "end": 1898.33, "word": " indices", "probability": 0.72216796875}, {"start": 1898.33, "end": 1899.43, "word": " ال", "probability": 0.9501953125}, {"start": 1899.43, "end": 1899.89, "word": " indices", "probability": 0.87109375}, {"start": 1899.89, "end": 1900.33, "word": " او", "probability": 0.967529296875}, {"start": 1900.33, "end": 1901.15, "word": " المؤشرات", "probability": 0.98779296875}, {"start": 1901.15, "end": 1901.39, "word": " لل", "probability": 0.748046875}, {"start": 1901.39, "end": 1901.65, "word": " sub", "probability": 0.720703125}, {"start": 1901.65, "end": 1902.23, "word": " sequence", "probability": 0.9013671875}, {"start": 1902.23, "end": 1903.03, "word": " هي", "probability": 0.685546875}, {"start": 1903.03, "end": 1903.53, "word": " عداد", "probability": 0.9309895833333334}, {"start": 1903.53, "end": 1904.21, "word": " طبيعية", "probability": 0.9385986328125}, {"start": 1904.21, "end": 1904.49, "word": " هذا", "probability": 0.1153564453125}, {"start": 1904.49, "end": 1904.71, "word": " هي", "probability": 0.75341796875}, {"start": 1904.71, "end": 1905.07, "word": " اتنين", "probability": 0.8741455078125}, {"start": 1905.07, "end": 1905.49, "word": " اربعة", "probability": 0.8819986979166666}, {"start": 1905.49, "end": 1906.07, "word": " تمانية", "probability": 0.9571940104166666}, {"start": 1906.07, "end": 1906.77, "word": " هي", "probability": 0.6025390625}, {"start": 1906.77, "end": 1907.17, "word": " عداد", "probability": 0.9833984375}, {"start": 1907.17, "end": 1907.85, "word": " طبيعية", "probability": 0.9896240234375}, {"start": 1907.85, "end": 1909.23, "word": " والعداد", "probability": 0.84619140625}, {"start": 1909.23, "end": 1909.89, "word": " الطبيعية", "probability": 0.971923828125}, {"start": 1909.89, "end": 1910.27, "word": " هذه", "probability": 0.69189453125}, {"start": 1910.27, "end": 1910.97, "word": " بتشكل", "probability": 0.9718424479166666}, {"start": 1910.97, "end": 1911.65, "word": " sequence", "probability": 0.8896484375}, {"start": 1911.65, "end": 1913.57, "word": " هذه", "probability": 0.73681640625}, {"start": 1913.57, "end": 1913.99, "word": " عبارة", "probability": 0.99560546875}, {"start": 1913.99, "end": 1914.17, "word": " عن", "probability": 0.99658203125}, {"start": 1914.17, "end": 1914.81, "word": " sequence", "probability": 0.97998046875}, {"start": 1914.81, "end": 1915.17, "word": " of", "probability": 0.96923828125}, {"start": 1915.17, "end": 1915.69, "word": " natural", "probability": 0.9091796875}, {"start": 1915.69, "end": 1916.19, "word": " numbers", "probability": 0.8876953125}], "temperature": 1.0}, {"id": 72, "seek": 194578, "start": 1917.84, "end": 1945.78, "text": "صح؟ و ال sequence هذه is strictly .. strictly increasing .. strictly increasing يعني متزايدة زيادة صحيحة يعني R واحد أصغر منه مش أصغر منه أو يساوي R اتنين و R اتنين أصغر منه و لا تساوي R تلاتة و هكذا مظبوط؟ صح؟", "tokens": [9381, 5016, 22807, 4032, 2423, 8310, 29538, 307, 20792, 4386, 20792, 5662, 4386, 20792, 5662, 37495, 22653, 44650, 11622, 995, 25708, 3660, 30767, 1829, 18513, 3660, 20328, 5016, 1829, 5016, 3660, 37495, 22653, 497, 36764, 24401, 5551, 9381, 17082, 2288, 9154, 3224, 37893, 5551, 9381, 17082, 2288, 9154, 3224, 34051, 7251, 3794, 995, 45865, 497, 1975, 2655, 1863, 9957, 4032, 497, 1975, 2655, 1863, 9957, 5551, 9381, 17082, 2288, 9154, 3224, 4032, 20193, 6055, 3794, 995, 45865, 497, 6055, 1211, 9307, 3660, 4032, 8032, 4117, 15730, 3714, 19913, 3555, 2407, 9566, 22807, 20328, 5016, 22807], "avg_logprob": -0.1554361948122581, "compression_ratio": 1.7081081081081082, "no_speech_prob": 0.0, "words": [{"start": 1917.84, "end": 1918.76, "word": "صح؟", "probability": 0.8531901041666666}, {"start": 1918.76, "end": 1919.68, "word": " و", "probability": 0.625}, {"start": 1919.68, "end": 1919.92, "word": " ال", "probability": 0.921875}, {"start": 1919.92, "end": 1920.26, "word": " sequence", "probability": 0.88818359375}, {"start": 1920.26, "end": 1920.74, "word": " هذه", "probability": 0.56298828125}, {"start": 1920.74, "end": 1921.12, "word": " is", "probability": 0.92138671875}, {"start": 1921.12, "end": 1921.88, "word": " strictly", "probability": 0.96240234375}, {"start": 1921.88, "end": 1922.12, "word": " ..", "probability": 0.65576171875}, {"start": 1922.12, "end": 1924.84, "word": " strictly", "probability": 0.89697265625}, {"start": 1924.84, "end": 1928.2, "word": " increasing", "probability": 0.50732421875}, {"start": 1928.2, "end": 1928.58, "word": " ..", "probability": 0.369384765625}, {"start": 1928.58, "end": 1929.22, "word": " strictly", "probability": 0.96435546875}, {"start": 1929.22, "end": 1929.82, "word": " increasing", "probability": 0.96630859375}, {"start": 1929.82, "end": 1931.18, "word": " يعني", "probability": 0.94921875}, {"start": 1931.18, "end": 1932.9, "word": " متزايدة", "probability": 0.96650390625}, {"start": 1932.9, "end": 1933.64, "word": " زيادة", "probability": 0.9822998046875}, {"start": 1933.64, "end": 1934.58, "word": " صحيحة", "probability": 0.99580078125}, {"start": 1934.58, "end": 1935.44, "word": " يعني", "probability": 0.947998046875}, {"start": 1935.44, "end": 1935.68, "word": " R", "probability": 0.72216796875}, {"start": 1935.68, "end": 1936.16, "word": " واحد", "probability": 0.71484375}, {"start": 1936.16, "end": 1936.66, "word": " أصغر", "probability": 0.952392578125}, {"start": 1936.66, "end": 1936.96, "word": " منه", "probability": 0.9638671875}, {"start": 1936.96, "end": 1937.14, "word": " مش", "probability": 0.68310546875}, {"start": 1937.14, "end": 1937.52, "word": " أصغر", "probability": 0.976806640625}, {"start": 1937.52, "end": 1937.76, "word": " منه", "probability": 0.94287109375}, {"start": 1937.76, "end": 1937.92, "word": " أو", "probability": 0.60986328125}, {"start": 1937.92, "end": 1938.24, "word": " يساوي", "probability": 0.9483642578125}, {"start": 1938.24, "end": 1938.48, "word": " R", "probability": 0.9560546875}, {"start": 1938.48, "end": 1938.86, "word": " اتنين", "probability": 0.9111328125}, {"start": 1938.86, "end": 1939.46, "word": " و", "probability": 0.9052734375}, {"start": 1939.46, "end": 1939.7, "word": " R", "probability": 0.76416015625}, {"start": 1939.7, "end": 1940.1, "word": " اتنين", "probability": 0.9666748046875}, {"start": 1940.1, "end": 1940.6, "word": " أصغر", "probability": 0.9716796875}, {"start": 1940.6, "end": 1941.1, "word": " منه", "probability": 0.994140625}, {"start": 1941.1, "end": 1941.42, "word": " و", "probability": 0.366455078125}, {"start": 1941.42, "end": 1941.54, "word": " لا", "probability": 0.7998046875}, {"start": 1941.54, "end": 1941.98, "word": " تساوي", "probability": 0.941650390625}, {"start": 1941.98, "end": 1942.18, "word": " R", "probability": 0.9736328125}, {"start": 1942.18, "end": 1942.64, "word": " تلاتة", "probability": 0.845703125}, {"start": 1942.64, "end": 1942.8, "word": " و", "probability": 0.85205078125}, {"start": 1942.8, "end": 1943.28, "word": " هكذا", "probability": 0.671142578125}, {"start": 1943.28, "end": 1945.32, "word": " مظبوط؟", "probability": 0.7838541666666666}, {"start": 1945.32, "end": 1945.78, "word": " صح؟", "probability": 0.9768880208333334}], "temperature": 1.0}, {"id": 73, "seek": 197741, "start": 1949.03, "end": 1977.41, "text": "إذا السب سيكوانس السب سيكوانس من أي سيكوانس هي مجموعة جزئية منها صح لأن حدودها هي حدود حدود السب سيكوانس هي عناصر أو حدود من السيكوانس العصلية لكن مش أي حدود لازم تكون مرتبة بحيث أن المؤشرات تبعتها اتشكل strictly increasing sequence of natural numbers", "tokens": [28814, 15730, 21136, 3555, 8608, 1829, 4117, 2407, 7649, 3794, 21136, 3555, 8608, 1829, 4117, 2407, 7649, 3794, 9154, 36632, 8608, 1829, 4117, 2407, 7649, 3794, 39896, 3714, 7435, 2304, 2407, 27884, 10874, 11622, 19986, 10632, 9154, 11296, 20328, 5016, 5296, 33456, 11331, 3215, 23328, 11296, 39896, 11331, 3215, 23328, 11331, 3215, 23328, 21136, 3555, 8608, 1829, 4117, 2407, 7649, 3794, 39896, 18871, 33546, 2288, 34051, 11331, 3215, 23328, 9154, 21136, 1829, 4117, 2407, 7649, 3794, 18863, 9381, 1211, 10632, 44381, 37893, 36632, 11331, 3215, 23328, 5296, 31377, 2304, 6055, 30544, 3714, 43500, 49401, 4724, 5016, 1829, 12984, 14739, 9673, 33604, 46309, 9307, 6055, 3555, 34268, 11296, 1975, 2655, 8592, 28820, 20792, 5662, 8310, 295, 3303, 3547], "avg_logprob": -0.14208157133247892, "compression_ratio": 1.8823529411764706, "no_speech_prob": 0.0, "words": [{"start": 1949.03, "end": 1949.41, "word": "إذا", "probability": 0.588134765625}, {"start": 1949.41, "end": 1949.79, "word": " السب", "probability": 0.52288818359375}, {"start": 1949.79, "end": 1950.37, "word": " سيكوانس", "probability": 0.833984375}, {"start": 1950.37, "end": 1950.85, "word": " السب", "probability": 0.6309814453125}, {"start": 1950.85, "end": 1951.55, "word": " سيكوانس", "probability": 0.94775390625}, {"start": 1951.55, "end": 1951.87, "word": " من", "probability": 0.955078125}, {"start": 1951.87, "end": 1952.11, "word": " أي", "probability": 0.91015625}, {"start": 1952.11, "end": 1952.85, "word": " سيكوانس", "probability": 0.964599609375}, {"start": 1952.85, "end": 1953.43, "word": " هي", "probability": 0.82275390625}, {"start": 1953.43, "end": 1953.89, "word": " مجموعة", "probability": 0.95380859375}, {"start": 1953.89, "end": 1954.37, "word": " جزئية", "probability": 0.9326171875}, {"start": 1954.37, "end": 1954.81, "word": " منها", "probability": 0.990966796875}, {"start": 1954.81, "end": 1955.41, "word": " صح", "probability": 0.817626953125}, {"start": 1955.41, "end": 1956.23, "word": " لأن", "probability": 0.627197265625}, {"start": 1956.23, "end": 1956.93, "word": " حدودها", "probability": 0.976806640625}, {"start": 1956.93, "end": 1957.39, "word": " هي", "probability": 0.7861328125}, {"start": 1957.39, "end": 1958.17, "word": " حدود", "probability": 0.8792317708333334}, {"start": 1958.17, "end": 1959.01, "word": " حدود", "probability": 0.7621256510416666}, {"start": 1959.01, "end": 1959.35, "word": " السب", "probability": 0.816650390625}, {"start": 1959.35, "end": 1960.01, "word": " سيكوانس", "probability": 0.967529296875}, {"start": 1960.01, "end": 1961.01, "word": " هي", "probability": 0.82666015625}, {"start": 1961.01, "end": 1962.25, "word": " عناصر", "probability": 0.9860026041666666}, {"start": 1962.25, "end": 1962.81, "word": " أو", "probability": 0.6728515625}, {"start": 1962.81, "end": 1963.43, "word": " حدود", "probability": 0.9861653645833334}, {"start": 1963.43, "end": 1963.91, "word": " من", "probability": 0.990234375}, {"start": 1963.91, "end": 1964.43, "word": " السيكوانس", "probability": 0.939697265625}, {"start": 1964.43, "end": 1965.01, "word": " العصلية", "probability": 0.78997802734375}, {"start": 1965.01, "end": 1966.13, "word": " لكن", "probability": 0.8115234375}, {"start": 1966.13, "end": 1966.39, "word": " مش", "probability": 0.98095703125}, {"start": 1966.39, "end": 1966.61, "word": " أي", "probability": 0.72607421875}, {"start": 1966.61, "end": 1967.15, "word": " حدود", "probability": 0.9915364583333334}, {"start": 1967.15, "end": 1968.07, "word": " لازم", "probability": 0.9632161458333334}, {"start": 1968.07, "end": 1968.35, "word": " تكون", "probability": 0.974365234375}, {"start": 1968.35, "end": 1969.09, "word": " مرتبة", "probability": 0.97412109375}, {"start": 1969.09, "end": 1970.33, "word": " بحيث", "probability": 0.981689453125}, {"start": 1970.33, "end": 1970.51, "word": " أن", "probability": 0.453125}, {"start": 1970.51, "end": 1971.31, "word": " المؤشرات", "probability": 0.957275390625}, {"start": 1971.31, "end": 1972.17, "word": " تبعتها", "probability": 0.9527587890625}, {"start": 1972.17, "end": 1973.27, "word": " اتشكل", "probability": 0.882080078125}, {"start": 1973.27, "end": 1974.45, "word": " strictly", "probability": 0.7470703125}, {"start": 1974.45, "end": 1975.43, "word": " increasing", "probability": 0.94970703125}, {"start": 1975.43, "end": 1976.25, "word": " sequence", "probability": 0.95556640625}, {"start": 1976.25, "end": 1976.47, "word": " of", "probability": 0.96533203125}, {"start": 1976.47, "end": 1976.89, "word": " natural", "probability": 0.62451171875}, {"start": 1976.89, "end": 1977.41, "word": " numbers", "probability": 0.9267578125}], "temperature": 1.0}, {"id": 74, "seek": 200634, "start": 1978.58, "end": 2006.34, "text": "تمام؟ زي هيك إذاً هذا هو تعريف الـ subsequence إذا لو في أنا عندي sequence XN واخدت strictly increasing sequence of natural numbers فال sequence اللي المؤشرات تبعتها هي ال sequence RN اللي هي هذه عناصرها بنسميها subsequence من ال sequence XN", "tokens": [2655, 15042, 2304, 22807, 30767, 1829, 39896, 4117, 11933, 15730, 14111, 23758, 31439, 37279, 16572, 5172, 2423, 39184, 13924, 655, 11933, 15730, 45164, 8978, 41850, 18871, 16254, 8310, 1783, 45, 4032, 47283, 3215, 2655, 20792, 5662, 8310, 295, 3303, 3547, 6156, 6027, 8310, 13672, 1829, 9673, 33604, 46309, 9307, 6055, 3555, 34268, 11296, 39896, 2423, 8310, 45702, 13672, 1829, 39896, 29538, 18871, 33546, 2288, 11296, 44945, 38251, 1829, 11296, 13924, 655, 9154, 2423, 8310, 1783, 45], "avg_logprob": -0.19521104437964304, "compression_ratio": 1.5841121495327102, "no_speech_prob": 0.0, "words": [{"start": 1978.58, "end": 1979.52, "word": "تمام؟", "probability": 0.648193359375}, {"start": 1979.52, "end": 1979.9, "word": " زي", "probability": 0.81982421875}, {"start": 1979.9, "end": 1980.34, "word": " هيك", "probability": 0.943115234375}, {"start": 1980.34, "end": 1983.48, "word": " إذاً", "probability": 0.5382486979166666}, {"start": 1983.48, "end": 1983.66, "word": " هذا", "probability": 0.68798828125}, {"start": 1983.66, "end": 1983.86, "word": " هو", "probability": 0.99365234375}, {"start": 1983.86, "end": 1984.3, "word": " تعريف", "probability": 0.98876953125}, {"start": 1984.3, "end": 1984.48, "word": " الـ", "probability": 0.7529296875}, {"start": 1984.48, "end": 1985.22, "word": " subsequence", "probability": 0.728759765625}, {"start": 1985.22, "end": 1986.14, "word": " إذا", "probability": 0.881591796875}, {"start": 1986.14, "end": 1986.36, "word": " لو", "probability": 0.7822265625}, {"start": 1986.36, "end": 1986.48, "word": " في", "probability": 0.677734375}, {"start": 1986.48, "end": 1986.62, "word": " أنا", "probability": 0.71484375}, {"start": 1986.62, "end": 1986.9, "word": " عندي", "probability": 0.866455078125}, {"start": 1986.9, "end": 1987.32, "word": " sequence", "probability": 0.92822265625}, {"start": 1987.32, "end": 1988.1, "word": " XN", "probability": 0.3221435546875}, {"start": 1988.1, "end": 1989.5, "word": " واخدت", "probability": 0.873291015625}, {"start": 1989.5, "end": 1990.14, "word": " strictly", "probability": 0.72509765625}, {"start": 1990.14, "end": 1990.84, "word": " increasing", "probability": 0.96044921875}, {"start": 1990.84, "end": 1991.72, "word": " sequence", "probability": 0.96533203125}, {"start": 1991.72, "end": 1992.06, "word": " of", "probability": 0.9453125}, {"start": 1992.06, "end": 1992.52, "word": " natural", "probability": 0.92236328125}, {"start": 1992.52, "end": 1993.12, "word": " numbers", "probability": 0.8916015625}, {"start": 1993.12, "end": 1994.94, "word": " فال", "probability": 0.949462890625}, {"start": 1994.94, "end": 1995.66, "word": " sequence", "probability": 0.82373046875}, {"start": 1995.66, "end": 1996.12, "word": " اللي", "probability": 0.914306640625}, {"start": 1996.12, "end": 1996.88, "word": " المؤشرات", "probability": 0.971435546875}, {"start": 1996.88, "end": 1997.62, "word": " تبعتها", "probability": 0.961669921875}, {"start": 1997.62, "end": 1998.78, "word": " هي", "probability": 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"word": " X", "probability": 0.6611328125}, {"start": 2213.97, "end": 2214.35, "word": " in", "probability": 0.3359375}, {"start": 2214.35, "end": 2214.95, "word": " فأي", "probability": 0.9192708333333334}, {"start": 2214.95, "end": 2215.31, "word": " M", "probability": 0.9189453125}, {"start": 2215.31, "end": 2215.71, "word": " تل", "probability": 0.83056640625}, {"start": 2215.71, "end": 2216.43, "word": " هو", "probability": 0.876953125}, {"start": 2216.43, "end": 2216.73, "word": " sub", "probability": 0.96435546875}, {"start": 2216.73, "end": 2217.23, "word": "-sequence", "probability": 0.9650065104166666}, {"start": 2217.23, "end": 2217.65, "word": " منها", "probability": 0.993896484375}], "temperature": 1.0}, {"id": 83, "seek": 224748, "start": 2220.14, "end": 2247.48, "text": "كذلك لو أخدت أي sequence xn فالـ sequence x اللي الحد اللي عم تبعها المؤشر تبعه 2 أس n هذي برضه subsequence زي ما شوفنا و x2n الحدود الزوجية لو أخدت الحدود الزوجية فقط فهذا بعطيني subsequence و لو أخدت الحدود الفردية تعطيني subsequence ثانية و لا كده", "tokens": [4117, 8848, 23275, 45164, 5551, 9778, 3215, 2655, 36632, 8310, 2031, 77, 6156, 6027, 39184, 8310, 2031, 13672, 1829, 21542, 3215, 13672, 1829, 6225, 2304, 6055, 3555, 3615, 11296, 9673, 33604, 46309, 6055, 3555, 3615, 3224, 568, 5551, 3794, 297, 8032, 8848, 1829, 4724, 43042, 3224, 13924, 655, 30767, 1829, 19446, 13412, 38688, 8315, 4032, 2031, 17, 77, 21542, 3215, 23328, 6024, 110, 29245, 10632, 45164, 5551, 9778, 3215, 2655, 21542, 3215, 23328, 6024, 110, 29245, 10632, 6156, 47432, 6156, 3224, 15730, 45030, 9566, 9957, 1829, 13924, 655, 4032, 45164, 5551, 9778, 3215, 2655, 21542, 3215, 23328, 27188, 2288, 3215, 10632, 37279, 9566, 9957, 1829, 13924, 655, 38637, 7649, 10632, 4032, 20193, 9122, 3215, 3224], "avg_logprob": -0.19706357270479202, "compression_ratio": 2.0307692307692307, "no_speech_prob": 0.0, "words": [{"start": 2220.14, "end": 2220.74, "word": "كذلك", "probability": 0.8513997395833334}, {"start": 2220.74, 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2232.74, "end": 2234.74, "word": " و", "probability": 0.404541015625}, {"start": 2234.74, "end": 2236.02, "word": " x2n", "probability": 0.7220052083333334}, {"start": 2236.02, "end": 2236.68, "word": " الحدود", "probability": 0.97412109375}, {"start": 2236.68, "end": 2237.52, "word": " الزوجية", "probability": 0.959716796875}, {"start": 2237.52, "end": 2239.04, "word": " لو", "probability": 0.861328125}, {"start": 2239.04, "end": 2239.34, "word": " أخدت", "probability": 0.990478515625}, {"start": 2239.34, "end": 2239.64, "word": " الحدود", "probability": 0.98876953125}, {"start": 2239.64, "end": 2240.2, "word": " الزوجية", "probability": 0.9830322265625}, {"start": 2240.2, "end": 2240.76, "word": " فقط", "probability": 0.99658203125}, {"start": 2240.76, "end": 2241.34, "word": " فهذا", "probability": 0.8097330729166666}, {"start": 2241.34, "end": 2241.78, "word": " بعطيني", "probability": 0.70843505859375}, {"start": 2241.78, "end": 2242.52, "word": " subsequence", "probability": 0.9931640625}, {"start": 2242.52, "end": 2243.08, "word": " و", "probability": 0.58447265625}, {"start": 2243.08, "end": 2243.22, "word": " لو", "probability": 0.90478515625}, {"start": 2243.22, "end": 2243.54, "word": " أخدت", "probability": 0.987548828125}, {"start": 2243.54, "end": 2243.96, "word": " الحدود", "probability": 0.9954427083333334}, {"start": 2243.96, "end": 2244.74, "word": " الفردية", "probability": 0.9747314453125}, {"start": 2244.74, "end": 2245.84, "word": " تعطيني", "probability": 0.8743896484375}, {"start": 2245.84, "end": 2246.52, "word": " subsequence", "probability": 0.98681640625}, {"start": 2246.52, "end": 2246.9, "word": " ثانية", "probability": 0.94384765625}, {"start": 2246.9, "end": 2247.06, "word": " و", "probability": 0.7998046875}, {"start": 2247.06, "end": 2247.16, "word": " لا", "probability": 0.280517578125}, {"start": 2247.16, "end": 2247.48, "word": " كده", "probability": 0.8806966145833334}], "temperature": 1.0}, {"id": 84, "seek": 226698, "start": 2248.15, "end": 2266.99, "text": "الان سؤال اللى بهمنا احنا ما هي علاقة ال sequence بال subsequence من حيث ال convergence و ال divergence؟", "tokens": [6027, 7649, 8608, 33604, 6027, 13672, 7578, 4724, 16095, 8315, 1975, 5016, 8315, 19446, 39896, 11203, 995, 28671, 2423, 8310, 20666, 13924, 655, 9154, 11331, 1829, 12984, 2423, 32181, 4032, 2423, 47387, 22807], "avg_logprob": -0.2743566307951422, "compression_ratio": 1.3636363636363635, "no_speech_prob": 0.0, "words": [{"start": 2248.15, "end": 2248.51, "word": "الان", "probability": 0.31915283203125}, {"start": 2248.51, "end": 2258.47, "word": " سؤال", "probability": 0.8645833333333334}, {"start": 2258.47, "end": 2258.65, "word": " اللى", "probability": 0.699951171875}, {"start": 2258.65, "end": 2259.11, "word": " بهمنا", "probability": 0.8855794270833334}, {"start": 2259.11, "end": 2259.49, "word": " احنا", "probability": 0.9532877604166666}, {"start": 2259.49, "end": 2260.43, "word": " ما", "probability": 0.52880859375}, {"start": 2260.43, "end": 2260.69, "word": " هي", "probability": 0.50732421875}, {"start": 2260.69, "end": 2261.23, "word": " علاقة", "probability": 0.97265625}, {"start": 2261.23, "end": 2261.47, "word": " ال", "probability": 0.84912109375}, {"start": 2261.47, "end": 2261.93, "word": " sequence", "probability": 0.607421875}, {"start": 2261.93, "end": 2262.19, "word": " بال", "probability": 0.96484375}, {"start": 2262.19, "end": 2263.77, "word": " subsequence", "probability": 0.702880859375}, {"start": 2263.77, "end": 2264.51, "word": " من", "probability": 0.88720703125}, {"start": 2264.51, "end": 2264.87, "word": " حيث", "probability": 0.9676106770833334}, {"start": 2264.87, "end": 2265.01, "word": " ال", "probability": 0.91845703125}, {"start": 2265.01, "end": 2265.49, "word": " convergence", "probability": 0.88916015625}, {"start": 2265.49, "end": 2265.87, "word": " و", "probability": 0.52294921875}, {"start": 2265.87, "end": 2265.93, "word": " ال", "probability": 0.88427734375}, {"start": 2265.93, "end": 2266.99, "word": " divergence؟", "probability": 0.912841796875}], "temperature": 1.0}, {"id": 85, "seek": 230317, "start": 2274.41, "end": 2303.17, "text": "يعني لو كانت ال sequence convergent لو في اندي سيكوانس xn convergent ل x واخدت أي sub sequence منها هل هذه ال sequence لازم تكون convergent زيها ولا divergent لازم تكون convergent و ال limit تبعتها نفس ال limit و لها نفس ال limit ماشي لحظة", "tokens": [40228, 22653, 45164, 25961, 2655, 2423, 8310, 9652, 6930, 45164, 8978, 16472, 16254, 8608, 1829, 4117, 2407, 7649, 3794, 2031, 77, 9652, 6930, 5296, 2031, 4032, 47283, 3215, 2655, 36632, 1422, 8310, 9154, 11296, 8032, 1211, 29538, 2423, 8310, 5296, 31377, 2304, 6055, 30544, 9652, 6930, 30767, 1829, 11296, 49429, 18558, 6930, 5296, 31377, 2304, 6055, 30544, 9652, 6930, 4032, 2423, 4948, 6055, 3555, 34268, 11296, 8717, 36178, 2423, 4948, 4032, 5296, 11296, 8717, 36178, 2423, 4948, 3714, 33599, 1829, 5296, 5016, 19913, 3660], "avg_logprob": -0.2790441134396721, "compression_ratio": 1.8833333333333333, "no_speech_prob": 0.0, "words": [{"start": 2274.41, "end": 2274.77, "word": "يعني", "probability": 0.712890625}, {"start": 2274.77, "end": 2274.93, "word": " لو", "probability": 0.93505859375}, {"start": 2274.93, "end": 2275.29, "word": " كانت", "probability": 0.978515625}, {"start": 2275.29, "end": 2275.41, "word": " ال", "probability": 0.87451171875}, {"start": 2275.41, "end": 2275.69, "word": " sequence", "probability": 0.6396484375}, {"start": 2275.69, "end": 2276.41, "word": " convergent", "probability": 0.643798828125}, {"start": 2276.41, "end": 2276.57, "word": " لو", "probability": 0.55810546875}, {"start": 2276.57, "end": 2276.73, "word": " في", "probability": 0.456298828125}, {"start": 2276.73, "end": 2276.95, "word": " اندي", "probability": 0.652587890625}, {"start": 2276.95, "end": 2277.45, "word": " سيكوانس", "probability": 0.766357421875}, {"start": 2277.45, "end": 2277.85, "word": " xn", "probability": 0.39794921875}, {"start": 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2283.61, "end": 2284.01, "word": " لازم", "probability": 0.9861653645833334}, {"start": 2284.01, "end": 2284.43, "word": " تكون", "probability": 0.97021484375}, {"start": 2284.43, "end": 2285.29, "word": " convergent", "probability": 0.945556640625}, {"start": 2285.29, "end": 2287.49, "word": " زيها", "probability": 0.892578125}, {"start": 2287.49, "end": 2287.69, "word": " ولا", "probability": 0.767578125}, {"start": 2287.69, "end": 2288.47, "word": " divergent", "probability": 0.75}, {"start": 2288.47, "end": 2289.79, "word": " لازم", "probability": 0.8767903645833334}, {"start": 2289.79, "end": 2290.01, "word": " تكون", "probability": 0.99267578125}, {"start": 2290.01, "end": 2290.81, "word": " convergent", "probability": 0.92578125}, {"start": 2290.81, "end": 2291.57, "word": " و", "probability": 0.740234375}, {"start": 2291.57, "end": 2291.69, "word": " ال", "probability": 0.38134765625}, {"start": 2291.69, "end": 2291.89, "word": " limit", "probability": 0.95751953125}, {"start": 2291.89, "end": 2292.65, "word": " تبعتها", "probability": 0.77789306640625}, {"start": 2292.65, "end": 2293.09, "word": " نفس", "probability": 0.9970703125}, {"start": 2293.09, "end": 2293.31, "word": " ال", "probability": 0.81103515625}, {"start": 2293.31, "end": 2293.69, "word": " limit", "probability": 0.97607421875}, {"start": 2293.69, "end": 2295.17, "word": " و", "probability": 0.4384765625}, {"start": 2295.17, "end": 2295.41, "word": " لها", "probability": 0.6591796875}, {"start": 2295.41, "end": 2295.81, "word": " نفس", "probability": 0.998046875}, {"start": 2295.81, "end": 2296.11, "word": " ال", "probability": 0.8759765625}, {"start": 2296.11, "end": 2296.45, "word": " limit", "probability": 0.9873046875}, {"start": 2296.45, "end": 2302.77, "word": " ماشي", "probability": 0.8802083333333334}, {"start": 2302.77, "end": 2303.17, "word": " لحظة", "probability": 0.5130615234375}], "temperature": 1.0}, {"id": 86, "seek": 230986, "start": 2309.06, "end": 2309.86, "text": "كثير من الناس", "tokens": [4117, 12984, 13546, 9154, 2423, 8315, 3794], "avg_logprob": -0.587890625, "compression_ratio": 0.6857142857142857, "no_speech_prob": 5.960464477539063e-08, "words": [{"start": 2309.06, "end": 2309.48, "word": "كثير", "probability": 0.6656087239583334}, {"start": 2309.48, "end": 2309.6, "word": " من", "probability": 0.4716796875}, {"start": 2309.6, "end": 2309.86, "word": " الناس", "probability": 0.9391276041666666}], "temperature": 1.0}, {"id": 87, "seek": 234533, "start": 2319.93, "end": 2345.33, "text": "إذا كمان مرة بهمني أنا أنه لو في عندي sequence نظرية هذه بتقول لو في عندي sequence xn of real numbers وكانت ال sequence هذه convergent ل x فأي subsequence منها بتكون convergent و ال limit تبعتها هي نفس ال limit لل sequence xn", "tokens": [28814, 15730, 9122, 2304, 7649, 3714, 25720, 4724, 16095, 22653, 41850, 14739, 3224, 45164, 8978, 18871, 16254, 8310, 8717, 19913, 2288, 10632, 29538, 39894, 39648, 45164, 8978, 18871, 16254, 8310, 2031, 77, 295, 957, 3547, 4032, 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{"start": 2504.94, "end": 2505.6, "word": " النظرية", "probability": 0.8145751953125}, {"start": 2505.6, "end": 2506.0, "word": " هذه", "probability": 0.8857421875}, {"start": 2506.0, "end": 2507.34, "word": " أو", "probability": 0.8486328125}, {"start": 2507.34, "end": 2507.72, "word": " نوجد", "probability": 0.7857259114583334}, {"start": 2507.72, "end": 2508.26, "word": " العلاقة", "probability": 0.9599609375}, {"start": 2508.26, "end": 2508.56, "word": " بين", "probability": 0.7568359375}, {"start": 2508.56, "end": 2509.1, "word": " كيف", "probability": 0.940185546875}, {"start": 2509.1, "end": 2509.6, "word": " نثبت", "probability": 0.991455078125}, {"start": 2509.6, "end": 2509.8, "word": " ال", "probability": 0.93310546875}, {"start": 2509.8, "end": 2510.3, "word": " convergence", "probability": 0.89697265625}, {"start": 2510.3, "end": 2510.66, "word": " لل", "probability": 0.73095703125}, {"start": 2510.66, "end": 2511.14, "word": " sequence", "probability": 0.85302734375}, 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"end": 2521.07, "word": " و", "probability": 0.7099609375}, {"start": 2521.07, "end": 2521.27, "word": " هيك", "probability": 0.892822265625}, {"start": 2521.27, "end": 2521.63, "word": " نكون", "probability": 0.61944580078125}, {"start": 2521.63, "end": 2521.89, "word": " يعني", "probability": 0.861572265625}, {"start": 2521.89, "end": 2523.23, "word": " تقريبا", "probability": 0.9953125}, {"start": 2523.23, "end": 2523.51, "word": " ..", "probability": 0.61474609375}, {"start": 2523.51, "end": 2524.23, "word": " هنكمل", "probability": 0.9593505859375}, {"start": 2524.23, "end": 2524.37, "word": " ان", "probability": 0.74169921875}, {"start": 2524.37, "end": 2524.49, "word": " شاء", "probability": 0.976318359375}, {"start": 2524.49, "end": 2524.55, "word": " الله", "probability": 0.96435546875}, {"start": 2524.55, "end": 2524.91, "word": " المرة", "probability": 0.846923828125}, {"start": 2524.91, "end": 2525.45, "word": " الجاية", "probability": 0.9078776041666666}, {"start": 2525.45, "end": 2525.85, "word": " و", "probability": 0.97705078125}, {"start": 2525.85, "end": 2527.11, "word": " ..", "probability": 0.6240234375}, {"start": 2527.11, "end": 2528.71, "word": " هن", "probability": 0.900390625}, {"start": 2528.71, "end": 2528.93, "word": " ..", "probability": 0.9482421875}, {"start": 2528.93, "end": 2529.61, "word": " هنوقف", "probability": 0.978759765625}, {"start": 2529.61, "end": 2530.15, "word": " استخدام", "probability": 0.9827880859375}, {"start": 2530.15, "end": 2530.29, "word": " ال", "probability": 0.9814453125}, {"start": 2530.29, "end": 2531.05, "word": " powerpoint", "probability": 0.7080078125}, {"start": 2531.05, "end": 2532.29, "word": " ابتداء", "probability": 0.7764892578125}, {"start": 2532.29, "end": 2532.45, "word": " من", "probability": 0.94384765625}, {"start": 2532.45, "end": 2533.01, "word": " المحاضرة", "probability": 0.9893798828125}, {"start": 2533.01, "end": 2533.43, "word": " الجاية", "probability": 0.9581705729166666}, 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\ No newline at end of file
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/Mk2487qcZF8_raw.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/Mk2487qcZF8_raw.srt
new file mode 100644
index 0000000000000000000000000000000000000000..b3dba70e28b6ad2c48e03bfeb13e1b8c446a0275
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/Mk2487qcZF8_raw.srt
@@ -0,0 +1,1760 @@
+1
+00:00:21,450 --> 00:00:27,950
+Okay إذا احنا في المحاضرة الأخيرة أخدنا الـ
+
+2
+00:00:27,950 --> 00:00:33,670
+monotone convergence theorem وشوفنا
+
+3
+00:00:33,670 --> 00:00:38,970
+أنه في النظرية هذه لو في عندي sequence و ال
+
+4
+00:00:38,970 --> 00:00:42,710
+sequence هذه monotone يعني increasing أو
+
+5
+00:00:42,710 --> 00:00:48,510
+decreasing فعشان فبتكون convergent if and only if
+
+6
+00:00:48,510 --> 00:00:53,060
+it is boundedإذا الـ monotone sequence converges
+
+7
+00:00:53,060 --> 00:01:01,360
+if and only if it is bounded طيب
+
+8
+00:01:01,360 --> 00:01:04,420
+ال monotone sequence نوعين إما increasing أو
+
+9
+00:01:04,420 --> 00:01:07,360
+decreasing فلو كانت ال sequence increasing و طبعا
+
+10
+00:01:07,360 --> 00:01:10,940
+bounded فشوفنا إنها بتكون convergent حسب ال
+
+11
+00:01:10,940 --> 00:01:14,040
+statement الأول و ال limit تبعتها بساوي ال
+
+12
+00:01:14,040 --> 00:01:17,280
+supremum اللي لها ك set و لو كانت ال sequence
+
+13
+00:01:17,280 --> 00:01:22,420
+decreasing و بالطبع boundedفحسب ال statement الأول
+
+14
+00:01:22,420 --> 00:01:28,720
+تطلع convergence ونهايتها هي ال infimum تبعها ك Z
+
+15
+00:01:28,720 --> 00:01:33,040
+وشوفنا
+
+16
+00:01:33,040 --> 00:01:35,560
+برهانة مغرية في المحاضرة السابقة
+
+17
+00:01:38,060 --> 00:01:43,060
+الان بنشوف كيف نطبق النظرية في إثبات إن certain
+
+18
+00:01:43,060 --> 00:01:47,200
+sequences are convergent أو divergent النظرية هذه
+
+19
+00:01:47,200 --> 00:01:51,180
+بالمناسبة ممكن نستخدمها في إثبات إنه monotone
+
+20
+00:01:51,180 --> 00:01:55,960
+sequence معينة إما convergent أو divergent عشان
+
+21
+00:01:55,960 --> 00:01:59,120
+أثبت إن ال monotone sequence إذا عندي أنا monotone
+
+22
+00:01:59,120 --> 00:02:02,760
+sequence عشان أثبت إنها convergent لازم أثبت إنها
+
+23
+00:02:02,760 --> 00:02:08,980
+bounded العكس لو في عنده monotone sequenceوبدي
+
+24
+00:02:08,980 --> 00:02:14,640
+اثبت انها divergent يكفي ان اثبت انها unbounded
+
+25
+00:02:14,640 --> 00:02:21,840
+not bounded فهي ان ال sequence xn بساوي واحد على n
+
+26
+00:02:21,840 --> 00:02:27,240
+هاد ال sequence معروف انه ال limit انها convergent
+
+27
+00:02:27,240 --> 00:02:34,630
+وits limit is zero زيها زي ال sequence واحد على nو
+
+28
+00:02:34,630 --> 00:02:38,270
+ممكن تبرهن أن الـ sequence هذه convergence و
+
+29
+00:02:38,270 --> 00:02:42,030
+نهايتها بالساعة و سفر باستخدام تعريف epsilon
+
+30
+00:02:42,030 --> 00:02:49,070
+capital N لل limit بالظبط زي ما عملنا في برهان ال
+
+31
+00:02:49,070 --> 00:02:52,830
+limit أن ال limit لل sequence واحد علي N بالساعة و
+
+32
+00:02:52,830 --> 00:02:57,810
+سفر باستخدام الarchimedean property فهذا برهان
+
+33
+00:02:57,810 --> 00:03:04,810
+ممكن أي واحدة فيكم تكتبهاللي هو باستخدام تعريف
+
+34
+00:03:04,810 --> 00:03:08,110
+epsilon capital N زائد الـarchimedean property
+
+35
+00:03:08,110 --> 00:03:13,290
+بالظبط زي ما أثبتنا limit 1 على N بساوية 0 اليوم
+
+36
+00:03:13,290 --> 00:03:16,550
+هنشوف برهان تاني باستخدام الـ monotone convergence
+
+37
+00:03:16,550 --> 00:03:16,990
+theorem
+
+38
+00:03:20,740 --> 00:03:25,460
+السيكوانس هي بالحد العام ال n term تبعها xn one
+
+39
+00:03:25,460 --> 00:03:28,960
+over square root of n طبعا square root of n أصغر
+
+40
+00:03:28,960 --> 00:03:32,720
+من square root of z واحد لأي عدد طبيعي وبالتالي
+
+41
+00:03:32,720 --> 00:03:37,680
+مقلوب لكبير أصغر من مقلوب لكبير هذا xn زاد واحد
+
+42
+00:03:37,680 --> 00:03:44,240
+وهذا xn أنا عندي sequence is such that xn زاد واحد
+
+43
+00:03:44,240 --> 00:03:48,560
+أصغر من xn هذا معناه أن ال sequence is increasing
+
+44
+00:03:49,820 --> 00:03:54,740
+كذلك الـ sequence هذه is bounded لأنه هي فيه عدد
+
+45
+00:03:54,740 --> 00:03:59,700
+موجب M بساوي واحد عدد موجب بحيث أنه absolute xn
+
+46
+00:03:59,700 --> 00:04:04,780
+أصغر من أو يساوي M لكل N لأن أنا فيها ال sequence
+
+47
+00:04:04,780 --> 00:04:12,000
+increasingdecreasing و bounded اذا by monotone
+
+48
+00:04:12,000 --> 00:04:16,620
+convergence theorem ال sequence هذه هتكون
+
+49
+00:04:16,620 --> 00:04:23,220
+convergent و ال limit تبعتها بساوي ال infimum طيب
+
+50
+00:04:23,220 --> 00:04:27,740
+ال infimum للمجموعة هذه بتساوي سفر
+
+51
+00:04:30,570 --> 00:04:35,710
+وبرهان ذلك شبيه ببرهان الـ infimum للـ sequence 1
+
+52
+00:04:35,710 --> 00:04:40,310
+على n بالساوي 0 باستخدام الـ Archimedean property
+
+53
+00:04:40,310 --> 00:04:44,350
+راجعوا برهان أن الـ infimum للـ sequence 1 على n
+
+54
+00:04:44,350 --> 00:04:48,610
+وهذا ثمتناه في المحاضرات السابقة راجعوا البرهان
+
+55
+00:04:48,610 --> 00:04:54,800
+وكتبوا برهان مشابه لهبنفس الطريقة نثبت ان الانثرام
+
+56
+00:04:54,800 --> 00:04:58,660
+لسيكوانس هادى او الست هادى سفر اذا حسب الـ
+
+57
+00:04:58,660 --> 00:05:01,400
+monotone convergence theorem ال sequence واحد على
+
+58
+00:05:01,400 --> 00:05:05,700
+جذر M is convergent و ال limit تبعتها بساوي
+
+59
+00:05:05,700 --> 00:05:10,240
+الانثرام تبعها اللى هو سفر اذا هى مثال على تطبيق
+
+60
+00:05:10,240 --> 00:05:15,570
+الـ monotone convergence theoremكذلك ممكن برضه زي
+
+61
+00:05:15,570 --> 00:05:18,810
+ما قلتلكم نستخدم الـ monotone convergence theorem
+
+62
+00:05:18,810 --> 00:05:26,750
+في أثبات أن سيكوانس زي هذه is divergent نشوف
+
+63
+00:05:26,750 --> 00:05:30,870
+مع بعض كيف .. إذا هنا هاي سيكوانس الحد العام تبعها
+
+64
+00:05:30,870 --> 00:05:37,490
+xn هذا الint partial sum بالمناسبة هذا الint
+
+65
+00:05:37,490 --> 00:05:43,330
+partial sum في ال harmonic seriesسيجما من K بساول
+
+66
+00:05:43,330 --> 00:05:50,210
+واحد to infinity لواحد على K وهد
+
+67
+00:05:50,210 --> 00:05:53,110
+ال harmonic series is divergent معروف في calculus
+
+68
+00:05:53,110 --> 00:06:00,190
+بقى ال series هد is divergent وهد الحد العام في ال
+
+69
+00:06:00,190 --> 00:06:04,730
+sequence of partial sums وفي calculus بقى درسنا
+
+70
+00:06:04,730 --> 00:06:10,330
+إذا بتذكروا إذا فاكرين حاجةمن الموضوع هذا انه a
+
+71
+00:06:10,330 --> 00:06:13,970
+series converges if and only if ال sequence of
+
+72
+00:06:13,970 --> 00:06:18,130
+partial sums is convergent فلو ال series is
+
+73
+00:06:18,130 --> 00:06:21,130
+divergent ال sequence of partial sums is divergent
+
+74
+00:06:21,130 --> 00:06:24,830
+هذه هي ال sequence of partial sums حدث بتنها
+
+75
+00:06:24,830 --> 00:06:31,150
+divergent بس مش باستخدام calculus بقى باستخدام ال
+
+76
+00:06:31,150 --> 00:06:37,300
+monotone convergence theoremطيب ال sequence هي
+
+77
+00:06:37,300 --> 00:06:43,920
+الحد العام xn إذا الحد رقم n زياد واحد هي بنضيف
+
+78
+00:06:43,920 --> 00:06:49,400
+زياد واحد على n زياد واحد للمجموع هذا اللي هو xn
+
+79
+00:06:49,400 --> 00:06:54,320
+صح؟ وبالتالي زي ما أنتوا شايفين الحد xn زياد واحد
+
+80
+00:06:54,320 --> 00:07:00,560
+هو الحد xn زائد حد موجب وبالتالي هذا أكبر من xn
+
+81
+00:07:02,310 --> 00:07:06,670
+الكلام هذا صحيح لكل n إذاً هذا معناه إن ال
+
+82
+00:07:06,670 --> 00:07:14,690
+sequence xn is increasing، متزايدة، الآن أنا في
+
+83
+00:07:14,690 --> 00:07:19,410
+ندي sequence xn، هذه الحد الآن تبعها، و ال
+
+84
+00:07:19,410 --> 00:07:25,600
+sequence هذه increasing، monotone يعنيالان الـ
+
+85
+00:07:25,600 --> 00:07:30,700
+monotone convergence بتقول لي عشان أثبت أن ال
+
+86
+00:07:30,700 --> 00:07:34,260
+sequence هذي convergent لازم أثبت أنها bounded
+
+87
+00:07:34,260 --> 00:07:39,860
+وعشان أثبت أنها divergent لازم أثبت أنها unbounded
+
+88
+00:07:39,860 --> 00:07:44,520
+فتعالوا نفحص هل هي bounded ولا لأ إذا طلعت bounded
+
+89
+00:07:44,520 --> 00:07:48,640
+بتكون convergent إذا طلعت unbounded بتكون
+
+90
+00:07:48,640 --> 00:07:49,360
+divergent
+
+91
+00:07:52,580 --> 00:07:57,960
+تعالوا نشوف الحد العام يعني هنا في حاخد ال
+
+92
+00:07:57,960 --> 00:08:04,200
+sequence بدل x in x اللي الحد العام تبعها two to n
+
+93
+00:08:04,200 --> 00:08:09,640
+هذه عبارة عن subsequence فهذه subsequence من ال
+
+94
+00:08:09,640 --> 00:08:10,820
+sequence x in
+
+95
+00:08:16,700 --> 00:08:21,480
+يعني هذه جزء من ال sequence هذه هذه أول حد فيها x
+
+96
+00:08:21,480 --> 00:08:27,400
+رقم مؤشر تبعه اتنين صح؟ يعني هذه حدود ال sequence
+
+97
+00:08:27,400 --> 00:08:32,620
+هذه x اتنين لما n بساوة واحد بعدين اللي بعده x
+
+98
+00:08:32,620 --> 00:08:40,100
+أربع بعدين x تمام يعني و هكذا طبعا هذه الحدود هذه
+
+99
+00:08:40,100 --> 00:08:44,980
+كلها موجودة هنا صح؟ فهذه بنسميها subsequence من ال
+
+100
+00:08:44,980 --> 00:08:49,090
+sequence الأصليالان انا بدي اخد الحد العام لل sub
+
+101
+00:08:49,090 --> 00:08:56,750
+sequence دي اللي المؤشر تبعه 2 أس n بدل n طيب
+
+102
+00:08:56,750 --> 00:09:01,170
+انا عند xn بساوي المجموعة هذا واحد زاد نص زاد تلت
+
+103
+00:09:01,170 --> 00:09:06,290
+اخر حد واحد على n طب لما بدي ال n ب 2 أس n هيطلع
+
+104
+00:09:06,290 --> 00:09:10,650
+عند المجموعة واحد زاد نص زاد تلت الى اخر حد واحد
+
+105
+00:09:10,650 --> 00:09:16,620
+على 2 أس n صح؟ هذا من تعريف ال sequenceالان الحدود
+
+106
+00:09:16,620 --> 00:09:25,340
+هذه تبع المجموع هعملها blocks، بلوكات فواحد لوحده
+
+107
+00:09:25,340 --> 00:09:31,320
+في block، نص لوحده، هاخد التلت والربع مع بعض،
+
+108
+00:09:31,320 --> 00:09:38,080
+بعدين ال block الرابع هتكون خمس و سدس و سبعة و
+
+109
+00:09:38,080 --> 00:09:44,840
+تمان، أربع حدود مع بعض، جمهم مع بعضو هكذا إلى ال
+
+110
+00:09:44,840 --> 00:09:51,220
+block الأخيرة اللي هي 1 على 2 أس N سالب 1 زاد 1
+
+111
+00:09:51,220 --> 00:09:56,660
+إلى 1 على 2 أس N طيب
+
+112
+00:09:56,660 --> 00:10:02,080
+هاي ال 1 نزل النص، الآن التلت هذا، التلت أكبر من
+
+113
+00:10:02,080 --> 00:10:06,820
+ربع إن علامة التساوي هذه صارت أكبر، يعني بدلت
+
+114
+00:10:06,820 --> 00:10:12,650
+التلت بربع، والتلت أكبر من ربعفصار مجموع ربعين
+
+115
+00:10:12,650 --> 00:10:16,090
+الان في ال block اللي بعديها في عندي خمس و سُدس و
+
+116
+00:10:16,090 --> 00:10:22,110
+سبعة تمن كل واحد منهم أكبر من أو يساوي التمن فعشان
+
+117
+00:10:22,110 --> 00:10:27,450
+يصير في عندي هنا تمن زي تمن زي تمن أربع مرات اللي
+
+118
+00:10:27,450 --> 00:10:34,390
+هو بطلع نص أربعة أتمان نص وهنا ربعين نص صح؟ وهنا
+
+119
+00:10:34,390 --> 00:10:39,910
+هذا الحد اللي هنا هذاأكبر من واحد على اتنين أسئن
+
+120
+00:10:39,910 --> 00:10:44,450
+لأنه اتنين أسئن أكبر من اتنين أسئن سالب واحد زائد
+
+121
+00:10:44,450 --> 00:10:49,050
+واحد لكل ان إذاً هذا أكبر من واحد على اتنين أسئن
+
+122
+00:10:49,050 --> 00:10:53,410
+والبعد أكبر من واحد على اتنين أسئن و هكذا إذاً هنا
+
+123
+00:10:53,410 --> 00:10:57,550
+عندي واحد على اتنين أسئن مجموعة على نفسه اتنين
+
+124
+00:10:57,550 --> 00:11:03,890
+أسئن سالب واحد من المراتفمجموهم بيساوي مجموها دول
+
+125
+00:11:03,890 --> 00:11:07,030
+بيساوي اتنين اص ان سالب واحد في واحد على اتنين اص
+
+126
+00:11:07,030 --> 00:11:11,650
+ان بيطلع نص اذا هذا المجموعة نص وهذا نص واللي بعده
+
+127
+00:11:11,650 --> 00:11:16,990
+نص كلهم نصارى ماعدا اول حد اذا واحد وهي نص وهذا نص
+
+128
+00:11:16,990 --> 00:11:23,670
+اللي بعده نص واخر واحد نص طب كام حد في هنا هاي حد
+
+129
+00:11:23,670 --> 00:11:29,950
+ودول عددهم nin من الحدود وهذا وعد هاي in زاد واحد
+
+130
+00:11:29,950 --> 00:11:34,870
+من الحدود طب هدول عددهم in لما أجمع عدد على نفسه
+
+131
+00:11:34,870 --> 00:11:38,810
+in من المرات بيطلع in في نص اللي هو in عتنين زاد
+
+132
+00:11:38,810 --> 00:11:42,770
+واحد طيب لما in تقول ل infinity in عتنين يقول ل
+
+133
+00:11:42,770 --> 00:11:46,570
+infinity وبالتالي واحد زاد in عتنين بيروح ل
+
+134
+00:11:46,570 --> 00:11:50,410
+infinity تمام؟
+
+135
+00:11:50,410 --> 00:11:54,690
+إذا أنا عندي الحد العام لل sequence هذه أو ال
+
+136
+00:11:54,690 --> 00:12:02,730
+subsequenceطولة أكبر من مقدار يقول لـ infinity هو
+
+137
+00:12:02,730 --> 00:12:14,510
+بالتالي إذا
+
+138
+00:12:14,510 --> 00:12:21,510
+أنا عندي x to two to n tends to infinity as n
+
+139
+00:12:21,510 --> 00:12:29,270
+tends to infinityوبالتالي هذا معناه أن X على 2 نص
+
+140
+00:12:29,270 --> 00:12:38,330
+M أكبر من M لكل M عدد موجب ما معناه أن ال limit
+
+141
+00:12:38,330 --> 00:12:42,730
+لحد
+
+142
+00:12:42,730 --> 00:12:47,650
+هذا أو ال sequence X المؤشرات تبقية 2 نص M تقول
+
+143
+00:12:47,650 --> 00:12:52,750
+infinity معناته هذا الحد العام لها أكبر من أي عدد
+
+144
+00:12:52,750 --> 00:12:58,800
+موجبوبالتالي إذا الـ subsequence هذه اللي هي الحد
+
+145
+00:12:58,800 --> 00:13:06,980
+العام تبعها X plus 2N is unbounded وبالتالي
+
+146
+00:13:06,980 --> 00:13:15,560
+فهذا بيقدي إن ال sequence XN نفسها is unbounded
+
+147
+00:13:15,560 --> 00:13:21,300
+لأنه لو كانت ال sequence boundedفأي sub-sequence
+
+148
+00:13:21,300 --> 00:13:25,420
+منها اللي هي جزء منها هتكون bounded صح هذا واضح
+
+149
+00:13:25,420 --> 00:13:30,740
+تمام الان by monotone convergence theorem ال
+
+150
+00:13:30,740 --> 00:13:37,340
+sequence x in unbounded وبالتالي it is divergent
+
+151
+00:13:37,340 --> 00:13:45,000
+لأن لو كانت convergent فلازم تكون bounded okay إذا
+
+152
+00:13:45,000 --> 00:13:49,600
+هاي استخدمنا ال monotone convergence theoremلإثبات
+
+153
+00:13:49,600 --> 00:13:52,140
+أن سيكوانس معينة مُعينة مُعينة مُعينة مُعينة
+
+154
+00:13:52,140 --> 00:13:52,840
+مُعينة مُعينة مُعينة مُعينة مُعينة مُعينة مُعينة
+
+155
+00:13:52,840 --> 00:13:56,680
+مُعينة مُعينة مُعينة مُعينة مُعينة مُعينة مُعينة
+
+156
+00:13:56,680 --> 00:13:59,900
+مُعينة مُعينة مُعينة مُعينة مُعينة مُعينة مُعينة
+
+157
+00:13:59,900 --> 00:14:00,660
+مُعينة مُعينة مُعينة مُعينة مُعينة مُعينة مُعينة
+
+158
+00:14:00,660 --> 00:14:05,360
+مُعينة مُعينة مُعينة مُعينة مُعينة مُعينة
+
+159
+00:14:05,360 --> 00:14:11,900
+مُعينة مُعينة مُعينة
+
+160
+00:14:11,900 --> 00:14:12,220
+مُعين
+
+161
+00:14:16,610 --> 00:14:21,230
+المثال التالت برضه هنستخدم .. هنطبق عليه اللي هو
+
+162
+00:14:21,230 --> 00:14:23,790
+الـ monotone convergence theorem
+
+163
+00:14:34,440 --> 00:14:40,520
+بس هذا المثال مختلف عن المثالين السابقين الان انا
+
+164
+00:14:40,520 --> 00:14:44,820
+بدي اعرف ال sequence x in inductively بطريقة
+
+165
+00:14:44,820 --> 00:14:52,860
+استقرائية شوفنا
+
+166
+00:14:52,860 --> 00:14:56,460
+احنا لما بدينا ال chapter هذا ان ال sequences can
+
+167
+00:14:56,460 --> 00:15:01,740
+be defined in two waysاما explicitly زي مثلا ال
+
+168
+00:15:01,740 --> 00:15:06,900
+sequence xn بالساوي واحد على n او recursively او
+
+169
+00:15:06,900 --> 00:15:11,520
+inductively بطريقة استقرائية بان انا اخد قيمة للحد
+
+170
+00:15:11,520 --> 00:15:16,440
+الاول او اول حدين اعطيهم قيم محددة و بعدين اعرف
+
+171
+00:15:16,440 --> 00:15:22,510
+الحد العام بدالة الحدود اللي قبلهفهي اندي الحد
+
+172
+00:15:22,510 --> 00:15:28,110
+الاول نفرض انه بساوي واحد الان بنعرف xn زياد واحد
+
+173
+00:15:28,110 --> 00:15:31,870
+على انه square root لاتنين ضرب الحد اللي جابناه
+
+174
+00:15:31,870 --> 00:15:35,970
+وهذا لكل n لان بالطريقة هذه ممكن اعرف ان هذا
+
+175
+00:15:35,970 --> 00:15:39,610
+بيعطينا sequence الان هذه ال sequence عايزين نثبت
+
+176
+00:15:39,610 --> 00:15:44,610
+انها convergent زائد ان ال limit تبقاتها بساوي
+
+177
+00:15:44,610 --> 00:15:45,350
+لعدد اتنين
+
+178
+00:15:48,640 --> 00:15:52,540
+لبرهان ذلك سنستخدم الـ monotone convergence
+
+179
+00:15:52,540 --> 00:15:57,940
+theorem عشان
+
+180
+00:15:57,940 --> 00:16:01,680
+أقدر استخدام الـ monotone convergence theorem ففي
+
+181
+00:16:01,680 --> 00:16:07,300
+عندي ال claim الأول يعني بدي أثبت في الإدعاء الأول
+
+182
+00:16:07,300 --> 00:16:14,260
+هذا ان ال sequence xn is increasing and bounded by
+
+183
+00:16:14,260 --> 00:16:14,640
+2
+
+184
+00:16:18,060 --> 00:16:24,180
+فلبرهان ذلك بنلاحظ
+
+185
+00:16:24,180 --> 00:16:31,920
+أن X1 من التعريف تبع ال sequence X1 بساوي 1 و X2
+
+186
+00:16:31,920 --> 00:16:34,660
+ممكن أجيبها من ال recursive formula أو ال
+
+187
+00:16:34,660 --> 00:16:39,500
+inductive formula إن أنا أاخد n بساوي 1 فبطلع X2
+
+188
+00:16:39,500 --> 00:16:49,840
+بساوي جدر 2 ل X1 و X1 1 إذا X2 بطلع جدر 2وبالتالي
+
+189
+00:16:49,840 --> 00:16:54,160
+من الحسابات هذه بيطلع اندي هاي X واحد X واحد
+
+190
+00:16:54,160 --> 00:16:59,080
+بيساوي واحد وبالتالي اكبر منها بيساوي واحد واسغر
+
+191
+00:16:59,080 --> 00:17:04,420
+من X اتنين لان X اتنين جدر اتنين الواحد اصغر من
+
+192
+00:17:04,420 --> 00:17:08,620
+جدر اتنين و
+
+193
+00:17:08,620 --> 00:17:12,960
+X اتنين اللي هو جدر اتنين اصغر من الاتنين لان كل
+
+194
+00:17:12,960 --> 00:17:13,760
+هذا صحيح
+
+195
+00:17:19,580 --> 00:17:25,920
+تمام؟ لسه ما خلصناهش لسه ما خلصناهش احنا ما فرضنا
+
+196
+00:17:25,920 --> 00:17:30,580
+انه صحيح احنا أثبتناه لسه
+
+197
+00:17:30,580 --> 00:17:35,220
+ما أثبتناش هذا ال claim لسه ما أثبتناه احنا لسه ده
+
+198
+00:17:35,220 --> 00:17:41,110
+بداية البرهانةالبرهان لـ claim بدأنا بما راحظنا ان
+
+199
+00:17:41,110 --> 00:17:47,150
+x1 من التعريف مقطع بساوي واحد و x2 حسبناها منها
+
+200
+00:17:47,150 --> 00:17:51,630
+بساوي جدر اتنين ل x1 اللي هو جدر اتنين وبالتالي
+
+201
+00:17:51,630 --> 00:17:58,270
+بطلع اندي هيك هاي x1 أكبر من أو ساوي واحد و أصغر
+
+202
+00:17:58,270 --> 00:18:03,850
+من جدر اتنين اللي هو x2 و x2 اللي هي جدر اتنين
+
+203
+00:18:03,850 --> 00:18:09,210
+أصغر من اتنينليش احنا عملنا هذا الكلام لان هيبين
+
+204
+00:18:09,210 --> 00:18:16,170
+الان now الان بدي اثبت بدي استخدم ال induction we
+
+205
+00:18:16,170 --> 00:18:28,150
+use induction لاثبات العبارة هذه وهي ان xn اصغر من
+
+206
+00:18:28,150 --> 00:18:32,970
+xn زائد واحد وهذا اصغر من اتنين وهذا اكبر من أوسع
+
+207
+00:18:32,970 --> 00:18:41,020
+الواحد لكلمن البرهن صحة العبارة هذه by induction
+
+208
+00:18:41,020 --> 00:18:47,240
+طيب الحالة اللي فيها خد n بساوية واحد الحالة اللي
+
+209
+00:18:47,240 --> 00:18:53,300
+فيها n بساوية واحد هي هاي x واحد أكبر من أو ساوية
+
+210
+00:18:53,300 --> 00:19:01,340
+واحد هدا هي و أصغر من x اتنين هدا هيو X2 أصغر من 2
+
+211
+00:19:01,340 --> 00:19:05,280
+إذا العبارة هذه صحيحة لما N بساوي 1 لأنه هنا
+
+212
+00:19:05,280 --> 00:19:10,280
+أثبتناها الآن
+
+213
+00:19:10,280 --> 00:19:18,920
+افرضي .. افرضي أن العبارة صحيحة عند N بساوي K يعني
+
+214
+00:19:18,920 --> 00:19:27,820
+عندك هنا XKاكبر من او ساول واحد اصغر من X K زايد
+
+215
+00:19:27,820 --> 00:19:34,100
+واحد اصغر من اتنين هنا فرضنا هذا ال induction high
+
+216
+00:19:34,100 --> 00:19:41,920
+precision وعايزين نثبت ان هذا بيقدي ان القبارة
+
+217
+00:19:41,920 --> 00:19:48,980
+صحيحة and N بيساوي K زايد واحد يعني بدي اثبت هذه
+
+218
+00:19:48,980 --> 00:19:50,660
+المتباينة
+
+219
+00:19:55,390 --> 00:20:02,590
+بدي أثبت المتباينة هذه طبعا فتعالى نشوف كيف نثبت
+
+220
+00:20:02,590 --> 00:20:21,690
+المتباينة هذه طيب
+
+221
+00:20:21,690 --> 00:20:29,980
+أنا عنديهي عندي المتباينة هذه احنا
+
+222
+00:20:29,980 --> 00:20:49,600
+فرضين ان المتباينة هذه صحيحة احنا
+
+223
+00:20:49,600 --> 00:20:52,640
+فرضين من induction hypothesis ان هذه المتباينة
+
+224
+00:20:52,640 --> 00:20:58,910
+صحيحةأضرب المتباينة هذه في اتنين هي اضرب كل
+
+225
+00:20:58,910 --> 00:21:02,710
+الأطراف في اتنين فبصير اتنين اصغر من اتنين XK اصغر
+
+226
+00:21:02,710 --> 00:21:09,290
+من اتنين XK زائد واحد اصغر من اربع وهذا بيقدي ان
+
+227
+00:21:09,290 --> 00:21:11,570
+واحد اصغر من جدر اتنين
+
+228
+00:21:15,730 --> 00:21:21,830
+و اذا انا الان باخد الجذر التربيعى لكل الأطراف هذه
+
+229
+00:21:21,830 --> 00:21:26,750
+اخد الجذر التربيعى فهي جذر اتنين طبعا اكبر من واحد
+
+230
+00:21:26,750 --> 00:21:34,350
+اصغر منه يساوي جذر اتنين XK اللى هو XK زاد واحد
+
+231
+00:21:34,350 --> 00:21:37,770
+هذا طبعا من التعريف تبع ال sequence من ال
+
+232
+00:21:37,770 --> 00:21:43,130
+inductive formula جذر اتنين XK حسب التعريف بساوي
+
+233
+00:21:43,130 --> 00:21:50,440
+XK زاد واحدوهذا أصغر من هنا جدر اتنين xk أصغر من
+
+234
+00:21:50,440 --> 00:21:56,840
+جدر اتنين xk زائد واحد وهذا أصغر من جدر الأربع
+
+235
+00:21:56,840 --> 00:22:01,180
+اللي هو الأتنين إذا هاي بيطلع عندي واحد أصغر من أو
+
+236
+00:22:01,180 --> 00:22:06,580
+يساوي xk زائد واحد وهذا برضه من ال inductive
+
+237
+00:22:06,580 --> 00:22:15,940
+formula الجدر التربيع هذا بيساوي xk زائد اتنينإذا
+
+238
+00:22:15,940 --> 00:22:21,620
+هي 1 أصغر من أو ساوي xk زاد 1 أصغر من xk زاد 2
+
+239
+00:22:21,620 --> 00:22:28,100
+أصغر من 2 وبالتالي إذا أثبتنا إحنا صحة العبارة هذه
+
+240
+00:22:28,100 --> 00:22:34,700
+عن k زاد 1 وبالتالي هيك بنكون كملنا ال induction
+
+241
+00:22:34,700 --> 00:22:43,060
+okay طبعا إذا ال claim تعالوا نشوف الآن ليش ال
+
+242
+00:22:43,060 --> 00:22:48,470
+sequenceأه ليه ال sequence تبعتنا بتطلع bounded
+
+243
+00:22:48,470 --> 00:22:55,530
+وincreasing فاكرين احنا أثبتنا by induction ان x
+
+244
+00:22:55,530 --> 00:23:01,810
+in أصغر من x in زايد واحد أصغر من اتنين أكبر من
+
+245
+00:23:01,810 --> 00:23:10,150
+أوي ساوي واحد لكل in من الجزء هذا نستنتج
+
+246
+00:23:10,150 --> 00:23:14,460
+ان ال sequence is increasing صح؟لأن هى عندى xn
+
+247
+00:23:14,460 --> 00:23:21,640
+أصغر من xn زاد واحد لكل n ومن المتباينة كلها يعني
+
+248
+00:23:21,640 --> 00:23:28,200
+اللى هى xn أصغر من اتنين أكبر من أوسع واحد لكل n
+
+249
+00:23:28,200 --> 00:23:32,080
+هذا معناته ال sequence bounded هى محصورة بين واحد
+
+250
+00:23:32,080 --> 00:23:37,160
+واتنين و bounded above by اتنين لذلك هذا يكمل
+
+251
+00:23:37,160 --> 00:23:42,800
+برهان ال claim الأول يعنيوهو انه sequence x in
+
+252
+00:23:42,800 --> 00:23:47,240
+increasing و bounded الان by monotone convergence
+
+253
+00:23:47,240 --> 00:23:53,140
+theorem ال sequence x in هتكون convergent دعينا
+
+254
+00:23:53,140 --> 00:23:56,840
+نسمي ال limit تبعتها x وطبعا حسب ال monotone
+
+255
+00:23:56,840 --> 00:23:59,480
+convergence theorem بما انه sequence increasing
+
+256
+00:23:59,480 --> 00:24:05,960
+اذا ال limit تبعتها بساوي ال suprem لها ك set اذا
+
+257
+00:24:05,960 --> 00:24:09,600
+انا في عندي الآن ال sequence تبعتي convergent هي
+
+258
+00:24:09,600 --> 00:24:17,620
+عندي limitx in convergent بيساوي x اللي هي طبعا
+
+259
+00:24:17,620 --> 00:24:21,580
+حسب النظرية بيساوي ال supremum الان بدي أجيب قيمة
+
+260
+00:24:21,580 --> 00:24:25,460
+ال x هذا طبعا
+
+261
+00:24:25,460 --> 00:24:30,560
+مش سهل أن أجيب ال supremum ل ال sequence فبجيبها
+
+262
+00:24:30,560 --> 00:24:35,600
+بطريقة تانية إذا
+
+263
+00:24:35,600 --> 00:24:38,560
+ال claim التاني بدي أثبت أن ال x ال limit ل ال
+
+264
+00:24:38,560 --> 00:24:40,720
+sequence اللي هي x بيساوي 2
+
+265
+00:24:43,730 --> 00:24:47,450
+طيب انا عندي من تعريف الـ sequence انا عندي xn زاد
+
+266
+00:24:47,450 --> 00:24:53,070
+واحد بساوي جدر اتنين xn و هذا الكلام صحيح for
+
+267
+00:24:53,070 --> 00:24:57,870
+every m ناخد ال limit لاتطرفين لما n تقول ل
+
+268
+00:24:57,870 --> 00:25:02,050
+infinity بتطلع limit xn زاد واحد بساوي limit جدر
+
+269
+00:25:02,050 --> 00:25:08,390
+اتنين ثابت في limit جدر ال xn مظبوط؟
+
+270
+00:25:09,940 --> 00:25:15,160
+طيب احنا فرضين او احنا استنتجنا احنا لسه مستنتجين
+
+271
+00:25:15,160 --> 00:25:19,340
+من ال monotone convergence ان limit xn بيساوي x
+
+272
+00:25:19,340 --> 00:25:25,400
+وبالتالي limit xn زاد واحد برضه بتساوي x وهي
+
+273
+00:25:25,400 --> 00:25:31,220
+بيساوي جذر اتنين و limit جذر xn بيساوي جذر ال x
+
+274
+00:25:31,220 --> 00:25:36,980
+حسب نظرية سابقة اذا ال limit هذه اذا هي x و جذر
+
+275
+00:25:36,980 --> 00:25:40,820
+اتنين في ال limit هذه بتطلع جذر ال xإذا أصبح أثيان
+
+276
+00:25:40,820 --> 00:25:46,380
+دي معادلة في مجهول واحد x ممكن أحلها و ذلك بتربيع
+
+277
+00:25:46,380 --> 00:25:53,740
+الطرفين فالمعادلة هذه بعد مربعها تصير هيك وهذه في
+
+278
+00:25:53,740 --> 00:25:59,340
+إلها حالين إما x بيطلع بساوي سفر أو x بساوي اتنين
+
+279
+00:25:59,340 --> 00:26:04,940
+احنا عايزين ال x ناخد x بساوي اتنين و نرفض x بساوي
+
+280
+00:26:04,940 --> 00:26:10,700
+سفر طب ليه نرفض x بساوي سفر؟لأن اثبتنا هنا by
+
+281
+00:26:10,700 --> 00:26:20,340
+induction أن xn أكبر من أوسع واحد أصغر من الأتنين
+
+282
+00:26:20,340 --> 00:26:25,960
+و أثبتنا أن ال sequence هذه convergent، إذا حسب
+
+283
+00:26:25,960 --> 00:26:27,200
+نظرية سابقة
+
+284
+00:26:30,490 --> 00:26:38,230
+إذن ال limit لل sequence xn هتطلع محصورة بين 2 و
+
+285
+00:26:38,230 --> 00:26:42,650
+بين 1 خدمة نظرية بتقول لو كانت ال sequence xn
+
+286
+00:26:42,650 --> 00:26:48,610
+convergent و xn أكبر من أو ساوي a أصغر من أو ساوي
+
+287
+00:26:48,610 --> 00:26:53,570
+b لكل n فال limit لل sequence xn بتطلع أيضا محصورة
+
+288
+00:26:53,570 --> 00:26:59,560
+بين a و bيعني طب هدى هى ال X فرضنا ان ال limit هدى
+
+289
+00:26:59,560 --> 00:27:04,060
+X اذا بطلع انا عندي X اكبر من او ساوي واحد اصغر من
+
+290
+00:27:04,060 --> 00:27:07,920
+الاتنين وبالتالي مش ممكن ال X اللى هى محصورة بين
+
+291
+00:27:07,920 --> 00:27:15,420
+واحد واتنين مش ممكن تساوي سفر مش ممكن تساوي سفر
+
+292
+00:27:15,420 --> 00:27:19,820
+اذا لازم الساوي .. وانا عندي سفر او اتنين اذا لازم
+
+293
+00:27:19,820 --> 00:27:25,570
+الساوي اتنينOkay إذا هين هيك استخدمنا الـ monotone
+
+294
+00:27:25,570 --> 00:27:31,030
+convergence بالمثل في أد التمرين زي هذه sequences
+
+295
+00:27:31,030 --> 00:27:36,290
+معرفة inductively و هتثبتوا أنها convergent و
+
+296
+00:27:36,290 --> 00:27:40,750
+تجيبوا قيمة ال limit بنفس الطريقة و بنفس الأسلوب
+
+297
+00:27:40,750 --> 00:27:46,250
+فحاولوا تستفيدوا من حل المثال الأخير هذا في حل مثل
+
+298
+00:27:46,250 --> 00:27:52,770
+هذه التمرين Okay تمام واضحإذن هنا أخدنا تطبيقات
+
+299
+00:27:52,770 --> 00:27:56,230
+متنوعة على الـ monotone convergence theorem وهي
+
+300
+00:27:56,230 --> 00:28:03,570
+التمرين ل section تلاتة تلاتة نبدأ
+
+301
+00:28:03,570 --> 00:28:09,230
+section أربعة أو تلاتة أربعة نعم بيقول إنه ممكن
+
+302
+00:28:09,230 --> 00:28:13,790
+نحل بحر ثاني و نثبت أنه الإثنان يصدرهم للإكسام
+
+303
+00:28:13,790 --> 00:28:17,770
+مظبوط صحيح الإثنان يتحركون على طريق اللملة اللي هي
+
+304
+00:28:17,770 --> 00:28:18,610
+الإكسام
+
+305
+00:28:21,690 --> 00:28:28,990
+والله انت فاكر فيه و بعدين قولي ليه هي
+
+306
+00:28:28,990 --> 00:28:34,050
+عندك sequence حدودها معروفة معرفة ممكن تكتب أول
+
+307
+00:28:34,050 --> 00:28:40,010
+اربع خمس عدود و تحاول تستنتجي ايه هي قيمة ال
+
+308
+00:28:40,010 --> 00:28:44,930
+supreme و تبرهنها طبعا فهذا متروك اليك
+
+309
+00:28:47,810 --> 00:28:52,030
+هذا يعني حل آخر فانا قلت ان ال suprem مش سهل ان
+
+310
+00:28:52,030 --> 00:28:56,070
+احنا نجيبه لمثل هذه ال sequences او ال sets
+
+311
+00:28:56,070 --> 00:28:59,230
+وبالتالي ال monotone convergence في الفيلم كان
+
+312
+00:28:59,230 --> 00:29:03,390
+ممكن يكون أسهل لأن هاي الكلام التاني هذا الأخير
+
+313
+00:29:03,390 --> 00:29:07,270
+مااخدش وجهة يعني أخدنا ال formula ال inductive
+
+314
+00:29:07,270 --> 00:29:11,570
+formula و أخدنا ال limit للطرفين و حلينا معادلة في
+
+315
+00:29:11,570 --> 00:29:16,800
+Xو ادركنا ان ال X ليس لازم تساوي سفر من هنا لان X
+
+316
+00:29:16,800 --> 00:29:20,820
+محصولة بين واحد و اتنين هذا أسهل من ان انا اجيب ال
+
+317
+00:29:20,820 --> 00:29:26,940
+suprem لكن هذا ممنعش ان ممكن حد معين يثبت ان ال
+
+318
+00:29:26,940 --> 00:29:33,060
+suprem هو اتنين اذا كان سهل فكان بيعني نستخدمه مش
+
+319
+00:29:33,060 --> 00:29:35,240
+سهل نستخدم ال monotone convergence
+
+320
+00:29:49,630 --> 00:29:56,070
+الـ sub-sequences and Bolzano-Weierstrass theorem،
+
+321
+00:29:56,070 --> 00:29:59,350
+ال sub-sequences شوفنا قبل شوية sub-sequence
+
+322
+00:30:11,180 --> 00:30:15,400
+شوفنا قبل لحظات في المثال التاني انه في عنده
+
+323
+00:30:15,400 --> 00:30:26,540
+sequence هي عنده sequence xn حدودها x1, x2, x3, x4
+
+324
+00:30:26,540 --> 00:30:34,160
+و هكذا و في كانت sequence تانية لحد الآن تبعها 2
+
+325
+00:30:34,160 --> 00:30:52,420
+أُس nالحدود هذي هتكون X2 X4 X8 و هكذا صح؟ لو سمنا
+
+326
+00:30:52,420 --> 00:31:01,340
+الاتنين هذي R1 والاربعة هذي سمنها R2 والتمانية R3
+
+327
+00:31:04,820 --> 00:31:10,940
+فبنلاحظ أن R1 أكبر من أو ساوء واحد، عدد طبيعي أكبر
+
+328
+00:31:10,940 --> 00:31:19,320
+من أو ساوء واحد وR2 أكبر من R1، اللي هو أربعة أكبر
+
+329
+00:31:19,320 --> 00:31:29,050
+من اتنين وR3 اللي هو تمانية أكبر من R2و هكذا اذا
+
+330
+00:31:29,050 --> 00:31:34,810
+ال sub sequence المؤشرات تبعتها او ال indices انا
+
+331
+00:31:34,810 --> 00:31:40,330
+بسميه index مجموعة index indices ال indices او
+
+332
+00:31:40,330 --> 00:31:44,710
+المؤشرات لل sub sequence هي عداد طبيعية هذا هي
+
+333
+00:31:44,710 --> 00:31:49,890
+اتنين اربعة تمانية هي عداد طبيعية والعداد الطبيعية
+
+334
+00:31:49,890 --> 00:31:55,170
+هذه بتشكل sequence هذه عبارة عن sequence of
+
+335
+00:31:55,170 --> 00:32:01,880
+natural numbersصح؟ و ال sequence هذه is strictly
+
+336
+00:32:01,880 --> 00:32:08,200
+.. strictly increasing
+
+337
+00:32:08,200 --> 00:32:14,580
+.. strictly increasing يعني متزايدة زيادة صحيحة
+
+338
+00:32:14,580 --> 00:32:18,860
+يعني R واحد أصغر منه مش أصغر منه أو يساوي R اتنين
+
+339
+00:32:18,860 --> 00:32:23,280
+و R اتنين أصغر منه و لا تساوي R تلاتة و هكذا
+
+340
+00:32:23,280 --> 00:32:25,780
+مظبوط؟ صح؟
+
+341
+00:32:29,030 --> 00:32:33,430
+إذا السب سيكوانس السب سيكوانس من أي سيكوانس هي
+
+342
+00:32:33,430 --> 00:32:39,350
+مجموعة جزئية منها صح لأن حدودها هي حدود حدود السب
+
+343
+00:32:39,350 --> 00:32:46,130
+سيكوانس هي عناصر أو حدود من السيكوانس العصلية لكن
+
+344
+00:32:46,130 --> 00:32:52,170
+مش أي حدود لازم تكون مرتبة بحيث أن المؤشرات تبعتها
+
+345
+00:32:52,170 --> 00:32:56,890
+اتشكل strictly increasing sequence of natural
+
+346
+00:32:56,890 --> 00:33:03,480
+numbersتمام؟ زي هيك إذاً
+
+347
+00:33:03,480 --> 00:33:06,900
+هذا هو تعريف الـ subsequence إذا لو في أنا عندي
+
+348
+00:33:06,900 --> 00:33:12,060
+sequence XN واخدت strictly increasing sequence of
+
+349
+00:33:12,060 --> 00:33:17,620
+natural numbers فال sequence اللي المؤشرات تبعتها
+
+350
+00:33:17,620 --> 00:33:24,060
+هي ال sequence RN اللي هي هذه عناصرها بنسميها
+
+351
+00:33:24,060 --> 00:33:30,640
+subsequence من ال sequence XNو هاي أمثلة هتابر هذه
+
+352
+00:33:30,640 --> 00:33:33,900
+الـ sequence x in الـ sequence of natural numbers
+
+353
+00:33:33,900 --> 00:33:40,860
+فهذه subsequence منها اتنين in الـ sequence of
+
+354
+00:33:40,860 --> 00:33:44,940
+even numbers او even natural numbers ده هي على
+
+355
+00:33:44,940 --> 00:33:54,800
+سرعة اتنين اربعة ستة و هكذا وهذه عبارة عن sequence
+
+356
+00:33:56,170 --> 00:34:03,930
+of odd national numbers واحد تلاتة خمسة و هكذا
+
+357
+00:34:03,930 --> 00:34:11,550
+وحدود ال sequence هذه هي X R واحد هذا X اتنين هذا
+
+358
+00:34:11,550 --> 00:34:18,430
+رقمه هذا رقم اتنين يعني R واحد بساوة اتنين طيب X R
+
+359
+00:34:18,430 --> 00:34:25,450
+اتنين اربع X R اتنينر2 هذا حد رقم أربعة، ر2 بيساوي
+
+360
+00:34:25,450 --> 00:34:31,610
+أربعة و ر1 بيساوي اتنين، و اتنين أصغر من أربعة، XR
+
+361
+00:34:31,610 --> 00:34:39,130
+تلاتة ستة، ر تلاتة ستة نفس الحاجة، يعني هذه
+
+362
+00:34:39,130 --> 00:34:44,020
+subsequence وهذه subsequence من ال sequence Xلأن
+
+363
+00:34:44,020 --> 00:34:48,280
+مأشراتهم كلهم بشكل strictly increasing sequences
+
+364
+00:34:48,280 --> 00:34:53,000
+of natural numbers بالمثل ال sequence 1 على 2 n
+
+365
+00:34:53,000 --> 00:35:03,540
+سالب 1 و ال sequence 1 على n factorial هدول
+
+366
+00:35:03,540 --> 00:35:07,840
+برضه أيضا sub sequences من ال sequence 1 على n
+
+367
+00:35:11,850 --> 00:35:16,490
+لكن الـ sequence اللي لحد الآن تبقى الحدودها واحد
+
+368
+00:35:16,490 --> 00:35:24,370
+على واحد، سفر، تلت، سفر، خمس، سفر، و هكذا هذه ليست
+
+369
+00:35:24,370 --> 00:35:32,450
+subsequence من الـ sequence واحد على انه لأن السفر
+
+370
+00:35:32,450 --> 00:35:37,150
+هذا هايلها، مش موجودة، ليست ثلاثا تاني لل sequence
+
+371
+00:35:37,150 --> 00:35:43,480
+هذه ومؤشرات الحدوديعني لا تشكل strictly increasing
+
+372
+00:35:43,480 --> 00:35:47,640
+sequence طيب
+
+373
+00:35:47,640 --> 00:35:52,780
+لو أخدت أي tail أي tail M tail حيث M fixed natural
+
+374
+00:35:52,780 --> 00:35:58,740
+number ف X أي tail ده M tail of any sequence X in
+
+375
+00:35:58,740 --> 00:36:03,640
+طبعا ال M tail ده حدوده عبارة عن sequence الحد
+
+376
+00:36:03,640 --> 00:36:10,190
+الأول تبعهاx capital M زاد واحد الحد التاني x
+
+377
+00:36:10,190 --> 00:36:16,870
+capital M زاد اتنين التالت x capital M زاد تلاتة و
+
+378
+00:36:16,870 --> 00:36:21,170
+هكذا فطبعا
+
+379
+00:36:21,170 --> 00:36:26,510
+هذه عبارة عن sub sequence من ال sequence الام لأن
+
+380
+00:36:26,510 --> 00:36:32,430
+كل أنصر في ال sub sequence هذه هي موجودة هنا صح؟
+
+381
+00:36:33,790 --> 00:36:39,710
+والمؤشرات تبعات ال sub-sequence هي M زاد واحد اصغر
+
+382
+00:36:39,710 --> 00:36:45,830
+من R اتنين اللي هو M زاد اتنين وR اتنين اصغر من R
+
+383
+00:36:45,830 --> 00:36:50,130
+تلاتة اللي هو M زاد تلاتة وكده هذا sub-sequence
+
+384
+00:36:50,130 --> 00:36:54,950
+ولا مش sub-sequence؟ لو أخدت أي sequence X in فأي
+
+385
+00:36:54,950 --> 00:37:02,220
+M تل هو sub-sequence منهاكذلك لو أخدت أي sequence
+
+386
+00:37:02,220 --> 00:37:09,400
+xn فالـ sequence x اللي الحد اللي عم تبعها المؤشر
+
+387
+00:37:09,400 --> 00:37:16,020
+تبعه 2 أس n هذي برضه subsequence زي ما شوفنا و x2n
+
+388
+00:37:16,020 --> 00:37:21,340
+الحدود الزوجية لو أخدت الحدود الزوجية فقط فهذا
+
+389
+00:37:21,340 --> 00:37:25,840
+بعطيني subsequence و لو أخدت الحدود الفردية تعطيني
+
+390
+00:37:25,840 --> 00:37:38,470
+subsequence ثانية و لا كدهالان سؤال
+
+391
+00:37:38,470 --> 00:37:42,190
+اللى بهمنا احنا ما هي علاقة ال sequence بال
+
+392
+00:37:42,190 --> 00:37:46,990
+subsequence من حيث ال convergence و ال divergence؟
+
+393
+00:37:54,410 --> 00:37:56,950
+يعني لو كانت ال sequence convergent لو في اندي
+
+394
+00:37:56,950 --> 00:38:01,250
+سيكوانس xn convergent ل x واخدت أي sub sequence
+
+395
+00:38:01,250 --> 00:38:07,490
+منها هل هذه ال sequence لازم تكون convergent زيها
+
+396
+00:38:07,490 --> 00:38:11,890
+ولا divergent لازم تكون convergent و ال limit
+
+397
+00:38:11,890 --> 00:38:22,770
+تبعتها نفس ال limit و لها نفس ال limit ماشي
+
+398
+00:38:22,770 --> 00:38:23,170
+لحظة
+
+399
+00:38:29,060 --> 00:38:29,860
+كثير من الناس
+
+400
+00:38:39,930 --> 00:38:46,370
+إذا كمان مرة بهمني أنا أنه لو في عندي sequence
+
+401
+00:38:46,370 --> 00:38:51,030
+نظرية هذه بتقول لو في عندي sequence xn of real
+
+402
+00:38:51,030 --> 00:38:56,350
+numbers وكانت ال sequence هذه convergent ل x فأي
+
+403
+00:38:56,350 --> 00:39:00,170
+subsequence منها بتكون convergent و ال limit
+
+404
+00:39:00,170 --> 00:39:05,330
+تبعتها هي نفس ال limit لل sequence xn
+
+405
+00:39:08,450 --> 00:39:15,870
+وهذا يعني ممكن ان احنا نثبته بسهولة عشان اثبت ان
+
+406
+00:39:15,870 --> 00:39:22,590
+ال subsequence XRN converge ل X فبستخدم تعريف Y
+
+407
+00:39:22,590 --> 00:39:27,930
+capital N فلو أخدت أي Y أكبر من السفر أنا عندي ال
+
+408
+00:39:27,930 --> 00:39:32,560
+sequence الأصلية هي convergent ل Xوبالتالي من
+
+409
+00:39:32,560 --> 00:39:36,720
+تعريف ال convergence لما أن XM converged ل X إذا
+
+410
+00:39:36,720 --> 00:39:39,940
+يوجد عدد طبيعي يعتمد على إبسلون بحيث أن المسافة
+
+411
+00:39:39,940 --> 00:39:45,700
+بين XM و X أصغر من إبسلون لكل M أكبر من أو يساوي
+
+412
+00:39:45,700 --> 00:39:52,980
+capital M طيب أنا عندي المؤشرات
+
+413
+00:39:52,980 --> 00:39:58,160
+تبع السب سيكوينس بتشكل increasing
+
+414
+00:39:58,160 --> 00:40:03,420
+sequenceوأول واحد .. أول عدد فيها طبعا هذا عدد
+
+415
+00:40:03,420 --> 00:40:09,800
+طبيعي وبالتالي أكبر من أو ساوي الواحد فبالتالي ال
+
+416
+00:40:09,800 --> 00:40:15,160
+Rn هدولة ال Rn ممكن اثبات باستخدام ال induction أن
+
+417
+00:40:15,160 --> 00:40:22,220
+Rn أكبر من أو ساوي N لكل N وبالتالي
+
+418
+00:40:22,220 --> 00:40:28,970
+لو أخدت N أكبر من أو ساوي capital N فعندي أنا Rnمن
+
+419
+00:40:28,970 --> 00:40:34,590
+هنا أكبر من أو ساوي small n و ال n أنا ماخده أكبر
+
+420
+00:40:34,590 --> 00:40:38,750
+من أو ساوي capital N إذا بيطلع عندي RN هاي بيطلع
+
+421
+00:40:38,750 --> 00:40:43,150
+عندي RN أكبر من أو ساوي capital N وبالتالي من ال
+
+422
+00:40:43,150 --> 00:40:48,810
+implication 13 ال implication 13 بتقوللي لأي عدد
+
+423
+00:40:49,980 --> 00:40:55,300
+أكبر من أو ساوية capital N المسافة بين X للعدد هذا
+
+424
+00:40:55,300 --> 00:41:02,090
+للمؤشر هذا سالب X أصغر من Yإذا أنا هيك أثبتت ..
+
+425
+00:41:02,090 --> 00:41:07,550
+أنا هيك أثبتت أنه ال .. لأي epsilon أكبر من السفر
+
+426
+00:41:07,550 --> 00:41:12,190
+في capital N يعتمد على epsilon بحيث لكل N أكبر منه
+
+427
+00:41:12,190 --> 00:41:16,830
+ساوي capital N المسافة بين XRN و X أصغر من epsilon
+
+428
+00:41:16,830 --> 00:41:21,320
+وبالتالي من تعريف epsilon capital N للنهايةأنا هيك
+
+429
+00:41:21,320 --> 00:41:27,640
+بكون أثبتت أنه limit xr n لما n تقول ل infinity
+
+430
+00:41:27,640 --> 00:41:35,720
+بساوي x وهذا هو المطلوب طبعا
+
+431
+00:41:35,720 --> 00:41:40,780
+في هنا أمثلة باقي شوية أمثلةفهذه الأمثلة يعني
+
+432
+00:41:40,780 --> 00:41:46,000
+حاولوا أنكم تقرؤوها في مثلين كيف نطبق النظرية هذه
+
+433
+00:41:46,000 --> 00:41:50,660
+أو نوجد العلاقة بين كيف نثبت ال convergence لل
+
+434
+00:41:50,660 --> 00:41:55,900
+sequence من خلال إثبات
+
+435
+00:41:55,900 --> 00:42:00,290
+ال convergence لل subsequences أو العكسفحاولوا
+
+436
+00:42:00,290 --> 00:42:04,490
+تقرؤوها و هيك نكون يعني تقريبا .. هنكمل ان شاء
+
+437
+00:42:04,490 --> 00:42:10,290
+الله المرة الجاية و .. هن .. هنوقف استخدام ال
+
+438
+00:42:10,290 --> 00:42:14,290
+powerpoint ابتداء من المحاضرة الجاية و هنشره على
+
+439
+00:42:14,290 --> 00:42:19,850
+اللغة okay انتهت المحاضرة نشوفكم ان شاء الله يوم
+
+440
+00:42:19,850 --> 00:42:20,250
+اتنين
+
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/Nztl0T85AIM_raw.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/Nztl0T85AIM_raw.srt
new file mode 100644
index 0000000000000000000000000000000000000000..3e038b0384c74d400ba11c725c6ecafe06e2693c
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/Nztl0T85AIM_raw.srt
@@ -0,0 +1,1752 @@
+1
+00:00:21,210 --> 00:00:28,030
+انراجع مع بعض ال order properties of R او خواص
+
+2
+00:00:28,030 --> 00:00:33,790
+الترتيب للأعداد الحقيقية احنا من بداية ال chapter
+
+3
+00:00:33,790 --> 00:00:37,630
+قلنا انه ال real number system نظام الأعداد
+
+4
+00:00:37,630 --> 00:00:43,250
+الحقيقية نظام
+
+5
+00:00:43,250 --> 00:00:52,230
+الأعداد الحقيقية يتكون من مجموعة R boldface Rمع
+
+6
+00:00:52,230 --> 00:00:57,370
+عمليتين فنائيتين واحدة عملية الجامعة واحدة عملية
+
+7
+00:00:57,370 --> 00:01:04,390
+الضرب وافترضنا ان العمليات هذه بتحقق خمس خواص اللي
+
+8
+00:01:04,390 --> 00:01:08,910
+هي خواص ال field اللي هو ال commutative law,
+
+9
+00:01:09,050 --> 00:01:17,690
+associative law, distributive laws, existence of
+
+10
+00:01:17,690 --> 00:01:23,340
+identities, existence of inversesبعدين ضفنا على
+
+11
+00:01:23,340 --> 00:01:28,300
+ذلك انه افترضنا انه ال real number system R بتحقق
+
+12
+00:01:28,300 --> 00:01:33,020
+برضه خاصية الترتيب او خواص الترتيب اللي هي الخاصية
+
+13
+00:01:33,020 --> 00:01:38,440
+السادسة هذه الخاصية السادسة هذه تجزأت يعني تنص على
+
+14
+00:01:38,440 --> 00:01:43,560
+ما يليه نفترض انه يوجد مجموعة جزئية من R غير خالية
+
+15
+00:01:43,560 --> 00:01:49,360
+و المجموعة الجزئية هذه بنسميها P اللي هو اول حرف
+
+16
+00:01:49,360 --> 00:01:56,080
+في positiveعشان نسميها بعد هيك the set of positive
+
+17
+00:01:56,080 --> 00:02:02,020
+real numbers ف ال set P هذه closed under addition
+
+18
+00:02:02,020 --> 00:02:08,540
+and under multiplication كمان نفترض أن ال set P
+
+19
+00:02:08,540 --> 00:02:13,180
+هذه بتحقق الخاصية الثلاثية ال trichotomy property
+
+20
+00:02:14,240 --> 00:02:18,920
+which means that for any real number a exactly one
+
+21
+00:02:18,920 --> 00:02:27,060
+of the three possibilities holds either a belongs
+
+22
+00:02:27,060 --> 00:02:34,420
+to p or a equals zero or negative a belongs to p
+
+23
+00:02:34,420 --> 00:02:42,230
+بناء على هذه الخاصية شفنا أن الأعداد الحقيقيةgets
+
+24
+00:02:42,230 --> 00:02:47,350
+partitioned to three mutually disjoint sets يعني
+
+25
+00:02:47,350 --> 00:02:54,450
+الخاصية هذه بتجزق بتخليني أجزق العداد الحقيقية إلى
+
+26
+00:02:54,450 --> 00:03:00,110
+تلت مجموعات منفصلة مثنى مثنى pair-wise disjoint
+
+27
+00:03:00,110 --> 00:03:05,650
+يعني إن لو أخدت أي مجموعتين عشوائيتين من التلاتة
+
+28
+00:03:05,650 --> 00:03:09,310
+تقطعهم بساوي فايل مافيش بينهم عناصر و مش تلتين
+
+29
+00:03:10,410 --> 00:03:15,170
+واتحادهم بساوي ال R، لأن هذا بشكل تجزء على ال R،
+
+30
+00:03:15,170 --> 00:03:19,630
+تجزء على ال R أو بنسميها في الرياضيات partition of
+
+31
+00:03:19,630 --> 00:03:25,210
+R ال set P هذه سمنها set of positive real numbers
+
+32
+00:03:25,210 --> 00:03:35,210
+وعرفنا negative P على إنها negative عناصر ال set P
+
+33
+00:03:44,210 --> 00:03:49,430
+Okay فهي معرفة negative P هي كل ال elements
+
+34
+00:03:49,430 --> 00:03:56,850
+negative A such that A element in P بعدين
+
+35
+00:03:56,850 --> 00:04:02,430
+عرفنا علاقة الترتيب، الآن بنعرف اللي هو order
+
+36
+00:04:02,430 --> 00:04:08,340
+relation على Rما معنى أنه a لو في ending two real
+
+37
+00:04:08,340 --> 00:04:12,720
+numbers ما معنى a أصغر من b أو b أكبر من a قولنا
+
+38
+00:04:12,720 --> 00:04:19,820
+معناها أن الفرق بين b و a is positive real number
+
+39
+00:04:19,820 --> 00:04:24,480
+أو ينتمي لمجموعة الأعداد المجتمعةطب ما معناه a
+
+40
+00:04:24,480 --> 00:04:28,760
+أصغر من أو ساوي b أو b أكبر من أو ساوي a؟ معناته
+
+41
+00:04:28,760 --> 00:04:33,240
+الفرق بين b و a ينتمي للأعداد الموجبة، يعني الفرق
+
+42
+00:04:33,240 --> 00:04:40,120
+موجب أو يساوي سفر أو يساوي سفر، إذن معناه تاني طيب
+
+43
+00:04:40,120 --> 00:04:50,940
+و أعتقد إن احنا بعد هيك أثبتنا أه
+
+44
+00:04:50,940 --> 00:04:52,780
+وقفنا عند النظرية هذه
+
+45
+00:04:57,810 --> 00:05:02,310
+نظرية واحد خمسة قلنا إنه لأي لو أخدت أي تلت أعداد
+
+46
+00:05:02,310 --> 00:05:08,730
+حقيقية فعند الخواص التالية تتحقق نجموعة الخواص هذه
+
+47
+00:05:08,730 --> 00:05:17,210
+تتحقق فالخواص
+
+48
+00:05:17,210 --> 00:05:21,630
+هذه هذا
+
+49
+00:05:21,630 --> 00:05:27,330
+هي أمامكم transitivity خاصية التعدىايه يعني
+
+50
+00:05:27,330 --> 00:05:35,310
+التعدى؟ يعني اذا انا في عندي تلت أعداد حقه في A و
+
+51
+00:05:35,310 --> 00:05:38,770
+B و C
+
+52
+00:05:43,970 --> 00:05:52,530
+وكان B هنا أكبر .. B أكبر من A and C أكبر من B
+
+53
+00:05:52,530 --> 00:05:55,790
+فهذا
+
+54
+00:05:55,790 --> 00:06:09,770
+بيؤدي أنه C أكبر من A خليني
+
+55
+00:06:09,770 --> 00:06:15,290
+أنا أكسهم عشان .. كليهم زي ..مهمة موجودة هناك هي a
+
+56
+00:06:15,290 --> 00:06:27,090
+أكبر من b هي a أكبر من b و b أكبر من c فبطلع
+
+57
+00:06:27,090 --> 00:06:38,470
+c أو a بطلع أكبر من cهذه a أكبر من b و b أكبر من c
+
+58
+00:06:38,470 --> 00:06:44,750
+إذا نقدر نتعدى و نقول a أكبر من c فهذه بيسموها في
+
+59
+00:06:44,750 --> 00:06:50,730
+الرياضيات transitivity أو خاصية التعدى الخاصية
+
+60
+00:06:50,730 --> 00:06:53,990
+التانية
+
+61
+00:06:53,990 --> 00:06:59,930
+بنسميها tricotomy برضه خاصية ثلاثية جاية من
+
+62
+00:06:59,930 --> 00:07:05,770
+الخاصية الثلاثية اللى شفناها قبل شويةفبتقول
+
+63
+00:07:05,770 --> 00:07:08,650
+exactly one of the following holds واحد من تلات
+
+64
+00:07:08,650 --> 00:07:19,190
+احتمالات بتحصل اما a اكبر من b او a بتساوي b او a
+
+65
+00:07:19,190 --> 00:07:28,260
+اصغر من bلأي عددين حقيقيين A وB واحد فقط من
+
+66
+00:07:28,260 --> 00:07:32,800
+الاحتمالات التلاتة بيكون صحيح وهو اما A أكبر من B
+
+67
+00:07:32,800 --> 00:07:38,860
+أو A بساوي B أو A أصغر من B الـ Antisymmetry
+
+68
+00:07:38,860 --> 00:07:43,640
+property علاقة أكبر من أو ساويها دي بنسميها
+
+69
+00:07:43,640 --> 00:07:48,400
+Antisymmetric يعني ايه؟ بتحقق خاصية تضاد التماثل
+
+70
+00:07:49,760 --> 00:07:54,960
+إيه يعني؟ مع أن لو كانت A على علاقة مع B و B على
+
+71
+00:07:54,960 --> 00:08:00,940
+علاقة مع A فلازم يطلع A بساوي B، A أكبر من أو ساوي
+
+72
+00:08:00,940 --> 00:08:05,300
+B و B أكبر من أو ساوي A فلازم A ساوي B، هذي
+
+73
+00:08:05,300 --> 00:08:11,640
+بنسميها Anti-symmetry propertyهنا الخاصية هذه لو
+
+74
+00:08:11,640 --> 00:08:18,140
+كان a أكبر من b وضفت للطرفين أي عدد c فالمتباينة
+
+75
+00:08:18,140 --> 00:08:22,040
+تبقى زي ما هي شريتها زي ما هي طيب لو في عندي
+
+76
+00:08:22,040 --> 00:08:26,100
+متباينة a أكبر من b لان نتحدث عن متباينات
+
+77
+00:08:26,100 --> 00:08:31,960
+inequalitiesلو كان a أكبر من b و c عدد موجب وضربت
+
+78
+00:08:31,960 --> 00:08:35,960
+الطرفين في عدد الموجب c فإشارة المتباينة تبقى كما
+
+79
+00:08:35,960 --> 00:08:40,380
+هي لكن لو ضربت المتباينة في عدد سالب إشارة
+
+80
+00:08:40,380 --> 00:08:46,900
+المتباينة تناكز الخاصية f بتقول أنه لأي عدد حقيقي
+
+81
+00:08:46,900 --> 00:08:51,300
+لا يساوي سفر مربع أي عدد حقيقي لا يساوي سفر دائما
+
+82
+00:08:51,300 --> 00:08:55,400
+بيكون عدد موجب الواحد
+
+83
+00:08:55,900 --> 00:08:59,780
+الـ Distinguished elements في R أو في الـ real
+
+84
+00:08:59,780 --> 00:09:02,940
+number system اللي هم السفر والواحد اللي هو ال
+
+85
+00:09:02,940 --> 00:09:07,280
+identity elements سمناهم بيحققوا ان واحد دايما
+
+86
+00:09:07,280 --> 00:09:13,400
+اكبر من السفر و سالب واحد اصغر من السفر كمان لأي
+
+87
+00:09:13,400 --> 00:09:16,840
+عدد طبيعي هذي the set of natural numbers اي عدد
+
+88
+00:09:16,840 --> 00:09:23,210
+طبيعي بيكون دايما موجب اي عدد طبيعي بيطلع موجبلو
+
+89
+00:09:23,210 --> 00:09:27,450
+كان a عدد حقيقي موجب فمقلوبه موجب لو كان a عدد
+
+90
+00:09:27,450 --> 00:09:38,990
+حقيقي سالب مقلوبه بيطلع سالب الخاصية
+
+91
+00:09:38,990 --> 00:09:45,610
+الأخيرة ال لو كان a أصغر من b و اتنين موجبين
+
+92
+00:09:45,610 --> 00:09:53,160
+فمقلوب لصغير أكبر من مقلوبالكبير أو مقلوب الكبير
+
+93
+00:09:53,160 --> 00:09:56,680
+أصغر من مقلوب الصغير بصرت اتنين اللي هم نفس
+
+94
+00:09:56,680 --> 00:10:00,980
+الإشارة لكن لو كان واحد موجة بواحد سالب فالكلام
+
+95
+00:10:00,980 --> 00:10:08,060
+هذا مش صحيح خدوا بالك طيب نشوف نمر بسرعة على
+
+96
+00:10:08,060 --> 00:10:15,500
+البرهين قرأته البرهين انتوا؟ طيب
+
+97
+00:10:30,910 --> 00:10:38,550
+خاصية التعدى خاصية التعدى انا كان عندي a أكبر من b
+
+98
+00:10:38,550 --> 00:10:44,710
+and b أكبر من c بدنا نثبت ان هذا يعدي ان a أكبر من
+
+99
+00:10:44,710 --> 00:10:52,250
+c فالبرهان ذلك يكفي نثبت ان الفرق بين c و a موجب
+
+100
+00:10:52,990 --> 00:10:56,990
+يعني ينتمي لل set P of positive real numbers
+
+101
+00:10:56,990 --> 00:11:02,450
+فتعالوا نثبت الكلام هذا أنا عندي من المعطيات او من
+
+102
+00:11:02,450 --> 00:11:08,370
+الفرض الفرق هذا موجب والفرق هذا موجب من المعطيات
+
+103
+00:11:09,240 --> 00:11:13,680
+طيب set P closed under addition مغلقة تحت عملية
+
+104
+00:11:13,680 --> 00:11:18,820
+الجمع إذا مجموعة أنصرين في P بيطلع أنصر تالت في P
+
+105
+00:11:18,820 --> 00:11:22,660
+هذا الأنصر التالت اللي بيقول المجموعة طلع A سالب C
+
+106
+00:11:22,660 --> 00:11:28,560
+هذا معناه مادام الفرخ هذا تملى P معناته الفرخ هذا
+
+107
+00:11:28,560 --> 00:11:33,900
+موجب أو A أكبر من C as required كما هو مطلوب،
+
+108
+00:11:33,900 --> 00:11:38,040
+مظبوط؟ واضح؟ طيب
+
+109
+00:11:42,860 --> 00:11:49,940
+أي عدد حقيقي له واحد من تلت احتمالات اما موجب او
+
+110
+00:11:49,940 --> 00:11:56,720
+سفر او سالب الان بناء على هذه الخاصية ممكن نثبت
+
+111
+00:11:56,720 --> 00:12:01,940
+الخاصية الثلاثية الخاصية
+
+112
+00:12:01,940 --> 00:12:05,600
+بي
+
+113
+00:12:09,230 --> 00:12:15,030
+قلنا إن لو كان لأي عددين حقيقيين لأي عددين حقيقيين
+
+114
+00:12:15,030 --> 00:12:19,750
+a و b، a أكبر من b أو a بساوي b أو a أصغر من b
+
+115
+00:12:19,750 --> 00:12:22,910
+فالبرهان
+
+116
+00:12:22,910 --> 00:12:27,350
+ذلك بيعتمد على ال try-cutting property اللي شفناها
+
+117
+00:12:27,350 --> 00:12:33,470
+قبل شوية فأنا
+
+118
+00:12:33,470 --> 00:12:33,870
+عندي
+
+119
+00:12:37,890 --> 00:12:41,310
+حسب الـ trichotomy property، لو أخدت الفرق هذا،
+
+120
+00:12:41,310 --> 00:12:46,850
+هذا real number فأي real number إما positive أو
+
+121
+00:12:46,850 --> 00:12:54,390
+بساوي سفر أو negative، صح؟ وهذا بكافئ، الكلام هذا
+
+122
+00:12:54,390 --> 00:13:01,210
+بكافئ A سالب B ينتمي ل B بكافئ أنه الـ A أكبر من B
+
+123
+00:13:02,260 --> 00:13:07,220
+طب وهذا ينتمي لـ 0 بكافة أن a بساوي b أو الفرق
+
+124
+00:13:07,220 --> 00:13:11,660
+بساوي 0 وبالتالي a بساوي b و الفرق هذا ينتمي ل
+
+125
+00:13:11,660 --> 00:13:16,180
+negative b معناته الفرق هذا سالب يعني معناه أن a
+
+126
+00:13:16,180 --> 00:13:21,760
+أصغر من b وهذا اللي بدنا إياه هذا اللي بدنا إياه
+
+127
+00:13:21,760 --> 00:13:25,620
+طيب
+
+128
+00:13:25,620 --> 00:13:32,390
+الجزء C قلنا اللي هو ال antisymmetry propertyالـ
+
+129
+00:13:32,390 --> 00:13:37,990
+Anti-symmetry property نفكركم فيها بتقول لو كان a
+
+130
+00:13:37,990 --> 00:13:46,590
+أكبر من أو يساوي b and b أكبر من أو يساوي a فهذا
+
+131
+00:13:46,590 --> 00:13:52,210
+بيعدي أن a بساوي b، بظبط؟ طيب
+
+132
+00:13:57,150 --> 00:14:02,570
+أنا بدأ أثبت أن A بساوي B، هذه النتيجة، فبدأ أعمل
+
+133
+00:14:02,570 --> 00:14:07,750
+برهان بالتناقض، فبرهان بالتناقض دائما نفرض مافيه
+
+134
+00:14:07,750 --> 00:14:12,670
+النتيجة هو الصح، وبنفسها إلى التناقض، ف assume أن
+
+135
+00:14:12,670 --> 00:14:21,500
+A لا تساوي Bإذا حسب الخاصية الفلاثية هذا بيقدّي ان
+
+136
+00:14:21,500 --> 00:14:30,800
+اما a أصغر من b or b أصغر من a، مظبوط؟ طيب إذا هنا
+
+137
+00:14:30,800 --> 00:14:39,720
+.. الآن لو أخدت .. لو أخدت ال a أكبر من b اللي هو
+
+138
+00:14:39,720 --> 00:14:46,880
+الاحتمال هذالو أخدت .. لو قلت أن a أكبر من b فهذا
+
+139
+00:14:46,880 --> 00:14:54,140
+بتناقض مع الفرض .. بتناقض مع الفرض أن a أصغر من ..
+
+140
+00:14:54,140 --> 00:15:01,160
+a أصغر من أوسع من b هدول اتنين بيعطون التناقض طيب
+
+141
+00:15:01,160 --> 00:15:06,400
+لو افترضت الاحتمال التاني أن a أصغر من b فهذا
+
+142
+00:15:06,400 --> 00:15:15,210
+بتناقض مع الفرض أن a أكبر منأو يساوي الـ B إذا في
+
+143
+00:15:15,210 --> 00:15:20,790
+الحالتين لو فرضت هذا صح بتناقض مع هذا الجزء لو
+
+144
+00:15:20,790 --> 00:15:25,410
+فرضت هذا صح بتناقض مع هذا الجزء اللي هو جزء من
+
+145
+00:15:25,410 --> 00:15:29,970
+الفرض وبالتالي في كلتا الحالتين الفرض أن A لا
+
+146
+00:15:29,970 --> 00:15:35,050
+يساوي B أدى إلى تناقض إذا الصح أن A لازم تساوي B
+
+147
+00:15:35,050 --> 00:15:40,600
+كما هو مطلوب okay هذا برهان بالتناقضواضح تمام
+
+148
+00:15:40,600 --> 00:15:47,740
+مفهوم فاهمين ولا هيك يعني أمور
+
+149
+00:15:47,740 --> 00:15:53,040
+سهلة وبسيطة وكلها يعني مبادئ رياضيات احنا هنا يعني
+
+150
+00:15:53,040 --> 00:15:58,840
+مراجعة لمبادئ رياضيات أو طرق البرهان في مبادئ
+
+151
+00:15:58,840 --> 00:16:03,680
+رياضيات طيب
+
+152
+00:16:03,680 --> 00:16:07,220
+الآن بنثبت القصية F
+
+153
+00:16:13,430 --> 00:16:18,050
+لأي عدد حقيقي لا يساوي سفر دائما مربع و بيطلع موجب
+
+154
+00:16:18,050 --> 00:16:22,330
+فعشان أثبت مربع ال a موجب لازم أثبت ان مربع ال a
+
+155
+00:16:22,330 --> 00:16:30,890
+ينتمي لفئة او مجموعة العداد الموجبة طيب احنا فرضين
+
+156
+00:16:30,890 --> 00:16:34,950
+a لايساوي سفر اذا by tricotomy property بالخاصية
+
+157
+00:16:34,950 --> 00:16:40,470
+التلاتية اما a موجب او سالب يعني معناه هذا او هذا
+
+158
+00:16:40,470 --> 00:16:51,220
+الانلو كانت ال A موجبة فمربع و ال P مغلقة تحت
+
+159
+00:16:51,220 --> 00:16:56,440
+عملية الضرب فحاصل ضرب A في A اللي هو A تربية بيطلع
+
+160
+00:16:56,440 --> 00:17:03,580
+ينتمي يعني هذا بيساوي A تربية ال
+
+161
+00:17:03,580 --> 00:17:07,660
+A ينتمي ل P إذا حاصل الضرب ينتمي ل P وبالتالي A
+
+162
+00:17:07,660 --> 00:17:12,080
+تربية موجبة okay وهذا اللي احنا عايزينهالحالة
+
+163
+00:17:12,080 --> 00:17:17,600
+التانية طب افرض انه negative A تنتمي ل P او A
+
+164
+00:17:17,600 --> 00:17:23,200
+تنتمي ل negative P يعني A سالم ففي الحالة هذه لو
+
+165
+00:17:23,200 --> 00:17:29,220
+ضربت هذا العنصر في نفسه بطلع ينتمي إلى ال P بطلع
+
+166
+00:17:29,220 --> 00:17:33,480
+ينتمي إلى ال P وهذا بطلع بساوي من الخواص اللي
+
+167
+00:17:33,480 --> 00:17:37,510
+أخدناها قبل هيكيعني هذا عبارة عن هذا سالب إيه
+
+168
+00:17:37,510 --> 00:17:40,970
+بكتبه سالب واحد في إيه و سالب إيه التاني نفس
+
+169
+00:17:40,970 --> 00:17:45,650
+الحاجة سالب واحد في إيه فبطلع سالب واحد في سالب
+
+170
+00:17:45,650 --> 00:17:50,170
+واحد في إيه تربية و هذا واحد فبطلع إيه تربية تنتمي
+
+171
+00:17:50,170 --> 00:17:54,830
+لدي وبالتالي إيه تربية موجبة إذا هنا أثبتنا إن أي
+
+172
+00:17:54,830 --> 00:17:59,690
+عدد حقيقي مختلف عن السفر دائما مربع موجب
+
+173
+00:18:13,230 --> 00:18:23,990
+خاصية جي الخاصية
+
+174
+00:18:23,990 --> 00:18:24,510
+جي
+
+175
+00:18:30,830 --> 00:18:36,810
+احنا بنفبط أن الواحد أكبر من السفر فبكل بساطة واحد
+
+176
+00:18:36,810 --> 00:18:42,710
+بساوي واحد ضرب نفسه وهذا بيطلع واحد تربية و قبل
+
+177
+00:18:42,710 --> 00:18:46,710
+شوية شوفنا و الواحد مختلف عن السفر إذا المربع
+
+178
+00:18:46,710 --> 00:18:54,870
+بيطلع موجب حسب الخاصية السابقة، أثبت؟هذا معناه إذا
+
+179
+00:18:54,870 --> 00:18:59,770
+هيثبتنا واحد أكبر من السفر وبالتالي واحد ينتمي لل
+
+180
+00:18:59,770 --> 00:19:04,670
+positive real numbers إذا سالب واحد ينتمي ل
+
+181
+00:19:04,670 --> 00:19:07,610
+negative two يعني negative واحد أصغر من السفر
+
+182
+00:19:07,610 --> 00:19:20,050
+عملية بسيطة طيب احنا الآن بدنا نثبت ان كل
+
+183
+00:19:22,970 --> 00:19:31,370
+عدد حقيقي موجب مقلوبه موجب اه فبنعمل برهان
+
+184
+00:19:31,370 --> 00:19:36,190
+بالتناقض اذا هنا هندي اللي عايز اثبته هنا بس هنذكر
+
+185
+00:19:36,190 --> 00:19:41,430
+ال statement اللي بدنا نثبته يعني ال statement
+
+186
+00:19:41,430 --> 00:19:46,710
+اللي عايز اثبته لو كان a موجب ف reciprocal تبعه
+
+187
+00:19:46,710 --> 00:19:51,920
+بطلع موجب او مقلوبه بطلع موجبلبرهان ذلك نعمل برهان
+
+188
+00:19:51,920 --> 00:20:01,560
+بالتناقض نفرض أن واحد على a أقل من السفر وطبعا
+
+189
+00:20:01,560 --> 00:20:06,380
+عندي انا من الفرض هذا الفرض لازال قائم a أكبر من
+
+190
+00:20:06,380 --> 00:20:13,520
+السفر عندي الفرابين هدول فعندي a أكبر من السفر و
+
+191
+00:20:13,520 --> 00:20:17,640
+واحد على a أصغر من السفر فهذا بيقدي
+
+192
+00:20:20,870 --> 00:20:28,190
+لو ضربت المتباينة هذه في a اللي هو عدد موجب فهيصير
+
+193
+00:20:28,190 --> 00:20:32,450
+اندي واحد على a في a أصغر من سفر في a اللي هو
+
+194
+00:20:32,450 --> 00:20:37,290
+بيساوي سفر طب هدف بيساوي واحد ان هك بيطلع واحد
+
+195
+00:20:37,290 --> 00:20:42,530
+أصغر من سفر وبالتالي هدف يعطيني تناقض لأن الواحد
+
+196
+00:20:42,530 --> 00:20:47,530
+أكبر من سفر لسه مثبتينه قبل شوية ان هدف بيأدي إلى
+
+197
+00:20:47,530 --> 00:20:54,640
+تناقض وبالتاليمقلوب الـ A لازم يكون موجب بالمثل لو
+
+198
+00:20:54,640 --> 00:21:00,980
+كان مقلوب الـ A سالب فممكن نثبت انه مقلوب و ايضا
+
+199
+00:21:00,980 --> 00:21:05,900
+بيطلع سالب فالبرهان مشابه حسيبكم انتوا تكتبوه
+
+200
+00:21:05,900 --> 00:21:07,720
+تمام؟
+
+201
+00:21:21,570 --> 00:21:23,670
+أنا مش عارف لسه أنا هيك بعمل
+
+202
+00:21:50,000 --> 00:21:59,840
+طيب ال .. الجزء هذا الأخير إيش كان هذا؟ إيش كنا
+
+203
+00:21:59,840 --> 00:22:14,980
+بدنا نثبت هناك؟
+
+204
+00:22:17,920 --> 00:22:26,800
+اه إذا كان a عدد موجب و أصغر من b فهذا بيقدّي أن
+
+205
+00:22:26,800 --> 00:22:34,100
+مقلوب الكبير أصغر من مقلوب الصغير بظبط
+
+206
+00:22:34,100 --> 00:22:39,080
+و طبعا هذا موجب فلإثبات
+
+207
+00:22:39,080 --> 00:22:43,360
+أن واحد على بي أصغر من واحد على ايه بتثبت أن الفرق
+
+208
+00:22:43,360 --> 00:22:52,860
+بين واحد على ايه واحد على ايهو 1 على D ينتمي إلى P
+
+209
+00:22:52,860 --> 00:22:59,900
+أو موجة طيب الان هاي ناخد 1 على A سلب 1 على B
+
+210
+00:22:59,900 --> 00:23:05,140
+فاخدنا خاصية ناخد مقام مشترك A B و بعدين بيصير
+
+211
+00:23:05,140 --> 00:23:11,160
+عندى هذا بتحول لحاصل ضرب الان هذا positive number
+
+212
+00:23:11,160 --> 00:23:15,420
+لان احنا فرضين ان ال B أكبر من A فالفرق هذا
+
+213
+00:23:15,420 --> 00:23:23,780
+positiveو A B فبطلع
+
+214
+00:23:23,780 --> 00:23:29,200
+هذا مقلوب ال positive بطلع positive فهذا
+
+215
+00:23:29,200 --> 00:23:33,120
+positive و هذا positive و ال 6 دي closed under
+
+216
+00:23:33,120 --> 00:23:36,400
+multiplication إذن حاصر الضربة ده بطلع positive
+
+217
+00:23:36,400 --> 00:23:45,000
+لكون حاصر الضرب هنا العناصر فيه موجبة وبالتالي
+
+218
+00:23:46,170 --> 00:23:53,450
+إذا .. إذا هذا بيطلع أكبر من الصفر هذا بيطلع الفرق
+
+219
+00:23:53,450 --> 00:23:58,030
+أكبر من أوم وجب وبالتالي واحد على أيه أكبر من واحد
+
+220
+00:23:58,030 --> 00:24:03,770
+على بيه okay الأجزاء المتبقية D وE وH ممكن برهانة
+
+221
+00:24:03,770 --> 00:24:10,030
+بالمثل فاحنا دايما بنسيب للطالب شوية حاجات يثبتها
+
+222
+00:24:11,300 --> 00:24:15,740
+يعني عشان ان الطالب يشارك شوية و إلا بيصير عملية
+
+223
+00:24:15,740 --> 00:24:20,020
+التدريس مملة لو احنا بدنا نشرحلكم كل حاجة و مانخلش
+
+224
+00:24:20,020 --> 00:24:25,880
+ولا إيش للطالب فبصير عملية مملة و بعدين الفهم بكون
+
+225
+00:24:25,880 --> 00:24:31,860
+ماخص كل ما انت شاركت أكتر كل ما شعرتي أو حسيتي
+
+226
+00:24:31,860 --> 00:24:36,900
+بالمعلومة أكتر و كل ما فهمتيها أكتر فالحاجات هذه
+
+227
+00:24:36,900 --> 00:24:40,500
+بالإضافة للتمارين اللي في نهاية كل section في
+
+228
+00:24:40,500 --> 00:24:47,560
+الكتابحالها كتير بساعد في فهم المادة بدون ذلك بظل
+
+229
+00:24:47,560 --> 00:24:57,420
+فهمكم نقص ننتقل إلى نظرية أخرى نظرية واحد ستة
+
+230
+00:24:57,420 --> 00:25:02,680
+نظرية هذه نظرية يعني بسيطة ومهمة
+
+231
+00:25:04,730 --> 00:25:09,830
+رغم بساطتها لكن مهمة إيش بتقول النظرية هذه بتقول
+
+232
+00:25:09,830 --> 00:25:16,250
+لو أخدت أي عددين حقيقين و a أكبر من b فلازم يكون a
+
+233
+00:25:16,250 --> 00:25:26,310
+أكبر من متوسط a و b و أكبر من b البرهان بسيط هي
+
+234
+00:25:26,310 --> 00:25:32,330
+عند الفرض أنا فارض أن a أكبر من bبتثبت أن a أكبر
+
+235
+00:25:32,330 --> 00:25:39,330
+من نص مجموعة a و b و نص مجموعة a و b أكبر من b طيب
+
+236
+00:25:39,330 --> 00:25:44,490
+نثبت المتباينة الأولى هذه نثبت المتباينة الأولى
+
+237
+00:25:44,490 --> 00:25:49,230
+الأول بعدين نثبت التانية
+
+238
+00:25:52,810 --> 00:25:59,530
+فالإثبات الجزء الأول فهي عندي a أكبر من b إذا لو
+
+239
+00:25:59,530 --> 00:26:07,110
+جمعت a على نفسها ده اتنين a لو جمعت على الطرفين a
+
+240
+00:26:07,110 --> 00:26:11,630
+فبطلع عندي a زائد a أكبر من b زاد a هذه خاصية
+
+241
+00:26:11,630 --> 00:26:15,810
+أخدناها قبل a إذا اتنين a بيطلع أكبر من a زائد b
+
+242
+00:26:16,680 --> 00:26:22,900
+كذلك لو جمعت على الطرفين هنا B فبطلع A زائد B أكبر
+
+243
+00:26:22,900 --> 00:26:26,320
+من B زائد B A زائد B أكبر من B زائد B اللي هو
+
+244
+00:26:26,320 --> 00:26:32,320
+اتنين B إذا أنا في عندي الآن متباينتين اتنين A
+
+245
+00:26:32,320 --> 00:26:39,960
+أكبر من A زائد B هيا اتنين A أكبر من A زائد B وA
+
+246
+00:26:39,960 --> 00:26:46,700
+زائد B أكبر من اتنين B إذا by transitivityخاصية
+
+247
+00:26:46,700 --> 00:26:52,740
+التعدى ممكن استنتج ان اتنين a اكبر من a زايد b
+
+248
+00:26:52,740 --> 00:26:59,440
+اكبر من اتنين b الان العدد اتنين عدد طبيعي وشوفنا
+
+249
+00:26:59,440 --> 00:27:03,360
+في الخاصية بتقول اي عدد طبيعي هو عدد موجب فى
+
+250
+00:27:03,360 --> 00:27:09,520
+النظرية اللى فاتت كذلك اي عدد موجب مقلوبه موجب اذا
+
+251
+00:27:09,520 --> 00:27:14,520
+النص عدد موجب الان لو ضربت المتباينة هذه فى النص
+
+252
+00:27:14,520 --> 00:27:19,340
+اللى هو عدد موجبإشاراتها تبقى زي ما هي هذه خاصية
+
+253
+00:27:19,340 --> 00:27:27,740
+خلناها في النظرية هذه تمام؟ إذا أنا حضرب الفنص هي
+
+254
+00:27:27,740 --> 00:27:33,960
+ضربت طبعا هذا بيساوي a وهذا بيساوي b وبالتالي نحصل
+
+255
+00:27:33,960 --> 00:27:40,520
+على المطلوب إذا يعني براهين سهلة وبسيطة النظرية
+
+256
+00:27:40,520 --> 00:27:46,960
+هذه مهمة لأن نتيجة اللي بعدهاأو أهميتها تظهر في
+
+257
+00:27:46,960 --> 00:27:53,580
+النتيجة اللي بعدها اللي هي corollary 171 corollary
+
+258
+00:27:53,580 --> 00:28:04,620
+171 بيقول أن
+
+259
+00:28:04,620 --> 00:28:12,040
+أي عدد موجب بيكون أكبر من نصه اللي هو موجب أي عدد
+
+260
+00:28:12,040 --> 00:28:18,520
+حقيقي موجب دايما أكبر من نصهوبالتالي هذا معناه في
+
+261
+00:28:18,520 --> 00:28:23,200
+رياضيات أن الأعداد الحقيقية الموجبة مالهاش
+
+262
+00:28:23,200 --> 00:28:27,960
+smallest element مافيش .. لو أخدت الأعداد الحقيقية
+
+263
+00:28:27,960 --> 00:28:35,200
+الموجبة اللي هي set P فهذا ال set ماقدرش أحط أصبعي
+
+264
+00:28:35,200 --> 00:28:42,360
+على أصغر عنصر فيها مالهاش أصغر عنصرhas no smallest
+
+265
+00:28:42,360 --> 00:28:48,580
+element لأن لو أخدت أي عنصر موجب و سميته a فبقدر
+
+266
+00:28:48,580 --> 00:28:54,200
+ألاقي عدد موجب أخر أصغر منه اللي هو نصف فبالتالي
+
+267
+00:28:54,200 --> 00:28:59,800
+ال set of positive numbers has no strictly
+
+268
+00:28:59,800 --> 00:29:04,500
+positive element تمام؟ البرهان تبع الكرولري هذا
+
+269
+00:29:04,500 --> 00:29:09,360
+بينتج من نظريةيعني خد بي بساوة سفر في النظرية اللى
+
+270
+00:29:09,360 --> 00:29:26,280
+فاتت نظرية واحد ستة هي تشوفها مع بعض نظرية
+
+271
+00:29:26,280 --> 00:29:30,860
+واحد ستة لو أخدت بي بساوة سفر فبطلع ا اكبر من نص ا
+
+272
+00:29:30,860 --> 00:29:37,970
+اكبر من سفر اذا هذه نتيجة سريعة مظبوطOkay إذا يعني
+
+273
+00:29:37,970 --> 00:29:45,050
+هذه بعض الحاجات السهلة والبسيطة، هنا في نظرية كتير
+
+274
+00:29:45,050 --> 00:29:49,950
+مهمة، هذه برضه نظرية هنستخدمها بكرا يعني في
+
+275
+00:29:49,950 --> 00:29:55,770
+المستقبل، نظرية واحد تمنع، نظرية كتير مهمة وأهمتها
+
+276
+00:29:55,770 --> 00:30:02,670
+هنشوفها في الشبات الرجايةإيش هذه النظرية بتقول؟ لو
+
+277
+00:30:02,670 --> 00:30:08,110
+في عندي عدد حقيقي غير سالب، غير سالب، و في نفس
+
+278
+00:30:08,110 --> 00:30:14,050
+الوقت أصغر من إبسلون لكل عدد موجب إبسلون، فهذا
+
+279
+00:30:14,050 --> 00:30:19,350
+العدد لازم يكون هو السفر، وهي برهان بالتناقض
+
+280
+00:30:22,990 --> 00:30:28,470
+كمان مرة العدد غير السالب اللى بيكون اي اصغر من اي
+
+281
+00:30:28,470 --> 00:30:33,590
+عدد موجب هو السفر مافيش غير السفر اللى بيحقق
+
+282
+00:30:33,590 --> 00:30:40,250
+لخاصية هذه لبرهان ذلك نعمل برهان بالتناقض افرض ان
+
+283
+00:30:40,250 --> 00:30:45,830
+ال a ان ال a هذا بيسويش السفر و في نفس الوجهة a
+
+284
+00:30:45,830 --> 00:30:48,850
+غير سالب اذا يعني a موجب صح؟
+
+285
+00:30:52,860 --> 00:30:57,980
+الان حسب نظرية الكورينة النتيجة واحد سبعة اذا a
+
+286
+00:30:57,980 --> 00:31:05,260
+بطلع اكبر من نص a فاخد epsilon zero هنا عدد موجب
+
+287
+00:31:05,260 --> 00:31:11,260
+بساوي a ع اتنين نص a هذا عدد موجب اذا هاني نجحت في
+
+288
+00:31:11,260 --> 00:31:18,670
+ايجاد عدد epsilon zero عدد موجب وال a اكبر منههذا
+
+289
+00:31:18,670 --> 00:31:22,570
+يتناقض مع الفرض أن a أصغر من إبسلون لكل إبسلون
+
+290
+00:31:22,570 --> 00:31:29,190
+أكبر من السفر أظبط؟ لأن هذا التناقض يثبت النظرية
+
+291
+00:31:29,190 --> 00:31:39,010
+واضح تمام؟ واضح البرهن؟ عيده طيب أنا عندي a عدد
+
+292
+00:31:39,010 --> 00:31:43,190
+حقيقي غير سالم وفي نفس الوجهة أصغر من كل الأعداد
+
+293
+00:31:43,190 --> 00:31:49,030
+الموجبة إبسلون بدا أثبت أن a بساوي سفربرهان
+
+294
+00:31:49,030 --> 00:31:53,630
+بالتناقض prove by contradiction assume or suppose
+
+295
+00:31:53,630 --> 00:31:58,910
+the contrary النقيض أو النفي تبع النتيجة يعني a ما
+
+296
+00:31:58,910 --> 00:32:03,390
+بيستويش صفر نفي a بيستوي صفر a لا تستوي صفر طب أنا
+
+297
+00:32:03,390 --> 00:32:07,470
+كاتب هنا ال contrary a أكبر من صفر هذا صح بناء على
+
+298
+00:32:07,470 --> 00:32:11,930
+أن الفرض a أكبر من أكبر من صفر وما بيستويش صفر إذن
+
+299
+00:32:11,930 --> 00:32:16,970
+أكبر من صفر صح طيب الآن
+
+300
+00:32:18,230 --> 00:32:23,270
+لو أخدت Epsilon Zero بساوي نص A و بما أنه A عدد
+
+301
+00:32:23,270 --> 00:32:27,650
+موجب فنتيجة واحدة السابعة بتقول لو كان A عدد موجب
+
+302
+00:32:27,650 --> 00:32:35,520
+فنص A بطلع عدد موجبإذا هيني و في نفس الوجد كمان ال
+
+303
+00:32:35,520 --> 00:32:40,820
+a أكبر من نص a ال a أكبر من نص a وبالتالي إذا هيني
+
+304
+00:32:40,820 --> 00:32:45,900
+لجحت في إيجاد epsilon zero عدد موجب و a أكبر منه
+
+305
+00:32:45,900 --> 00:32:54,700
+هذا بتناقض مع الفرض أنه بتناقض مع الفرض أنه a أصغر
+
+306
+00:32:54,700 --> 00:33:03,090
+من epsilon لكل epsilon موجبة صح؟ الإبارة هذه هينفي
+
+307
+00:33:03,090 --> 00:33:07,710
+هذه نفي هذه نفي هذه نفي هذه نفي هذه نفي هذه نفي
+
+308
+00:33:07,710 --> 00:33:10,950
+هذه نفي هذه نفي هذه نفي هذه نفي هذه نفي هذه نفي
+
+309
+00:33:10,950 --> 00:33:10,950
+هذه نفي هذه نفي هذه نفي هذه نفي هذه نفي هذه نفي
+
+310
+00:33:10,950 --> 00:33:11,090
+هذه نفي هذه نفي هذه نفي هذه نفي هذه نفي هذه نفي
+
+311
+00:33:11,090 --> 00:33:11,090
+هذه نفي هذه نفي هذه نفي هذه نفي هذه نفي هذه نفي
+
+312
+00:33:11,090 --> 00:33:11,430
+هذه نفي هذه نفي هذه نفي هذه نفي هذه نفي هذه نفي
+
+313
+00:33:11,430 --> 00:33:11,430
+هذه نفي هذه نفي هذه نفي هذه نفي هذه نفي هذه نفي
+
+314
+00:33:11,430 --> 00:33:11,430
+هذه نفي هذه نفي هذه نفي هذه نفي هذه نفي هذه نفي
+
+315
+00:33:11,430 --> 00:33:11,430
+هذه نفي هذه نفي هذه نفي هذه نفي هذه نفي هذه نفي
+
+316
+00:33:11,430 --> 00:33:12,810
+هذه نفي هذه نفي هذه نفي هذه نفي هذه نفي هذه نفي
+
+317
+00:33:12,810 --> 00:33:13,730
+هذه نفي هذه نفي هذه نفي هذه نفي هذه نفي هذه نفي
+
+318
+00:33:13,730 --> 00:33:21,570
+هذه نفي هذه نفي هذه نفي هذه نفي هذه نفي هذه نفي
+
+319
+00:33:21,570 --> 00:33:25,250
+هذه نفي هذه نفي
+
+320
+00:33:28,550 --> 00:33:32,050
+Okay، إذا إحنا لحد الآن يعني كل شغلنا مبادئ
+
+321
+00:33:32,050 --> 00:33:37,270
+رياضيات، صح؟ طيب، طب ما هي مبادئ رياضيات هي أساس
+
+322
+00:33:37,270 --> 00:33:46,390
+ال .. اسمها أساسية الرياضيات، فاسم على مسمة فبختل
+
+323
+00:33:46,390 --> 00:33:50,870
+فهمة المادة هذه، جابت علينا منيحة يعني، هترتاح في
+
+324
+00:33:50,870 --> 00:33:57,380
+المستجبل كتير Bernoulli inequalityبرنول
+
+325
+00:33:57,380 --> 00:34:01,760
+الانيقوليتي هذه يعني في شوية متباينات طبعا مهمة في
+
+326
+00:34:01,760 --> 00:34:06,860
+الكتاب انا اختارت واحدة منهم لكن في بعض المتباينات
+
+327
+00:34:06,860 --> 00:34:13,180
+الأخرى موجودة في الكتاب وارجو انكم تقراوها فبرنول
+
+328
+00:34:13,180 --> 00:34:15,960
+الانيقوليتي هذه واحدة منهم متباينة برنول يعني
+
+329
+00:34:15,960 --> 00:34:23,230
+بيقول لو كان X عدد حقيقي أكبر من سالب واحدفمجموعة
+
+330
+00:34:23,230 --> 00:34:28,750
+واحد و X to the power N دايما أكبر من أو ساوي واحد
+
+331
+00:34:28,750 --> 00:34:38,850
+زائد N ضرب X وهذا صحيح لكل الأعداد الطبيعية نعم
+
+332
+00:34:38,850 --> 00:34:46,290
+في نظرية جاب الهاجم ماخلنهاش شوف
+
+333
+00:34:46,290 --> 00:34:46,890
+مع بعض
+
+334
+00:34:50,730 --> 00:35:07,610
+أه صحيح نشوف النظرية واحد تسعة نظرية
+
+335
+00:35:07,610 --> 00:35:10,870
+واحد تسعة بتقول لو كان أندي عددين حقيقين حاصل
+
+336
+00:35:10,870 --> 00:35:15,610
+ضربهم موجب فيا إما اتنين موجبين يا إما اتنين
+
+337
+00:35:15,610 --> 00:35:20,450
+سالبين صح؟ممكن يكون الاتنين مختلفين في الإشارة و
+
+338
+00:35:20,450 --> 00:35:25,530
+حصل ضربهم موجب إذا حصل ضرب عددين موجب بيقدر انه
+
+339
+00:35:25,530 --> 00:35:33,670
+اما اتنين موجبين او اتنين سالبين فالبرهان نشوف كيف
+
+340
+00:35:33,670 --> 00:35:39,640
+افرض الفرض تبعنا ان حصل ضرب A وB موجبفهذا أكيد
+
+341
+00:35:39,640 --> 00:35:42,780
+بيقدّي ان لا ال a بيساوي سفر ولا ال b بيساوي سفر
+
+342
+00:35:42,780 --> 00:35:46,520
+لأن لو واحد منهم بيساوي سفر فحاصل الدرب هيطلع
+
+343
+00:35:46,520 --> 00:35:50,720
+بيساوي سفر contradiction تناقض صح؟ لأن هذا
+
+344
+00:35:50,720 --> 00:36:02,020
+الاستنتاج منطقي طيب الان احنا ال a ناخد ناخد الجزء
+
+345
+00:36:02,020 --> 00:36:09,250
+هذا الان انا عند a لا يساوي سفربقى اتراي كاتومي
+
+346
+00:36:09,250 --> 00:36:14,650
+property حسب الخلصية التي هي اما a أكبر من سفر أو
+
+347
+00:36:14,650 --> 00:36:23,650
+a أصغر من سفر صح؟ بقى في احتمالين طيب ناخد ال a لو
+
+348
+00:36:23,650 --> 00:36:30,370
+كان افرض ان a أكبر من سفر فهذا بيدى ان واحد على a
+
+349
+00:36:30,370 --> 00:36:42,620
+أكبر من سفرهذا يعني 1 على a أكبر من 0 بيؤدي
+
+350
+00:36:42,620 --> 00:36:48,700
+أيضًا إلى بي اللي هو بساوي ال
+
+351
+00:36:48,700 --> 00:36:53,720
+بي ممكن اكتبها واحد في بي والواحد ممكن ابدله بواحد
+
+352
+00:36:53,720 --> 00:36:57,800
+على a في a واستخدم ال associative law واكتب هذا
+
+353
+00:36:57,800 --> 00:37:04,910
+على صورة واحد على a في a بي الان هذا موجبوهذا موجب
+
+354
+00:37:04,910 --> 00:37:12,330
+إذا حصلت ضرب بيطلع موجب إذا هذه أثبتت أن ال a أكبر
+
+355
+00:37:12,330 --> 00:37:19,950
+من ال b أكبر من السفر لأ احنا أخدنا ال a أكبر من
+
+356
+00:37:19,950 --> 00:37:24,290
+السفر فأدت
+
+357
+00:37:24,290 --> 00:37:28,690
+إلى أن ال b
+
+358
+00:37:28,690 --> 00:37:32,430
+أكبر من السفر وبالتالي بيطلع ال a و ال b موجبين
+
+359
+00:37:34,090 --> 00:37:40,810
+بالمثل لو افترضت .. اخدت لو افترضت ان a سالب فطبعا
+
+360
+00:37:40,810 --> 00:37:46,410
+مقلوب العدد السالب بيطلع سالب وبالتالي ال b اللي
+
+361
+00:37:46,410 --> 00:37:52,830
+هي بتساوي واحد على a في a b زي ما عملنا هنا ال b
+
+362
+00:37:52,830 --> 00:37:56,810
+بتطلع بتساوي واحد على a في a b ف ..
+
+363
+00:38:00,670 --> 00:38:05,850
+فهذا بيطلع الحاصل بضرب سالب لأن عندي انا هي هذه
+
+364
+00:38:05,850 --> 00:38:12,290
+المتباينة هذه هي واحد على ا سالب لو ضربت المتباينة
+
+365
+00:38:12,290 --> 00:38:18,370
+هذه في العدد الموجب a,b اللي هو عدد موجب فبصير
+
+366
+00:38:18,370 --> 00:38:21,910
+المتباينة هذه عبارة عن واحد على a في a,b الطرف
+
+367
+00:38:21,910 --> 00:38:29,070
+الشمال وضربتها في عدد موجب فبطلع أصغر من سفر في a
+
+368
+00:38:29,070 --> 00:38:30,030
+,b اللي هو سفر
+
+369
+00:38:32,730 --> 00:38:38,730
+وبالتالي بيطلع عندي الـ B بيطلع عند الـ B التي هي
+
+370
+00:38:38,730 --> 00:38:46,390
+أصغر للسفرإذا مرة تانية لو فرضنا أن a,b أكبر من 0
+
+371
+00:38:46,390 --> 00:38:50,670
+فشوفنا أن لا ال a بالساوية 0 ولا ال b بالساوية 0
+
+372
+00:38:50,670 --> 00:38:56,670
+وبالتالي أما بطلع a أكبر من 0 أو a أصغر من 0 في
+
+373
+00:38:56,670 --> 00:39:01,010
+الحالة الأولى لو كان a أكبر من 0 بطلع b أكبر من 0
+
+374
+00:39:01,010 --> 00:39:05,170
+وبالتالي a وb موجبين في الاحتمال التاني أو الحالة
+
+375
+00:39:05,170 --> 00:39:09,940
+التانية لو كان a سالب فشوفنا أن بطلع b سالبو
+
+376
+00:39:09,940 --> 00:39:18,480
+بالتالي اتنين سالبين okay تمام نشوف
+
+377
+00:39:18,480 --> 00:39:27,240
+الان Bernoulli inequality اليوم
+
+378
+00:39:27,240 --> 00:39:32,620
+هناخد برهان by induction برضه مبادئ الرياضيات خلنا
+
+379
+00:39:32,620 --> 00:39:39,360
+برهان by contradiction و direct proofوهنشوف move
+
+380
+00:39:39,360 --> 00:39:44,880
+by induction نمسح
+
+381
+00:39:44,880 --> 00:39:49,300
+اللوح بيرنول
+
+382
+00:39:49,300 --> 00:39:52,420
+ال equality زي ما قلنا لو كان x عدد حقيقي أكبر من
+
+383
+00:39:52,420 --> 00:39:58,000
+سالب واحد فلمّا أضيف عليه واحد وارفع لقوة n هذا
+
+384
+00:39:58,000 --> 00:40:02,380
+بيطلع أكبر من أو سالب واحد زائد n في x وهذا صحيح
+
+385
+00:40:02,380 --> 00:40:07,290
+لكل الأعداد الطبيعيةالبرغم by induction لو كانت n
+
+386
+00:40:07,290 --> 00:40:12,690
+بساوي واحد بثبت صحة العبارة عند n بساوي واحد لأن
+
+387
+00:40:12,690 --> 00:40:20,350
+ان تبدأ من واحد فلو كان n بساوي واحد فالطرف
+
+388
+00:40:20,350 --> 00:40:23,530
+الشمال
+
+389
+00:40:23,530 --> 00:40:31,900
+بطلع واحد زاد x صح؟الطرف الشمال واحد زائد X والطرف
+
+390
+00:40:31,900 --> 00:40:37,240
+اليمين برضه واحد زائد X فبطلع مساواة وطبعا
+
+391
+00:40:37,240 --> 00:40:42,720
+المساواة بقدر بدلها بأكبر من أوسعه مافي مشكلة
+
+392
+00:40:42,720 --> 00:40:46,020
+تمام؟
+
+393
+00:40:46,020 --> 00:40:52,700
+إذا العبارة هذه صحيحة عند N بالساوي واحد الآن نفرض
+
+394
+00:40:52,700 --> 00:40:57,780
+أن العبارة صحيحة عند N بالساوي K حيث K أكبر من
+
+395
+00:40:57,780 --> 00:41:05,800
+واحدهذا ما نسميه induction hypothesis الفرض تبع ال
+
+396
+00:41:05,800 --> 00:41:12,740
+induction نفرض صحة العبارة عند N بساوة K حيث K
+
+397
+00:41:12,740 --> 00:41:18,080
+أكبر من 1 هذا معناه أن 1 زاد X to K bigger than or
+
+398
+00:41:18,080 --> 00:41:24,510
+equal to 1 plus K Xطيب الان نريد نكمل ال induction
+
+399
+00:41:24,510 --> 00:41:32,790
+عايزين نثبت صحة العبارة واحد اللي هي هذه العبارة
+
+400
+00:41:32,790 --> 00:41:38,730
+واحد مش عارف من الواحد رايح العبارة واحد هذه نثب
+
+401
+00:41:38,730 --> 00:41:45,650
+الصحة عندنا بساوة K زياد واحد طيب from اتنين هذه
+
+402
+00:41:45,650 --> 00:41:47,870
+العبارة اتنين اللي هي induction hypothesis
+
+403
+00:41:52,040 --> 00:41:59,640
+بتدفع في الـ type هاي العبارة هذه لما n ساوي k
+
+404
+00:41:59,640 --> 00:42:05,880
+زائد واحد هصير واحد زائد x الكل أس k زائد واحد
+
+405
+00:42:05,880 --> 00:42:12,410
+أكبر من أو ساوي واحد زائد k زائد واحدفي X هذه
+
+406
+00:42:12,410 --> 00:42:18,390
+العبارة and N بساوي K زي 1 نبدأ بالطرف الشمال و
+
+407
+00:42:18,390 --> 00:42:22,670
+نثبت أنه أكبر من أو يساوي الطرف اليمين هاي الطرف
+
+408
+00:42:22,670 --> 00:42:27,950
+الشمال بقدر أجزئه حسب قوانين الأسس ل1 plus K to K
+
+409
+00:42:27,950 --> 00:42:34,410
+و 1 زي X to K ضرب 1 زي X الآن من اتنين من العبارة
+
+410
+00:42:34,410 --> 00:42:38,070
+التانية one plus X to K اللي هو induction
+
+411
+00:42:38,070 --> 00:42:43,340
+hypothesisحسب اتنين هذا اكبر من او يساوي واحد زياد
+
+412
+00:42:43,340 --> 00:42:48,220
+ك اكس مضروب في واحد زياد اكس بنضرب هدول في بعض و
+
+413
+00:42:48,220 --> 00:42:55,440
+بنرتب فبطلع حاصل الضرب هذا هو واحد زياد ك زياد
+
+414
+00:42:55,440 --> 00:43:01,340
+واحد في اكس زياد ك في اكس تربيه الآن
+
+415
+00:43:01,340 --> 00:43:08,860
+هذا هذا عدد موجب هذا عدد موجبلأن K عدد طبيعي و X
+
+416
+00:43:08,860 --> 00:43:16,060
+تربيه عدد موجب لما أشيل هذا أشطبه فبصغر المقدار
+
+417
+00:43:16,060 --> 00:43:20,580
+لما أشيل عدد موجب من عدد أو أنقص من عدد عدد موجب
+
+418
+00:43:20,580 --> 00:43:27,580
+بصغر فبالتالي هذا أكبر من واحد زاد K زاد واحد في X
+
+419
+00:43:28,490 --> 00:43:33,310
+وهذا هو الطرف اليمين للمتباينة 1 اللي احنا عايزين
+
+420
+00:43:33,310 --> 00:43:38,050
+نثبت صحتها عند m بساوي k زيادة واحدة اذا this
+
+421
+00:43:38,050 --> 00:43:42,830
+completes the induction هذا بيكمل البرهان بال
+
+422
+00:43:42,830 --> 00:43:50,050
+induction مظبوط صح تمام واضح اذا هاي صار في ان
+
+423
+00:43:50,050 --> 00:43:54,170
+متباينة
+
+424
+00:43:54,170 --> 00:44:01,080
+Bernoulli زي ما قلنا لكم في في الفي ال section هذا
+
+425
+00:44:01,080 --> 00:44:08,240
+بعض المتباينات الأخرى فبإمكانكم تقرؤوها ال
+
+426
+00:44:08,240 --> 00:44:12,260
+homework الآن خلصنا احنا section اتنين واحد اعتقد
+
+427
+00:44:12,260 --> 00:44:20,040
+فالمسائل المطلوب انكم تحلوها اللي هي موجودة مرسوصة
+
+428
+00:44:20,040 --> 00:44:23,280
+هنا وبرضه
+
+429
+00:44:23,280 --> 00:44:26,500
+زي ما قلتلكم في syllabus موجود على الصفحه تبعتي
+
+430
+00:44:27,940 --> 00:44:32,720
+فبارضه في ال homework هذا موجود ل .. مش ل ال
+
+431
+00:44:32,720 --> 00:44:38,400
+section هذا لكل ال .. المنهج اذا نبدأ نحل المسائل
+
+432
+00:44:38,400 --> 00:44:44,180
+هذه و ان شاء الله لسبوع الجاي بنعمل مناقشة فانا
+
+433
+00:44:44,180 --> 00:44:49,200
+هعمل مناقشة .. انا اللي هكون مناقشة لكم اليوم لأ
+
+434
+00:44:49,200 --> 00:44:52,960
+مافيش مناقشة لأنه لسه احنا يعني ماخدناش material
+
+435
+00:44:52,960 --> 00:45:00,110
+كافيةاو اللي لسه يعني مش مهيئين او مش محضرين
+
+436
+00:45:00,110 --> 00:45:06,170
+فهنواصل ونحاول ان شاء الله أسبوع الجاى ناخد كل
+
+437
+00:45:06,170 --> 00:45:10,450
+ساعة هذه الساعة
+
+438
+00:45:10,450 --> 00:45:12,930
+الأخيرة هذه المتأخرة نعملها مناقشة
+
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/QCtISTGMQww_raw.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/QCtISTGMQww_raw.srt
new file mode 100644
index 0000000000000000000000000000000000000000..2eace87e661b03a412f449c699bc283c97fa8a5b
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/QCtISTGMQww_raw.srt
@@ -0,0 +1,1400 @@
+1
+00:00:20,770 --> 00:00:26,750
+السلام عليكم بسم الله الرحمن الرحيم اليوم طبعا
+
+2
+00:00:26,750 --> 00:00:33,950
+فينا لقائن اللقاء الأول هناخد فيه مناقشة زي ما ..
+
+3
+00:00:33,950 --> 00:00:40,390
+زي ما احنا متعودين هنناقش اليوم section تلاتة
+
+4
+00:00:40,390 --> 00:00:47,170
+أربعة و تلاتة خمسة فإذا في عندكم أي أسئلة معينة في
+
+5
+00:00:47,170 --> 00:00:53,870
+section تلاتة أربعةأو تلاتة خمسة فهذه فرصة عندكم
+
+6
+00:00:53,870 --> 00:01:01,690
+أنكم يعني تسألوا الأسئلة و نحاول نحلها مع بعض فمين
+
+7
+00:01:01,690 --> 00:01:04,470
+في عندك أي سؤال في ال section تلاتة أربعة الأول
+
+8
+00:01:04,470 --> 00:01:09,010
+السؤال الرابع حصل دار الجامعة التلانية كوشيسي بصرش
+
+9
+00:01:09,010 --> 00:01:13,190
+زيت كادر في ال section تلاتة أربعة
+
+10
+00:01:16,800 --> 00:01:22,480
+الأول في section تلاتة اربعة فيها أي أسئلة؟ سؤال
+
+11
+00:01:22,480 --> 00:01:23,020
+تمانية
+
+12
+00:01:45,600 --> 00:01:57,660
+هذه السؤال تمانية D section تلاتة أربعة determine
+
+13
+00:01:57,660 --> 00:02:05,760
+the following limits determine
+
+14
+00:02:05,760 --> 00:02:13,200
+يعني حد دي أو أوج دي ال limit لل sequence اللي
+
+15
+00:02:13,200 --> 00:02:25,120
+الحد العام تبعهاواحد زائد اتنين واحد زائد
+
+16
+00:02:25,120 --> 00:02:33,200
+واحد على اتنين N الكل أس تلاتة N لما
+
+17
+00:02:33,200 --> 00:02:40,640
+N تقول ل Infinity احنا
+
+18
+00:02:40,640 --> 00:02:42,040
+عندنا ال ..
+
+19
+00:02:48,210 --> 00:02:54,670
+We know احنا فيه ان ال limit المعروفة limit واحد
+
+20
+00:02:54,670 --> 00:03:04,010
+زايد X على M الكل أس M as M tends to infinity
+
+21
+00:03:04,010 --> 00:03:10,850
+بساوي E أس X ال limit هذه معروفة موجودة هنا
+
+22
+00:03:14,080 --> 00:03:26,660
+باستخدام ال limit هذه ممكن ان احنا وبالتالي
+
+23
+00:03:26,660 --> 00:03:29,900
+تعالى
+
+24
+00:03:29,900 --> 00:03:36,380
+نشوف واحد زائد واحد على اتنين in الكل اصلا تلاتة
+
+25
+00:03:36,380 --> 00:03:42,060
+inنحاول نكتب هذا على صورة واحد زائد X على M الكل
+
+26
+00:03:42,060 --> 00:03:48,280
+قص M فهذا ممكن كتابته على صورة المقدار هذا على
+
+27
+00:03:48,280 --> 00:03:57,120
+الصورة واحد زائد نص على M نص
+
+28
+00:03:57,120 --> 00:04:05,600
+على M الكل قص M الكل
+
+29
+00:04:05,600 --> 00:04:06,160
+تكيين
+
+30
+00:04:09,900 --> 00:04:19,240
+صحيح وبالتالي therefore limit لما انتقل ل infinity
+
+31
+00:04:19,240 --> 00:04:28,160
+ل ال sequence اللي لحد تبعها هذا بيساوي limit لما
+
+32
+00:04:28,160 --> 00:04:36,570
+انتقل ل infinity ل الكلام هذا للمفضار اللي هناو هي
+
+33
+00:04:36,570 --> 00:04:43,330
+عندي واحد زي ال X على M او M الكل اص M ف limit
+
+34
+00:04:43,330 --> 00:04:53,370
+المقدار اللي جوا بساوي E اص نص او E اص X الكل ده
+
+35
+00:04:53,370 --> 00:04:57,150
+كئيب انا مقدر ادخل ال limit جوا القصير المربعين
+
+36
+00:04:57,150 --> 00:05:02,830
+limit المقدار اللي جوا جوا القصير المربعينحسب
+
+37
+00:05:02,830 --> 00:05:08,030
+القاعدة او القانون هذا E أُس X اللي هو نص وبعدين 2
+
+38
+00:05:08,030 --> 00:05:15,610
+بالكلتك A فبطلع E أُس 3 على 2 إذا هذه هي قيمة ال
+
+39
+00:05:15,610 --> 00:05:19,990
+limit okay تمام في
+
+40
+00:05:19,990 --> 00:05:25,090
+أسلة تانية في ال section هذا تلاتة أربعة
+
+41
+00:05:32,810 --> 00:05:48,550
+PIS love section تلاتة أربعة طيب
+
+42
+00:05:48,550 --> 00:05:54,230
+section تلاتة خمسة أنا في السؤال الرابع الشخصي
+
+43
+00:05:54,230 --> 00:05:54,650
+التاني
+
+44
+00:06:02,810 --> 00:06:07,190
+سيكشن تلاتة خمسة الرابع ولا التالت قصدك؟ الرابع
+
+45
+00:06:07,190 --> 00:06:17,710
+سؤال
+
+46
+00:06:17,710 --> 00:06:23,810
+رقم أربعة سيكشن تلاتة خمسة طيب
+
+47
+00:06:23,810 --> 00:06:27,890
+سؤال
+
+48
+00:06:27,890 --> 00:06:33,710
+أربعة سيكشن تلاتة خمسةالسؤال هذا بتقول show
+
+49
+00:06:33,710 --> 00:06:37,750
+directly from the definition
+
+50
+00:07:01,910 --> 00:07:17,610
+إذا XIN و YIN هم سيكونات كوشي هم سيكونات
+
+51
+00:07:17,610 --> 00:07:21,070
+كوشي هم
+
+52
+00:07:21,070 --> 00:07:27,550
+سيكونات كوشي هم سيكونات كوشي
+
+53
+00:07:30,650 --> 00:07:34,270
+فبرهان هذه زي البرهان اللي أخدناها قبل هيك في
+
+54
+00:07:34,270 --> 00:07:41,290
+قوانين نهايات proof
+
+55
+00:07:41,290 --> 00:07:50,350
+similar to proof of
+
+56
+00:07:50,350 --> 00:07:53,470
+the fact أنه
+
+57
+00:08:18,370 --> 00:08:23,890
+بساوي limit xn ضرب limit yn
+
+58
+00:08:27,620 --> 00:08:31,000
+و أعتقد هذا برهنها بالتفصيل using epsilon over two
+
+59
+00:08:31,000 --> 00:08:40,220
+argument تذكره؟ أه فحاولي تكتبي برهان مشابه و إذا
+
+60
+00:08:40,220 --> 00:08:50,280
+كان ماقدرتيش ممكن بعديها أعطيكي برهان تفصيلي okay؟
+
+61
+00:08:50,280 --> 00:08:52,180
+في أي أسئلة تانية؟
+
+62
+00:08:58,920 --> 00:09:04,620
+section تلقيتها خمسة
+
+63
+00:09:04,620 --> 00:09:15,820
+سؤال
+
+64
+00:09:15,820 --> 00:09:23,200
+عشرة ناس
+
+65
+00:09:23,200 --> 00:09:24,000
+سؤال عشر
+
+66
+00:09:28,420 --> 00:09:38,620
+section تلاتة خمسة if
+
+67
+00:09:38,620 --> 00:09:46,600
+x one less than x two are
+
+68
+00:09:46,600 --> 00:09:51,660
+arbitrary
+
+69
+00:09:55,690 --> 00:10:09,210
+real numbers and
+
+70
+00:10:09,210 --> 00:10:14,530
+xn
+
+71
+00:10:14,530 --> 00:10:19,470
+بيساوي نص
+
+72
+00:10:21,890 --> 00:10:37,690
+xn-2 plus xn-1 for n أكبر من 2 show in the
+
+73
+00:10:37,690 --> 00:10:42,470
+sequence xn is convergent
+
+74
+00:10:50,120 --> 00:10:55,760
+what is its limit what is
+
+75
+00:10:55,760 --> 00:11:03,520
+its limit ده هي النهاية تبعتنا بدنا نوجد نهايتها
+
+76
+00:11:03,520 --> 00:11:15,440
+في
+
+77
+00:11:15,440 --> 00:11:25,450
+حد فيكم فكر فيحلل السؤال هذاأي حد يعني حاول
+
+78
+00:11:25,450 --> 00:11:37,510
+ال burn in clever في أي أفكار أي شيء مافيش عندكم
+
+79
+00:11:37,510 --> 00:11:45,650
+أي فكرة عن الحل نحلوا
+
+80
+00:11:45,650 --> 00:11:46,890
+مع بعض نشوف
+
+81
+00:11:49,810 --> 00:11:58,450
+أنا عندي من الفرض x1 أصغر من x2 أعداد حقيقية فهعرف
+
+82
+00:11:58,450 --> 00:12:07,890
+let L بساوي x2 negative x1 هذا بيطلع عدد موجب لأن
+
+83
+00:12:07,890 --> 00:12:16,770
+x2 أكبر من x1 الآن باستخدام ال induction use
+
+84
+00:12:16,770 --> 00:12:17,570
+induction
+
+85
+00:12:21,840 --> 00:12:29,220
+on n بيمكنكم
+
+86
+00:12:29,220 --> 00:12:35,820
+تثبته انه المعادلة
+
+87
+00:12:35,820 --> 00:12:44,080
+التالية absolute xn plus one negative xn بساوي L
+
+88
+00:12:44,080 --> 00:12:49,960
+over two to n negative one and this is true for
+
+89
+00:12:49,960 --> 00:12:50,820
+every L
+
+90
+00:12:59,660 --> 00:13:08,000
+إذا كان n بساوي لكل
+
+91
+00:13:08,000 --> 00:13:14,300
+n أكبر من أو يساوي اتنين عشان هنا يكون هذا المعنى
+
+92
+00:13:14,300 --> 00:13:21,290
+فابدى ب n بساوي اتنين و اثبت ان المعادلة هذه صحقفل
+
+93
+00:13:21,290 --> 00:13:24,910
+بصحيته عند n بساوي k حيث k أكبر من اتنين أعداد
+
+94
+00:13:24,910 --> 00:13:29,410
+طبيعي وثبت صحيته عند n بساوي k زاد واحد طبعا في
+
+95
+00:13:29,410 --> 00:13:35,070
+الإثباتات بدك تستخدم التعريف تبع xn بدلالة الحدود
+
+96
+00:13:35,070 --> 00:13:35,670
+اللي جابله
+
+97
+00:13:41,020 --> 00:13:45,480
+و التعريف اللي انا اخدته ان ال هو عبارة عن الفرط
+
+98
+00:13:45,480 --> 00:13:51,340
+بين X2 و X1 اذا هذا سهل ممكن تبعته by induction
+
+99
+00:13:51,340 --> 00:14:01,740
+الان بعد ما تتبت هذا by induction now use this to
+
+100
+00:14:01,740 --> 00:14:02,540
+show that
+
+101
+00:14:08,470 --> 00:14:20,450
+بنستخدم المعادلة هذه to show that if M أكبر من N
+
+102
+00:14:20,450 --> 00:14:33,250
+أعداد طبيعية then absolute XM negative XM ال
+
+103
+00:14:33,250 --> 00:14:40,070
+absolute value لفرق XM و XMطبعا هذا هنخليه اصغر من
+
+104
+00:14:40,070 --> 00:14:48,610
+او ساوي absolute xn minus xn plus one زي absolute
+
+105
+00:14:48,610 --> 00:14:57,510
+xn plus one minus xn plus two و هكذا
+
+106
+00:15:02,520 --> 00:15:09,560
+absolute xm negative one negative xm
+
+107
+00:15:09,560 --> 00:15:13,840
+okay
+
+108
+00:15:13,840 --> 00:15:24,240
+تمام انا ايش عملت طرحت xn زيادة واحد ورجعتها طرحت
+
+109
+00:15:24,240 --> 00:15:29,290
+xn زيادة اتنين ورجعتهاو بعدين أخدت الأزواج هذه مع
+
+110
+00:15:29,290 --> 00:15:34,530
+بعض و استخدمت ل triangle inequality ال absolute
+
+111
+00:15:34,530 --> 00:15:39,390
+value ل مجموعة كبيرة أصغر من أوسع مجموعة absolute
+
+112
+00:15:39,390 --> 00:15:47,250
+values فالان
+
+113
+00:15:47,250 --> 00:15:54,530
+باستخدام لو سمينا المعادلة هاد ال star فباستخدام
+
+114
+00:15:54,530 --> 00:16:04,180
+ال starالـ absolute value هذه بساوي ال over two to
+
+115
+00:16:04,180 --> 00:16:12,840
+n minus واحد okay هذه نفسها دي و ال absolute value
+
+116
+00:16:12,840 --> 00:16:24,240
+اللي بعدها بساوي ال على two to n و هكذا و آخر حد
+
+117
+00:16:24,240 --> 00:16:33,390
+هيكون ال overtwo to m negative two لان
+
+118
+00:16:33,390 --> 00:16:39,170
+استخدام السارق مظبوط الان ممكن ناخد عامل مشترك
+
+119
+00:16:39,170 --> 00:16:52,230
+هاخد
+
+120
+00:16:52,230 --> 00:16:54,750
+عامل مشترك L over
+
+121
+00:17:16,500 --> 00:17:17,900
+تمام؟
+
+122
+00:17:20,140 --> 00:17:25,760
+الان هذا اصغر من واحد على اتنين قص ان نجاتف واحد
+
+123
+00:17:25,760 --> 00:17:34,980
+والمجموع هذا اصغر من اتنين
+
+124
+00:17:34,980 --> 00:17:41,660
+لان هذا جزء من الهارمونيك جيومتريكس سيريز اللي هي
+
+125
+00:17:41,660 --> 00:17:46,820
+واحد على اتنين قص ان من ان equal zero to infinity
+
+126
+00:17:46,820 --> 00:17:51,550
+هذه مجموعة اتنينفهذا جزء منها وبالتالي المجموعة
+
+127
+00:17:51,550 --> 00:17:55,650
+هذا أصغر من المجموعة الـ geometric series هذه
+
+128
+00:17:55,650 --> 00:18:01,490
+المجموعة بساوة اتنين okay فإذا هذا المجموعة أصغر
+
+129
+00:18:01,490 --> 00:18:06,030
+من اتنين إذا هذا أصغر من واحد على اتنين قص ان سالب
+
+130
+00:18:06,030 --> 00:18:14,990
+واحد في اتنين اللي هو عبارة عن واحد على اتنين قص
+
+131
+00:18:14,990 --> 00:18:17,410
+ان سالب اتنين
+
+132
+00:18:28,770 --> 00:18:35,790
+الان هذا بيروح للسفر as n tends to infinityهو
+
+133
+00:18:35,790 --> 00:18:40,770
+بالتالي ده بتطلع عندي أنا ال limit ل absolute xn
+
+134
+00:18:40,770 --> 00:18:49,670
+minus xm as m as n tenths of infinity و طبعاً m
+
+135
+00:18:49,670 --> 00:18:55,410
+تقول infinity تساوي سفر therefore ال sequence xn
+
+136
+00:18:55,410 --> 00:18:57,270
+is Cauchy
+
+137
+00:19:02,500 --> 00:19:09,620
+و بالتالي therefore by cauchy criterion by cauchy
+
+138
+00:19:09,620 --> 00:19:17,660
+criterion x in convergence صح؟ say
+
+139
+00:19:26,770 --> 00:19:35,110
+دعينا نسمي ال limit ل x in بساوي x حيث x is real
+
+140
+00:19:35,110 --> 00:19:41,110
+number الان مطلوب ان احنا نجد قيمة ال x قيمة ال
+
+141
+00:19:41,110 --> 00:19:48,470
+limit لل sequence اللى هى سمنها x فبنرجع للمعادلة
+
+142
+00:19:48,470 --> 00:19:49,730
+هناك
+
+143
+00:20:07,880 --> 00:20:13,020
+to find x
+
+144
+00:20:13,020 --> 00:20:18,980
+take limits
+
+145
+00:20:18,980 --> 00:20:23,960
+of
+
+146
+00:20:23,960 --> 00:20:26,940
+both sides
+
+147
+00:20:32,640 --> 00:20:49,540
+في المعادلة هذه اللي هنا and then
+
+148
+00:20:49,540 --> 00:20:56,480
+take limit of both sides in this equation to get
+
+149
+00:20:56,480 --> 00:21:09,330
+انه limitx in بساوي نص في limit x in negative two
+
+150
+00:21:09,330 --> 00:21:21,850
+زاد limit x in negative one as intensity هذا
+
+151
+00:21:21,850 --> 00:21:28,510
+بيقدي ان limit x in عبارة عن x بساوي نص هذه
+
+152
+00:21:28,510 --> 00:21:33,460
+subsequenceمن الـ xn، وبالتالي، الـ limit تبعتها
+
+153
+00:21:33,460 --> 00:21:37,880
+برضه x وهذا الـ subsequence من الـ xn والـ limit
+
+154
+00:21:37,880 --> 00:21:45,400
+تبعتها x لإن كانت الـ sequence convergent لـ x فأي
+
+155
+00:21:45,400 --> 00:21:50,080
+subsequence بتكون convergent لنفس الـ x وهذا بيطلع
+
+156
+00:21:50,080 --> 00:21:51,100
+بساوية
+
+157
+00:22:12,850 --> 00:22:18,570
+فاحنا هيك ماطلعش عندنا إش جديد طالع X بساوي X
+
+158
+00:22:18,570 --> 00:22:25,950
+فاحنا هيك مازمناش ال limit مازمناش
+
+159
+00:22:25,950 --> 00:22:26,570
+ال limit
+
+160
+00:22:50,820 --> 00:22:57,180
+فاحنا هيك ماجبناش ال limit وهدي مشكلة أو يعني ال
+
+161
+00:22:57,180 --> 00:22:57,600
+..
+
+162
+00:23:14,130 --> 00:23:21,570
+طب خلينا نرجع للمعادلة اللي اثبتناها اثبتناها
+
+163
+00:23:21,570 --> 00:23:22,410
+by induction
+
+164
+00:24:17,920 --> 00:24:22,600
+طيب خلّي موضوع ال limit هذا لمرة تانية
+
+165
+00:24:31,270 --> 00:24:33,370
+سي حد عنده أي فكرة؟
+
+166
+00:24:55,010 --> 00:25:03,350
+خلّي ال limit نجيبها المرة الجاية يبدو
+
+167
+00:25:03,350 --> 00:25:09,630
+أن الطريقة تباعتنا ما .. ما زبطتش فنحاول نفكر فيها
+
+168
+00:25:09,630 --> 00:25:14,810
+مرة تانية و نجيبها، مين عنده سؤال تاني؟ و أنتوا
+
+169
+00:25:14,810 --> 00:25:17,830
+كمان طبعا فكروا في إيجاب قيمة ال limit
+
+170
+00:25:25,090 --> 00:25:33,030
+في أي أسئلة تانية في ال section تلاتة خمسة سؤال
+
+171
+00:25:33,030 --> 00:25:46,130
+تمانية سؤال تمانية سؤال
+
+172
+00:25:46,130 --> 00:25:54,030
+تمانية section تلاتة خمسة show directly
+
+173
+00:25:57,190 --> 00:26:05,590
+مش هو directly that
+
+174
+00:26:05,590 --> 00:26:12,550
+a bounded a
+
+175
+00:26:12,550 --> 00:26:24,370
+bounded a bounded monotone a bounded monotone
+
+176
+00:26:27,280 --> 00:26:36,500
+ماراتون ان كريزم ماراتون
+
+177
+00:26:36,500 --> 00:26:42,560
+ان كريزم الاختصار هو
+
+178
+00:26:42,560 --> 00:26:43,300
+كوشي
+
+179
+00:26:56,170 --> 00:27:01,310
+السيكوينس دي bounded و monotone increasing فاحنا
+
+180
+00:27:01,310 --> 00:27:10,100
+خدنا نظرية بتطلع لازم تكونby monotone convergence
+
+181
+00:27:10,100 --> 00:27:13,100
+theorem it is convergent عسب ال monotone
+
+182
+00:27:13,100 --> 00:27:17,380
+convergence و بالتالي ممكن نقول by Cauchy
+
+183
+00:27:17,380 --> 00:27:20,780
+criterion بما انها convergent اذا Cauchy هذا برهان
+
+184
+00:27:20,780 --> 00:27:26,180
+و صحيح بس الكتاب مش عايز هذا البرهان عايز برهان
+
+185
+00:27:26,180 --> 00:27:30,500
+مباشر باستخدام ال definition تبع ال Cauchy
+
+186
+00:27:30,500 --> 00:27:38,260
+sequenceOkay فالبرهان بسيط وسهل يعني ما هوش صعب
+
+187
+00:27:38,260 --> 00:27:44,060
+proof since
+
+188
+00:27:44,060 --> 00:27:49,820
+طبعا
+
+189
+00:27:49,820 --> 00:27:57,400
+ال sequence خليني أسميها x in since
+
+190
+00:27:57,400 --> 00:28:02,980
+ال sequence تبعتنا x inis bounded بما ان ال
+
+191
+00:28:02,980 --> 00:28:08,980
+sequence is bounded then
+
+192
+00:28:08,980 --> 00:28:25,680
+لو أخدت U بساوي Supremum ل XN-N ينتمي لإن Supremum
+
+193
+00:28:25,680 --> 00:28:28,060
+هذا exists in R
+
+194
+00:28:32,840 --> 00:28:39,620
+by supremum property او by completeness property
+
+195
+00:28:39,620 --> 00:28:48,520
+اي bounded set has supremum فخلينا نفرض ان ال U هو
+
+196
+00:28:48,520 --> 00:28:54,300
+supremum فهذا بيطلع عدد حقيقي exist يعني عدد حقيقي
+
+197
+00:28:54,300 --> 00:29:01,120
+الان من ال definition by
+
+198
+00:29:01,120 --> 00:29:08,250
+definitionof supremum
+
+199
+00:29:08,250 --> 00:29:16,030
+هو بالأحرى مش من التعريف من لمّة تأتي بعد التعريف
+
+200
+00:29:16,030 --> 00:29:19,630
+والمّة
+
+201
+00:29:19,630 --> 00:29:23,110
+من
+
+202
+00:29:23,110 --> 00:29:29,990
+هنا by Aprilius
+
+203
+00:29:40,440 --> 00:29:44,220
+النظرية هذه خلينا نرجعها أو نجمها او نجمها او
+
+204
+00:29:44,220 --> 00:29:48,460
+نجمها او نجمها U of a set
+
+205
+00:30:02,180 --> 00:30:07,180
+of a set S
+
+206
+00:30:07,180 --> 00:30:11,960
+is
+
+207
+00:30:11,960 --> 00:30:16,440
+the supremum is
+
+208
+00:30:16,440 --> 00:30:17,420
+the supremum
+
+209
+00:30:22,570 --> 00:30:32,010
+S F and only F لكل أبسلون أكبر من السفر يوجد عنصر
+
+210
+00:30:32,010 --> 00:30:39,950
+S يعتبد على أبسلون في S بحيث أنه لو ضفنا .. لو
+
+211
+00:30:39,950 --> 00:30:45,250
+طرحنا من ال supremum أبسلون فبيبطل أكبر دعوة بيصير
+
+212
+00:30:45,250 --> 00:30:55,520
+أصغر من Sباستخدام هذه النظرية السابقة لو أخدت
+
+213
+00:30:55,520 --> 00:31:01,500
+given
+
+214
+00:31:01,500 --> 00:31:04,880
+epsilon
+
+215
+00:31:04,880 --> 00:31:09,620
+أكبر من السفر لو أخدت epsilon عشوائية أكبر من
+
+216
+00:31:09,620 --> 00:31:17,320
+السفر يوجد capital N في N
+
+217
+00:31:22,170 --> 00:31:35,390
+عدد طبيعي بحيث ان ال .. ال .. ال
+
+218
+00:31:35,390 --> 00:31:43,690
+U نيجاتيب XN عرفوا ال U نيجاتيب إبسلون بيطلع أصغر
+
+219
+00:31:43,690 --> 00:31:50,270
+من S Y S Y حقول ده X اللي ال index سبقه capital N
+
+220
+00:31:51,310 --> 00:31:54,710
+إذا هذا العنصر اللي في الـ sequence أو في ال set
+
+221
+00:31:54,710 --> 00:32:01,950
+هذه اللي هو S Y يعتمد على إبسلون ما ال N هذه تعتمد
+
+222
+00:32:01,950 --> 00:32:09,310
+على إبسلون وهي U إترح منه إبسلون بيطلع أصغر من
+
+223
+00:32:09,310 --> 00:32:15,150
+عنصر S Y اللي هو موجود هنا إذا هذا حسب النظرية
+
+224
+00:32:15,150 --> 00:32:20,050
+تمام؟ الآن since
+
+225
+00:32:22,680 --> 00:32:29,060
+لاحظوا انتوا ان ال XIN هذا عنصر في الست وبالتالي و
+
+226
+00:32:29,060 --> 00:32:32,660
+ال U upper bound ال U اللي هو ال supreme دايما
+
+227
+00:32:32,660 --> 00:32:37,180
+بيكون upper bound فهذا بيطلع أصغر من أو ساوي ال U
+
+228
+00:32:37,180 --> 00:32:43,420
+since
+
+229
+00:32:43,420 --> 00:32:50,740
+ال sequence XIN is increasing متزايدة
+
+230
+00:32:55,230 --> 00:32:59,870
+we get نحصل
+
+231
+00:32:59,870 --> 00:33:08,330
+على التالي فنحصل
+
+232
+00:33:08,330 --> 00:33:17,590
+على التالي لو كان M أكبر من أو ساوي N أكبر من أو
+
+233
+00:33:17,590 --> 00:33:26,720
+ساوي capital Nفهذا هيقدّي إلى أن u negative
+
+234
+00:33:26,720 --> 00:33:35,200
+epsilon أصغر من x capital N من هنا و x capital N
+
+235
+00:33:35,200 --> 00:33:42,260
+أصغر من أو سوى x small n لأن ال sequence is
+
+236
+00:33:42,260 --> 00:33:46,730
+increasing و small n أكبر من أو سوى capital Nو هذا
+
+237
+00:33:46,730 --> 00:33:54,070
+أصغر من أوي ساوي xm لأن ال sequence increasing و
+
+238
+00:33:54,070 --> 00:33:59,690
+xm أصغر من أوي ساوي u لأن ال u upper bound لكل
+
+239
+00:33:59,690 --> 00:34:06,670
+عناصر ال sequence تمام؟ و ال u
+
+240
+00:34:15,980 --> 00:34:22,260
+فهذا بيقدّي .. هذا
+
+241
+00:34:22,260 --> 00:34:30,320
+بيقدّي ان xn minus
+
+242
+00:34:30,320 --> 00:34:37,000
+epsilon أصغر من xn لأن الepsilon عدد موجب فلما
+
+243
+00:34:37,000 --> 00:34:44,570
+اطلع عدد موجب العدد xn بصغروعندي xn أصغر من أو
+
+244
+00:34:44,570 --> 00:34:50,490
+ساوي xm لأن ال m أكبر من أو ساوي n و ال sequence
+
+245
+00:34:50,490 --> 00:34:57,110
+تبعتي increasing و xm أصغر من أو ساوي ال u
+
+246
+00:35:11,700 --> 00:35:21,020
+من هنا، من المتباينة هذه واضح ان انا عندي ال
+
+247
+00:35:21,020 --> 00:35:31,020
+U أصغر من XN زاد إبسلون، صح؟ هل عندي U negative
+
+248
+00:35:31,020 --> 00:35:39,160
+إبسلون أصغر من XN فاضيف إبسلون فبطلع U أصغر من XN
+
+249
+00:35:40,390 --> 00:35:49,470
+زايد ابسلون اذا انا بطلع عندى xn
+
+250
+00:35:49,470 --> 00:36:02,450
+او ال .. ال xm تطلع اكبر من xn minus ابسلون اصغر
+
+251
+00:36:02,450 --> 00:36:04,730
+من xn زايد ابسلون
+
+252
+00:36:08,370 --> 00:36:19,010
+وهذا معناه ان absolute xn او xn minus xn اصغر من
+
+253
+00:36:19,010 --> 00:36:28,730
+epsilon اصبت؟ صح؟ وهذا صحيح لكل M أكبر من أو ساوي
+
+254
+00:36:28,730 --> 00:36:34,030
+M أكبر من أو ساوي capital M تمام؟ اذا بنقول هنا
+
+255
+00:36:34,030 --> 00:36:46,340
+sinceepsilon أكبر من السفر was arbitrary it
+
+256
+00:36:46,340 --> 00:36:50,640
+follows بينتج
+
+257
+00:36:50,640 --> 00:36:58,820
+من تعريف الكوشي sequence that sequence xm is
+
+258
+00:36:58,820 --> 00:36:59,620
+cauchy
+
+259
+00:37:03,200 --> 00:37:11,000
+وهنا هيك طبقنا التعريف وهذا برهان مباشر okay تمام
+
+260
+00:37:11,000 --> 00:37:22,940
+واضح في
+
+261
+00:37:22,940 --> 00:37:26,520
+أي سؤال سؤال تانية واضح البرهان في أي سفصار
+
+262
+00:37:32,980 --> 00:37:38,320
+في أي سؤال تاني section تلاتة خمسة أو تلاتة ستة
+
+263
+00:37:38,320 --> 00:37:41,600
+لبعده
+
+264
+00:37:41,600 --> 00:37:50,060
+الناس
+
+265
+00:37:50,060 --> 00:37:50,920
+اللي بدرسوا
+
+266
+00:38:03,440 --> 00:38:10,060
+التلاتة ستة؟ سؤال أربعة في رياضينا.
+
+267
+00:38:23,080 --> 00:38:24,980
+هذا ما حللناه إزاي؟
+
+268
+00:38:44,990 --> 00:38:54,970
+السكتشن تلاتة ستة السؤال الرابع ترى بي establish
+
+269
+00:38:54,970 --> 00:38:58,370
+establish
+
+270
+00:38:58,370 --> 00:39:01,490
+the
+
+271
+00:39:01,490 --> 00:39:05,790
+proper divergence
+
+272
+00:39:05,790 --> 00:39:10,270
+of
+
+273
+00:39:10,270 --> 00:39:12,950
+sequence
+
+274
+00:39:16,730 --> 00:39:25,730
+يجيب الان زي الواحد ان الان اكوار الواحد تنتهي ان
+
+275
+00:39:25,730 --> 00:39:33,910
+انا اثبت ان انا اثبت
+
+276
+00:39:33,910 --> 00:39:39,250
+ان انا اثبت ان انا اثبت
+
+277
+00:39:39,250 --> 00:39:45,220
+ان انا اثبت ان انا اثبتلا ال sequence اللي ال inf
+
+278
+00:39:45,220 --> 00:39:49,820
+term تبعها و الحد العام تبعها جدر انزعي الواحد لما
+
+279
+00:39:49,820 --> 00:39:58,300
+M تقوى ل infinity بساوي infinity استاذ ماعرفش طلقة
+
+280
+00:39:58,300 --> 00:40:07,900
+D ماعرفش بيه بيه عرفت حلوة دي طلقة دي
+
+281
+00:40:07,900 --> 00:40:13,460
+استاذ بدك دي طرقة ديالفرقة دي
+
+282
+00:40:40,070 --> 00:40:46,330
+يعني هيطلع عندي هنا limit in على جدر in زي الواحد
+
+283
+00:40:46,330 --> 00:40:53,390
+لما in تقول infinity بساوي infinity ممكن تستخدم
+
+284
+00:40:53,390 --> 00:41:03,190
+limit comparison test او comparison test اه فال
+
+285
+00:41:07,800 --> 00:41:12,180
+فهل فكرتي انك تستخدمي comparison test او limit
+
+286
+00:41:12,180 --> 00:41:16,260
+comparison test؟ فكرت عن نظري يعني برنامج ده يفير
+
+287
+00:41:16,260 --> 00:41:20,140
+لأ و أنت عندك التعريف و عندك في limit comparison
+
+288
+00:41:20,140 --> 00:41:29,620
+test و comparison test لأ
+
+289
+00:41:29,620 --> 00:41:36,110
+هاي موجود في تلاتة ستة في ..مافيش limit comparison
+
+290
+00:41:36,110 --> 00:41:49,030
+test؟ فيك طيب هاي انا شايف في ..
+
+291
+00:41:49,030 --> 00:41:55,870
+انت
+
+292
+00:41:55,870 --> 00:42:01,970
+عندك .. ياعمل المقارنة انت
+
+293
+00:42:01,970 --> 00:42:03,010
+عندك ..
+
+294
+00:42:10,490 --> 00:42:19,450
+يعني قارنة ال .. جدر ال N بدك تقارن هذه مع
+
+295
+00:42:19,450 --> 00:42:26,430
+N على جدر ال N لما N تكون كبيرة ابنهم الواحد فهذه
+
+296
+00:42:26,430 --> 00:42:30,090
+بيصير زيها زي ال sequence اللي لحد الآن تبعها N
+
+297
+00:42:30,090 --> 00:42:34,390
+على جدر ال N وهذه بيساوي جدر ال N
+
+298
+00:42:39,240 --> 00:42:43,420
+فبتعمل limit comparison use limit comparison test
+
+299
+00:42:43,420 --> 00:42:51,280
+هاي عندي an ولا إيش مسميهم الكتاب نعم
+
+300
+00:42:51,280 --> 00:42:59,780
+يعني xn وهي yn وتعالى نحسب ال limit
+
+301
+00:43:08,200 --> 00:43:13,480
+تعالى نحسب ال limit ل
+
+302
+00:43:13,480 --> 00:43:21,100
+xn على yn as n tends to infinity بساوي ال limit ل
+
+303
+00:43:21,100 --> 00:43:31,820
+n على الجذر n زي الواحد تقسيم
+
+304
+00:43:37,120 --> 00:43:43,740
+في واحد على جدر ال N لما N تقول ال infinity بيساوي
+
+305
+00:43:43,740 --> 00:43:47,840
+limit لجدر
+
+306
+00:43:47,840 --> 00:43:55,000
+التربية A ل N على N زي واحد مظبوط؟
+
+307
+00:43:55,000 --> 00:44:02,380
+بندخل ال limit جوه فبقلع واحد اكبر من صفته الان
+
+308
+00:44:02,380 --> 00:44:09,580
+sinceاحنا اثبتنا ان ال series sigma او ال sequence
+
+309
+00:44:09,580 --> 00:44:16,700
+since ال sequence ان الحد العام تبعها جدر ال n او
+
+310
+00:44:16,700 --> 00:44:21,720
+since limit yn
+
+311
+00:44:21,720 --> 00:44:27,480
+as n tends to infinity بالساوية limit جدر ال n as
+
+312
+00:44:27,480 --> 00:44:31,500
+n tends to infinity بالساوية infinity we get
+
+313
+00:44:34,620 --> 00:44:41,340
+from limit comparison test from limit comparison
+
+314
+00:44:41,340 --> 00:44:47,100
+test from limit
+
+315
+00:44:47,100 --> 00:44:48,480
+comparison test from limit comparison test from
+
+316
+00:44:48,480 --> 00:44:48,480
+limit comparison test from limit comparison test
+
+317
+00:44:48,480 --> 00:44:48,480
+from limit comparison test from limit comparison
+
+318
+00:44:48,480 --> 00:44:48,480
+test from limit comparison test from limit
+
+319
+00:44:48,480 --> 00:44:48,480
+comparison test from limit comparison test from
+
+320
+00:44:48,480 --> 00:44:48,480
+limit comparison test from limit comparison test
+
+321
+00:44:48,480 --> 00:44:48,480
+from limit comparison test from limit comparison
+
+322
+00:44:48,480 --> 00:44:48,480
+test from limit comparison test from limit
+
+323
+00:44:48,480 --> 00:44:48,480
+comparison test from limit comparison test from
+
+324
+00:44:48,480 --> 00:44:48,480
+limit comparison test from limit comparison test
+
+325
+00:44:48,480 --> 00:44:48,480
+from limit comparison test from limit comparison
+
+326
+00:44:48,480 --> 00:44:48,480
+test from limit comparison test from limit
+
+327
+00:44:48,480 --> 00:44:48,480
+comparison test from limit comparison test from
+
+328
+00:44:48,480 --> 00:44:48,480
+limit comparison test from limit comparison test
+
+329
+00:44:48,480 --> 00:44:48,500
+from limit comparison test from limit comparison
+
+330
+00:44:48,500 --> 00:44:48,500
+test from limit comparison test from limit
+
+331
+00:44:48,500 --> 00:44:48,500
+comparison test from limit comparison test from
+
+332
+00:44:48,500 --> 00:44:48,500
+limit comparison test from limit comparison test
+
+333
+00:44:48,500 --> 00:44:48,500
+from limit comparison test from limit comparison
+
+334
+00:44:48,500 --> 00:44:48,500
+test from limit comparison test from limit
+
+335
+00:44:48,500 --> 00:44:48,500
+comparison test from limit comparison test from
+
+336
+00:44:48,500 --> 00:44:48,500
+limit comparison test from limit comparison test
+
+337
+00:44:48,500 --> 00:44:48,600
+from limit comparison test from limit comparison
+
+338
+00:44:48,600 --> 00:44:48,600
+test from limit comparison test from limit
+
+339
+00:44:48,600 --> 00:44:48,640
+comparison test from limit comparison test from
+
+340
+00:44:48,640 --> 00:44:48,640
+limit comparison test from limit comparison test
+
+341
+00:44:48,640 --> 00:44:48,640
+from limit comparison test from limit comparison
+
+342
+00:44:48,640 --> 00:44:48,640
+test from limit comparison test from limit
+
+343
+00:44:48,640 --> 00:44:50,520
+comparison test from limit comparison test from
+
+344
+00:44:50,520 --> 00:44:57,100
+limit comparison test from limit
+
+345
+00:44:57,100 --> 00:45:00,980
+comparison test from limit comparison test
+
+346
+00:45:13,250 --> 00:45:18,150
+تمام؟ okay اذا ممكن نستخدم ال limit comparison
+
+347
+00:45:18,150 --> 00:45:23,290
+test او ال diet comparison test اذا نفع و نحل باج
+
+348
+00:45:23,290 --> 00:45:28,010
+التمارين في .. او الأجزاء الأخرى في exercise أربعة
+
+349
+00:45:28,010 --> 00:45:33,430
+رقم .. section تلاتة ستة okay بنوقف هنا و بنكمل
+
+350
+00:45:33,430 --> 00:45:38,190
+المرة الجاية نكتفي بهذا القدر
+
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/SrQnjpF43P0_postprocess.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/SrQnjpF43P0_postprocess.srt
new file mode 100644
index 0000000000000000000000000000000000000000..af36ba5d169ce347186a07abc2ec4cccad08445c
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/SrQnjpF43P0_postprocess.srt
@@ -0,0 +1,1612 @@
+1
+00:00:22,080 --> 00:00:28,960
+بسم الله الرحمن الرحيم ال .. هنواصل اليوم ان شاء
+
+2
+00:00:28,960 --> 00:00:39,080
+الله ال .. ال .. الشئ اللي بدأناه بخصوص المقدمة عن
+
+3
+00:00:39,080 --> 00:00:46,520
+ال infinite series بعتقد كانت أخر نظرية أخدناها هي
+
+4
+00:00:46,520 --> 00:00:47,700
+Cauchy criterion
+
+5
+00:00:51,790 --> 00:00:56,710
+كوشي كريتيريا for
+
+6
+00:00:56,710 --> 00:01:10,250
+infinite series a
+
+7
+00:01:10,250 --> 00:01:10,810
+series
+
+8
+00:01:22,490 --> 00:01:31,530
+converges if and only if الشرط التالي بتحقق لكل
+
+9
+00:01:31,530 --> 00:01:38,430
+epsilon أكبر من 0 يوجد capital N يعتمد على epsilon
+
+10
+00:01:38,430 --> 00:01:46,200
+عدد طبيعي بحيث أنهلو كان M أكبر من N أكبر من أو
+
+11
+00:01:46,200 --> 00:01:56,560
+ساوي capital M فهذا بيقدي أنه absolute SM minus SN
+
+12
+00:01:56,560 --> 00:02:05,840
+بساوي absolute XN زاد واحد زاد XN زاد اتنين و هكذا
+
+13
+00:02:05,840 --> 00:02:07,300
+إلى XM
+
+14
+00:02:11,640 --> 00:02:15,100
+النظرية هذه تبعت كوشي أخدناها المرة اللي فاتت و
+
+15
+00:02:15,100 --> 00:02:22,080
+برهناها مظبوط اليوم هناخد نظرية تانية و مهمة و
+
+16
+00:02:22,080 --> 00:02:31,660
+النظرية هذه بتقول انه let x in be sequence of non
+
+17
+00:02:31,660 --> 00:02:39,620
+-negative real numbers then
+
+18
+00:02:39,620 --> 00:02:40,420
+series
+
+19
+00:02:43,080 --> 00:02:49,340
+xn converges if
+
+20
+00:02:49,340 --> 00:02:55,680
+and only if الsequence of partial sums its
+
+21
+00:02:55,680 --> 00:03:05,600
+sequence of partial sums اللي
+
+22
+00:03:05,600 --> 00:03:07,540
+هي sn is bounded
+
+23
+00:03:16,760 --> 00:03:24,080
+proof we have sn
+
+24
+00:03:24,080 --> 00:03:36,020
+بساوي او sn زايد واحد بساوي sn
+
+25
+00:03:36,020 --> 00:03:42,070
+زايد xn زايد واحدالان زاد فرص partial sum بيساوي
+
+26
+00:03:42,070 --> 00:03:46,150
+الانف partial sum وبنضيف عليه لحد رقم ان زاد واحد
+
+27
+00:03:58,050 --> 00:04:02,070
+و طبعا ال Xn زاد واحد هذا احنا فرضين ان حدود ال
+
+28
+00:04:02,070 --> 00:04:08,530
+sequence Xn كلها غير سالبة فهذا غير سالب وبالتالي
+
+29
+00:04:08,530 --> 00:04:14,790
+المجموع هذا بالتأكيد اكبر من او سوى Sn لكل N في N
+
+30
+00:04:14,790 --> 00:04:17,870
+فهذا
+
+31
+00:04:17,870 --> 00:04:24,530
+معناه ان ال sequence Sn is increasing متزايدة
+
+32
+00:04:26,960 --> 00:04:30,480
+بالتالي، من خلال الوضع الوضع الوضع الوضع الوضع
+
+33
+00:04:30,480 --> 00:04:36,420
+الوضع الوضع
+
+34
+00:04:36,420 --> 00:04:40,960
+الوضع
+
+35
+00:04:40,960 --> 00:04:48,560
+الوضع الوضع
+
+36
+00:05:21,390 --> 00:05:23,410
+وهو المطلوب
+
+37
+00:05:25,890 --> 00:05:29,190
+أحنا أخدنا قبل هيك أن أي infinite series بتكون
+
+38
+00:05:29,190 --> 00:05:32,890
+convergent if and only if the sequence of partial
+
+39
+00:05:32,890 --> 00:05:36,810
+sums is convergent، صح؟ طيب، ال sequence of
+
+40
+00:05:36,810 --> 00:05:42,650
+partial sums بمعنها increasing فهي convergent by
+
+41
+00:05:42,650 --> 00:05:48,050
+monotone convergence theorem بتكون convergent if
+
+42
+00:05:48,050 --> 00:05:52,130
+and only if it is boundedوبالتالي الـ series
+
+43
+00:05:52,130 --> 00:05:54,770
+converges if and only if the sequence of partial
+
+44
+00:05:54,770 --> 00:06:02,210
+sums is bounded وهذا يثبت النظرية تمام؟ إذن هذه
+
+45
+00:06:02,210 --> 00:06:09,410
+النظرية أهميتها في أننا يعني تطبيقها زي ما هنشوف
+
+46
+00:06:09,410 --> 00:06:12,650
+في
+
+47
+00:06:12,650 --> 00:06:14,590
+لما هنا صغيرة لما
+
+48
+00:06:22,840 --> 00:06:28,020
+إذا .. إذا
+
+49
+00:06:28,020 --> 00:06:35,460
+Sn هي عملية عملية عملية عملية عملية عملية عملية
+
+50
+00:06:35,460 --> 00:06:40,200
+عملية عملية عملية عملية عملية عملية عملية عملية
+
+51
+00:06:40,200 --> 00:06:41,360
+عملية عملية عملية عملية عملية عملية عملية عملية
+
+52
+00:06:41,360 --> 00:06:41,460
+عملية عملية عملية عملية عملية عملية عملية عملية
+
+53
+00:06:41,460 --> 00:06:41,480
+عملية عملية عملية عملية عملية عملية عملية عملية
+
+54
+00:06:41,480 --> 00:06:43,900
+عملية عملية عملية عملية عملية عملية عملية عملية
+
+55
+00:06:43,900 --> 00:06:50,560
+عملية
+
+56
+00:06:50,560 --> 00:07:06,360
+عإذا كان هناك تجارب سكن
+
+57
+00:07:06,360 --> 00:07:15,180
+من سن التي مرتبطة
+
+58
+00:07:15,180 --> 00:07:22,480
+فالنتيجة
+
+59
+00:07:25,450 --> 00:07:34,870
+ثم سيكوانس SN نفسها مرتبط يعني
+
+60
+00:07:34,870 --> 00:07:37,450
+لو كان في عندي سيكوانس of non-negative real
+
+61
+00:07:37,450 --> 00:07:43,790
+numbers و السيكوانس هذه حدودها غير سالبة و
+
+62
+00:07:43,790 --> 00:07:50,030
+increase متزايدة و لو وجد subsequence من السيكوانس
+
+63
+00:07:50,030 --> 00:07:57,220
+SN و السيكوانس مرتبطالسيكوانس الأم أو السيكوانس
+
+64
+00:07:57,220 --> 00:08:02,260
+الأصلية بتكون أيضا bounded فالبرهان
+
+65
+00:08:02,260 --> 00:08:05,500
+تبع النظرية اللي هسيبكم هي أن تتبرهنوا ك exercise
+
+66
+00:08:05,500 --> 00:08:09,980
+for easy
+
+67
+00:08:09,980 --> 00:08:20,880
+exercise اذا انا هنسيبكم اتبرهنوا كتمرين في عندي
+
+68
+00:08:20,880 --> 00:08:21,580
+الآن
+
+69
+00:08:27,800 --> 00:08:44,940
+firm ال P series test let
+
+70
+00:08:44,940 --> 00:08:51,480
+P عدد موجب أصغر let P
+
+71
+00:08:55,440 --> 00:09:01,680
+هي عدد موجب اي عدد موجب ده
+
+72
+00:09:01,680 --> 00:09:09,640
+P series ده P series اللي هي سيجما from N equals
+
+73
+00:09:09,640 --> 00:09:17,100
+zero to infinity لواحد على N أُس P ال series هذه
+
+74
+00:09:17,100 --> 00:09:23,160
+بنسميها P series ال P series هذه ايش مالها؟ واحد
+
+75
+00:09:25,240 --> 00:09:37,540
+converges if P أكبر من واحد and name by virges if
+
+76
+00:09:37,540 --> 00:09:45,620
+P أصغر من أو يساوي واحد prove
+
+77
+00:09:45,620 --> 00:09:52,580
+نثبت الجزء الأول assume
+
+78
+00:09:54,990 --> 00:10:02,570
+إن P أكبر من 1 let
+
+79
+00:10:02,570 --> 00:10:07,510
+R بيساوي
+
+80
+00:10:07,510 --> 00:10:14,490
+واحد على اتنين أس P سالب واحد ال P أكبر من واحد
+
+81
+00:10:14,490 --> 00:10:21,330
+فالأس هذا موجب واحد على اتنين أس موجب طبعا هذا
+
+82
+00:10:21,330 --> 00:10:21,950
+بيطلع
+
+83
+00:10:24,540 --> 00:10:34,800
+عدد موجب و أصغر من واحد then ال R العدد R هذا أكبر
+
+84
+00:10:34,800 --> 00:10:44,300
+من سفر أصغر من واحد طيب four K بساوي اتنين اقصى
+
+85
+00:10:44,300 --> 00:10:52,860
+واحد سالب واحد اللي هو واحد MDS
+
+86
+00:10:58,390 --> 00:11:04,950
+عندي ال partial sum رقم
+
+87
+00:11:04,950 --> 00:11:10,130
+K اللي
+
+88
+00:11:10,130 --> 00:11:18,970
+هو بيساوي اتنين خليني اسمي هذا K واحد فال
+
+89
+00:11:18,970 --> 00:11:26,370
+partial sum S رقم K واحدبساوي طبعا k1 بساوي واحد
+
+90
+00:11:26,370 --> 00:11:36,190
+هنا هذا هو s1 بساوي واحد ال
+
+91
+00:11:36,190 --> 00:11:39,490
+first partial sum اللي هو مجموعة الحد الأول الحد
+
+92
+00:11:39,490 --> 00:11:47,470
+الأول هنا واحد صح طيب
+
+93
+00:11:47,470 --> 00:11:49,810
+four لو أخدت k2
+
+94
+00:11:52,090 --> 00:11:57,230
+لو أخدت K2 بيساوي اتنين اقصى اتنين سالب واحد هذا
+
+95
+00:11:57,230 --> 00:12:01,570
+بيطلع تلاتة ففي
+
+96
+00:12:01,570 --> 00:12:09,290
+الحالة هذه بيطلع عندي ال
+
+97
+00:12:09,290 --> 00:12:14,710
+partial sum رقم K2 هو نفس ال partial sum رقم تلاتة
+
+98
+00:12:14,710 --> 00:12:18,950
+فطبعا هذا هيساوي مجموع اول تلاتة حدود
+
+99
+00:12:22,250 --> 00:12:33,350
+فعندي هنا هيطلع أول حد واحد، التاني واحد على اتنين
+
+100
+00:12:33,350 --> 00:12:36,530
+أسقي
+
+101
+00:12:36,530 --> 00:12:42,670
+و الحد التالت واحد على تلاتة أسبيل
+
+102
+00:12:54,470 --> 00:12:58,710
+خلّينا هذه ال N مفروض تكون من واحد لان ات من واحد
+
+103
+00:12:58,710 --> 00:13:05,550
+لان ات ف S K2 هو S تلاتة S تلاتة بساوي واحد على
+
+104
+00:13:05,550 --> 00:13:10,110
+واحد أس P اللي هو واحد زائد واحد على اتنين أس P
+
+105
+00:13:10,110 --> 00:13:17,620
+زائد واحد على تلاتة أس Pطيب أنا عندي هذا أصغر من
+
+106
+00:13:17,620 --> 00:13:23,360
+واحد زايد واحد على اتنين أُس P زايد واحد على اتنين
+
+107
+00:13:23,360 --> 00:13:34,820
+أُس P since لأنه اتنين أُس P أصغر من تلاتة أُس P
+
+108
+00:13:34,820 --> 00:13:42,400
+فبتعني مقلوب تلاتة أُس P أصغر من مقلوب اتنين أُس P
+
+109
+00:13:42,400 --> 00:13:51,330
+صح؟و هذا بيساوي واحد زائد واحد على اتنين اصفى زائد
+
+110
+00:13:51,330 --> 00:13:58,850
+اتنين اتنين في واحد على اتنين اصفى هزبوت هدول حدين
+
+111
+00:13:58,850 --> 00:14:03,870
+متشابهين وهذا بيطلع بيساوي واحد زائد واحد على
+
+112
+00:14:03,870 --> 00:14:12,590
+اتنين اصفى سالب واحد وهذا بيساوي واحد زائد اربع
+
+113
+00:14:14,460 --> 00:14:20,460
+معرف ال R على إنها 1 على 2 أُس P سالب واحد
+
+114
+00:14:20,460 --> 00:14:24,420
+Similarly
+
+115
+00:14:24,420 --> 00:14:29,820
+بالمثل لو
+
+116
+00:14:29,820 --> 00:14:39,060
+كانت K تلاتة بساوي اتنين أس تلاتة سالب واحد يعني
+
+117
+00:14:39,060 --> 00:14:43,140
+سبعة then
+
+118
+00:14:44,850 --> 00:14:53,850
+هيكون SK3 هيطلع بساوي SK2
+
+119
+00:14:53,850 --> 00:14:58,350
+زائد
+
+120
+00:14:58,350 --> 00:15:05,210
+واحد على أربعة أس P زائد واحد على خمسة أس P زائد
+
+121
+00:15:05,210 --> 00:15:10,990
+واحد على ستة أس P زائد واحد على سبعة أس P مظبوط
+
+122
+00:15:10,990 --> 00:15:11,990
+هيك صح؟
+
+123
+00:15:14,290 --> 00:15:18,410
+لأن S K 2 هو واحد زاد واحد على اتنين أسفل زاد واحد
+
+124
+00:15:18,410 --> 00:15:23,430
+على تلاتة أسفل وهذا
+
+125
+00:15:23,430 --> 00:15:37,510
+أصغر من .. طبعا S K 2 أصغر من واحد قلنا زائد R
+
+126
+00:15:41,610 --> 00:15:46,750
+و بعدين الحدود هذه .. كل واحد من الحدود هذه أصغر
+
+127
+00:15:46,750 --> 00:15:52,250
+من أو يساوي واحد على أربعة أس P واحد على أربعة أس
+
+128
+00:15:52,250 --> 00:15:57,570
+P .. واحد على أربعة أس P .. واحد على أربعة أس P
+
+129
+00:15:57,570 --> 00:16:02,690
+لأن كل واحد من المقامات هذه أكبر من أو يساوي أربعة
+
+130
+00:16:02,690 --> 00:16:09,870
+أس Pطب أقول جديش عددهم أربعة هذا بيساوي واحد زايد
+
+131
+00:16:09,870 --> 00:16:17,730
+R زايد أربعة على أربعة أس P هو يساوي واحد زايد R
+
+132
+00:16:17,730 --> 00:16:27,810
+زايد واحد على أربعة أس P سالب واحد إذن بيطلع عندي
+
+133
+00:16:27,810 --> 00:16:38,880
+أنا S K تلاتة أصغر من واحد زايد Rزائد واحد على
+
+134
+00:16:38,880 --> 00:16:43,480
+اتنين اتنين
+
+135
+00:16:43,480 --> 00:16:49,820
+قص اتنين في P سالب واحد وهذا
+
+136
+00:16:49,820 --> 00:16:58,520
+هو R تربية هذا بيساوي واحد زائد R زائد R تربية لو
+
+137
+00:16:58,520 --> 00:17:04,120
+سمرنا في العملية هذه continuing
+
+138
+00:17:07,150 --> 00:17:16,370
+inductively بطريقة استقرائية continuing
+
+139
+00:17:16,370 --> 00:17:22,590
+inductively يعني by induction باستخدام
+
+140
+00:17:22,590 --> 00:17:28,350
+ال induction we
+
+141
+00:17:28,350 --> 00:17:32,570
+get نحصل على التالي for
+
+142
+00:17:40,680 --> 00:17:50,100
+for K J بيساوي اتنين اقصد J سالب واحد we
+
+143
+00:17:50,100 --> 00:17:55,340
+have S
+
+144
+00:17:55,340 --> 00:18:08,260
+K J هيطلع اصغر من واحد زائد R زائد R ترمية زائد
+
+145
+00:18:10,870 --> 00:18:22,010
+R أُس J سالب واحد و
+
+146
+00:18:22,010 --> 00:18:27,830
+هذا صحيح لكل J في N هزبط
+
+147
+00:18:27,830 --> 00:18:36,550
+هيك صح هيتعالى نشوف لما J كانت بالسلب واحد فطلع
+
+148
+00:18:36,550 --> 00:18:43,660
+عندي Sك واحد بيساوي واحد وأصغر من أو يساوي الواحد
+
+149
+00:18:43,660 --> 00:18:47,600
+لما
+
+150
+00:18:47,600 --> 00:18:54,920
+ك ج بيساوي اتنين لما
+
+151
+00:18:54,920 --> 00:19:02,980
+ج بيساوي اتنينفطلع SK2 أصغر من واحد زائد R واحد
+
+152
+00:19:02,980 --> 00:19:09,660
+زائد هاي اتنين سالب واحد الأسف لما J بساوية تلاتة
+
+153
+00:19:09,660 --> 00:19:15,800
+عندي طلع ال SK تلاتة أصغر من واحد زائد R زائد R أس
+
+154
+00:19:15,800 --> 00:19:21,280
+تلاتة سالب واحد إذا S sub KJ أصغر من واحد زائد R
+
+155
+00:19:21,280 --> 00:19:30,540
+إلى R أس J minus واحدالان هذا المجموع أصغر من
+
+156
+00:19:30,540 --> 00:19:39,280
+مجموع ال infinite series اللي هي summation من j
+
+157
+00:19:39,280 --> 00:19:48,980
+بساوي سفر إلى ملا نهاية لR أس J صح؟
+
+158
+00:19:50,160 --> 00:19:56,760
+هذا عبارة
+
+159
+00:19:56,760 --> 00:20:01,440
+عن finite sum أصغر من مجموعة infinite series هذه
+
+160
+00:20:01,440 --> 00:20:10,700
+عبارة عن geometric series with first term واحد وال
+
+161
+00:20:10,700 --> 00:20:18,140
+ratio تبعها R والـ R أكبر من صفر أصغر من واحدإذا
+
+162
+00:20:18,140 --> 00:20:23,080
+الـ geometric series هذه converges ومجموعة بساوي
+
+163
+00:20:23,080 --> 00:20:30,820
+واحد على واحد minus R الكلام
+
+164
+00:20:30,820 --> 00:20:37,320
+هذا صحيح for all j belong to M وطبعا ال SKJ هذا
+
+165
+00:20:37,320 --> 00:20:42,760
+عبارة عن مجموعة partial sum partial sum لأعداد
+
+166
+00:20:42,760 --> 00:20:44,560
+موجبة وبالتالي موجبة
+
+167
+00:20:47,700 --> 00:20:52,720
+إذا أنا هيك أثبتت إن ال sub
+
+168
+00:20:52,720 --> 00:20:58,700
+sequence SKJ bounded below by 0 و bounded above by
+
+169
+00:20:58,700 --> 00:21:05,420
+العدد الموجب 1 على 1 minus R وبالتالي،
+
+170
+00:21:05,420 --> 00:21:10,160
+إذا هيك نستنتج، therefore the subsequence
+
+171
+00:21:13,370 --> 00:21:18,210
+ده sub-sequence اللي هي SKJ هاد ال sub-sequence من
+
+172
+00:21:18,210 --> 00:21:24,370
+مين؟ من ال sequence of partial sums SN اشملها is
+
+173
+00:21:24,370 --> 00:21:31,490
+bounded أثبتنا أنها ايه؟ bounded، مصدقوط؟ طلعنا
+
+174
+00:21:31,490 --> 00:21:35,090
+احنا عملنا construction ل sub-sequenceمن الـ
+
+175
+00:21:35,090 --> 00:21:37,370
+sequence of partial sums وهي طلعت bounded هي
+
+176
+00:21:37,370 --> 00:21:41,670
+bounded below by 0 bounded above by 1 over 1 minus
+
+177
+00:21:41,670 --> 00:21:53,610
+r لأن حسب اللمّة اللي فاتت so by above لمّة ال
+
+178
+00:21:53,610 --> 00:21:56,730
+sequence of partial sums نفسها is bounded
+
+179
+00:22:02,950 --> 00:22:08,570
+وبالتالي إذا by above theorem مدام ال sequence of
+
+180
+00:22:08,570 --> 00:22:11,910
+partial sums is bounded إذا ال series converges
+
+181
+00:22:11,910 --> 00:22:15,710
+okay
+
+182
+00:22:15,710 --> 00:22:23,790
+إذا بعديكم نقول so
+
+183
+00:22:23,790 --> 00:22:29,930
+by above theorem ال
+
+184
+00:22:29,930 --> 00:22:46,080
+series sigma1 على N أكبر من 1 فهذا
+
+185
+00:22:46,080 --> 00:22:53,300
+يثبت الجزء الأول من النظرية خلّينا
+
+186
+00:22:53,300 --> 00:22:54,640
+نثبت الجزء التاني
+
+187
+00:23:10,080 --> 00:23:17,200
+using induction you
+
+188
+00:23:17,200 --> 00:23:21,660
+can show أنه
+
+189
+00:23:21,660 --> 00:23:26,800
+for P
+
+190
+00:23:26,800 --> 00:23:29,140
+أكبر من صفر أصغر من واحد
+
+191
+00:23:32,370 --> 00:23:37,970
+لو كانت ال P طبعا أكبر من صفر وأصغر من أو ساوي
+
+192
+00:23:37,970 --> 00:23:46,110
+الواحد ف N أوس P بطلع أصغر من أو ساوي N لكل N في N
+
+193
+00:23:46,110 --> 00:23:49,670
+صح؟
+
+194
+00:23:49,670 --> 00:23:57,430
+انا ممكن اثباته by inductionSo, 1 على n أصغر لو
+
+195
+00:23:57,430 --> 00:24:08,470
+ساقى 1 على n أُس في for all n ينتمي إلى n فإن
+
+196
+00:24:08,470 --> 00:24:12,710
+هذا بيقدّي هذا بيقدّي
+
+197
+00:24:30,450 --> 00:24:36,590
+this implies أن ال summation from k بساوي واحد to
+
+198
+00:24:36,590 --> 00:24:44,150
+n لواحد على k أصغر لو ساوي summation من k بساوي
+
+199
+00:24:44,150 --> 00:24:55,850
+واحد إلى n لواحد على n على k أوس Pطب ما هذا عبارة
+
+200
+00:24:55,850 --> 00:25:02,270
+عن ال partial sum لسمي S N لل harmonic series ..
+
+201
+00:25:02,270 --> 00:25:09,410
+لل .. لل harmonic series و هذا عبارة عن ال partial
+
+202
+00:25:09,410 --> 00:25:17,870
+sum سمي S N star لل P series صح؟ إن أنا أصبح عندي
+
+203
+00:25:17,870 --> 00:25:21,850
+أنا S N أصغر من أو ساوي S N star
+
+204
+00:25:24,550 --> 00:25:29,430
+where لكل n where
+
+205
+00:25:29,430 --> 00:25:36,530
+sn بساوي sigma واحد على k من k بساوي واحد إلى n و
+
+206
+00:25:36,530 --> 00:25:42,230
+sn star بساوي sigma من k بساوي واحد إلى n لواحد
+
+207
+00:25:42,230 --> 00:25:51,630
+على k أُس P طيب since أثبتنا احنا قبل هيك انه ال
+
+208
+00:25:51,630 --> 00:25:53,230
+sequence of partial sums
+
+209
+00:25:56,310 --> 00:26:01,790
+السيكوانس SN هذا عبارة عن السيكوانس of partial
+
+210
+00:26:01,790 --> 00:26:02,290
+sums
+
+211
+00:26:12,510 --> 00:26:17,510
+of the harmonic series sigma 1 على n ايش مالهم؟
+
+212
+00:26:17,510 --> 00:26:23,070
+اثبتنا انهم unbounded ال sequence هذه is unbounded
+
+213
+00:26:23,070 --> 00:26:30,670
+بنالنا ذلك في مثال سابق unbounded
+
+214
+00:26:30,670 --> 00:26:35,070
+فلو سمينا المتباين
+
+215
+00:26:35,070 --> 00:26:39,530
+هذا star then
+
+216
+00:26:39,530 --> 00:26:40,370
+it follows
+
+217
+00:26:44,900 --> 00:26:50,900
+from star ينتج من المتباينة star إذا كانت ال
+
+218
+00:26:50,900 --> 00:26:55,600
+sequence هذه unbounded
+
+219
+00:26:55,600 --> 00:27:00,620
+لصغيرة unbounded فالحدود أكبر is unbounded ال
+
+220
+00:27:00,620 --> 00:27:11,140
+sequence SM star is unbounded وبالتالي
+
+221
+00:27:11,140 --> 00:27:13,600
+حسب النظرية أعلى so by
+
+222
+00:27:20,890 --> 00:27:26,890
+السيريز هي سيكوينس باشا سمسا بايتالي P Series
+
+223
+00:27:26,890 --> 00:27:32,930
+سيجما من K بساوة واحد وإنفينيتي لواحد على K أوس P
+
+224
+00:27:32,930 --> 00:27:36,290
+is divergent
+
+225
+00:27:41,610 --> 00:27:48,170
+حسب النظرية اللى ذكرناها سابقًا، طيب، في عندي هاي
+
+226
+00:27:48,170 --> 00:27:51,750
+series of positive numbers أو non-negative real
+
+227
+00:27:51,750 --> 00:27:55,890
+numbers و ال sequence of partial sums أبعدتها
+
+228
+00:27:55,890 --> 00:28:00,250
+unbounded، إذا حسب النظرية ال series is not
+
+229
+00:28:00,250 --> 00:28:05,260
+convergent أو divergent، صح؟إذن هذا بتثبت الجزء
+
+230
+00:28:05,260 --> 00:28:09,880
+التاني وبالتاني هيك بيكون بارهاننا اختبار ال P
+
+231
+00:28:09,880 --> 00:28:15,200
+-series test او ال P-series test إذن
+
+232
+00:28:15,200 --> 00:28:20,960
+أي P-series بتكون convergent إذا ال P أكبر من واحد
+
+233
+00:28:20,960 --> 00:28:25,620
+و divergent إذا P أصغر من أول ساول واحد في أي
+
+234
+00:28:25,620 --> 00:28:26,220
+سؤال؟
+
+235
+00:28:52,320 --> 00:28:55,800
+زي ما أخرنا comparison و limit comparison test
+
+236
+00:28:55,800 --> 00:29:00,920
+للsequences في comparison test لل series
+
+237
+00:29:00,920 --> 00:29:08,900
+comparison
+
+238
+00:29:08,900 --> 00:29:13,960
+test
+
+239
+00:29:13,960 --> 00:29:17,020
+for series
+
+240
+00:29:33,680 --> 00:29:40,800
+لت XIN و YIN بـ sequences of non-negative real
+
+241
+00:29:40,800 --> 00:29:41,380
+numbers
+
+242
+00:29:49,780 --> 00:29:58,060
+يكون such that xn أصغر من أو ساوي yn أكبر من أو
+
+243
+00:29:58,060 --> 00:30:06,540
+ساوي سفر for all n أكبر من أو ساوي k for some k
+
+244
+00:30:06,540 --> 00:30:14,660
+ينتمي إلى n أنا
+
+245
+00:30:14,660 --> 00:30:19,950
+لدي two sequences of real numberالتنتين حدودهم غير
+
+246
+00:30:19,950 --> 00:30:26,790
+سالبة من capital .. من capital K و انت طالع ممكن
+
+247
+00:30:26,790 --> 00:30:30,350
+الحدود اللي جابل .. اللي رقمهم أصغر من K بتكون
+
+248
+00:30:30,350 --> 00:30:36,150
+سالبة مش مشكلة أو مايكونش xn أصغر .. ممكن يكون xn
+
+249
+00:30:36,150 --> 00:30:41,830
+أكبر من yn مش مشكلة لكن من عند capital Kلكل مؤشر
+
+250
+00:30:41,830 --> 00:30:45,810
+لكل index because on an equal key أنا بدي xn أصغر
+
+251
+00:30:45,810 --> 00:30:51,990
+من وسائر yn واتنين يكونوا غير سلبين الآن إذا كانت
+
+252
+00:30:51,990 --> 00:30:58,190
+ال series sigma
+
+253
+00:30:58,190 --> 00:31:06,970
+xn إذا كانت ال series الأولى convergent أو الكبيرة
+
+254
+00:31:09,920 --> 00:31:17,540
+convergent بتقدي ان ال series الأصغر converge and
+
+255
+00:31:17,540 --> 00:31:29,520
+لو كانت ال series الأكبر الأصغر diverge فبتقدي
+
+256
+00:31:29,520 --> 00:31:36,700
+أن ال series الأكبر بالتأكيد diverge وهي
+
+257
+00:31:36,700 --> 00:31:45,380
+البرهان البرهن الجزء الأولassume أنه ال series
+
+258
+00:31:45,380 --> 00:31:56,180
+sigma yn converges then
+
+259
+00:31:56,180 --> 00:32:00,000
+by cauchy
+
+260
+00:32:00,000 --> 00:32:10,000
+criterion for series given
+
+261
+00:32:12,200 --> 00:32:18,900
+epsilon أكبر من الصفر يوجد capital N يعتمد على
+
+262
+00:32:18,900 --> 00:32:26,920
+epsilon عدد طبيعي بحيث أنه لكل M أكبر من small n
+
+263
+00:32:26,920 --> 00:32:31,620
+أكبر من أو ساوي capital N هيطلع عندي absolute
+
+264
+00:32:39,200 --> 00:32:51,200
+YNZ1Z YNZ2 و هكذا الى YM أصغر من إبسن هذا من Koshi
+
+265
+00:32:51,200 --> 00:32:51,760
+criterion
+
+266
+00:33:15,050 --> 00:33:23,830
+طيب انا عندي let
+
+267
+00:33:23,830 --> 00:33:34,730
+M بساوي ال maximum الأكبر ب N العدد K هذا الطبيعي
+
+268
+00:33:34,730 --> 00:33:40,010
+K و العدد الطبيعي capital N اللي بيعتمد على
+
+269
+00:33:40,010 --> 00:33:40,330
+epsilon
+
+270
+00:33:43,140 --> 00:33:49,360
+فاكيد طبعا هذا عدد طبيعي وهذا عدد طبيعي فان ام
+
+271
+00:33:49,360 --> 00:33:54,200
+هيطلع عدد طبيعي وام يعتمد على ابسلون لأن الام
+
+272
+00:33:54,200 --> 00:34:01,500
+يعتمد على capital ان وcapital ان تعتمد على ابسلون
+
+273
+00:34:01,500 --> 00:34:09,900
+اذا لو اخدت انا ان اكبر من ام او ام اكبر من ان وان
+
+274
+00:34:09,900 --> 00:34:18,560
+اكبر من او ساوي capital انفهذا بالتأكيد هيقدّي انه
+
+275
+00:34:18,560 --> 00:34:23,480
+ال ..
+
+276
+00:34:23,480 --> 00:34:29,520
+ال .. من هنا من الفرض نسمي الفرض هذا star
+
+277
+00:34:34,930 --> 00:34:45,790
+فمن star هيطلع عندي مفروض xn زائد واحد زائد xn
+
+278
+00:34:45,790 --> 00:34:51,130
+زائد اتنين زائد و هكذا الى xm
+
+279
+00:35:01,640 --> 00:35:06,300
+الـ absolute هذا الان كله موجب هذه كل حدود موجبة
+
+280
+00:35:06,300 --> 00:35:12,160
+أو غير سالبة لأن ال N أكبر من أوي ساوي كابتل N
+
+281
+00:35:12,160 --> 00:35:17,380
+وبالتالي هذا بيقدي ان N أكبر من أوي ساوي كابتل K
+
+282
+00:35:17,380 --> 00:35:25,520
+صح؟ إذا باستخدام star كابتل
+
+283
+00:35:25,520 --> 00:35:28,260
+عفوا ان هذه المفروضة تكون M
+
+284
+00:35:31,530 --> 00:35:36,470
+الان لكل M أكبر من M أكبر من أو ساوي capital M
+
+285
+00:35:36,470 --> 00:35:41,110
+capital M هذه أكبر من أو ساوي K وبالتالي M أكبر من
+
+286
+00:35:41,110 --> 00:35:46,030
+أو ساوي K إذن الحدود هذه كلها موجبة أو غير سالبة
+
+287
+00:35:46,030 --> 00:35:58,830
+صح من star و أصغر من أو ساوي اللي هو YM زاد واحد
+
+288
+00:35:58,830 --> 00:36:10,810
+زاد YMزائد اتنين زائد و هكذا الى الى
+
+289
+00:36:10,810 --> 00:36:19,870
+YM برضه هذه كلها حدود غير سالبة وبالتالي هذه هي
+
+290
+00:36:19,870 --> 00:36:25,370
+نفس ال absolute YN زائد واحد زائد YN زائد اتنين
+
+291
+00:36:25,370 --> 00:36:25,910
+زائد
+
+292
+00:36:29,190 --> 00:36:37,590
+ym ومن هنا لاحظوا ال n أكبر من أو ساوي capital M و
+
+293
+00:36:37,590 --> 00:36:42,870
+ال M أكبر من أو ساوي capital N فبطلع عندي كمان هنا
+
+294
+00:36:42,870 --> 00:36:50,630
+ال N أكبر من أو ساوي capital M لأن ال M أكبر من أو
+
+295
+00:36:50,630 --> 00:36:56,650
+ساوي capital N وبالتالي من ال implication هذههذا
+
+296
+00:36:56,650 --> 00:37:04,470
+بيطلع أصغر من إبسلم إن أنا طلع عندي absolute xn
+
+297
+00:37:04,470 --> 00:37:11,050
+زائد واحد زائد xn زائد اتنين زائد إلى آخرى إلى xn
+
+298
+00:37:11,050 --> 00:37:17,380
+أصغر من إبسلمهذا عدد غير سالب المجموعة ده غير سالب
+
+299
+00:37:17,380 --> 00:37:20,800
+لإن كل ال X غير سالب وبالتالي ال absolute value هي
+
+300
+00:37:20,800 --> 00:37:24,680
+نفسها هذا هو ال absolute value العدد غير سالب نفسه
+
+301
+00:37:24,680 --> 00:37:31,060
+لإن هنا أثبتنا الكلام هذا صحيح لكل M أكبر من N
+
+302
+00:37:31,060 --> 00:37:36,880
+أكبر من أو يساوي capital M so بستخدم كوشي
+
+303
+00:37:36,880 --> 00:37:42,780
+criterion كمان مرة by koshi criterion
+
+304
+00:37:45,130 --> 00:37:50,250
+for series الـ
+
+305
+00:37:50,250 --> 00:37:57,310
+series sigma xn converges لأن
+
+306
+00:37:57,310 --> 00:38:02,130
+هاي شرط كوشي متحقق، صح؟
+
+307
+00:38:02,130 --> 00:38:07,390
+هاي لأي إبسلون given إبسلون أكبر من السفر أثناء أن
+
+308
+00:38:07,390 --> 00:38:13,490
+يوجد capital Mيعني يعتمد على إبسلون بحيث لكل M
+
+309
+00:38:13,490 --> 00:38:18,350
+أكبر من N أكبر من أو ساوي كابتال N فلاندي أبسليوت
+
+310
+00:38:18,350 --> 00:38:23,150
+XN زاد واحد زائد إلى XM أصغر من إبسلون إذا حسب
+
+311
+00:38:23,150 --> 00:38:27,050
+كوشي criterion ال series Sigma XN converges هذا
+
+312
+00:38:27,050 --> 00:38:36,790
+بكمل برهان الجزء الأول الجزء التاني بينتج
+
+313
+00:38:36,790 --> 00:38:37,890
+من الجزء الأول
+
+314
+00:38:42,270 --> 00:38:52,390
+this is the contrapositive .. the contrapositive
+
+315
+00:38:52,390 --> 00:39:00,330
+أو the contraposition .. this is the
+
+316
+00:39:00,330 --> 00:39:04,930
+contraposition of ال statement واحد
+
+317
+00:39:08,170 --> 00:39:12,270
+أنا في عندي قانون في ال logic بيقول إذا كان P فأدي
+
+318
+00:39:12,270 --> 00:39:19,370
+ل Q فال statement هذا بكافئ ال counter positive مش
+
+319
+00:39:19,370 --> 00:39:21,950
+النفي تبعه ال counter positive معناه المعاكس
+
+320
+00:39:21,950 --> 00:39:30,550
+الإيجابي فهذا بكافئ not Q implies not P فذا
+
+321
+00:39:30,550 --> 00:39:37,130
+أثبتنا إن هذا true فهذا بيكون trueطب تعالى نشوف
+
+322
+00:39:37,130 --> 00:39:39,810
+هذا هذا اللى انا اثبتنا انه true اللى هو الجزء ا
+
+323
+00:39:39,810 --> 00:39:43,590
+او الجزء واحد الجزء التانى هو ال counter positive
+
+324
+00:39:44,960 --> 00:39:50,320
+هل هذا صحيح تعالوا نقرأ الجزء التاني او تعالوا
+
+325
+00:39:50,320 --> 00:39:55,420
+نجيب ال contrapositive للعبارة في الجزء الأول ال
+
+326
+00:39:55,420 --> 00:39:59,800
+contrapositive للعبارة في الجزء الأول نفي هذا اللي
+
+327
+00:39:59,800 --> 00:40:04,160
+هو ال series sigma x in by dirge بيقدي الى نفي هذا
+
+328
+00:40:04,160 --> 00:40:08,420
+اللي هو ال series y in by dirgeوبالتالي هيك بنكون
+
+329
+00:40:08,420 --> 00:40:17,700
+كملنا البرران تمام؟ واضح؟ في أي سؤال؟ أي استفسار؟
+
+330
+00:40:17,700 --> 00:40:25,180
+أحيانا بيكون صعب أن احنا نعملمقارنة بين حدود ال
+
+331
+00:40:25,180 --> 00:40:32,500
+series sigma xn و حدود ال series sigma yn بالطريقة
+
+332
+00:40:32,500 --> 00:40:37,480
+المباشرة زي ما في ال star أحيانا مش سهل نعمل
+
+333
+00:40:37,480 --> 00:40:43,020
+مقارنة زي هذه وبالتالي بنلجأ إلى اختبار اخر بنسميه
+
+334
+00:40:43,020 --> 00:40:49,320
+limit comparison testفنكتب ال test هذا دكتور؟ نعم
+
+335
+00:40:49,320 --> 00:40:53,560
+هلأ عبس النظرية برضه خطأ يعني نفس نحو اللي كنا
+
+336
+00:40:53,560 --> 00:40:58,360
+نعمله بال sequence نعمله بال series هى الفترة
+
+337
+00:40:58,360 --> 00:41:01,940
+التالية؟ اه طبعا يعني انت لو كان مثلا هذه الصحية
+
+338
+00:41:01,940 --> 00:41:08,160
+كلامك بالظبط يعني لو كانت ال series ال series مثلا
+
+339
+00:41:08,160 --> 00:41:14,190
+هذه ال sigma yn diverseهل هذا بيقعد ان sigma xn
+
+340
+00:41:14,190 --> 00:41:18,530
+diverge؟ مش بالضرورة ممكن diverge و ممكن converge
+
+341
+00:41:18,530 --> 00:41:25,510
+نفس الحاجة لو كانت ال series sigma xn converge هل
+
+342
+00:41:25,510 --> 00:41:31,030
+sigma yn converge؟ ممكن او ممكن لأ اكيد و ممكن
+
+343
+00:41:31,030 --> 00:41:34,690
+نجيب counter examples لازم تفكري في إيجاد counter
+
+344
+00:41:34,690 --> 00:41:36,850
+examples و انت بتدرسيه
+
+345
+00:41:40,810 --> 00:41:53,630
+نشوف ال limit comparison test limit
+
+346
+00:41:53,630 --> 00:42:02,030
+comparison test
+
+347
+00:42:18,300 --> 00:42:27,120
+لت Xn وYn بيكونوا سيكوانس من حدود حقيقية مفتوحة
+
+348
+00:42:27,120 --> 00:42:36,560
+بيكون حدود حقيقية مفتوحة بيكون حدود حقيقية مفتوحة
+
+349
+00:42:36,560 --> 00:42:38,340
+بيكون حدود حقيقية مفتوحة بيكون حدود حقيقية مفتوحة
+
+350
+00:42:38,340 --> 00:42:40,480
+بيكون حدود حقيقية مفتوحة بيكون حدود حقيقية مفتوحة
+
+351
+00:42:47,730 --> 00:42:56,850
+و ال R هذا يعني عدد حقيقي طبعا ينتمي إلى R ففي
+
+352
+00:42:56,850 --> 00:43:04,670
+عندي برضه نتيجتين النتيجة الأولى إذا كان ال R
+
+353
+00:43:04,670 --> 00:43:09,630
+بسويش سفر then
+
+354
+00:43:09,630 --> 00:43:15,330
+ال series sigma X M converges if and only if
+
+355
+00:43:24,150 --> 00:43:43,650
+الجزء التاني من النظرية لو
+
+356
+00:43:43,650 --> 00:43:52,220
+كان R بساوي سفربعد ذلك لو الار بيسوي سفر يعني ان
+
+357
+00:43:52,220 --> 00:44:01,220
+الار بيسويش سفر then ال series sigma y in دي درجز
+
+358
+00:44:01,220 --> 00:44:07,840
+بيقدي ان ال series sigma x in دي درجز
+
+359
+00:44:11,380 --> 00:44:19,300
+او لأ convergence افضل لو كانت series sigma yn
+
+360
+00:44:19,300 --> 00:44:24,480
+convergence بيقدي ان series sigma xn ايبان
+
+361
+00:44:24,480 --> 00:44:27,620
+convergence فقط اتجاه واحد لكن الاكس مش شرط تكون
+
+362
+00:44:27,620 --> 00:44:37,460
+صحيح okay تمام هو البرهان
+
+363
+00:44:37,460 --> 00:44:38,520
+يعني كتير سهل
+
+364
+00:44:49,520 --> 00:45:02,300
+بنسمي الشرط هذا star الجزء
+
+365
+00:45:02,300 --> 00:45:12,940
+الأول assume ان R لا يساوي سفر طبعا
+
+366
+00:45:12,940 --> 00:45:21,670
+في الحالة هذه ال R بطلع موجدلأن لحظة انتوا ان xn
+
+367
+00:45:21,670 --> 00:45:27,810
+على yn هذه كلها أعداد موجبة و ال limit ل sequence
+
+368
+00:45:27,810 --> 00:45:31,930
+أعداء كل حدودها موجبة بيطلع المفروض يطلع ال limit
+
+369
+00:45:31,930 --> 00:45:37,970
+تبعتها موجبة إذا كانت ال limit موجودة واضح؟ إذن
+
+370
+00:45:37,970 --> 00:45:42,090
+هذا من نظرية سابقة لما أن حدود ال sequence xn على
+
+371
+00:45:42,090 --> 00:45:45,930
+yn هذه ال sequence حدودها كلها موجبة إذا نهايتها
+
+372
+00:45:45,930 --> 00:45:58,820
+تطلع أيضا موجبةطيب وبالتالي take epsilon بساوي R ع
+
+373
+00:45:58,820 --> 00:46:09,180
+2 طبعا هذا بالتأكيد عدد موجب الان by star since
+
+374
+00:46:09,180 --> 00:46:18,240
+XN على YN converges to R as N tends to infinity
+
+375
+00:46:20,590 --> 00:46:26,850
+بقدر نلاقي capital N يعتمد على إبسلون عدد طبيعي
+
+376
+00:46:26,850 --> 00:46:32,830
+بحيث أنه absolute لكل N أكبر من أو يساوي capital N
+
+377
+00:46:32,830 --> 00:46:39,730
+بيطلع absolute xn على yn minus r أصغر من إبسلون
+
+378
+00:46:39,730 --> 00:46:49,430
+اللي هي بيساوي R ع 2 وهذا بيقديلو فكنا وعملنا ال
+
+379
+00:46:49,430 --> 00:46:56,730
+absolute value هيطلع عندي xn
+
+380
+00:46:56,730 --> 00:47:03,910
+على yn أكبر من أو ساوي R ع 2 أصغر من أو ساوي 3R ع
+
+381
+00:47:03,910 --> 00:47:11,530
+2 وبما أنه هذا صحيح لكل N أكبر من أو ساوي capital
+
+382
+00:47:11,530 --> 00:47:21,570
+Nالان اضرب في YN YN طبعا عدد موجب فبطلع XN أصغر من
+
+383
+00:47:21,570 --> 00:47:27,370
+أو ساوي تلاتة R ع اتنين في YN أكبر من أو ساوي R ع
+
+384
+00:47:27,370 --> 00:47:33,510
+اتنين في YN وهذا صحيح لكل N أكبر من أو ساوي
+
+385
+00:47:33,510 --> 00:47:37,330
+capital N تمام الان now
+
+386
+00:47:45,120 --> 00:47:53,940
+نسمي هذه double star فالان
+
+387
+00:47:53,940 --> 00:48:03,820
+if sigma x and converge then
+
+388
+00:48:03,820 --> 00:48:12,540
+by double star and comparison test and comparison
+
+389
+00:48:12,540 --> 00:48:22,450
+testواختبار المقارنة إذا كانت هذه convergent فبطلع
+
+390
+00:48:22,450 --> 00:48:31,810
+هذه ال series convergent صح وبالتالي بطلع sigma yn
+
+391
+00:48:31,810 --> 00:48:38,370
+convergence هذا ثابت موجة ممكن نضرب في مقلوب و
+
+392
+00:48:38,370 --> 00:48:42,490
+نتخلص منه إذا كانت هذه convergent فهذه convergent
+
+393
+00:48:46,950 --> 00:48:55,330
+Also إذا كانت ال series sigma y in converge then
+
+394
+00:48:55,330 --> 00:49:00,810
+برضه by المتباينة double star and ال comparison
+
+395
+00:49:00,810 --> 00:49:04,890
+test إذا
+
+396
+00:49:04,890 --> 00:49:06,070
+ناخد الجزء هذا
+
+397
+00:49:09,440 --> 00:49:12,720
+إذا كانت ال series هذي convergent و الكبيرة هذي
+
+398
+00:49:12,720 --> 00:49:17,840
+convergent ففي ثابت موجب convergent فهذه تطلع
+
+399
+00:49:19,770 --> 00:49:25,650
+بطلع sigma xn conversion وبالتالي هذا بكمل برهان
+
+400
+00:49:25,650 --> 00:49:30,110
+الجزء الأول طبعا برهان الجزء التاني هيكون يعني
+
+401
+00:49:30,110 --> 00:49:36,530
+مشابه فلأن الواجهة انتهى هنوقف و هسيبكم تقرؤ برهان
+
+402
+00:49:36,530 --> 00:49:41,890
+الجزء الأول من الكتاب فنكتفي بهذا القدر و ان شاء
+
+403
+00:49:41,890 --> 00:49:43,730
+الله نكمل المحاضرة الجاية
+
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/Sym_17KvBqE_raw.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/Sym_17KvBqE_raw.srt
new file mode 100644
index 0000000000000000000000000000000000000000..61e545be400f3faf5643b9aa0cdce87099352988
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/Sym_17KvBqE_raw.srt
@@ -0,0 +1,1728 @@
+1
+00:00:20,830 --> 00:00:26,410
+بسم الله الرحمن الرحيم احنا في المحاضرة اللى فاتت
+
+2
+00:00:26,410 --> 00:00:32,690
+اتحدثنا عن ال limit comparison test وبرهننا
+
+3
+00:00:32,690 --> 00:00:37,470
+الجزء الاول منه فنرجع مع بعض ال limit comparison
+
+4
+00:00:37,470 --> 00:00:43,190
+test for infinite series طبعا طبعا في limit
+
+5
+00:00:43,190 --> 00:00:47,450
+comparison test for sequences الان هذا الافتبار
+
+6
+00:00:47,450 --> 00:00:52,340
+قصد ال infinite seriesلو في عندي two sequences of
+
+7
+00:00:52,340 --> 00:00:57,760
+positive real numbers بحيث ان limit ال quotient
+
+8
+00:00:57,760 --> 00:01:05,700
+تبعهم exist بساوي عدد R ففي عندي نتيجتين، لو كان
+
+9
+00:01:05,700 --> 00:01:12,170
+العدد R أو limit R هذه لا تساوي 0ففي الحالة هذه
+
+10
+00:01:12,170 --> 00:01:18,130
+sigma x in series sigma x in convergence if and
+
+11
+00:01:18,130 --> 00:01:21,550
+only if ال series sigma y in convergence يعني
+
+12
+00:01:21,550 --> 00:01:24,910
+اتنين اما اتنين بيكونوا convergence زي بعض او
+
+13
+00:01:24,910 --> 00:01:28,630
+اتنين بيكونوا divergence زي بعض الجزء التاني بيقول
+
+14
+00:01:28,630 --> 00:01:32,010
+لو كانت ال R اللي هي limit لل quotient مساوة سفر
+
+15
+00:01:32,010 --> 00:01:37,110
+وإذا كانت ال series اللي الحد العام تبع Y in
+
+16
+00:01:37,110 --> 00:01:41,770
+convergenceفال series هذا بيقدر ال series اللي هي
+
+17
+00:01:41,770 --> 00:01:48,510
+sigma xn كلها يعني اعتقد ان احنا برهن الجزء الأول
+
+18
+00:01:48,510 --> 00:01:55,750
+برا اللي فاتت بظبط و خلينا نبرهن الجزء التاني طبعا
+
+19
+00:01:55,750 --> 00:02:06,370
+since اذا هنا let assume r
+
+20
+00:02:06,370 --> 00:02:07,650
+بساوي سفر
+
+21
+00:02:18,190 --> 00:02:24,490
+أما لو أخدت إبسلون أنا بساوي العدد واحد فهذا
+
+22
+00:02:24,490 --> 00:02:29,910
+إبسلون موجبة إحنا
+
+23
+00:02:29,910 --> 00:02:38,070
+لدينا من الفرض sense limit xn over yn as n tends
+
+24
+00:02:38,070 --> 00:02:45,640
+to infinityبساوي R اللي هو سفر الآن فمن تعريف by
+
+25
+00:02:45,640 --> 00:02:51,260
+definition of limit for epsilon positive زي هذه
+
+26
+00:02:51,260 --> 00:02:57,420
+يوجد capital N يعتمد على epsilon اللي هو الواحد
+
+27
+00:02:57,420 --> 00:03:03,900
+natural number بحيث انه لكل N أكبر من أو ساوي
+
+28
+00:03:03,900 --> 00:03:11,260
+capital Nهذا بيدّي أن ال absolute value ل xn على
+
+29
+00:03:11,260 --> 00:03:18,120
+yn minus zero بيطلع أصغر من ال epsilon اللي احنا
+
+30
+00:03:18,120 --> 00:03:26,660
+ماخدينها واحد طب xn عدد موجب و yn عدد موجبفال
+
+31
+00:03:26,660 --> 00:03:33,760
+quotient هذا كسر هذا موجب سالد سفر فهذا بيقدي ان
+
+32
+00:03:33,760 --> 00:03:42,880
+xn over yn أصغر من واحد لو ضربنا الطرفين العدد
+
+33
+00:03:42,880 --> 00:03:58,110
+الموجب yn فهذا هيقدي ان xn أصغر من ynوهذا صحيح لكل
+
+34
+00:03:58,110 --> 00:04:05,550
+N أكبر من أو يستوي capital N now
+
+35
+00:04:05,550 --> 00:04:09,090
+if
+
+36
+00:04:09,090 --> 00:04:20,550
+sigma yn converges then
+
+37
+00:04:21,920 --> 00:04:26,140
+by direct comparison test اللي أخدناها المرة اللي
+
+38
+00:04:26,140 --> 00:04:30,420
+فاتت إذا ال series الحد اللي عام تبعها أكبر
+
+39
+00:04:30,420 --> 00:04:34,480
+convergent فالأصغر
+
+40
+00:04:34,480 --> 00:04:42,920
+ال series الأصغر converges وهذا هو المطلوب هذا
+
+41
+00:04:42,920 --> 00:04:46,340
+اللي احنا عايزين نتبته إنه لو كانت ال series yn
+
+42
+00:04:46,340 --> 00:04:50,690
+convergent فلازم هذا يطلع convergent هذا صحيحby
+
+43
+00:04:50,690 --> 00:04:55,110
+direct comparison test لذلك هذا يكمل برهان الجزء
+
+44
+00:04:55,110 --> 00:05:02,230
+التالي نرجع الأن ناخد أمثلة على تطبيقات على ال
+
+45
+00:05:02,230 --> 00:05:08,590
+direct comparison test و على limit comparison test
+
+46
+00:05:11,980 --> 00:05:15,680
+كيف نستخدم ال comparison tests الاختبارين هدول
+
+47
+00:05:15,680 --> 00:05:27,780
+فيثبات ان ال series معينة is convergent discuss
+
+48
+00:05:27,780 --> 00:05:38,840
+.. discuss the convergence of
+
+49
+00:05:38,840 --> 00:05:40,360
+the following series
+
+50
+00:06:00,990 --> 00:06:07,110
+فناخد series sigma from n equals one to infinity ل
+
+51
+00:06:07,110 --> 00:06:17,370
+one over n squared plus n بالمناسبة
+
+52
+00:06:17,370 --> 00:06:18,450
+ال series هذه
+
+53
+00:06:23,110 --> 00:06:29,010
+ممكن نقارنها، الحد العام تبعها هذا، لما N تكون
+
+54
+00:06:29,010 --> 00:06:36,410
+large فممكن نهمل ال N بالنسبة ل N تربية و نعتبر أن
+
+55
+00:06:36,410 --> 00:06:42,730
+هذه ال series شبيهة أو behaves like تتصرف زي ال
+
+56
+00:06:42,730 --> 00:06:45,650
+series sigma 1 على N تربية
+
+57
+00:06:50,030 --> 00:06:54,610
+الان بنشوف إذا ممكن نطبق اختبار المقارنة المباشرة
+
+58
+00:06:54,610 --> 00:06:58,670
+ال direct comparison test بنطبقه وإذا ما اقدرناش
+
+59
+00:06:58,670 --> 00:07:06,950
+بنلجأ لاختبار تبع ال limit comparison test
+
+60
+00:07:24,400 --> 00:07:37,040
+فهنا ممكن يعني من السهل أن احنا نستخدم ال
+
+61
+00:07:37,040 --> 00:07:43,040
+direct comparison test لأنه انا عندي ال N تربيع
+
+62
+00:07:43,040 --> 00:07:50,400
+زائد N أكبر من أو يساوي Nأكبر من أو ساوي N تربية
+
+63
+00:07:50,400 --> 00:08:00,220
+لكل N ينتمي ل N هذا بيقدي أنه مقلوب N تربية زايد N
+
+64
+00:08:00,220 --> 00:08:08,680
+أصغر من أو ساوي مقلوب N تربية لكل N ك N الان
+
+65
+00:08:08,680 --> 00:08:13,020
+ال series
+
+66
+00:08:13,020 --> 00:08:15,360
+sigma واحد على N تربية
+
+67
+00:08:18,710 --> 00:08:29,730
+a P series is P series صح؟ with P
+
+68
+00:08:29,730 --> 00:08:40,270
+بيساوي اتنين اكبر من واحد so
+
+69
+00:08:40,270 --> 00:08:49,230
+it convergesby .. it is convergent by P series
+
+70
+00:08:49,230 --> 00:08:56,550
+test في ال P series test بيقوللي إذا كان أي P
+
+71
+00:08:56,550 --> 00:09:02,890
+series زي هذه بتكون convergent إذا كان P أكبر من
+
+72
+00:09:02,890 --> 00:09:08,530
+واحد و divergent إذا كان P أصغر من أوسع و أعلى و
+
+73
+00:09:08,530 --> 00:09:14,510
+برهننا الكلام هذا في المحاضرة السابقة أو الجبلةإذا
+
+74
+00:09:14,510 --> 00:09:20,250
+أنا في عندى two series واحدة الحد العام تبعها واحد
+
+75
+00:09:20,250 --> 00:09:23,790
+على انتر بيه وهذا الconversion وواحدة الحد العام
+
+76
+00:09:23,790 --> 00:09:28,090
+تبعها واحد على انتر بيه الزادة وهذا الحد العام
+
+77
+00:09:28,090 --> 00:09:31,250
+أصغر من أو ساوي الحد العام لهذه الconversion إذا
+
+78
+00:09:31,250 --> 00:09:35,630
+ممكن استخدم so
+
+79
+00:09:35,630 --> 00:09:38,550
+by direct comparison test
+
+80
+00:09:42,520 --> 00:09:46,740
+السيريز اللي هي sigma من n equals one to infinity
+
+81
+00:09:46,740 --> 00:09:56,860
+لواحد على n squared plus n converges
+
+82
+00:09:56,860 --> 00:10:03,340
+إذا السيريز هذه أتباعنا هي انها convergence by
+
+83
+00:10:03,340 --> 00:10:07,140
+direct comparison استخدمنا ال direct comparison
+
+84
+00:10:07,140 --> 00:10:09,160
+test مفهوم واضح؟
+
+85
+00:10:12,050 --> 00:10:13,950
+ناخد مثال تاني
+
+86
+00:10:36,080 --> 00:10:39,580
+بتاعة اتنين لو أخدنا series sigma from n equals
+
+87
+00:10:39,580 --> 00:10:47,780
+one to infinity لواحد على n تربية سالف n زائد
+
+88
+00:10:47,780 --> 00:10:54,180
+واحد بما نفحص هل ال series هذي convergent ولا
+
+89
+00:10:54,180 --> 00:10:57,520
+divergent طبعا
+
+90
+00:10:59,200 --> 00:11:04,280
+أول شيء بنفكر فيه، بنشوف كيف ال series هذه بتتصرف،
+
+91
+00:11:04,280 --> 00:11:07,740
+ما هي ال series القريبة منها، و اللي احنا عارفين
+
+92
+00:11:07,740 --> 00:11:12,600
+أنها أو ممكن نحكم عليها بسهولة، ن be convergent أو
+
+93
+00:11:12,600 --> 00:11:15,980
+divergent، يعني بدي أقارن ال series هذه ب series
+
+94
+00:11:15,980 --> 00:11:20,520
+تانيةمن السهل اني احكم عليها هل هي convergent او
+
+95
+00:11:20,520 --> 00:11:27,160
+divergent فلما N تكون كبيرة و ان N is sufficiently
+
+96
+00:11:27,160 --> 00:11:32,900
+large لما N تقول infinity ممكن اهمل N و اهمل 1
+
+97
+00:11:32,900 --> 00:11:41,080
+وبالتالي ال series هذه behaves تتصرف زي ال series
+
+98
+00:11:41,080 --> 00:11:42,880
+1 على N ترمية
+
+99
+00:11:45,470 --> 00:11:55,230
+اللي هي احنا عارفين which is كل بيت واحد طبعا by P
+
+100
+00:11:55,230 --> 00:12:01,270
+seriousness زي ما شرحنا في المثال الأول الآن
+
+101
+00:12:01,270 --> 00:12:10,120
+السؤال اللي بيطرح نفسه is it true هل واحد علىإن
+
+102
+00:12:10,120 --> 00:12:15,000
+تربية سالف إن زاد واحد أصغر من أو يساوي واحد على
+
+103
+00:12:15,000 --> 00:12:20,640
+إن تربية عشان نستخدم .. هل هذا الكلام صحيح لكل إن؟
+
+104
+00:12:20,640 --> 00:12:25,920
+لأ مش فاكرش أنا فللأسف هذا مش صحيح وبالتالي
+
+105
+00:12:25,920 --> 00:12:29,940
+مابقدرش أستخدم إن هذا not true
+
+106
+00:12:34,430 --> 00:12:41,410
+for example على سبيل المثال take m بساوي اتنين
+
+107
+00:12:41,410 --> 00:12:50,310
+هنجد المتباين هذه مش صح اذا مقدرش انا استخدم ال
+
+108
+00:12:50,310 --> 00:12:54,310
+direct comparison test اذا في الحالة هذه لازم
+
+109
+00:12:54,310 --> 00:12:59,190
+استخدم ال limit comparison test او ابحث عن مقارنة
+
+110
+00:12:59,190 --> 00:13:01,310
+تانية however
+
+111
+00:13:06,140 --> 00:13:17,200
+you can show بإمكانكم تخبطه أنه الواحد على n تربية
+
+112
+00:13:17,200 --> 00:13:23,500
+negative n زائد واحد هذا أصغر من أو ساوي اتنين على
+
+113
+00:13:23,500 --> 00:13:31,280
+n تربية وهذا صحيح لكل n في n إذن هذه المتباينة
+
+114
+00:13:31,280 --> 00:13:34,420
+صحيحة وبالتالي ممكن الآن
+
+115
+00:13:39,990 --> 00:13:46,330
+الان بإمكانك استخدام
+
+116
+00:13:46,330 --> 00:13:53,030
+تجارة مقارنة مباشرة للتأكيد
+
+117
+00:13:53,030 --> 00:14:02,650
+عشان تستنتجوا ان سيريز سيجما واحد على إنتر بيه
+
+118
+00:14:02,650 --> 00:14:08,980
+نيجاتيب ن بلس واحدconvergent لأنه ال series هذه
+
+119
+00:14:08,980 --> 00:14:15,920
+لأنه since ال series اللي الحد العام تبعها اتنين
+
+120
+00:14:15,920 --> 00:14:20,740
+على انتر بيها هي نفسها اتنين ضارب ال series sigma
+
+121
+00:14:20,740 --> 00:14:26,600
+واحد على انتر بيها و ال series هذه قلنا convergent
+
+122
+00:14:26,600 --> 00:14:29,660
+لأنها في series نضربها في عدد موجب بتضلها
+
+123
+00:14:29,660 --> 00:14:31,700
+convergent
+
+124
+00:14:34,390 --> 00:14:38,990
+لازم نثبت على ذلك الكلام هذا الكلام لازم تثبتيه صح
+
+125
+00:14:38,990 --> 00:14:45,970
+المشكلة في الحل هذا ان انا او انتوا كيف نبيه يخطر
+
+126
+00:14:45,970 --> 00:14:53,170
+على بالكم ان المتباين هذا صح اه it is not easy to
+
+127
+00:14:53,170 --> 00:14:57,030
+figure out this inequality مش سهل ان يختر على
+
+128
+00:14:57,030 --> 00:15:04,110
+بالنا او نستنتج ال .. او يعني ..بنعرف إنه في
+
+129
+00:15:04,110 --> 00:15:09,870
+متباينة زي هذه صحيحة هذا مش سهل وبالتالي ممكن
+
+130
+00:15:09,870 --> 00:15:14,090
+نستخدم ال limit comparison test ونرايح رأسنا ال
+
+131
+00:15:14,090 --> 00:15:16,690
+limit comparison test في الحالة هذه أسهل من إن أنا
+
+132
+00:15:16,690 --> 00:15:21,550
+يعني أخمن
+
+133
+00:15:21,550 --> 00:15:25,950
+.. أخمن يعني حاجة زي هذه okay فتعالوا نشوف كيف
+
+134
+00:15:25,950 --> 00:15:28,070
+نستخدم ال limit comparison test
+
+135
+00:15:31,920 --> 00:15:40,220
+أذا هنا we use limit
+
+136
+00:15:40,220 --> 00:15:45,160
+comparison test with
+
+137
+00:15:45,160 --> 00:15:54,640
+a n بساوي واحد على n تربيع minus n زايد واحد أو xn
+
+138
+00:15:54,640 --> 00:15:55,720
+فالبسامينات
+
+139
+00:15:57,840 --> 00:16:07,600
+و Yn بساوية واحد على M تربية فاني
+
+140
+00:16:07,600 --> 00:16:13,340
+ايجي نحسب ال limit ل Xn over Yn as N tenths of
+
+141
+00:16:13,340 --> 00:16:21,720
+infinity بساوية limit هاي Xn تقسيم Yn بتطلع M
+
+142
+00:16:21,720 --> 00:16:28,990
+تربية على M تربية negative M plus oneو ال limit
+
+143
+00:16:28,990 --> 00:16:36,930
+هذا عشان نحسبها بالجسم bust مقام على n تربية ففي
+
+144
+00:16:36,930 --> 00:16:41,750
+ال bust واحد واحد سالب واحد على n موجب واحد على n
+
+145
+00:16:41,750 --> 00:16:47,210
+تربية لإن تقول ال infinity وهذا بطلع واحد على واحد
+
+146
+00:16:47,210 --> 00:16:54,410
+سالب صفر موجب صفر ويساوي واحد لايساوي صفر إذن ال R
+
+147
+00:16:55,770 --> 00:16:59,910
+الـ R في ال limit comparison test طلعت بالساوي
+
+148
+00:16:59,910 --> 00:17:07,630
+واحد لا يساوي سفر وانا عندى اذا since وانا عندى ال
+
+149
+00:17:07,630 --> 00:17:13,050
+series sigma yn اللى هى sigma واحد على انتر بيان
+
+150
+00:17:13,050 --> 00:17:17,830
+is convergent then
+
+151
+00:17:17,830 --> 00:17:26,700
+by limit comparison test ال series sigma xnاللي هو
+
+152
+00:17:26,700 --> 00:17:32,960
+الحد اللي عم تبعها واحد على انتر بيه minus ان زاد
+
+153
+00:17:32,960 --> 00:17:41,560
+واحد كون بيعجز وهو مطلوب okay إذا هنا استخدمنا ال
+
+154
+00:17:41,560 --> 00:17:46,020
+limit كون .. لما يعجز أو يفشل ال comparison أو ال
+
+155
+00:17:46,020 --> 00:17:49,920
+direct comparison test بنرجع إلى limit comparison
+
+156
+00:17:49,920 --> 00:17:57,220
+testهنا لازم يجب ملاحظة انه اي سؤال بنحل بال
+
+157
+00:17:57,220 --> 00:18:02,660
+comparison test ممكن حله او نطبق عليه ال limit
+
+158
+00:18:02,660 --> 00:18:07,800
+comparison test لكن العكس ليس صحيح وبالتالي ال
+
+159
+00:18:07,800 --> 00:18:12,240
+limit comparison test اشمل و اعام من ال direct
+
+160
+00:18:12,240 --> 00:18:17,460
+comparison test ناخد مثال تالت واضح الحل في اي
+
+161
+00:18:17,460 --> 00:18:22,470
+سؤال او استفسار؟إذا دائما في مخرج يعني إذا انت مش
+
+162
+00:18:22,470 --> 00:18:26,390
+عارف تعمل direct comparison فاستخدم ال limit
+
+163
+00:18:26,390 --> 00:18:30,670
+comparison test وهذا مش صعب تشوفي دائما ال series
+
+164
+00:18:30,670 --> 00:18:35,730
+اللي قدامك behaves like some familiar series تتصرف
+
+165
+00:18:35,730 --> 00:18:41,130
+زي series معروفة لدينا و احنا عارف نقدر من السهل
+
+166
+00:18:41,130 --> 00:18:43,530
+نحكم عليها هل convergent او divergent
+
+167
+00:18:49,830 --> 00:18:57,370
+فلو أخدنا مثلا ال series هذه summation from
+
+168
+00:18:57,370 --> 00:19:03,630
+n equals one to infinity ل one over square root of
+
+169
+00:19:03,630 --> 00:19:08,530
+n plus one ف
+
+170
+00:19:08,530 --> 00:19:11,910
+ال series .. this series behaves طبعا لما n ..
+
+171
+00:19:11,910 --> 00:19:19,330
+when n gets large we neglect الواحد نهم الواحدوهذه
+
+172
+00:19:19,330 --> 00:19:29,190
+السيريز تتصرف من حيث التقارب والتباعد مثل سيجما
+
+173
+00:19:29,190 --> 00:19:31,210
+واحد على جذر الان
+
+174
+00:19:38,390 --> 00:19:46,390
+طيب can we السؤال يتفرج نفسه can we use direct
+
+175
+00:19:46,390 --> 00:19:51,670
+comparison test للإجابة
+
+176
+00:19:51,670 --> 00:19:57,770
+على السؤال هذا بنلاحظ أن n زائد 1 أكبر منها ويساوي
+
+177
+00:19:57,770 --> 00:20:06,310
+n لكل n هذا بيقدر أن واحدوبالتالي الجدر التربيعي ل
+
+178
+00:20:06,310 --> 00:20:10,110
+N زائد واحد أكبر من أو ساوي جدر ال N لكل N
+
+179
+00:20:10,110 --> 00:20:17,910
+وبالتالي هذا بيقدي أن واحد على الجدر التربيعي ل N
+
+180
+00:20:17,910 --> 00:20:26,950
+زائد واحد أقل من أو ساوي واحد على جدر ال N لكل Nو
+
+181
+00:20:26,950 --> 00:20:31,850
+احنا عارفين ان ال series هذه divergent لأنها P
+
+182
+00:20:31,850 --> 00:20:36,850
+series و ال P بساوي نص أصغر من واحد و هاد ال
+
+183
+00:20:36,850 --> 00:20:42,770
+series أصغر منها أو أصغر منها و يساويهافال direct
+
+184
+00:20:42,770 --> 00:20:46,970
+comparison test بيعطينيش نتيجة، بيعطينيش نتيجة إذا
+
+185
+00:20:46,970 --> 00:20:50,550
+الكبيرة divergent فالصغيرة ممكن تكون convergent
+
+186
+00:20:50,550 --> 00:20:57,150
+وممكن تكون divergent إذا هنا ال direct comparison
+
+187
+00:20:57,150 --> 00:21:01,050
+test fails،
+
+188
+00:21:01,050 --> 00:21:09,290
+fails يعني يفشل، يفشل وبالتالي مافيش أمامنا خيار
+
+189
+00:21:09,290 --> 00:21:13,350
+اللي احنا .. اللي .. اللي هالنا نستعملأو نستخدم
+
+190
+00:21:13,350 --> 00:21:27,350
+limit comparison test نستخدم
+
+191
+00:21:27,350 --> 00:21:29,970
+limit comparison test
+
+192
+00:21:34,290 --> 00:21:39,970
+with xn بيساوي واحد على ال square root of n plus
+
+193
+00:21:39,970 --> 00:21:48,230
+one و yn بيساوي one over square root of n نحسم ال
+
+194
+00:21:48,230 --> 00:21:53,990
+limit ل xn over yn as n tends to infinity بيساوي
+
+195
+00:21:53,990 --> 00:21:57,770
+ال limit هاي
+
+196
+00:21:57,770 --> 00:22:06,100
+جسم xn على yn بيطلع الجدر التربيعيلان على ان plus
+
+197
+00:22:06,100 --> 00:22:11,500
+one لما ان تقول ال infinity دخل ال limit تحت الجدر
+
+198
+00:22:11,500 --> 00:22:15,740
+لأن ال square root function is continuous فاندخل
+
+199
+00:22:15,740 --> 00:22:21,660
+ال limit و limit المقدار تحت الجدر بطلع واحد
+
+200
+00:22:21,660 --> 00:22:28,920
+وبالتالي واحد لا يساوي سوى إذا ال R في limit
+
+201
+00:22:28,920 --> 00:22:34,540
+comparison first طلعتdifferent from zero لأ تساوي
+
+202
+00:22:34,540 --> 00:22:44,020
+سفر و since ال series sigma من n equals one to
+
+203
+00:22:44,020 --> 00:22:52,860
+infinity لواحد على جدر ال n يعبر عن sigma واحد على
+
+204
+00:22:52,860 --> 00:22:58,160
+n أصمص is a p-series with
+
+205
+00:23:03,240 --> 00:23:11,940
+P بساوي نص أصغر من واحد it diverges
+
+206
+00:23:11,940 --> 00:23:24,780
+يعني بتطلع divergent by P series test ال series
+
+207
+00:23:24,780 --> 00:23:28,560
+يعني divergent وبالتالي
+
+208
+00:23:31,020 --> 00:23:34,760
+by limit comparison test حسب ال limit comparison
+
+209
+00:23:34,760 --> 00:23:44,020
+test هيعندي sigma x in و sigma y in sigma y in ده
+
+210
+00:23:44,020 --> 00:23:51,200
+هي طلعت divergent و ال R limit لرئيسه لا يساوي سفر
+
+211
+00:23:51,200 --> 00:23:56,100
+لان التانية زيها divergent ده is sigma x in اللي
+
+212
+00:23:56,100 --> 00:24:02,990
+هو واحد على الجذر التربيهي ال N زي واحدby agents
+
+213
+00:24:02,990 --> 00:24:17,470
+حسب ال limit comparison test okay تمام واضح طيب
+
+214
+00:24:17,470 --> 00:24:18,790
+ناخد كمان مثال
+
+215
+00:24:30,910 --> 00:24:37,470
+مثال رقم أربعة خلّينا نفحص ال series اللي هي
+
+216
+00:24:37,470 --> 00:24:44,770
+summation from n equals one to infinity ل one over
+
+217
+00:24:44,770 --> 00:24:52,070
+n factorial طبعا
+
+218
+00:24:52,070 --> 00:25:00,050
+هذه مش واضحممكن تقارنها لأن N factorial N
+
+219
+00:25:00,050 --> 00:25:04,810
+factorial بالساوي N نقش واحد N negative واحد N
+
+220
+00:25:04,810 --> 00:25:11,890
+negative اتنين إلى تلاتة في اتنين في واحد فمش
+
+221
+00:25:11,890 --> 00:25:21,350
+عارفين ايش نقارنها اه فهذا مش واضح لكن by trial
+
+222
+00:25:21,350 --> 00:25:31,990
+انا بتقوله بالتجريبنقدر احنا نحاول يعني نقرر او
+
+223
+00:25:31,990 --> 00:25:37,330
+يعني نشوف ان هنا عند عشان n في n سالب واحد في n
+
+224
+00:25:37,330 --> 00:25:43,510
+سالب اتنين فممكن نقارن ال series هذه بواحد على n
+
+225
+00:25:43,510 --> 00:25:51,750
+ترمية نشوف كيف ممكن نعمل المقارنة اذا هنا في حالين
+
+226
+00:25:51,750 --> 00:26:01,200
+هناSolution واحد نحن نحاول نقارن بالإيه فال
+
+227
+00:26:01,200 --> 00:26:08,740
+solution الأول أو الحل الأول بيعتمد use
+
+228
+00:26:08,740 --> 00:26:17,460
+induction to show that ممكن
+
+229
+00:26:17,460 --> 00:26:24,210
+نثبت بال induction أنهN تربية أصغر من N factorial
+
+230
+00:26:24,210 --> 00:26:30,290
+لكل N أكبر من أو ساوي أربعة المتباينة هذه صحيحة
+
+231
+00:26:30,290 --> 00:26:34,050
+لكل الأعداد الطبيعية أكبر من أو ساوي أربعة هذا
+
+232
+00:26:34,050 --> 00:26:38,390
+ممكن نثبته by induction زي ما اتعلمته هذا سؤال في
+
+233
+00:26:38,390 --> 00:26:44,670
+مبادئ رياضياتنشوف مع بعض الهدى صح نشوف أول حالة
+
+234
+00:26:44,670 --> 00:26:48,990
+لحظة ال N بتبدأ من أربعة مش من واحد ف N بساوي واحد
+
+235
+00:26:48,990 --> 00:26:52,390
+هنا هصير N بساوي أربعة و الباقى ال induction زي ما
+
+236
+00:26:52,390 --> 00:26:57,210
+اتعلمنا فلو N بساوي أربعة أربعة تربيه ستة عشر أصغر
+
+237
+00:26:57,210 --> 00:27:01,070
+من أربعة فاكتوريا الأربعة و عشرين ستة عشر أصغر من
+
+238
+00:27:01,070 --> 00:27:05,050
+أربعة و عشرين صحيحإذا العبارة صحيحة عند n بالساوية
+
+239
+00:27:05,050 --> 00:27:09,410
+أربعة افرض صحيتها عند n بالساوية k حيث k أي عدد
+
+240
+00:27:09,410 --> 00:27:13,570
+طبيعي أكبر من أربعة وثبت صحيتها عند n بالساوية k
+
+241
+00:27:13,570 --> 00:27:18,830
+زادة، أعتقد هذه مثلة في أخدت زيها في مبادئ رياضية،
+
+242
+00:27:18,830 --> 00:27:23,050
+رح نسيب .. سيبقى لكم .. ليه؟ ايه شو بتهارفنا مثلا
+
+243
+00:27:23,050 --> 00:27:25,610
+نختار الأربعة؟ ليش ما هو مثلا تلاتة أو واحد، سيبقى
+
+244
+00:27:25,610 --> 00:27:29,150
+احنا متعودين في ال induction؟أه لأنه انت ال ..
+
+245
+00:27:29,150 --> 00:27:34,030
+يعني نضل نجرب لحد ما نصر نصر بره صح اه من أربعة و
+
+246
+00:27:34,030 --> 00:27:37,690
+انت طالع تصير صحيحة أما قبل أربعة بتكون خطأ
+
+247
+00:27:37,690 --> 00:27:42,210
+وبالتالي مالهاش معناه أما من أربعة و أنت طالع
+
+248
+00:27:42,210 --> 00:27:49,830
+هتكون صحيحة فبنهم الأول تلت قيم لهم okay اذا و
+
+249
+00:27:49,830 --> 00:27:58,220
+بالتاليهذا بيقدي ان واحد على n factorial أصغر من
+
+250
+00:27:58,220 --> 00:28:04,040
+واحد على n تردية لكل n أكبر من أو ساوية أربعة
+
+251
+00:28:04,040 --> 00:28:11,960
+وبالتالي و ال series طبعا وبالتالي ممكن نستخدم ال
+
+252
+00:28:11,960 --> 00:28:15,800
+direct comparison test يعني الحالة هذه
+
+253
+00:28:24,200 --> 00:28:28,020
+و نستخدم الاختصار الوحيد الوحيد الوحيد الوحيد
+
+254
+00:28:28,020 --> 00:28:43,880
+الوحيد الوحيد الوحيد الوحيد
+
+255
+00:28:45,400 --> 00:28:50,520
+هذه الـ series هي ال key series بس بتبدأ من أربعة
+
+256
+00:28:50,520 --> 00:28:55,240
+فكأني حدث يتأول تلات حدود منها فهذا بيأثرش على ال
+
+257
+00:28:55,240 --> 00:28:59,420
+divergence أو ال convergence لل series إذا حدث
+
+258
+00:28:59,420 --> 00:29:04,980
+omitting أو deleting finite number of terms from
+
+259
+00:29:04,980 --> 00:29:09,000
+an infinite series does not affect the convergence
+
+260
+00:29:09,000 --> 00:29:13,240
+or the divergence of the series حدث عدد منتهي من
+
+261
+00:29:13,240 --> 00:29:19,540
+حدود ال seriesأو إضافة عدد منتهي كمان إلى حدود ال
+
+262
+00:29:19,540 --> 00:29:24,180
+series لا يؤثر لا على التقارب ولا على التباعد تبع
+
+263
+00:29:24,180 --> 00:29:34,900
+ال series هذا حقيقة سهل لو يعني و بدهاش برهان لأن
+
+264
+00:29:34,900 --> 00:29:40,300
+الحدود المنتهية هذه مجموعة بيطلع عدد منتهي فما
+
+265
+00:29:40,300 --> 00:29:47,230
+بأثرش على التقاربمن series بفرش على التقارب او
+
+266
+00:29:47,230 --> 00:29:52,310
+التباعد او اضافة عدد لان بما ان ال series
+
+267
+00:29:52,310 --> 00:29:58,110
+converges then ال series sigma واحد على n
+
+268
+00:29:58,110 --> 00:30:03,990
+factorial converges
+
+269
+00:30:03,990 --> 00:30:11,160
+من n بالساوية اربعة الى ملامية طبعا هذا بقدرإن أنا
+
+270
+00:30:11,160 --> 00:30:15,360
+لو ضفت لل series الحدود المتبقية من n بالساعة واحد
+
+271
+00:30:15,360 --> 00:30:22,160
+إلى تلاتة وبتصير من infinity هنا لواحد
+
+272
+00:30:22,160 --> 00:30:28,160
+على n factorial تطلع
+
+273
+00:30:28,160 --> 00:30:33,280
+conversion وهذا اللي بدنا يعني، إذن هذا أحد
+
+274
+00:30:33,280 --> 00:30:38,100
+الحلولة، okay؟ زي ما زملتكم يعني اخترحت، بتقول طب
+
+275
+00:30:38,100 --> 00:30:44,050
+و أنا إيش بدي أختار على بالي؟إن هذا المتباينة
+
+276
+00:30:44,050 --> 00:30:47,890
+الصحيحة اللي اعتمد عليها الحل أو اعتمدت عليها
+
+277
+00:30:47,890 --> 00:30:53,430
+المقارنة فمعاكم حاجة ممكن أنك .. يعني ماحدش يقدر
+
+278
+00:30:53,430 --> 00:30:57,970
+يعني يصل إلى ال .. أو ال percentage المتباينة هذه
+
+279
+00:30:57,970 --> 00:31:03,710
+اللي عليها بيرتكز الحل ففي حل تاني آخر نشوف الحل
+
+280
+00:31:03,710 --> 00:31:05,930
+التاني ال direct limit
+
+281
+00:31:09,330 --> 00:31:14,150
+الحل التاني solution
+
+282
+00:31:14,150 --> 00:31:18,430
+2 احنا
+
+283
+00:31:18,430 --> 00:31:26,430
+عارفين انه لو جسمت ناخد
+
+284
+00:31:26,430 --> 00:31:35,130
+xn بسعر واحد على n factorialبساوي واحد على ال
+
+285
+00:31:35,130 --> 00:31:41,770
+تربية كويس؟ زي ما عملناه في الحل الأول و بده قارن
+
+286
+00:31:41,770 --> 00:31:47,320
+التنتين هدول بس المقارنة المرة هذه هتكونبطريقة
+
+287
+00:31:47,320 --> 00:31:53,760
+مختلفة فلو أخدت xn و جسمتها على yn فطبعا هذا أكبر
+
+288
+00:31:53,760 --> 00:32:00,140
+من السبب لأن xn عدد موجب دايما لكل n و yn عدد موجب
+
+289
+00:32:00,140 --> 00:32:06,640
+فقسمت على دين موجبين بطلعة موجب وهذا بساوي n تربية
+
+290
+00:32:06,640 --> 00:32:13,720
+على yn اللي هو n factorial على n factorial
+
+291
+00:32:18,080 --> 00:32:26,220
+و هدا بساوي تاي n تربية على n factorial عبارة عن
+
+292
+00:32:26,220 --> 00:32:34,160
+واحد في اتنين في تلاتة الى n سالب اتنين في n سالب
+
+293
+00:32:34,160 --> 00:32:45,700
+واحد في n مظبوط؟ ممكن اختصر n مع n و هيبقى عندي
+
+294
+00:32:54,210 --> 00:33:05,130
+فهيبقى عندي n على واحد في اتنين الى n سالب اتنين
+
+295
+00:33:05,130 --> 00:33:14,650
+في n سالب واحد الان ممكن اثبات ان المقام هذا اكبر
+
+296
+00:33:14,650 --> 00:33:17,570
+من اتنين
+
+297
+00:33:20,110 --> 00:33:28,310
+إثنين في N سالب إثنين في N سالب واحد وهذا أكبر من
+
+298
+00:33:28,310 --> 00:33:38,410
+إثنين في N سالب واحد في N وهذا صحيح ليس لكل الـ N
+
+299
+00:33:38,410 --> 00:33:48,270
+مش لكل الأعداد الطبيعية N هذا أكبر من N سالب إثنين
+
+300
+00:33:48,270 --> 00:33:56,730
+في Nو هذا صحيح فقط لكل n أكبر من أو يساوي خمسة
+
+301
+00:33:56,730 --> 00:34:01,990
+يعني عند الأربعة مش صح و عند التلاتة و اتنين و
+
+302
+00:34:01,990 --> 00:34:08,580
+الواحد مش صحOkay؟ إذن N تربية على N factorial
+
+303
+00:34:08,580 --> 00:34:14,940
+بتطلع .. الآن هذا المقام أكبر من العدد هذا
+
+304
+00:34:14,940 --> 00:34:23,280
+وبالتالي المقلوب بتطلع أصغر من N على N في N سالب 2
+
+305
+00:34:23,280 --> 00:34:32,100
+طبعا N بتروح مع Nبيبقى عندي واحد على n ساوى اتنين
+
+306
+00:34:32,100 --> 00:34:37,140
+ويقول الكلام هذا صحيح لكل n أكبر من أو ساوى خمسة
+
+307
+00:34:48,530 --> 00:34:55,770
+xn على yn أصغر من واحد على n ثالث اتنين طبعا أكبر
+
+308
+00:34:55,770 --> 00:35:01,690
+من سفر أو أكبر من أو يساوي سفر وهذا صحيح لكل n
+
+309
+00:35:01,690 --> 00:35:06,890
+أكبر من أو يساوي خمسة الان هذا لما انتقل ل
+
+310
+00:35:06,890 --> 00:35:11,950
+infinity هذا بيروح لسفر لما انتقل ل infinity هذا
+
+311
+00:35:11,950 --> 00:35:16,610
+بيروح لسفر اذا by sandwich theorem
+
+312
+00:35:23,770 --> 00:35:30,910
+بطل عند ال limit ل xn over yn as n tends to
+
+313
+00:35:30,910 --> 00:35:36,570
+infinity بساوي سفر هاد هى ال R في ال limit
+
+314
+00:35:36,570 --> 00:35:42,150
+comparison test طيب since
+
+315
+00:35:44,540 --> 00:35:49,640
+سيجما واي ان اللي هي سيجما واحد على ان تربيعي
+
+316
+00:35:49,640 --> 00:35:58,740
+converges حسب الجزء الثاني من limit comparison
+
+317
+00:35:58,740 --> 00:36:02,160
+test limit comparison test بيقول إذا كان limit ال
+
+318
+00:36:02,160 --> 00:36:07,600
+ratio بساوي سفر وكانت سيجما واي ان convergent إذا
+
+319
+00:36:07,600 --> 00:36:10,400
+هذا بيقدر
+
+320
+00:36:13,430 --> 00:36:19,310
+سيجما اكس ام اللي هي سيجما وان اوبر ام فاكتوريال
+
+321
+00:36:19,310 --> 00:36:23,490
+convergence رغم المفهوم
+
+322
+00:36:26,800 --> 00:36:29,680
+واحد استخدم ال direct comparison test، التاني
+
+323
+00:36:29,680 --> 00:36:33,660
+استخدم ال limit comparison test، اتنين كان فيهم
+
+324
+00:36:33,660 --> 00:36:39,940
+شوية شغل مش سهل، لكن هذا هو الموجود، مفيش أسهل من
+
+325
+00:36:39,940 --> 00:36:46,400
+هذا فعلى أي حال يعني ال .. الأسئلة في الكتاب هتكون
+
+326
+00:36:46,400 --> 00:36:50,780
+معظمها سهلة إما في الحل بال limit comparison test
+
+327
+00:36:50,780 --> 00:36:55,520
+أو بال direct comparison test، في أي سؤال أو
+
+328
+00:36:55,520 --> 00:37:00,800
+استفسار؟الامور واضحة الحل واضح انا عارف انه كيف
+
+329
+00:37:00,800 --> 00:37:05,580
+يخطر على بالنا نعمل المقارنات هذه وهذا كلامكم صحيح
+
+330
+00:37:05,580 --> 00:37:12,980
+هذا يعني شيء مش سهل لكن في بعض المسائل ال .. يعني
+
+331
+00:37:12,980 --> 00:37:21,740
+ال .. مش سهل ان احنا نعمل المقارنة لكن بنحاول ..
+
+332
+00:37:21,740 --> 00:37:33,030
+بنحاول اللي بيحاولبيصل إلى حل خليني يعني احنا مش
+
+333
+00:37:33,030 --> 00:37:37,710
+عايزين نبدأ section جديد الصحيح ان هيك يعني ال
+
+334
+00:37:37,710 --> 00:37:42,550
+chapter خلص فعشان مابداش يعني نبدأ المرة الجاية
+
+335
+00:37:42,550 --> 00:37:49,210
+chapter جديد فخليني اخد احل سؤال من ال homework
+
+336
+00:37:49,210 --> 00:37:53,590
+problems السؤال هنا question
+
+337
+00:37:57,590 --> 00:38:06,030
+exercise رقم خمسة section تلاتة سبعة لأن هذا تمرين
+
+338
+00:38:06,030 --> 00:38:10,190
+خمسة في section تلاتة سبعة اللي هو آخر section في
+
+339
+00:38:10,190 --> 00:38:17,010
+chapter تلاتة السؤال بيقول can
+
+340
+00:38:17,010 --> 00:38:24,030
+you السؤال كتير يعني مهم و interesting can you
+
+341
+00:38:24,030 --> 00:38:31,770
+giveيعني كتاب بخاطب الطالب بيقوله can you give an
+
+342
+00:38:31,770 --> 00:38:44,810
+example هل بإمكانك تعطي مثال of a convergent of a
+
+343
+00:38:44,810 --> 00:38:53,550
+convergent series sigma xn and a divergent
+
+344
+00:39:03,070 --> 00:39:11,470
+بحيث ان المجموعة تبع ال two series يكون
+
+345
+00:39:11,470 --> 00:39:20,010
+convergent is convergent explain
+
+346
+00:39:20,010 --> 00:39:29,830
+وضحي الإجابةهتكون يا yes يا no و في كل تلحالتين بن
+
+347
+00:39:29,830 --> 00:39:37,710
+.. نعطيك تفسر ال yes او انه تبعتك فانا بقول انه
+
+348
+00:39:37,710 --> 00:39:44,450
+خلينا نعطيلكم يعني تشوفكم تفكروا نعطيكم دقيقة
+
+349
+00:39:44,450 --> 00:39:53,230
+تفكروا و تحاولوا تجيبوا مثال زي ما هو مطلوب إذا
+
+350
+00:39:53,230 --> 00:39:53,990
+كده إذا أمكن
+
+351
+00:39:57,650 --> 00:40:04,570
+فمين عندها مثال؟ كمان مرة بنجيب مثال ل two series
+
+352
+00:40:04,570 --> 00:40:10,370
+واحدة convergent اللي هي هذه الأولى والتانية
+
+353
+00:40:10,370 --> 00:40:16,190
+divergent بحيث أن مجموعهم يكون convergent هل هذا
+
+354
+00:40:16,190 --> 00:40:23,190
+ممكن؟ إذا ممكن طيب ممكن تعطيني مثال على ذلك يعني
+
+355
+00:40:23,190 --> 00:40:27,890
+اعطيني مثاليوضح صحة ال .. الكلام هذه ال example
+
+356
+00:40:27,890 --> 00:40:30,930
+مثلا نخدها هي أسهل إيش الواحد على الأن أو الأول و
+
+357
+00:40:30,930 --> 00:40:36,130
+أنت الرابعين خليني لحظة شوية لو سمحت هاي أخبرتكم
+
+358
+00:40:36,130 --> 00:40:37,350
+طرح example
+
+359
+00:40:41,280 --> 00:40:47,500
+أيه وقتك؟ ال XN قبل عن الواحد على الان تربية واحد
+
+360
+00:40:47,500 --> 00:40:55,520
+على ان تربية فطبعا هذا بقدر سيجما XN كل ذات يسار
+
+361
+00:40:55,520 --> 00:41:02,740
+سيجما واحد على ان تربية كل بيرزلأن هذه P series و
+
+362
+00:41:02,740 --> 00:41:07,380
+ال P بيساوي اتنين اكبر من واحد، صح؟ والتانية
+
+363
+00:41:07,380 --> 00:41:13,120
+الواحدة الجدر الأن نخدها YM بيساوي واحد على الجدر
+
+364
+00:41:13,120 --> 00:41:19,860
+الأن بتصير أص نص، طيب، بتصير سماشة للواحدالان
+
+365
+00:41:19,860 --> 00:41:25,920
+sigma yn بيساوي sigma 1 على n اصلا اصلا بي سيريز
+
+366
+00:41:25,920 --> 00:41:30,100
+هادي divergent بي بي سيريز هادي بي بي بي بي بي بي
+
+367
+00:41:30,100 --> 00:41:30,100
+بي بي بي بي بي بي بي بي بي بي بي بي بي بي بي بي بي
+
+368
+00:41:30,100 --> 00:41:30,400
+بي بي بي بي بي بي بي بي بي بي بي بي بي بي بي بي بي
+
+369
+00:41:30,400 --> 00:41:34,760
+بي بي بي بي بي بي بي بي بي بي بي بي بي بي بي
+
+370
+00:41:34,760 --> 00:41:34,800
+بي بي بي بي بي بي بي بي بي بي بي بي بي بي بي بي بي
+
+371
+00:41:34,800 --> 00:41:35,040
+بي بي بي بي بي بي بي بي بي بي بي بي بي بي بي بي بي
+
+372
+00:41:35,040 --> 00:41:44,900
+بي
+
+373
+00:41:44,900 --> 00:41:53,900
+بيهي sigma واحد على N تربية زائد واحد على N أص نص
+
+374
+00:41:53,900 --> 00:42:02,420
+صح؟ وهذا بيساوي summation ناخد مقام مشترك N تربية
+
+375
+00:42:02,420 --> 00:42:10,140
+فبطلع واحد زائد N أص .. أص تلاتة عشان .. أص تلاتة
+
+376
+00:42:10,140 --> 00:42:14,040
+عشان .. مظبوط؟
+
+377
+00:42:21,390 --> 00:42:30,430
+هل هذه convergent؟ لما n تكون كبيرة .. اه لما n
+
+378
+00:42:30,430 --> 00:42:37,790
+تكون كبيرة هذه بتكون behaves like sigma
+
+379
+00:42:39,040 --> 00:42:45,440
+واحد لأ مش واحد ع انتر بياني مهم للواحد وفضل
+
+380
+00:42:45,440 --> 00:42:50,420
+عندي N أس ثلاثة ع اتنين ع انتر بيان اللي بيساوي
+
+381
+00:42:50,420 --> 00:42:54,740
+سيجما واحد ع ن أس نص
+
+382
+00:42:58,550 --> 00:43:03,570
+و ممكن الأن نستخدم ال limit comparison test نثبت
+
+383
+00:43:03,570 --> 00:43:07,430
+أن هذه divergent لأن هذه divergent باستخدام ال
+
+384
+00:43:07,430 --> 00:43:11,630
+limit comparison testزيادة .. زيادة .. زيادة ..
+
+385
+00:43:11,630 --> 00:43:14,670
+زيادة .. زيادة .. زيادة .. زيادة .. زيادة .. زيادة
+
+386
+00:43:14,670 --> 00:43:19,910
+.. زيادة
+
+387
+00:43:19,910 --> 00:43:30,990
+.. زيادة .. زيادة .. زيادة .. زيادة
+
+388
+00:43:30,990 --> 00:43:31,310
+..
+
+389
+00:43:34,540 --> 00:43:43,240
+another example طيب xn بساوي سالب واحد و سالب ن
+
+390
+00:43:43,240 --> 00:43:51,160
+مثلا yn بساوي واحد yn بساوي واحداه ف ال series
+
+391
+00:43:51,160 --> 00:43:58,520
+sigma x n diverge و sigma y n diverge فتنتهي ال
+
+392
+00:43:58,520 --> 00:44:01,540
+diverge، مانفعش، مانفعش، مانفعش، مانفعش، مانفعش،
+
+393
+00:44:01,540 --> 00:44:01,880
+مانفعش، مانفعش، مانفعش، مانفعش، مانفعش، مانفعش،
+
+394
+00:44:01,880 --> 00:44:02,160
+مانفعش، مانفعش، مانفعش، مانفعش، مانفعش، مانفعش،
+
+395
+00:44:02,160 --> 00:44:03,340
+مانفعش، مانفعش، مانفعش، مانفعش، مانفعش، مانفعش،
+
+396
+00:44:03,340 --> 00:44:08,340
+مانفعش، مانفعش، مانفعش، مانفعش، مانفعش، مانفعش،
+
+397
+00:44:08,340 --> 00:44:14,220
+مانفعش، مانفعش، مانفعش، مانفعش، مانعلى مدرسة الأرض
+
+398
+00:44:14,220 --> 00:44:18,380
+ان هو من من أنتوا ساوي أربع على مالة نهاية حكينا
+
+399
+00:44:18,380 --> 00:44:20,820
+انه من أنتوا ساوي أربع على مالة نهاية هذا converge
+
+400
+00:44:20,820 --> 00:44:25,300
+بس اللي جابل حكينا انه converge انه diverge اللي
+
+401
+00:44:25,300 --> 00:44:28,280
+جابلنا طيب احنا عشان بسم ان احنا بنحكي على السؤال
+
+402
+00:44:28,280 --> 00:44:33,860
+هذا خليني احنا في هذا المثال في عندك مثال؟ خلاص
+
+403
+00:44:33,860 --> 00:44:38,440
+طبعا ال .. ال .. السابق هذا بعدين بنتناقش فيه
+
+404
+00:44:38,440 --> 00:44:43,270
+خليني أجرب عشان أنا مافيش وجهة على السؤال هذالو
+
+405
+00:44:43,270 --> 00:44:46,950
+كلكم حاولتوا .. كل واحدة حاولت تجيب مثال، كل أمثلة
+
+406
+00:44:46,950 --> 00:44:51,370
+أبقاتكم هتكون غلطة أو هتفشل، ليه؟ لأن مافيش ولا
+
+407
+00:44:51,370 --> 00:44:56,890
+مثال، لأن مافيش مثال، فانت قاعدين بتجيبوا .. تعطوا
+
+408
+00:44:56,890 --> 00:45:01,790
+حاجة مستحيلة، مش موجودة، إذا الإجابة على هذا
+
+409
+00:45:01,790 --> 00:45:02,330
+السؤال
+
+410
+00:45:08,820 --> 00:45:19,880
+إن ال answer ال answer is no لا يمكن يعطى مثال على
+
+411
+00:45:19,880 --> 00:45:22,700
+two series واحدة convergent والتانية divergent
+
+412
+00:45:22,700 --> 00:45:26,020
+مجموعة بتطلع convergent مستحيل this is impossible
+
+413
+00:45:26,020 --> 00:45:34,660
+لبرهان أو لثبات ذلك if
+
+414
+00:45:34,660 --> 00:45:47,190
+if thisإذا كان هذا صحيح أو إذا كان هذا صحيح يعني
+
+415
+00:45:47,190 --> 00:45:52,230
+لو اقدرت النجيب series convergent و series
+
+416
+00:45:52,230 --> 00:45:57,710
+divergent و مجموعة convergent then
+
+417
+00:45:57,710 --> 00:46:01,610
+we would have
+
+418
+00:46:03,890 --> 00:46:08,590
+إنه الـ series sigma yn اللي احنا فرضين انها
+
+419
+00:46:08,590 --> 00:46:15,650
+divergent اللي هي بساوي sigma xn زائد yn minus
+
+420
+00:46:15,650 --> 00:46:23,290
+sigma xn احنا قلنا لو هذا كان true معناته ال
+
+421
+00:46:23,290 --> 00:46:27,690
+series هذه convergent معناته هذه convergent ومن
+
+422
+00:46:27,690 --> 00:46:32,610
+الفرض هذه convergentوالفرق بين two convergent
+
+423
+00:46:32,610 --> 00:46:38,330
+series is convergent، إذن هذا هتطلع .. إذن الفرق
+
+424
+00:46:38,330 --> 00:46:43,830
+هيكون convergent وبالتالي إذن ال series sigma yn
+
+425
+00:46:43,830 --> 00:46:48,930
+is convergent، وهذا contradiction لإن احنا فرضين
+
+426
+00:46:48,930 --> 00:46:56,040
+أنها divergentهذا مش ممكن يكون true عشان هي كانت
+
+427
+00:46:56,040 --> 00:47:01,600
+كتبت if it were true مستحيل ..مستحيل مش if it was
+
+428
+00:47:01,600 --> 00:47:08,280
+true okay طبعا اذا انا ساطيع اعطاء مثال يعطيني
+
+429
+00:47:08,280 --> 00:47:13,800
+المواصفات هذه بالمرةتمام؟ إذا بنوقف هنا و هيك
+
+430
+00:47:13,800 --> 00:47:17,120
+بنكون خلصنا ال chapter تلاتة المرة الجاية ان شاء
+
+431
+00:47:17,120 --> 00:47:22,620
+الله هنبقى في chapter أربعة فشكرا لكم و نشوفكم ان
+
+432
+00:47:22,620 --> 00:47:23,640
+شاء الله يوم السبت
+
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/VKBf-GBS8EU.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/VKBf-GBS8EU.srt
new file mode 100644
index 0000000000000000000000000000000000000000..5b09768430a5eb4a8ea3a52469f93c98ea2d415e
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/VKBf-GBS8EU.srt
@@ -0,0 +1,1419 @@
+1
+00:00:21,140 --> 00:00:25,840
+احنا المرة اللي فاتت أخذنا موضوع الـ sub sequences و
+
+2
+00:00:25,840 --> 00:00:30,840
+آخر نظرية أخذناها في الموضوع هذا كان في نظرية 2.16
+
+3
+00:00:30,840 --> 00:00:36,380
+النظرية هذه بتنص على إنه لو كان في عندي sequence 
+
+4
+00:00:36,380 --> 00:00:41,340
+of real numbers وكانت convergent فأي subsequence 
+
+5
+00:00:41,340 --> 00:00:47,680
+منها بتكون convergent و ليها نفس الـ limit تمام؟
+
+6
+00:00:53,530 --> 00:00:59,850
+الآن بدنا نطبق النظرية هذه من خلال الأمثلة التالية
+
+7
+00:00:59,850 --> 00:01:06,250
+فالمثال
+
+8
+00:01:06,250 --> 00:01:13,510
+الأول لو كان 1 أصغر من أو لو كان صفر أصغر من B 
+
+9
+00:01:13,510 --> 00:01:19,410
+أصغر من 1 فبدنا نثبت أن هذا بيؤدي أن limit الـ
+
+10
+00:01:19,410 --> 00:01:30,240
+sequence bn بساوي صفر برهان ذلك بنعرف
+
+11
+00:01:30,240 --> 00:01:34,680
+الـ sequence Xn اللي الحد العام تبعها بساوي B أس 
+
+12
+00:01:34,680 --> 00:01:42,220
+N تمام؟ الآن من الفرض أنا عندي صفر أصغر من B أصغر
+
+13
+00:01:42,220 --> 00:01:50,360
+من 1 هذا بيؤدي أن Xn اللي هي بساوي B أس N  الـ B 
+
+14
+00:01:50,360 --> 00:01:54,360
+هذا عدد موجب أصغر من 1 فكل ما كبرت القوة تبعته كل 
+
+15
+00:01:54,360 --> 00:01:59,670
+ما بتصغر يعني هذا أكبر من B أس n زائد 1 اللي هو
+
+16
+00:01:59,670 --> 00:02:04,410
+Xn زائد 1 الكلام هذا صحيح لكل الأعداد الطبيعية
+
+17
+00:02:04,410 --> 00:02:11,350
+n فهذا بيؤدي ان الـ sequence xn is decreasing
+
+18
+00:02:11,350 --> 00:02:25,510
+متناقصة كذلك احنا في عندنا since بما أنه الـ Xn 
+
+19
+00:02:25,510 --> 00:02:31,790
+تبعتي اللي احنا عرفناها على عينها B أس N إذا كان B
+
+20
+00:02:31,790 --> 00:02:36,410
+أكبر من 0 أصغر من 1 فـ B أس N بيطلع أكبر من أو يساوي 
+
+21
+00:02:36,410 --> 00:02:44,210
+0 أصغر من أو يساوي 1 وهذا صحيح لكل N هذا معناه أن
+
+22
+00:02:44,210 --> 00:02:48,690
+الصفر حد أدنى للـ sequence BN والواحد حد أعلى
+
+23
+00:02:49,370 --> 00:02:53,530
+وبالتالي sequence bn is bounded من أسفل ومن أعلى
+
+24
+00:02:53,530 --> 00:02:59,450
+وبالتالي bounded إذا الـ sequence xn is bounded 
+
+25
+00:02:59,450 --> 00:03:03,170
+الآن
+
+26
+00:03:03,170 --> 00:03:06,370
+أنا في عندي sequence xn decreasing وبالتالي
+
+27
+00:03:06,370 --> 00:03:11,670
+monotone و bounded إذا by monotone convergence تطلع
+
+28
+00:03:11,670 --> 00:03:12,170
+convergent
+
+29
+00:03:15,900 --> 00:03:28,260
+by monotone convergence theorem xn converges say
+
+30
+00:03:28,260 --> 00:03:39,320
+دعنا خلّينا نسمي الـ limit تبعتها x say limit xn 
+
+31
+00:03:39,320 --> 00:03:40,880
+بساوي x
+
+32
+00:03:43,880 --> 00:03:50,460
+الآن بدنا نثبت انها هيثبت لنا أن الـ sequence xn اللي
+
+33
+00:03:50,460 --> 00:03:55,320
+الحد العام تبعها B أس N تطلع convergent إلى عدد x
+
+34
+00:03:55,320 --> 00:04:03,660
+الآن بدنا نثبت ان الـ x هذا هو صفر  أكلم الـ
+
+35
+00:04:03,660 --> 00:04:17,420
+x بساوي صفر طيب by الـ theorem اثنين ستة عشر الـ
+
+36
+00:04:17,420 --> 00:04:25,240
+subsequence لو أخدت الـ subsequence اللي حدودها
+
+37
+00:04:25,240 --> 00:04:31,280
+زوجية لو أخدت الحدود الزوجية من الـ sequence xn هذه
+
+38
+00:04:31,280 --> 00:04:38,620
+فهذه subsequence من xn فهذه أيضا converges لـ x
+
+39
+00:04:41,360 --> 00:04:47,460
+حسب نظرية 2.16 الـ sequence xn converge لـ x x2
+
+40
+00:04:47,460 --> 00:04:50,960
+in subsequence من xn وبالتالي convergent by 
+
+41
+00:04:50,960 --> 00:05:02,480
+theorem 16 إلى x طيب، الآن so as بما أنه 
+
+42
+00:05:06,320 --> 00:05:13,660
+x2n بيساوي B أس اتنين n x2n بدها باتنين n 
+
+43
+00:05:13,660 --> 00:05:18,700
+بيساوي B أس اتنين n وهذه ممكن كتبتها على صورة B 
+
+44
+00:05:18,700 --> 00:05:28,700
+أس n الكل تربيع وهذا عبارة عن xn تربيع الكلام هذا
+
+45
+00:05:28,700 --> 00:05:33,280
+صحيح لكل n خدوا الـ limit للطرفين لما n تؤول لـ
+
+46
+00:05:33,280 --> 00:05:43,650
+infinity إذا الـ limit لـ x2n لما n تؤول infinity
+
+47
+00:05:43,650 --> 00:05:52,630
+بساوي limit xn تربيع لما n تؤول infinity وهذا
+
+48
+00:05:52,630 --> 00:06:00,730
+بساوي limit xn الكل تربيع طيب limit .. أنا عندي
+
+49
+00:06:00,730 --> 00:06:04,270
+limit xn بساوي x إذا هذا x تربيع
+
+50
+00:06:07,550 --> 00:06:13,890
+و limit x2n بساوي x إن أنا أصبح في عندي معادلة x 
+
+51
+00:06:13,890 --> 00:06:19,730
+بساوي x تربيع حل المعادلة هذه في x فبطلع x بساوي
+
+52
+00:06:19,730 --> 00:06:29,830
+صفر أو x بساوي 1 تمام؟
+
+53
+00:06:36,360 --> 00:06:41,500
+طيب مين أخذ الصفر ولا الواحد؟
+
+54
+00:06:41,500 --> 00:06:46,620
+أنا عندي الـ sequence تبعتي decreasing متناقصة أنا 
+
+55
+00:06:46,620 --> 00:06:53,480
+عندي .. أنا عندي الـ X since 
+
+56
+00:06:53,480 --> 00:07:00,200
+Xn is decreasing متناقصة
+
+57
+00:07:04,980 --> 00:07:11,740
+و الـ limit تبعتها و x اللي هي بساوي limit xn 
+
+58
+00:07:11,740 --> 00:07:19,660
+من هنا limit xn هتطلع أكبر من أو يساوي صفر أصغر
+
+59
+00:07:19,660 --> 00:07:24,900
+من أو يساوي الواحد و الـ x إما بساوي صفر أو 1 و 
+
+60
+00:07:24,900 --> 00:07:32,020
+متناقصة فلازم الـ x الـ limit تبعتها x يساوي صفر
+
+61
+00:07:35,220 --> 00:07:40,360
+لأنها بتتناقص مش بتزايد okay إذا الـ X بساوي صفر
+
+62
+00:07:40,360 --> 00:07:44,120
+برضه
+
+63
+00:07:44,120 --> 00:07:50,740
+ممكن نحن نقول إن الـ sequence الـ X بساوي الـ infimum
+
+64
+00:07:50,740 --> 00:07:58,780
+لـ XN حيث N ينتمي لـ N حسب الـ monotone convergence 
+
+65
+00:07:58,780 --> 00:08:03,420
+theorem وهي الـ XN bounded below by صفر والصفر هو
+
+66
+00:08:03,420 --> 00:08:12,190
+الـ infimum لها إذاً هذا بيساوي الصفر لأن 
+
+67
+00:08:12,190 --> 00:08:18,290
+الصفر عبارة عن lower bound هو أكبر ممكن إثبات إنه
+
+68
+00:08:18,290 --> 00:08:25,090
+أكبر lower bound طيب واضح المثال هذا؟ كيف طبقنا 
+
+69
+00:08:25,090 --> 00:08:31,170
+نظرية 2.16 لإيجاد limit للـ convergent sequence لأن 
+
+70
+00:08:31,170 --> 00:08:35,250
+احنا أثبتنا إن الـ sequence convergent أخذنا sequence
+
+71
+00:08:35,250 --> 00:08:38,530
+الحد اللي عام تبعها B أس n أثبتنا إنها
+
+72
+00:08:38,530 --> 00:08:42,990
+convergent by monotone convergence theorem وجبنا 
+
+73
+00:08:42,990 --> 00:08:48,650
+قيمة الـ limit باستخدام نظرية 2.16 أو ممكن باستخدام
+
+74
+00:08:48,650 --> 00:08:52,790
+الـ monotone convergence theorem تمام؟ ناخد مثال
+
+75
+00:08:52,790 --> 00:08:53,250
+تاني
+
+76
+00:09:04,470 --> 00:09:09,990
+لو كان عندي c عدد حقيقي أكبر من 1 فهذا بيؤدي ان
+
+77
+00:09:09,990 --> 00:09:15,550
+الـ limit لـ c أس 1 على n لما n تؤول infinity
+
+78
+00:09:15,550 --> 00:09:21,030
+بيساوي 1 البرهان
+
+79
+00:09:21,030 --> 00:09:27,430
+بنفس الطريقة اللي برهننا فيها المثال الأول let
+
+80
+00:09:27,430 --> 00:09:33,610
+المرة هذه yn نعرّف sequence yn الـ nth term
+
+81
+00:09:33,610 --> 00:09:42,570
+تبقى yn بساوي c أس 1 على n لكل n عدد طبيعي then 
+
+82
+00:09:42,570 --> 00:09:49,230
+واضح أن yn زائد 1 بساوي c أس 1 على n زائد
+
+83
+00:09:49,230 --> 00:09:58,530
+1 و الـ c عدد أكبر من 1 وهذا الجذر رقم n زائد
+
+84
+00:09:58,530 --> 00:10:11,230
+1 له هذا بيطلع أكبر من أو أصغر من الجذر رقم n أو 
+
+85
+00:10:11,230 --> 00:10:17,690
+الجذر النوني لـ c كل ما كبر الجذر كل ما العدد
+
+86
+00:10:17,690 --> 00:10:23,720
+بيصغر إذا كان العدد أكبر من 1 وهذا بساوي yn وهذا 
+
+87
+00:10:23,720 --> 00:10:29,280
+صحيح لكل n هذا معناه yn زائد 1 أصغر من yn 
+
+88
+00:10:29,280 --> 00:10:39,160
+معناته الـ sequence yn is decreasing متناقصة also
+
+89
+00:10:39,160 --> 00:10:48,180
+أيضا أنا عندي في الـ sequence هذه y1 أكبر من أو
+
+90
+00:10:48,180 --> 00:10:56,200
+يساوي yn لأن الـ sequence متناقصة صح؟
+
+91
+00:10:56,200 --> 00:11:03,900
+و Yn من هنا Yn بساوي C أس N الـ C أكبر من 1 إذا 
+
+92
+00:11:03,900 --> 00:11:07,580
+الجذر النوني لـ C عدد أكبر من 1 بيبقى أكبر من 
+
+93
+00:11:07,580 --> 00:11:16,040
+1 إذا هذا أكبر من أو يساوي 1 تمام؟ وهذا
+
+94
+00:11:16,040 --> 00:11:22,810
+الكلام صحيح لكل N؟ إذن هي الـ sequence تبعتي yn
+
+95
+00:11:22,810 --> 00:11:28,230
+bounded below by one and bounded above by y1 y 
+
+96
+00:11:28,230 --> 00:11:36,370
+one عدد حقيقي موجب أكبر من 1 إذن
+
+97
+00:11:36,370 --> 00:11:43,550
+هذا معناه أن الـ sequence yn is bounded صح is
+
+98
+00:11:43,550 --> 00:11:52,170
+bounded so by monotone convergence theorem a
+
+99
+00:11:52,170 --> 00:12:04,010
+sequence yn converges converge say الـ limit تبعتها 
+
+100
+00:12:04,010 --> 00:12:11,970
+بساوي عدد y افترضوا ان الـ limit تبعتها بساوي
+
+101
+00:12:11,970 --> 00:12:21,450
+واحد الآن بنثبت ان الـ limit 
+
+102
+00:12:21,450 --> 00:12:30,950
+y بساوي واحد الـ claim ان الـ limit y بساوي
+
+103
+00:12:30,950 --> 00:12:34,550
+واحد كمان مرة نفس السلسلة اللي بستخدمها برنامج 
+
+104
+00:12:34,550 --> 00:12:41,680
+اتنين ستة عشر الـ subsequence اللي هي متتالية الحدود
+
+105
+00:12:41,680 --> 00:12:51,880
+الزوجية y2n هذي 
+
+106
+00:12:51,880 --> 00:12:56,260
+المفروض تكون convergent لنفس الـ limit تبعت الـ
+
+107
+00:12:56,260 --> 00:13:02,280
+sequence yn اللي هي y تمام طيب
+
+108
+00:13:02,280 --> 00:13:02,400
+but
+
+109
+00:13:08,660 --> 00:13:17,520
+Y2N شو بيساوي؟ C أس 1 على اتنين N وهذا بيساوي C
+
+110
+00:13:17,520 --> 00:13:24,680
+أس 1 على N الكل أس 1 على اتنين وهذا بيساوي C
+
+111
+00:13:24,680 --> 00:13:32,250
+أس 1 على N عبارة عن Yn الكل أس نصف الكلام هذا
+
+112
+00:13:32,250 --> 00:13:37,270
+صحيح لكل n إذا لو أخدت الـ limit للطرفين لما n تؤول 
+
+113
+00:13:37,270 --> 00:13:43,950
+infinity فبطلع عندي limit y2n as n times infinity
+
+114
+00:13:43,950 --> 00:13:48,330
+بساوي limit yn
+
+115
+00:13:48,330 --> 00:13:56,210
+لما n تؤول infinity الكل أس نصف وهذا بساوي limit الكل أس نصف 
+
+116
+00:14:00,780 --> 00:14:08,440
+طيب limit yn قلنا بتساوي y إذن هذا y أس نصف و
+
+117
+00:14:08,440 --> 00:14:14,920
+limit y2n قلنا بتساوي y إذن أنا أصبح في 
+
+118
+00:14:14,920 --> 00:14:20,700
+عندي معادلة y بساوي y أس نصف لو حلينا المعادلة هذه
+
+119
+00:14:20,700 --> 00:14:28,660
+في y فـ y تلبية بساوي 1 ومنها بطلع y بساوي صفر or y 
+
+120
+00:14:28,660 --> 00:14:29,940
+بساوي 1
+
+121
+00:14:32,490 --> 00:14:38,990
+احنا عايزين الـ y تساوي المثال التاني 1 عايزين
+
+122
+00:14:38,990 --> 00:14:49,090
+الـ y تساوي 1 تمام فأنا عندي since limit أنا 
+
+123
+00:14:49,090 --> 00:14:49,930
+عندي من هنا 
+
+124
+00:14:53,290 --> 00:15:01,650
+أنا عندي yn أكبر من أو يساوي 1 لكل n بيؤدي انه 
+
+125
+00:15:01,650 --> 00:15:10,350
+limit yn اللي هي y هي قاعدة نظرية بتقول لو y الـ 
+
+126
+00:15:10,350 --> 00:15:15,350
+sequence bounded below by a فـ limit yn تطلع أكبر 
+
+127
+00:15:15,350 --> 00:15:19,350
+من أو يساوي الواحد
+
+128
+00:15:24,190 --> 00:15:29,090
+طيب y أكبر من أو يساوي الواحد واحنا قلنا انه 
+
+129
+00:15:29,090 --> 00:15:33,430
+لازم تطلع إما صفر أو 1 فمين الـ .. الـ .. الإجابة
+
+130
+00:15:33,430 --> 00:15:40,090
+المنطقية إذا الـ y لازم يساوي 1 وبالتالي هيك 
+
+131
+00:15:40,090 --> 00:15:44,130
+ممكن اثبتنا ان الـ sequence اللي الـ instance تبعها 
+
+132
+00:15:44,130 --> 00:15:48,850
+c to one over n is convergent و الـ limit تبعتها
+
+133
+00:15:48,850 --> 00:15:51,430
+بساوي 1 تمام واضح؟
+
+134
+00:15:54,740 --> 00:15:59,300
+في أي سؤال؟ طيب
+
+135
+00:15:59,300 --> 00:16:01,360
+النظرية اللي بعد النظرية هذه
+
+136
+00:16:23,610 --> 00:16:28,650
+نظرية السبعة عشر divergence
+
+137
+00:16:28,650 --> 00:16:35,370
+criterion
+
+138
+00:16:51,000 --> 00:16:58,100
+let xn be sequence in R لو 
+
+139
+00:16:58,100 --> 00:17:02,200
+كانت xn sequence of real numbers then the
+
+140
+00:17:02,200 --> 00:17:07,700
+following statements are equivalent العبارات
+
+141
+00:17:07,700 --> 00:17:13,960
+التالية متكافئة xn does not converge to x ينتمي إلى 
+
+142
+00:17:13,960 --> 00:17:14,400
+R
+
+143
+00:17:18,590 --> 00:17:25,790
+ثنين يوجد ε₀ أكبر من صفر بحيث أنه such
+
+144
+00:17:25,790 --> 00:17:36,290
+that for any k عدد طبيعي يوجد
+
+145
+00:17:36,290 --> 00:17:44,790
+عدد طبيعي rk ينتمي إلى N with
+
+146
+00:17:46,090 --> 00:17:54,350
+rk أكبر من أو يساوي k and 
+
+147
+00:17:54,350 --> 00:18:00,930
+|x<sub>rk</sub> - x|
+
+148
+00:18:00,930 --> 00:18:07,550
+أكبر من أو يساوي ε₀
+
+149
+00:18:07,550 --> 00:18:10,870
+العبارة الثالثة
+
+150
+00:18:14,170 --> 00:18:21,010
+يوجد ε₀ أكبر من الصفر and a subsequence
+
+151
+00:18:21,010 --> 00:18:34,660
+... a subsequence x<sub>rk</sub> or x<sub>rn</sub> of the sequence x in
+
+152
+00:18:34,660 --> 00:18:42,080
+such that |x<sub>rn</sub>|
+
+153
+00:18:42,080 --> 00:18:54,540
+- x| أكبر من أو يساوي ε₀ لكل n تمام؟
+
+154
+00:18:56,650 --> 00:19:05,210
+لإثبات النظرية هذه عشان أثبت ثلاث عبارات متكافئة
+
+155
+00:19:05,210 --> 00:19:11,790
+حسب الlogic حسب المنطق أو مبادئ الرياضيات لازم
+
+156
+00:19:11,790 --> 00:19:17,490
+نثبت أن واحد بكافئ اثنين واثنين بكافئ ثلاثة وهذا
+
+157
+00:19:17,490 --> 00:19:22,330
+ممكن إثباته بأن احنا نثبت واحد بيؤدي لاثنين واثنين
+
+158
+00:19:22,330 --> 00:19:26,530
+بيؤدي لثلاثة وثلاثة بيؤدي لواحد هيك بنغلق الدائرة
+
+159
+00:19:26,530 --> 00:19:32,490
+فهذا اللي هنعمله فنبدأ بالبرهان نثبت الأول أن
+
+160
+00:19:32,490 --> 00:19:41,710
+العبارة الأولى implies الثانية بتؤدي للثانية ف
+
+161
+00:19:41,710 --> 00:19:42,390
+assume
+
+162
+00:19:45,130 --> 00:19:51,890
+العبارة الأولى صحيحة وهو x<sub>m</sub> does not converge to x
+
+163
+00:19:54,980 --> 00:20:00,680
+طيب ارجعوا لتعريف ε N definition of 
+
+164
+00:20:00,680 --> 00:20:04,200
+convergence ما معناه أن ال sequence x<sub>n</sub> converge ل
+
+165
+00:20:04,200 --> 00:20:08,560
+x معناه لكل ε أكبر من الصفر يوجد N
+
+166
+00:20:08,560 --> 00:20:12,280
+يعتمد على ε بحيث لكل n أكبر من أو يساوي
+
+167
+00:20:12,280 --> 00:20:17,040
+N المسافة بين x<sub>n</sub> و x أصغر من ε طب
+
+168
+00:20:17,040 --> 00:20:20,480
+مايعني x<sub>n</sub> لا تتقارب ل x معناه نفي كل الكلام هذا
+
+169
+00:20:20,480 --> 00:20:24,000
+اللي حكيناه بيحصل بدل لكل ε أكبر من الصفر
+
+170
+00:20:24,000 --> 00:20:29,780
+يوجد N بصير يوجد ε واحدة ε
+
+171
+00:20:29,780 --> 00:20:41,960
+₀ عدد موجب بحيث such that بحيث أنه لكل
+
+172
+00:20:43,760 --> 00:20:50,280
+k أو n عدد طبيعي the implication
+
+173
+00:20:57,890 --> 00:21:00,870
+ال implication تبع التعريف ε N ال
+
+174
+00:21:00,870 --> 00:21:06,070
+implication اللي هي لكل n أكبر من أو يساوي N
+
+175
+00:21:06,070 --> 00:21:13,970
+لازم يطلع المسافة بين x<sub>n</sub> و x أصغر من ε
+
+176
+00:21:13,970 --> 00:21:22,830
+₀ ال implication هذه is false ليست 
+
+177
+00:21:22,830 --> 00:21:27,430
+صحيحة طب ما هذا معناه هذا معناه
+
+178
+00:21:31,850 --> 00:21:41,490
+this means هذا يعني this means أنه لكل K
+
+179
+00:21:41,490 --> 00:21:48,590
+عدد طبيعي يوجد لكل
+
+180
+00:21:48,590 --> 00:21:54,630
+K عدد طبيعي مايعني أن هذا غلط معناه يوجد لكل K عدد
+
+181
+00:21:54,630 --> 00:21:59,250
+طبيعي K يوجد عدد واحد مش لكل ال N الأكبر
+
+182
+00:21:59,250 --> 00:22:06,030
+منه يوجد عدد طبيعي واحد أكبر من أو يساوي يوجد عدد
+
+183
+00:22:06,030 --> 00:22:13,690
+طبيعي سمه n أو r<sub>k</sub> يعتمد على K عدد
+
+184
+00:22:13,690 --> 00:22:17,750
+طبيعي بحيث أنه
+
+185
+00:22:21,710 --> 00:22:27,850
+بحيث أنه طبعا
+
+186
+00:22:27,850 --> 00:22:33,390
+ال r<sub>k</sub> هذا هيكون
+
+187
+00:22:33,390 --> 00:22:41,190
+أكبر من أو يساوي k and r<sub>k</sub>
+
+188
+00:22:41,190 --> 00:22:50,050
+أكبر من أو يساوي k and |x<sub>rk</sub>| or x<sub>r<sub>k</sub></sub>
+
+189
+00:22:50,050 --> 00:22:55,590
+- x| أكبر من أو يساوي بدل أصغر من ε₀
+
+190
+00:22:55,590 --> 00:23:05,410
+النفي تبعها أكبر من أو يساوي ε₀ now
+
+191
+00:23:05,410 --> 00:23:09,610
+replace
+
+192
+00:23:09,610 --> 00:23:18,450
+badly replace K by k
+
+193
+00:23:22,130 --> 00:23:25,970
+to get العبارة
+
+194
+00:23:25,970 --> 00:23:32,250
+اثنين صح؟
+
+195
+00:23:32,250 --> 00:23:38,950
+هاي بدلنا K بـ k فهنا أثبتنا أن يوجد يوجد
+
+196
+00:23:38,950 --> 00:23:46,350
+ε₀ أكبر من صفر بحيث لكل k يوجد
+
+197
+00:23:46,350 --> 00:23:53,150
+r<sub>k</sub> أكبر من أو يساوي k والمسافة بين x<sub>r<sub>k</sub></sub>
+
+198
+00:23:53,150 --> 00:23:56,750
+- x| أكبر من أو يساوي ε₀
+
+199
+00:24:04,730 --> 00:24:14,690
+الآن نثبت اثنين بيؤدي لثلاثة إذا two implies
+
+200
+00:24:14,690 --> 00:24:18,530
+three assume
+
+201
+00:24:18,530 --> 00:24:27,110
+two holds افترض أن العبارة الثانية صحيحة بني
+
+202
+00:24:27,110 --> 00:24:30,690
+نثبت أن العبارة الثالثة صحيحة طيب؟
+
+203
+00:24:37,940 --> 00:24:48,160
+then for k يساوي واحد يعني ينتمي إلى N الآن احنا
+
+204
+00:24:48,160 --> 00:24:53,320
+فترضين اثنين العبارة اثنين صحيحة إذا احنا فترضين أن 
+
+205
+00:24:53,320 --> 00:24:58,840
+يوجد ε₀ بحيث الكلام هذا بتحقق الآن لو
+
+206
+00:24:58,840 --> 00:25:04,420
+أخذت k هذه يساوي واحد فيوجد
+
+207
+00:25:06,750 --> 00:25:15,070
+r₁ عدد طبيعي وطبعا r₁ بالتأكيد أكبر من أو يساوي
+
+208
+00:25:15,070 --> 00:25:24,510
+واحد such that |x<sub>r₁</sub> - x| أكبر من أو
+
+209
+00:25:24,510 --> 00:25:33,250
+يساوي ε₀ صح؟ next for
+
+210
+00:25:34,680 --> 00:25:45,900
+k يساوي r₁ زائد واحد مش
+
+211
+00:25:45,900 --> 00:25:51,380
+هذا عدد طبيعي لو أخذت k يساوي r₁ زائد واحد r
+
+212
+00:25:51,380 --> 00:25:58,020
+واحد عدد طبيعي زائد واحد عدد طبيعي يوجد r₂ عدد
+
+213
+00:25:58,020 --> 00:26:08,800
+طبيعي و r₂ أكبر من أو يساوي r
+
+214
+00:26:08,800 --> 00:26:16,480
+واحد زائد واحد such that |x<sub>r₂</sub> - x|
+
+215
+00:26:16,480 --> 00:26:24,960
+أكبر من أو يساوي ε₀ صح طيب
+
+216
+00:26:24,960 --> 00:26:30,060
+كمان برضه لو استمرينا في العملية هذه now
+
+217
+00:26:32,620 --> 00:26:40,620
+for r₂ زائد واحد مش هذا عدد طبيعي لو أخذت k
+
+218
+00:26:40,620 --> 00:26:46,440
+يساوي اه لو أخذت k يساوي r₂ زائد واحد هذا
+
+219
+00:26:46,440 --> 00:26:51,680
+عدد طبيعي هنا اثنين اثنين لو أخذت k يساوي r₂
+
+220
+00:26:51,680 --> 00:27:01,160
+زائد واحد إذا حسب اثنين يوجد r₃ عدد طبيعي و r
+
+221
+00:27:01,160 --> 00:27:06,280
+₃ أكبر من أو يساوي ال k اللي هو r₂ زائد
+
+222
+00:27:06,280 --> 00:27:13,360
+واحد بحيث أن المسافة بين x<sub>r₃</sub> - x| أكبر
+
+223
+00:27:13,360 --> 00:27:18,400
+من أو يساوي ε₀ طب لو استمرينا في العملية
+
+224
+00:27:18,400 --> 00:27:27,040
+هذه شو هنحصل؟ ايه اللي هيحصل؟ continuing this
+
+225
+00:27:27,040 --> 00:27:27,860
+process
+
+226
+00:27:32,720 --> 00:27:35,460
+this process اللي هو استمرينا في العملية دي اللي
+
+227
+00:27:35,460 --> 00:27:49,200
+عملية تطبيق العبارة الثانية we obtain هنحصل على we
+
+228
+00:27:49,200 --> 00:27:54,960
+obtain strictly increasing
+
+229
+00:27:54,960 --> 00:28:01,700
+increasing sequence
+
+230
+00:28:06,220 --> 00:28:13,140
+r<sub>k</sub> من k يساوي واحد إلى ∞ هذه عبارة عن sequence
+
+231
+00:28:13,140 --> 00:28:20,620
+of natural numbers in N such
+
+232
+00:28:20,620 --> 00:28:28,940
+that and hence وبالتالي وبالتالي نحصل على a
+
+233
+00:28:28,940 --> 00:28:33,600
+subsequence x<sub>rk</sub>
+
+234
+00:28:34,700 --> 00:28:39,240
+من k يساوي واحد إلى ∞ هذه عبارة عن 
+
+235
+00:28:39,240 --> 00:28:45,980
+subsequence من ال sequence x<sub>n</sub> بحيث such that
+
+236
+00:28:45,980 --> 00:28:55,680
+|x<sub>rk</sub> - x| أكبر من أو يساوي ε₀
+
+237
+00:28:55,680 --> 00:29:01,160
+والكلام هذا صحيح لكل k ينتمي إلى N
+
+238
+00:29:03,880 --> 00:29:10,500
+هي في الخطوة الأولى حصلنا على r₁ وبالتالي على x<sub>r₁</sub>
+
+239
+00:29:10,500 --> 00:29:16,440
+بحيث |x<sub>r₁</sub> - x| أكبر من أو يساوي x نزيلة
+
+240
+00:29:16,440 --> 00:29:23,010
+في الخطوة الثانية حصلنا على r₂ وبالتالي x<sub>r₂</sub> لاحظوا
+
+241
+00:29:23,010 --> 00:29:30,030
+r₂ أكبر من r₁ و r₃ أكبر من r₂، إذن هذه sequence of
+
+242
+00:29:30,030 --> 00:29:33,830
+natural numbers strictly increasing، إذن ال
+
+243
+00:29:33,830 --> 00:29:39,030
+sequence، المؤشرات تبعها هي الأعداد الطبيعية، هذه
+
+244
+00:29:39,030 --> 00:29:44,110
+subsequence حسب التعريف من sequence x و بتحقق في
+
+245
+00:29:44,110 --> 00:29:49,510
+الخطوة الثانية |x<sub>r₂</sub> - x| أكبر من أو يساوي ε₀
+
+246
+00:29:49,510 --> 00:29:55,590
+الخطوة الثالثة لما k يساوي ثلاثة هي |x<sub>r₃</sub> - x|
+
+247
+00:29:55,590 --> 00:29:59,510
+أكبر من أو يساوي ε₀ وهكذا إذن هنا عملنا
+
+248
+00:29:59,510 --> 00:30:04,470
+construction عملنا بناء بنينا subsequence اللي هي
+
+249
+00:30:04,470 --> 00:30:09,650
+subsequence هذه من ال sequence x<sub>n</sub> بطريقة استقرائية
+
+250
+00:30:10,920 --> 00:30:15,420
+وهذه الـ subsequence بتحقق المتباينة هذه وهذه
+
+251
+00:30:15,420 --> 00:30:21,800
+بالضبط العبارة ثلاثة إذا three العبارة الثالثة whole
+
+252
+00:30:21,800 --> 00:30:24,960
+تمام؟
+
+253
+00:30:24,960 --> 00:30:30,460
+إذا هيك أثبتنا أن اثنين تؤدي لثلاثة باقي إثبات
+
+254
+00:30:30,460 --> 00:30:36,400
+أن العبارة الثالثة تعني واحدة
+
+255
+00:30:39,780 --> 00:30:48,460
+ف assume .. assume العبارة الثالثة صحيحة يعني يوجد
+
+256
+00:30:48,460 --> 00:30:57,260
+ε₀ أكبر من صفر and a subsequence x<sub>rk</sub>
+
+257
+00:30:57,260 --> 00:31:10,090
+of the sequence x in such that |x<sub>rk</sub> - x|
+
+258
+00:31:10,090 --> 00:31:18,510
+أكبر من أو يساوي ε₀ لكل k طيب
+
+259
+00:31:18,510 --> 00:31:29,170
+هذا معناه أو هذا بيؤدي أن x<sub>rk</sub>
+
+260
+00:31:29,170 --> 00:31:43,760
+أو x<sub>rn</sub> أو x<sub>rk</sub> لا تنتمي لـ (x - ε₀ , x + ε₀) x زائد ε₀ لا تنتمي
+
+261
+00:31:43,760 --> 00:31:45,220
+للفترة المفتوحة هذه
+
+262
+00:31:49,030 --> 00:31:53,890
+اللي هو هذه الفترة المفتوحة سميناها قبل هيك ε
+
+263
+00:31:53,890 --> 00:31:59,670
+₀ neighborhood لـ x صح؟ هذه فترة مفتوحة مركزها x
+
+264
+00:31:59,670 --> 00:32:04,330
+ونصف قطرها ε₀ المتباينة هذه بتقول إن هذا
+
+265
+00:32:04,330 --> 00:32:10,470
+الكلام لكل k لكل k لو حلت .. لو حلت المتباينة هذه
+
+266
+00:32:10,470 --> 00:32:14,990
+في x بيطلع في x لو حلت المتباينة هذه في x<sub>rk</sub> بيطلع
+
+267
+00:32:14,990 --> 00:32:23,320
+x<sub>rk</sub> لا ينتمي للفترة المفتوحة وبالتالي
+
+268
+00:32:23,320 --> 00:32:27,460
+hence by
+
+269
+00:32:27,460 --> 00:32:37,860
+definition by ال neighborhood definition of
+
+270
+00:32:37,860 --> 00:32:41,740
+limit
+
+271
+00:32:44,750 --> 00:32:49,850
+فاكرين احنا اخذنا تعريف ال limit لل sequence اول
+
+272
+00:32:49,850 --> 00:32:53,190
+تعريف كان neighborhood definition وبعدين اثبتنا
+
+273
+00:32:53,190 --> 00:32:58,470
+انه بكافئ في نظرية 2.2 اثبتنا انه بكافئ ال ε
+
+274
+00:32:58,470 --> 00:33:01,010
+N definition لل limit
+
+275
+00:33:10,150 --> 00:33:15,910
+x<sub>n</sub> converge to x معناه لأي
+
+276
+00:33:15,910 --> 00:33:21,390
+neighborhood لـ x زي هذا لازم
+
+277
+00:33:21,390 --> 00:33:29,210
+عشان
+
+278
+00:33:29,210 --> 00:33:32,550
+ال subsequence هذه converge لـ x لازم أي
+
+279
+00:33:32,550 --> 00:33:37,180
+neighborhood لـ x يحتوي كل حدود ال sequence من
+
+280
+00:33:37,180 --> 00:33:41,660
+N وانت طالع أو من K وانت طالع لكل 
+
+281
+00:33:41,660 --> 00:33:46,920
+small k أكبر من أو يساوي capital K هذا لازم يكون صحيح
+
+282
+00:33:46,920 --> 00:33:50,260
+لكل neighborhood طب أنا في .. لأن في عندي there
+
+283
+00:33:50,260 --> 00:33:55,740
+exists epsilon zero neighborhood لـ X وكل حدود الـ
+
+284
+00:33:55,740 --> 00:34:02,770
+subsequence مش موجودة فيه، هذا بالظبط نفي تعريف الـ
+
+285
+00:34:02,770 --> 00:34:05,230
+neighborhood definition للـ convergence وبالتالي
+
+286
+00:34:05,230 --> 00:34:09,350
+هذا معناه حسب تعريف الـ neighborhood definition أن
+
+287
+00:34:09,350 --> 00:34:15,550
+الـ subsequence هذه does not converge لـ X، طب احنا 
+
+288
+00:34:15,550 --> 00:34:19,970
+عايزين نثبت، عشان نثبت أن العبارة واحد صحيحة، عايزين
+
+289
+00:34:19,970 --> 00:34:23,810
+نثبت أن الـ sequence نفسها، مش الـ subsequence، الـ
+
+290
+00:34:23,810 --> 00:34:27,650
+sequence نفسها does not converge لـ X، إذا أنا بدي 
+
+291
+00:34:27,650 --> 00:34:38,290
+أكتب هنا claim لبرهان
+
+292
+00:34:38,290 --> 00:34:46,830
+العبارة الأولى، باقي اثبات الـ claim، وهو أن الـ
+
+293
+00:34:46,830 --> 00:34:55,150
+sequence x n نفسها does not converge لـ x، فنشوف 
+
+294
+00:34:55,150 --> 00:35:01,370
+مع بعض، assume ببرهان بالتناقض، assume on the contrary
+
+295
+00:35:01,370 --> 00:35:05,230
+أن
+
+296
+00:35:05,230 --> 00:35:10,990
+الـ sequence x n converge لـ x، okay، برهان بالتناقض
+
+297
+00:35:10,990 --> 00:35:22,050
+افرض أن الـ sequence converge لـ x، by a theorem اثنين 
+
+298
+00:35:22,050 --> 00:35:32,850
+بيقول the subsequence، the subsequence اللي هي X n k
+
+299
+00:35:32,850 --> 00:35:37,490
+الـ subsequence مش هاد الـ subsequence، هاد المفروض
+
+300
+00:35:37,490 --> 00:35:44,020
+تطلع convergent لـ X، وهدا ده ديني contradiction، لأن
+
+301
+00:35:44,020 --> 00:35:47,260
+أنا عندي sub sequence هنا استنتجنا أنها does not
+
+302
+00:35:47,260 --> 00:35:53,060
+converge لـ X، إذا في عندي تناقض، التناقض هذا سببه أن
+
+303
+00:35:53,060 --> 00:35:58,680
+احنا فرضنا أن X n converge لـ X، إذا بطلع عندي X n
+
+304
+00:35:58,680 --> 00:36:04,200
+does not converge لـ X، وبالتالي إذا one holds، إذا
+
+305
+00:36:04,200 --> 00:36:10,120
+one holds، وبالتالي هيك بنكون كملنا برهان النظرية
+
+306
+00:36:10,120 --> 00:36:15,580
+okay، تمام، إذا هيك اثبتنا أن التلاتة بيعد لواحد
+
+307
+00:36:15,580 --> 00:36:20,560
+وبالتالي العبارات التلاتة هذه متكافئة، احنا بهمنا
+
+308
+00:36:20,560 --> 00:36:26,140
+في التطبيق اللي هو الجزء الأخير، يعني عشان أنا اثبت
+
+309
+00:36:27,620 --> 00:36:32,400
+إنه sequence معينة does not converge to any real
+
+310
+00:36:32,400 --> 00:36:36,360
+number X، يكفي
+
+311
+00:36:36,360 --> 00:36:42,920
+إثبات أن يوجد Y0، يوجد subsequence بحيث أن المسافة
+
+312
+00:36:42,920 --> 00:36:47,780
+دي أكبر من أو يساوي Y0 لكل M، هنشوف الكلام هذا في
+
+313
+00:36:47,780 --> 00:36:58,230
+أمثلة لاحقة، لكن خلينا بس ناخد مثالا على النظرية هذه
+
+314
+00:36:58,230 --> 00:37:15,210
+إذا
+
+315
+00:37:15,210 --> 00:37:23,470
+ناخد examples هاي
+
+316
+00:37:23,470 --> 00:37:24,410
+مثال واحد
+
+317
+00:37:28,440 --> 00:37:32,300
+الـ sequence اللي الحد العام تبعها سالب واحد plus
+
+318
+00:37:32,300 --> 00:37:40,560
+n is divergent، طبعا
+
+319
+00:37:40,560 --> 00:37:43,620
+احنا اثبتنا قبل هيك أن الـ sequence هي divergent
+
+320
+00:37:43,620 --> 00:37:47,640
+عملنا proof by contradiction، فرضنا أن أنا 
+
+321
+00:37:47,640 --> 00:37:55,040
+convergent ووصلنا إلى تناقض، صح؟ اليوم هناخد برهان
+
+322
+00:37:55,040 --> 00:38:04,780
+ثاني، باستخدام نظرية 16 أو نظرية التانية، يعني نشوف
+
+323
+00:38:04,780 --> 00:38:12,820
+مع بعض، prove if
+
+324
+00:38:12,820 --> 00:38:25,060
+it were convergent، say
+
+325
+00:38:30,030 --> 00:38:38,350
+-1-N converges to X ينتمي إلى R، لو فرضنا إن
+
+326
+00:38:38,350 --> 00:38:44,970
+سيكوانس هذه convergent هنثبت إنها divergent ببرهان
+
+327
+00:38:44,970 --> 00:38:51,350
+بالتناقض، لو فرضنا إنها convergent to some X، إذا
+
+328
+00:38:51,350 --> 00:38:56,570
+كانت convergent، إن اسمها لمات، then
+
+329
+00:39:00,730 --> 00:39:07,130
+الـ sub sequences اللي
+
+330
+00:39:07,130 --> 00:39:18,390
+هم سالب واحد أس اثنين n and سالب واحد أس اثنين n plus واحد
+
+331
+00:39:18,390 --> 00:39:25,470
+سالب واحد، هذه الـ subsequence هي الحدود الزوجية من
+
+332
+00:39:25,470 --> 00:39:31,150
+هنا، وهذه الحدود الفردية، إذا كانت الـ sequence
+
+333
+00:39:31,150 --> 00:39:36,430
+نفسها converged لـ X، فالتنتين هذول both converged لـ
+
+334
+00:39:36,430 --> 00:39:45,110
+X، و
+
+335
+00:39:45,110 --> 00:39:48,670
+بالتالي، so X
+
+336
+00:39:51,100 --> 00:40:00,080
+بتساوي limit سالب واحد قو اثنين n، صح؟ وهذه بتساوي
+
+337
+00:40:00,080 --> 00:40:06,400
+limit سالب واحد قو اثنين n زائد واحد، الـ sequence هذه
+
+338
+00:40:06,400 --> 00:40:15,620
+ثابت واحد بتساوي واحد، صح؟ and برضه احنا قلنا أن الـ
+
+339
+00:40:15,620 --> 00:40:23,400
+X بتساوي limit الـ subsequence للحدود الفردية اللي
+
+340
+00:40:23,400 --> 00:40:28,580
+هي هذه، طيب
+
+341
+00:40:28,580 --> 00:40:36,140
+سالب واحد قو عدد فردي بطلع سالب واحد، إذن هذه الـ
+
+342
+00:40:36,140 --> 00:40:41,760
+sequence حدودها فردية، إذن هي عبارة عن sequence
+
+343
+00:40:41,760 --> 00:40:50,260
+ثابت سالب واحد، وبالتالي limit لثابت بطلع ثابت، إذا
+
+344
+00:40:50,260 --> 00:40:56,180
+أنا أطلع عندي واحد بتساوي x من المعادلة الأولى
+
+345
+00:40:56,180 --> 00:41:01,120
+وكذلك الـ x بتساوي سالب واحد، يعني معناه واحد بتساوي
+
+346
+00:41:01,120 --> 00:41:10,130
+سالب واحد، وهذا contradiction، تمام؟ إذا مستحيل أن الـ
+
+347
+00:41:10,130 --> 00:41:13,510
+sequence هذه تكون convergent لأنها لازم تكون
+
+348
+00:41:13,510 --> 00:41:21,050
+divergent، okay، تمام؟ إذا هنا كلمة were الدلالة
+
+349
+00:41:21,050 --> 00:41:26,470
+على الاستحالة، كان ممكن اسمها الـ sequence هذه مفرد
+
+350
+00:41:26,470 --> 00:41:32,400
+واحدة، مفروض أقول if it was convergent لكن أنا عارف
+
+351
+00:41:32,400 --> 00:41:35,400
+أنه مستحيل أنها تكون convergent فلدلالة على
+
+352
+00:41:35,400 --> 00:41:41,880
+استحالة بستخدم it were زي if I were a king مش if I
+
+353
+00:41:41,880 --> 00:41:47,140
+was a king، لكن أنا مش king، okay، تمام؟ إذا بنوقف
+
+354
+00:41:47,140 --> 00:41:50,880
+عند هذا المثال، المحاضرة هي انتهت، وبنكمل إن شاء
+
+355
+00:41:50,880 --> 00:41:51,720
+الله، سبوع جديد
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/VKBf-GBS8EU_postprocess.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/VKBf-GBS8EU_postprocess.srt
new file mode 100644
index 0000000000000000000000000000000000000000..6620d81cd11549259ab81877a8fef638e4b91a70
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/VKBf-GBS8EU_postprocess.srt
@@ -0,0 +1,1420 @@
+1
+00:00:21,140 --> 00:00:25,840
+أحنا المرة اللي فات أخدنا موضوع ال sub sequences و
+
+2
+00:00:25,840 --> 00:00:30,840
+أخر نظرية أخدناها في الموضوع هذا كان في نظرية 2.16
+
+3
+00:00:30,840 --> 00:00:36,380
+النظرية هذه بتنص على إنه لو كان في عندي sequence
+
+4
+00:00:36,380 --> 00:00:41,340
+of real numbers و كانت convergent فأي subsequence
+
+5
+00:00:41,340 --> 00:00:47,680
+منها بتكون convergent ويلها نفس ال limit تمام؟
+
+6
+00:00:53,530 --> 00:00:59,850
+الان بدنا نطبق النظرية هذه من خلال الأمثلة التالية
+
+7
+00:00:59,850 --> 00:01:06,250
+فالمثال
+
+8
+00:01:06,250 --> 00:01:13,510
+الأول لو كان واحد أصغر من أو لو كان سفر أصغر من بي
+
+9
+00:01:13,510 --> 00:01:19,410
+أصغر من واحد فبدنا نثبت أن هذا بيقدي أن limit ال
+
+10
+00:01:19,410 --> 00:01:30,240
+sequence bn بساوي سفرفلبورحان ذلك بنعرف
+
+11
+00:01:30,240 --> 00:01:34,680
+الـ sequence Xn اللي الحد العام تبعها بساوي B أُس
+
+12
+00:01:34,680 --> 00:01:42,220
+N تمام؟ الآن من الفرض أنا عندي صفر أصغر من B أصغر
+
+13
+00:01:42,220 --> 00:01:50,360
+من 1 هذا بيقدّي أن Xn اللي هي بساوي B أُس N الـ B
+
+14
+00:01:50,360 --> 00:01:54,360
+هذا عدد موجب أصغر من 1 فكل ما كبرت القوة تبعته كل
+
+15
+00:01:54,360 --> 00:01:59,670
+ما زغريعني هذا أكبر من b أُس n زايد واحد اللي هو
+
+16
+00:01:59,670 --> 00:02:04,410
+xn زايد واحد الكلام هذا صحيح لكل الأعداد الطبيعية
+
+17
+00:02:04,410 --> 00:02:11,350
+n فهذا بيقدي ان ال sequence xn is decreasing
+
+18
+00:02:11,350 --> 00:02:25,510
+متناقصة كذلك احنا في عندنا since بما انهالـ Xn
+
+19
+00:02:25,510 --> 00:02:31,790
+تبعتي اللي احنا عرفناها على عينها B أس N إذا كان B
+
+20
+00:02:31,790 --> 00:02:36,410
+أكبر من 0 أصغر من 1 ف B أس N بيطلع أكبر من أو ساوي
+
+21
+00:02:36,410 --> 00:02:44,210
+0 أصغر من أو ساوي 1 وهذا صحيح لكل N هذا معناه أن
+
+22
+00:02:44,210 --> 00:02:48,690
+السفر حد أدنى لل sequence BN والواحد حد أعلى
+
+23
+00:02:49,370 --> 00:02:53,530
+وبالتالي سيكوانس b in is bounded من أسفل ومن أعلى
+
+24
+00:02:53,530 --> 00:02:59,450
+وبالتالي bounded إذا السيكوانس x in is bounded
+
+25
+00:02:59,450 --> 00:03:03,170
+الآن
+
+26
+00:03:03,170 --> 00:03:06,370
+أنا في اندي سيكوانس x in decreasing وبالتالي
+
+27
+00:03:06,370 --> 00:03:11,670
+monotone وbounded إذا by monotone convergence تطلع
+
+28
+00:03:11,670 --> 00:03:12,170
+conversion
+
+29
+00:03:15,900 --> 00:03:28,260
+by monotone convergence theorem xn converges say
+
+30
+00:03:28,260 --> 00:03:39,320
+دعنا خلّينا نسمي ال limit تبعتها x say limit xn
+
+31
+00:03:39,320 --> 00:03:40,880
+بساوي x
+
+32
+00:03:43,880 --> 00:03:50,460
+الان بدنا نثبت انها هيثبتنا ان ال sequence xn اللي
+
+33
+00:03:50,460 --> 00:03:55,320
+الحد العام تبعها بيوس ن تطلع convergent إلى عدد x
+
+34
+00:03:55,320 --> 00:04:03,660
+الان بدنا نثبت ان ال x هذا هو سفر اكلم ال
+
+35
+00:04:03,660 --> 00:04:17,420
+x بساوي سفر طيب by ال theoremاثنين ستة عشر ال
+
+36
+00:04:17,420 --> 00:04:25,240
+sub sequence لو أخدت ال sub sequence اللي حدودها
+
+37
+00:04:25,240 --> 00:04:31,280
+زوجية لو أخدت الحدود الزوجية من ال sequence xn هذه
+
+38
+00:04:31,280 --> 00:04:38,620
+فهذه sub sequence من xn فهذه أيضا converges ل x
+
+39
+00:04:41,360 --> 00:04:47,460
+حسب نظرية 2.16 الـ sequence x in converge ل x x 2
+
+40
+00:04:47,460 --> 00:04:50,960
+in subsequence من x in وبالتالي convergent by
+
+41
+00:04:50,960 --> 00:05:02,480
+theorem 16 إلى x طيب، الآن so as بما أنه
+
+42
+00:05:06,320 --> 00:05:13,660
+x2n بيساوي بي أس اتنين n x اتنين n بد ان باتنين n
+
+43
+00:05:13,660 --> 00:05:18,700
+بيساوي بي أس اتنين n وهذه ممكن كتبتها على صورة بي
+
+44
+00:05:18,700 --> 00:05:28,700
+أس ان لكل تربية وهذا عبارة عن xn تربية الكلام هذا
+
+45
+00:05:28,700 --> 00:05:33,280
+صحيح لكل n خدوا ال limit للطرفين لما n تقول ل
+
+46
+00:05:33,280 --> 00:05:43,650
+infinityإذا ال limit ل x2n لما n تقول infinity
+
+47
+00:05:43,650 --> 00:05:52,630
+بساوي limit xn تربيع لما n تقول infinity وهذا
+
+48
+00:05:52,630 --> 00:06:00,730
+بساوي limit xn لكل تربيع طيب limit .. أنا عندي
+
+49
+00:06:00,730 --> 00:06:04,270
+limit xn بساوي x إذا هذا x تربيع
+
+50
+00:06:07,550 --> 00:06:13,890
+و limit x2 in بساوي x إن أنا أصبح في عندي معادلة x
+
+51
+00:06:13,890 --> 00:06:19,730
+بساوي x تلبيا حل المعادلة هذه في x فبطلع x بساوي
+
+52
+00:06:19,730 --> 00:06:29,830
+سفر أو x بساوي واحد تمام؟
+
+53
+00:06:36,360 --> 00:06:41,500
+طيب مين أخد السفر و لا الواحد؟
+
+54
+00:06:41,500 --> 00:06:46,620
+أنا عندي ال sequence تبعتي decreasing متناقصة أنا
+
+55
+00:06:46,620 --> 00:06:53,480
+عندي .. أنا عندي ال X since
+
+56
+00:06:53,480 --> 00:07:00,200
+X in is decreasing متناقصة
+
+57
+00:07:04,980 --> 00:07:11,740
+و ال limit تبعتها و x اللي هي بالساوي limit x in
+
+58
+00:07:11,740 --> 00:07:19,660
+من هنا limit x in هتطلع أكبر من أو ساوي سفر أصغر
+
+59
+00:07:19,660 --> 00:07:24,900
+من أو ساوي الواحد و ال x إما بالساوي سفر أو واحد و
+
+60
+00:07:24,900 --> 00:07:32,020
+متناقصة فلازم ال x ال limit تبعتها x ساوي سفر
+
+61
+00:07:35,220 --> 00:07:40,360
+لأنها بتتناقص مش بتزايد okay إذا ال X بساوي سفر
+
+62
+00:07:40,360 --> 00:07:44,120
+برضه
+
+63
+00:07:44,120 --> 00:07:50,740
+ممكن نحن نقول إن ال sequence ال X بساوي ال infimum
+
+64
+00:07:50,740 --> 00:07:58,780
+ل XN حيث N ينتمي ل N حسب ال monotone convergence
+
+65
+00:07:58,780 --> 00:08:03,420
+theorem وهي ال XN bounded below by سفر والسفر هو
+
+66
+00:08:03,420 --> 00:08:12,190
+ال infimum لهاإذاً هذا بيساوي السفر لأن
+
+67
+00:08:12,190 --> 00:08:18,290
+السفر عبارة عن lower bound هو أكبر ممكن إثبات إنه
+
+68
+00:08:18,290 --> 00:08:25,090
+أكبر lower bound طيب واضح المثال هذا؟ كيف طبقنا
+
+69
+00:08:25,090 --> 00:08:31,170
+نظرية 2.16 لإيجاد limit لل-convergent sequence لأن
+
+70
+00:08:31,170 --> 00:08:35,250
+احنا أثبتنا إن ال sequence convergentأخذنا سيكوينس
+
+71
+00:08:35,250 --> 00:08:38,530
+الحد اللي عام تبعها بي أس ان أثبتنا إنها
+
+72
+00:08:38,530 --> 00:08:42,990
+convergence by monotone convergence theorem وجبنا
+
+73
+00:08:42,990 --> 00:08:48,650
+قيمة ال limit باستخدام نظرية 2.16 أو ممكن باستخدام
+
+74
+00:08:48,650 --> 00:08:52,790
+ال monotone convergence theorem تمام؟ ناخد مثال
+
+75
+00:08:52,790 --> 00:08:53,250
+تاني
+
+76
+00:09:04,470 --> 00:09:09,990
+لو كان عندي c عدد حقيقي أكبر من واحد فهذا بيؤدي ان
+
+77
+00:09:09,990 --> 00:09:15,550
+ال limit ل c أس واحد على n لما n تقول infinity
+
+78
+00:09:15,550 --> 00:09:21,030
+بيساوي واحد البرهان
+
+79
+00:09:21,030 --> 00:09:27,430
+بنفس الطريقة اللي برهننا فيها المثال الأول let
+
+80
+00:09:27,430 --> 00:09:33,610
+المرة هذه y in انعرف sequence y inالـ nth term
+
+81
+00:09:33,610 --> 00:09:42,570
+تبقى yn بساوي c أس واحد على n لكل n عدد طبيعي then
+
+82
+00:09:42,570 --> 00:09:49,230
+واضح أن yn زائد واحد بساوي c أس واحد على n زائد
+
+83
+00:09:49,230 --> 00:09:58,530
+واحد و ال c عدد أكبر من واحدوهذا الجذر رقم n زاد
+
+84
+00:09:58,530 --> 00:10:11,230
+واحد له هذا بطلع أكبر من أو أصغر من الجذر رقم n أو
+
+85
+00:10:11,230 --> 00:10:17,690
+الجذر اللوني ل ال c كل ما كبر الجذر كل ما العدد
+
+86
+00:10:17,690 --> 00:10:23,720
+زغر إذا كان العدد أكبر من واحد وهذا بساوي ynو هذا
+
+87
+00:10:23,720 --> 00:10:29,280
+صحيح لكل n هذا معناه yn زياد واحد أصغر من yn
+
+88
+00:10:29,280 --> 00:10:39,160
+معناته ال sequence yn is decreasing متناقصة also
+
+89
+00:10:39,160 --> 00:10:48,180
+أيضا أنا عندي في ال sequence هذه y واحد أكبر من أو
+
+90
+00:10:48,180 --> 00:10:56,200
+ساوي ynلأن الـ sequence متناقصة صح؟
+
+91
+00:10:56,200 --> 00:11:03,900
+و YN من هنا YN بساوي C أس N الـ C أكبر من واحد إذا
+
+92
+00:11:03,900 --> 00:11:07,580
+الجذر النوني لـ C عدد أكبر من واحد بيبقى أكبر من
+
+93
+00:11:07,580 --> 00:11:16,040
+واحد إذا هذا أكبر من أو ساوي واحد تمام؟ وهذا
+
+94
+00:11:16,040 --> 00:11:22,810
+الكلام صحيح لكل N؟إذن هي ال sequence تبعتي y in
+
+95
+00:11:22,810 --> 00:11:28,230
+bounded below by one and bounded above by y one y
+
+96
+00:11:28,230 --> 00:11:36,370
+one عدد حفيفي موجب أكبر من واحد إذن
+
+97
+00:11:36,370 --> 00:11:43,550
+هذا معناه أن ال sequence y inis bounded صح is
+
+98
+00:11:43,550 --> 00:11:52,170
+bounded so by monotone convergence theorem a
+
+99
+00:11:52,170 --> 00:12:04,010
+sequence yn converges converge say ال limit تبعتها
+
+100
+00:12:04,010 --> 00:12:11,970
+بساوي عدد yافترضوا ان ال limit تبعتها بالساعة
+
+101
+00:12:11,970 --> 00:12:21,450
+واحدة الان بنثبت ان ال limit
+
+102
+00:12:21,450 --> 00:12:30,950
+y بالساعة واحدة ال claim ان ال limit y بالساعة
+
+103
+00:12:30,950 --> 00:12:34,550
+واحدة كمان مرة نفس السلسلة اللي بستخدمها برنامج
+
+104
+00:12:34,550 --> 00:12:41,680
+اتنين ستاشر ال subsequence اللي هيمتتالية الحدود
+
+105
+00:12:41,680 --> 00:12:51,880
+الزوجية y2n هذي
+
+106
+00:12:51,880 --> 00:12:56,260
+المفروض تكون convergent لنفس ال limit تبعت ال
+
+107
+00:12:56,260 --> 00:13:02,280
+sequence yn اللي هي y تمام طيب
+
+108
+00:13:02,280 --> 00:13:02,400
+but
+
+109
+00:13:08,660 --> 00:13:17,520
+Y2N شو بيساوي؟ C أس واحد على اتنين N وهذا بيساوي C
+
+110
+00:13:17,520 --> 00:13:24,680
+أس واحد على N الكل أس واحد على اتنين وهذا بيساوي C
+
+111
+00:13:24,680 --> 00:13:32,250
+أس واحد على N عبارة عن YN الكل أس نصالكلام هذا
+
+112
+00:13:32,250 --> 00:13:37,270
+صحيح لكل n إذا لو أخدت ال limit للطرفين لما n تقول
+
+113
+00:13:37,270 --> 00:13:43,950
+infinity فبطلع عندي limit y2n as n times infinity
+
+114
+00:13:43,950 --> 00:13:48,330
+بساوي limit yn
+
+115
+00:13:48,330 --> 00:13:56,210
+لما n تقول infinity أص نص وهذا بساوي limit أص نص
+
+116
+00:14:00,780 --> 00:14:08,440
+طيب limit y in قلنا بتساوي y إذن هذا y أص نص و
+
+117
+00:14:08,440 --> 00:14:14,920
+limit y اتنين in قلنا بتساوي y إذن أنا أصبح في
+
+118
+00:14:14,920 --> 00:14:20,700
+عندي معادلة y بساوي y أص نص لو حلينا المعادلة هذه
+
+119
+00:14:20,700 --> 00:14:28,660
+في y فy تلبية بساوي y ومنها بطلع y بساوي سفر or y
+
+120
+00:14:28,660 --> 00:14:29,940
+بساوي واحدة
+
+121
+00:14:32,490 --> 00:14:38,990
+أحنا عايزين ال y تساوي المثال التاني واحد عايزين
+
+122
+00:14:38,990 --> 00:14:49,090
+ال y تساوي واحد تمام فأنا عندي since limit أنا
+
+123
+00:14:49,090 --> 00:14:49,930
+عندي من هنا
+
+124
+00:14:53,290 --> 00:15:01,650
+أنا عندي yn أكبر من أو ساوي واحد لكل n بيقدي انه
+
+125
+00:15:01,650 --> 00:15:10,350
+limit yn اللي هي y هي خانة نظرية بتقول لو y ال
+
+126
+00:15:10,350 --> 00:15:15,350
+sequence bounded below by a ف limit yn تطلع أكبر
+
+127
+00:15:15,350 --> 00:15:19,350
+من أو ساوي الواحد
+
+128
+00:15:24,190 --> 00:15:29,090
+طيب why أكبر من أو ساوي الواحد و احنا قلنا انه
+
+129
+00:15:29,090 --> 00:15:33,430
+لازم تطلع اما سفر او واحد فمين ال .. ال .. الإجابة
+
+130
+00:15:33,430 --> 00:15:40,090
+المنطقية اذا ال why لازم الساوي واحد وبالتالي هيك
+
+131
+00:15:40,090 --> 00:15:44,130
+ممكن اثبتنا ان ال sequence اللي ال instance تبعها
+
+132
+00:15:44,130 --> 00:15:48,850
+c to one over n is convergent و ال limit تبعتها
+
+133
+00:15:48,850 --> 00:15:51,430
+بالساوي واحد تمام واضح؟
+
+134
+00:15:54,740 --> 00:15:59,300
+في اي سؤال؟ طيب
+
+135
+00:15:59,300 --> 00:16:01,360
+النظرية اللي بعد النظرية هذه
+
+136
+00:16:23,610 --> 00:16:28,650
+نظرية السبعتاش divergence
+
+137
+00:16:28,650 --> 00:16:35,370
+criterion
+
+138
+00:16:51,000 --> 00:16:58,100
+let xn be sequence in R لو
+
+139
+00:16:58,100 --> 00:17:02,200
+كانت xn sequence of real numbers then the
+
+140
+00:17:02,200 --> 00:17:07,700
+following statements are equivalent الأبارات
+
+141
+00:17:07,700 --> 00:17:13,960
+التالية متكافئة xn does not converge ل x ينتمي إلى
+
+142
+00:17:13,960 --> 00:17:14,400
+R
+
+143
+00:17:18,590 --> 00:17:25,790
+تنين يوجد epsilon zero اكبر من سفر بحيث انه such
+
+144
+00:17:25,790 --> 00:17:36,290
+that for any k عدد طبيعي يوجد
+
+145
+00:17:36,290 --> 00:17:44,790
+عدد طبيعي rk ينتمي الى n with
+
+146
+00:17:46,090 --> 00:17:54,350
+rk أكبر من أو يساوي k and and
+
+147
+00:17:54,350 --> 00:18:00,930
+absolute x x
+
+148
+00:18:00,930 --> 00:18:07,550
+رقم rk minus x أكبر من أو يساوي epsilon zero
+
+149
+00:18:07,550 --> 00:18:10,870
+الأبارة التالتة
+
+150
+00:18:14,170 --> 00:18:21,010
+يوجد epsilon zero أكبر من الصفر and a subsequence
+
+151
+00:18:21,010 --> 00:18:34,660
+.. a subsequence xrk أو xrn of ال sequence x in
+
+152
+00:18:34,660 --> 00:18:42,080
+such that absolute xrn
+
+153
+00:18:42,080 --> 00:18:54,540
+minus x أكبر من أو يساوي epsilon zero لكل n تمام؟
+
+154
+00:18:56,650 --> 00:19:05,210
+لبرهان النظرية هذه عشان اثبت تلات عبارات متكافئة
+
+155
+00:19:05,210 --> 00:19:11,790
+حسب ال logic حسب المنطق أو مبادئ الرياضيات لازم
+
+156
+00:19:11,790 --> 00:19:17,490
+نثبت ان واحد بكافئ اتنين و اتنين بكافئ تلاتة وهذا
+
+157
+00:19:17,490 --> 00:19:22,330
+ممكن اثباتهبإن احنا نثبت واحد بيقدي لاتنين و اتنين
+
+158
+00:19:22,330 --> 00:19:26,530
+بيقدي لتلاتة و تلاتة بيقدي لواحد هيك بنغلق الدائرة
+
+159
+00:19:26,530 --> 00:19:32,490
+فهذا اللي هنعمله فنبدأ بالبرهان نثبت الأول ان
+
+160
+00:19:32,490 --> 00:19:41,710
+العبارة الأولى implies التانية بتقدي للتانية ف
+
+161
+00:19:41,710 --> 00:19:42,390
+assume
+
+162
+00:19:45,130 --> 00:19:51,890
+العبارة الأولى صحيحة وهو xm does not converge لx
+
+163
+00:19:54,980 --> 00:20:00,680
+طيب ارجعوا لتعريف epsilon capital N definition of
+
+164
+00:20:00,680 --> 00:20:04,200
+convergence ما معناه ان ال sequence xn converge ل
+
+165
+00:20:04,200 --> 00:20:08,560
+x معناه لكل epsilon أكبر من السفر يوجد capital N
+
+166
+00:20:08,560 --> 00:20:12,280
+يعتمد على epsilon بحيث لكل N أكبر من أو ساوي
+
+167
+00:20:12,280 --> 00:20:17,040
+capital N المسافة بين xn و x أصغر من epsilonطب
+
+168
+00:20:17,040 --> 00:20:20,480
+مايعنى x in لا تتقارب ل x معناه نفي كل الكلام هذا
+
+169
+00:20:20,480 --> 00:20:24,000
+اللي حاكيناه بيحصل بدل لكل epsilon أكبر من الصفر
+
+170
+00:20:24,000 --> 00:20:29,780
+يوجد capital N بصير يوجد epsilon واحدة epsilon
+
+171
+00:20:29,780 --> 00:20:41,960
+zero عدد موجب بحيث such that بحيث انه لكل
+
+172
+00:20:43,760 --> 00:20:50,280
+كبت أو n عدد طبيعي the implication
+
+173
+00:20:57,890 --> 00:21:00,870
+ال implication تبع التعريف epsilon capital n ال
+
+174
+00:21:00,870 --> 00:21:06,070
+implication اللي هي لكل n أكبر من أو ساوي capital
+
+175
+00:21:06,070 --> 00:21:13,970
+K لازم يطلع المسافة بين xn و x أصغر من epsilon
+
+176
+00:21:13,970 --> 00:21:22,830
+zero ال implication هذه is false is false ليست
+
+177
+00:21:22,830 --> 00:21:27,430
+صحيحة طب ما هذا معناه هذا معناه
+
+178
+00:21:31,850 --> 00:21:41,490
+this means هذا يعني this means أنه لكل capital K
+
+179
+00:21:41,490 --> 00:21:48,590
+عدد طبيعي يوجد لكل
+
+180
+00:21:48,590 --> 00:21:54,630
+K عدد طبيعي مايعني أن هذا غلطمعناه يوجد لكل K عدد
+
+181
+00:21:54,630 --> 00:21:59,250
+طبيعي capital K يوجد عدد واحد مش لكل ال N الأكبر
+
+182
+00:21:59,250 --> 00:22:06,030
+منه يوجد عدد طبيعي واحد أكبر من أو ساوي يوجد عدد
+
+183
+00:22:06,030 --> 00:22:13,690
+طبيعي سميه N أو R sub capital K يعتمد على K عدد
+
+184
+00:22:13,690 --> 00:22:17,750
+طبيعي بحيث أنه
+
+185
+00:22:21,710 --> 00:22:27,850
+بحيث أنه طبعا
+
+186
+00:22:27,850 --> 00:22:33,390
+ال RK هذا هيكون
+
+187
+00:22:33,390 --> 00:22:41,190
+أكبر من أو ساوي K and RK
+
+188
+00:22:41,190 --> 00:22:50,050
+أكبر من أو ساوي K and absolute XN أو XRcapital k
+
+189
+00:22:50,050 --> 00:22:55,590
+minus x أكبر من أو يساوي بدل أصغر من epsilon zero
+
+190
+00:22:55,590 --> 00:23:05,410
+النفي تبعها أكبر من أو يساوي epsilon zero now
+
+191
+00:23:05,410 --> 00:23:09,610
+replace
+
+192
+00:23:09,610 --> 00:23:18,450
+badly replace capital k by small k
+
+193
+00:23:22,130 --> 00:23:25,970
+to get الأبارع
+
+194
+00:23:25,970 --> 00:23:32,250
+اتنين holes صح؟
+
+195
+00:23:32,250 --> 00:23:38,950
+هاي بدلي كابتال K بsmall kفهنا أثبتنا أن يوجد يوجد
+
+196
+00:23:38,950 --> 00:23:46,350
+εمسلون زيرو أكبر من سفر بحيث لكل لكل small k يوجد
+
+197
+00:23:46,350 --> 00:23:53,150
+R small k أكبر من أو ساوي small k والمسافة بين X R
+
+198
+00:23:53,150 --> 00:23:56,750
+small k minus X أكبر من أو ساوي امسلون زيرو
+
+199
+00:24:04,730 --> 00:24:14,690
+الأن نثبت اتنين بيقدي لواحد لتلاتة اذا two implies
+
+200
+00:24:14,690 --> 00:24:18,530
+three assume
+
+201
+00:24:18,530 --> 00:24:27,110
+two holds افرضه ان العبارة التانية صحيحة بيبني
+
+202
+00:24:27,110 --> 00:24:30,690
+نثبت ان العبارة التالتة صحيحة طيب؟
+
+203
+00:24:37,940 --> 00:24:48,160
+then for k بساوي واحد يعني تم إلى n الان احنا
+
+204
+00:24:48,160 --> 00:24:53,320
+فرضين اتنين العبارة اتنين صحيحة اذا احنا فرضين ان
+
+205
+00:24:53,320 --> 00:24:58,840
+يوجد epsilon zero بحيث الكلام هذا بتحقق الان لو
+
+206
+00:24:58,840 --> 00:25:04,420
+اخدت k هذه بساوي واحد فيوجد
+
+207
+00:25:06,750 --> 00:25:15,070
+R1 عدد طبيعي وطبعا R1 بالتأكيد أكبر من أو ساوي
+
+208
+00:25:15,070 --> 00:25:24,510
+واحد such that absolute X R1 سالب X أكبر من أو
+
+209
+00:25:24,510 --> 00:25:33,250
+ساوي epsilon zero صح؟ next for
+
+210
+00:25:34,680 --> 00:25:45,900
+ك بساوي R واحد زاد واحد مش
+
+211
+00:25:45,900 --> 00:25:51,380
+هاد عدد طبيعي لو أخدت ك بساوي R واحد زاد واحد R
+
+212
+00:25:51,380 --> 00:25:58,020
+واحد عدد طبيعي زاد واحد عدد طبيعي يوجد R اتنين عدد
+
+213
+00:25:58,020 --> 00:26:08,800
+طبيعيو R اتنين اكبر من او يساوي R
+
+214
+00:26:08,800 --> 00:26:16,480
+واحد زاد واحد such that absolute X R اتنين minus X
+
+215
+00:26:16,480 --> 00:26:24,960
+اكبر من او يساوي epsilon zero صح طيب
+
+216
+00:26:24,960 --> 00:26:30,060
+كمان برضه لو سمرنا في العملية هذه now
+
+217
+00:26:32,620 --> 00:26:40,620
+for R اتنين زائد واحد مش هاد عدد طبيعي لو أخدت K
+
+218
+00:26:40,620 --> 00:26:46,440
+لان بساوي اه لو أخدت K بساوي R اتنين زائد واحد هاد
+
+219
+00:26:46,440 --> 00:26:51,680
+عدد طبيعي هنا اتنين اتنين لو أخدت K بساوي R اتنين
+
+220
+00:26:51,680 --> 00:27:01,160
+زائد واحد اذا حسب اتنين يوجد R تلاتة عدد طبيعيوR
+
+221
+00:27:01,160 --> 00:27:06,280
+تلاتة أكبر من أو ساوي ال K اللي هو R اتنين زائد
+
+222
+00:27:06,280 --> 00:27:13,360
+واحد بحيث انه المسافة بين X R تلاتة minus X أكبر
+
+223
+00:27:13,360 --> 00:27:18,400
+من أو ساوي epsilon zero طب لو استمرنا في العملية
+
+224
+00:27:18,400 --> 00:27:27,040
+هذه شو هنحصل؟ ايه اللي هيحصل؟ continuing this
+
+225
+00:27:27,040 --> 00:27:27,860
+process
+
+226
+00:27:32,720 --> 00:27:35,460
+this process اللي هو سمرنا في العملية دي اللي
+
+227
+00:27:35,460 --> 00:27:49,200
+عملية تطبيق العبارة التانية we obtain هنحصل على we
+
+228
+00:27:49,200 --> 00:27:54,960
+obtain strictly strictly
+
+229
+00:27:54,960 --> 00:28:01,700
+increasing increasing sequence
+
+230
+00:28:06,220 --> 00:28:13,140
+RK من K بساوة واحد لإنفينتي هذه عبارة عن sequence
+
+231
+00:28:13,140 --> 00:28:20,620
+of natural numbers in N such
+
+232
+00:28:20,620 --> 00:28:28,940
+that and hence وبالتالي وبالتالي نحصل على a
+
+233
+00:28:28,940 --> 00:28:33,600
+subsequence XRK
+
+234
+00:28:34,700 --> 00:28:39,240
+من k بساوي واحد في infinity هذه عبارة عن
+
+235
+00:28:39,240 --> 00:28:45,980
+subsequence من ال sequence xn بحيث such that
+
+236
+00:28:45,980 --> 00:28:55,680
+absolute xr k minus x أكبر من أو ساوي epsilon zero
+
+237
+00:28:55,680 --> 00:29:01,160
+والكلام هذا صحيح لكل k ينتمي إلى n
+
+238
+00:29:03,880 --> 00:29:10,500
+هي في الخطوة الأولى حصلنا على R1 وبالتالي على XR1
+
+239
+00:29:10,500 --> 00:29:16,440
+بحيث absolute XR1 minus X أكبر نوى ساوية X نزيلة
+
+240
+00:29:16,440 --> 00:29:23,010
+في الخطوة التانية حصلنا على R2 وبالتالي XR2لاحظوا
+
+241
+00:29:23,010 --> 00:29:30,030
+R2 أكبر من R1 وR3 أكبر من R2، إذن هذه sequence of
+
+242
+00:29:30,030 --> 00:29:33,830
+natural numbers strictly increasing، إذن ال
+
+243
+00:29:33,830 --> 00:29:39,030
+sequence، المؤشرات تبعتها هي الأعداد الطبيعية، هذه
+
+244
+00:29:39,030 --> 00:29:44,110
+subsequence حسب التعريف من sequence Xوبتحقق في
+
+245
+00:29:44,110 --> 00:29:49,510
+الخطوة التانية absolute XR2-X أكبر من أو ساوي Y0
+
+246
+00:29:49,510 --> 00:29:55,590
+خطوة التالتة لما K بساوي تلاتة هي absolute XR3-X
+
+247
+00:29:55,590 --> 00:29:59,510
+أكبر من أو ساوي Y0 وهكذا إذن هنا عملنا
+
+248
+00:29:59,510 --> 00:30:04,470
+construction عملنا بناء بنينا subsequence اللي هي
+
+249
+00:30:04,470 --> 00:30:09,650
+subsequence هذه من ال sequence XN بطريقة استقرائية
+
+250
+00:30:10,920 --> 00:30:15,420
+و هذه الـ subsequence بتحقق المتباينة هذه وهذه
+
+251
+00:30:15,420 --> 00:30:21,800
+بالضبط العبارة تلاتة اذا three العبارة تلاتة whole
+
+252
+00:30:21,800 --> 00:30:24,960
+تمام؟
+
+253
+00:30:24,960 --> 00:30:30,460
+إذا هيك أثبتنا أن اتنين يعدي إلى تلاتة باقي أثبات
+
+254
+00:30:30,460 --> 00:30:36,400
+أن العبارة تلاتة تعني واحدة
+
+255
+00:30:39,780 --> 00:30:48,460
+ف assume .. assume العبارة تلاتة صحيحة يعني يوجد
+
+256
+00:30:48,460 --> 00:30:57,260
+epsilon zero أكبر من صفر and a subsequence xrk
+
+257
+00:30:57,260 --> 00:31:10,090
+of a sequence x in such thatabsolute xrk minus x
+
+258
+00:31:10,090 --> 00:31:18,510
+أكبر من أو ساوي y0 لكل k طيب
+
+259
+00:31:18,510 --> 00:31:29,170
+هذا معناه أو هذا بيقدّي ان xrk
+
+260
+00:31:29,170 --> 00:31:43,760
+او xrn او xrkلا تنتمي لـ X-Y0 X زاد Y0 لا تنتمي
+
+261
+00:31:43,760 --> 00:31:45,220
+للفترة المفتوحة هذه
+
+262
+00:31:49,030 --> 00:31:53,890
+اللي هو هذه الفترة المفتوحة سمناها قبل هيك ابسلون
+
+263
+00:31:53,890 --> 00:31:59,670
+زيرو neighborhood ل X صح؟ هذه فترة مفتوحة مركزها X
+
+264
+00:31:59,670 --> 00:32:04,330
+و نص قطرها ابسلون زيرو المتباينة هذه بتقول إن هذا
+
+265
+00:32:04,330 --> 00:32:10,470
+الكلام لكل K لكل K لو حلت .. لو حلت المتباينة هذه
+
+266
+00:32:10,470 --> 00:32:14,990
+في X بيطلع في X لو حلت المتباينة هذه في XRK بيطلع
+
+267
+00:32:14,990 --> 00:32:23,320
+XRK لا ينتميلأ الفترة مفتوحة وبالتالي
+
+268
+00:32:23,320 --> 00:32:27,460
+hence by
+
+269
+00:32:27,460 --> 00:32:37,860
+definition by ال neighborhood definition of
+
+270
+00:32:37,860 --> 00:32:41,740
+limit
+
+271
+00:32:44,750 --> 00:32:49,850
+فاكرين احنا اخدنا تعريف ال limit لل sequence اول
+
+272
+00:32:49,850 --> 00:32:53,190
+تعريف كان neighborhood definition و بعدين اثبتنا
+
+273
+00:32:53,190 --> 00:32:58,470
+انه بكافئ في نظرية 2.2 اثبتنا انه بكافئ ال epsilon
+
+274
+00:32:58,470 --> 00:33:01,010
+capital N definition لل limit
+
+275
+00:33:10,150 --> 00:33:15,910
+X in converge ل X X in converge ل X معناه لأي
+
+276
+00:33:15,910 --> 00:33:21,390
+neighborhood ل X زي هذا لازم
+
+277
+00:33:21,390 --> 00:33:29,210
+عشان
+
+278
+00:33:29,210 --> 00:33:32,550
+ال subsequence هذه converge ل X لازم أي
+
+279
+00:33:32,550 --> 00:33:37,180
+neighborhood ل X يحتويكل حدود الـ sequence من
+
+280
+00:33:37,180 --> 00:33:41,660
+كابتل N وانت طالع أو من كابتل K وانت طالع لكل
+
+281
+00:33:41,660 --> 00:33:46,920
+small k أكبر من أو ساوي كابتل K هذا لازم يكون صحيح
+
+282
+00:33:46,920 --> 00:33:50,260
+لكل neighborhood طب أنا في .. لأن في عندي there
+
+283
+00:33:50,260 --> 00:33:55,740
+exists epsilon zero neighborhood ل X وكل حدود ال
+
+284
+00:33:55,740 --> 00:34:02,770
+subsequence مش موجودة فيههذا بالظبط نفي تعريف الـ
+
+285
+00:34:02,770 --> 00:34:05,230
+neighborhood definition للـ convergence وبالتالي
+
+286
+00:34:05,230 --> 00:34:09,350
+هذا معناه حسب تعريف الـ neighborhood definition أن
+
+287
+00:34:09,350 --> 00:34:15,550
+الـ subsequence هذه does not converge ل Xطب احنا
+
+288
+00:34:15,550 --> 00:34:19,970
+عايزين نثبت عشان نثبت ان العبارة واحد صحيحة عايزين
+
+289
+00:34:19,970 --> 00:34:23,810
+نثبت ان ال sequence نفسها مش ال subsequence ال
+
+290
+00:34:23,810 --> 00:34:27,650
+sequence نفسها does not converge لأكس اذا انا بدي
+
+291
+00:34:27,650 --> 00:34:38,290
+اكتب هنا claim لبرهان
+
+292
+00:34:38,290 --> 00:34:46,830
+العبارة الأولى باقي اثبات ال claimوهو ان ال
+
+293
+00:34:46,830 --> 00:34:55,150
+sequence x in نفسها does not converge ل x فنشوف
+
+294
+00:34:55,150 --> 00:35:01,370
+مع بعض assume بورهان بالتناقض assume on contrary
+
+295
+00:35:01,370 --> 00:35:05,230
+ان
+
+296
+00:35:05,230 --> 00:35:10,990
+ال sequence x in converge ل x okay بورهان بالتناقض
+
+297
+00:35:10,990 --> 00:35:22,050
+افرض ان ال sequence converge ل xby a theorem اتنين
+
+298
+00:35:22,050 --> 00:35:32,850
+ستاش the subsequence the subsequence اللي هي X R K
+
+299
+00:35:32,850 --> 00:35:37,490
+ال subsequence مش هاد ال subsequence هاد المفروض
+
+300
+00:35:37,490 --> 00:35:44,020
+تطلع convergent ل X وهدا ده ديني contradictionلأن
+
+301
+00:35:44,020 --> 00:35:47,260
+أنا عندي sub sequence هنا استنتجنا أنها does not
+
+302
+00:35:47,260 --> 00:35:53,060
+converge ل X إذا في عندي تناقض التناقض هذا سببه أن
+
+303
+00:35:53,060 --> 00:35:58,680
+احنا فرضنا أن X in converge ل X إذا بطلع عندي X in
+
+304
+00:35:58,680 --> 00:36:04,200
+does not converge ل X وبالتالي إذا one holds إذا
+
+305
+00:36:04,200 --> 00:36:10,120
+one holdsوبالتالي هيك بنكون كملنا برهان النظرية
+
+306
+00:36:10,120 --> 00:36:15,580
+okay تمام اذا هيك اثبتنا ان التلاتة بيعد لواحد
+
+307
+00:36:15,580 --> 00:36:20,560
+وبالتالي العبارات التلاتة هذه متكافئة احنا بهمنا
+
+308
+00:36:20,560 --> 00:36:26,140
+في التطبيق اللي هو الجزء الأخير يعني عشان انا اثبت
+
+309
+00:36:27,620 --> 00:36:32,400
+إنه sequence معينة does not converge to any real
+
+310
+00:36:32,400 --> 00:36:36,360
+number X يكفي
+
+311
+00:36:36,360 --> 00:36:42,920
+اثبات أن يوجد Y0 يوجد subsequence بحيث أن المسافة
+
+312
+00:36:42,920 --> 00:36:47,780
+دي أكبر من أو ساوى Y0 لكل M هنشوف الكلام هذا في
+
+313
+00:36:47,780 --> 00:36:58,230
+أمثلة لاحقة لكن خلينا بس ناخد مثالعلى النظرية هذه
+
+314
+00:36:58,230 --> 00:37:15,210
+إذا
+
+315
+00:37:15,210 --> 00:37:23,470
+ناخد examples هاي
+
+316
+00:37:23,470 --> 00:37:24,410
+مثال واحد
+
+317
+00:37:28,440 --> 00:37:32,300
+الـ sequence اللي الحد العام تبعها سالب واحد plus
+
+318
+00:37:32,300 --> 00:37:40,560
+n is divergent طبعا
+
+319
+00:37:40,560 --> 00:37:43,620
+احنا اثبتنا قبل هيك ان ال sequence هي divergent
+
+320
+00:37:43,620 --> 00:37:47,640
+عملنا proof by contradiction فرضنا ان انا
+
+321
+00:37:47,640 --> 00:37:55,040
+convergent ووصلنا إلى تناقض صح اليوم هناخد برهان
+
+322
+00:37:55,040 --> 00:38:04,780
+تانيباستخدام نظرية 16 أو نظرية التانية يعني نشوف
+
+323
+00:38:04,780 --> 00:38:12,820
+مع بعض prove if
+
+324
+00:38:12,820 --> 00:38:25,060
+it were convergent say
+
+325
+00:38:30,030 --> 00:38:38,350
+-1-N converges to X ينتمي إلى R لو فرضنا إن
+
+326
+00:38:38,350 --> 00:38:44,970
+سيكوانس هذه convergent هنثبت إنها divergent بورحان
+
+327
+00:38:44,970 --> 00:38:51,350
+بالتناقض لو فرضنا إنها convergent to some X إذا
+
+328
+00:38:51,350 --> 00:38:56,570
+كانت convergent إن اسمها لمات then
+
+329
+00:39:00,730 --> 00:39:07,130
+الـ sub sequences اللي
+
+330
+00:39:07,130 --> 00:39:18,390
+هم سالب واحد أس اتنين in and سالب واحد أس اتنين in
+
+331
+00:39:18,390 --> 00:39:25,470
+سالب واحدهذه الـ subsequence هي الحدود الزوجية من
+
+332
+00:39:25,470 --> 00:39:31,150
+هنا و هذه الحدود الفردية إذا كانت ال sequence
+
+333
+00:39:31,150 --> 00:39:36,430
+نفسها converged ل X فالتنتين هذول both converged ل
+
+334
+00:39:36,430 --> 00:39:45,110
+X و
+
+335
+00:39:45,110 --> 00:39:48,670
+بالتالي so X
+
+336
+00:39:51,100 --> 00:40:00,080
+بتساوي limit سالب واحد قص اتنين in صح؟ وهذه بساوي
+
+337
+00:40:00,080 --> 00:40:06,400
+limit سالب واحد قص اتنين in واحد ال sequence هذه
+
+338
+00:40:06,400 --> 00:40:15,620
+ثابت واحد بساوي واحد صح؟ and برضه احنا قلنا ان ال
+
+339
+00:40:15,620 --> 00:40:23,400
+Xبتساوي limit ال subsequence للحدود الفردية اللي
+
+340
+00:40:23,400 --> 00:40:28,580
+هي هذه طيب
+
+341
+00:40:28,580 --> 00:40:36,140
+سالب واحد قص عدد فردي بطلع سالب واحد إذن هذه ال
+
+342
+00:40:36,140 --> 00:40:41,760
+sequence حدودها فردية إذن هي عبارة عن sequence
+
+343
+00:40:41,760 --> 00:40:50,260
+ثابت سالب واحد وبالتالي limit لثابت بطلع ثابتإذا
+
+344
+00:40:50,260 --> 00:40:56,180
+أنا أطلع عندي واحد بساوي x من المعادلة الأولى
+
+345
+00:40:56,180 --> 00:41:01,120
+وكذلك ال x بساوي سالب واحد يعني معناه واحد بساوي
+
+346
+00:41:01,120 --> 00:41:10,130
+سالب واحد وهذا contradictionتمام؟ إذا مستحيل أن ال
+
+347
+00:41:10,130 --> 00:41:13,510
+sequence هذه تكون convergent لأنها لازم تكون
+
+348
+00:41:13,510 --> 00:41:21,050
+divergent okay تمام؟ إذا هنا كلمة where الدلالة
+
+349
+00:41:21,050 --> 00:41:26,470
+على الاستحالة كان ممكن اسمها ال sequence هذه مفرد
+
+350
+00:41:26,470 --> 00:41:32,400
+واحدةمفروض اقول if it was convergent لكن انا عارف
+
+351
+00:41:32,400 --> 00:41:35,400
+انه مستحيل انها تكون convergent فلدلالة على
+
+352
+00:41:35,400 --> 00:41:41,880
+استحالة بستخدم it were زي if I were a king مش if I
+
+353
+00:41:41,880 --> 00:41:47,140
+was a king لكن انا مش king okay تمام؟ اذا بنوقف
+
+354
+00:41:47,140 --> 00:41:50,880
+عند هذا المثال المحاضرة هي انتهت و بنكمل ان شاء
+
+355
+00:41:50,880 --> 00:41:51,720
+الله سبوع جديد
+
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/XhLWrN2SkOQ.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/XhLWrN2SkOQ.srt
new file mode 100644
index 0000000000000000000000000000000000000000..5d758affbdd53e6517bb297ba4a6b31fd4815788
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/XhLWrN2SkOQ.srt
@@ -0,0 +1,1389 @@
+1
+00:00:20,890 --> 00:00:26,630
+أحنا هنكمل الموضوع لـ Properly Divergent Sequences
+
+2
+00:00:26,630 --> 00:00:31,690
+اللي بدأناه المحاضرة السابقة، فشوفنا في المحاضرة
+
+3
+00:00:31,690 --> 00:00:36,530
+السابقة تعريف ما معنى أنه limit لـ sequence xn 
+
+4
+00:00:36,530 --> 00:00:41,450
+بساوي infinity وما معنى أنه limit لـ sequence xn
+
+5
+00:00:41,450 --> 00:00:46,490
+بساوي negative infinity، طبعاً الـ sequence بتكون
+
+6
+00:00:46,490 --> 00:00:49,470
+properly divergent إذا كانت ال limit تبعتها
+
+7
+00:00:49,470 --> 00:00:54,030
+بساوي infinity أو سالب infinity، في عندي 
+
+8
+00:00:54,030 --> 00:00:58,550
+comparison test لـ .. لـ properly divergent
+
+9
+00:00:58,550 --> 00:01:01,790
+sequences، هذا ال test بيقول لي لو في عندي two
+
+10
+00:01:01,790 --> 00:01:06,330
+sequences xn و yn، two sequences of real numbers
+
+11
+00:01:06,330 --> 00:01:10,370
+بيحققوا الشرط star، satisfy the condition star، وهو
+
+12
+00:01:10,370 --> 00:01:15,400
+أن كل حد في xn أصغر من أو ساوي الحد اللي بنظره في
+
+13
+00:01:15,400 --> 00:01:22,080
+ال sequence التانية yn، هذا صحيح لكل n، فإذا كانت ال
+
+14
+00:01:22,080 --> 00:01:26,140
+limit of the bigger sequence or the smaller
+
+15
+00:01:26,140 --> 00:01:30,240
+sequence is infinity، then the limit of the bigger
+
+16
+00:01:30,240 --> 00:01:36,040
+sequence is infinity، and if the limit of the big
+
+17
+00:01:36,040 --> 00:01:39,580
+the bigger sequence is negative infinity، then the
+
+18
+00:01:39,580 --> 00:01:40,720
+limit of the smaller
+
+19
+00:01:50,070 --> 00:01:55,680
+الجزء بي هو نقاط نقاط نقاط نقاط نقاط نقاط an
+
+20
+00:01:55,680 --> 00:01:58,620
+application of the definition، طبقنا التعريف
+
+21
+00:01:58,620 --> 00:02:03,740
+بالبرهان زي ما أنتم شايفينه، برهان الجزء A similar
+
+22
+00:02:03,740 --> 00:02:09,840
+مشابه لجزء B، فحنسيبوا تمرين لكم، اتحاولوا يعني
+
+23
+00:02:09,840 --> 00:02:15,020
+اتبرهنوا بنفس الطريقة، okay تمام، فلو سمحتوا حاولوا
+
+24
+00:02:15,020 --> 00:02:21,720
+انكم اتبرهنوا الجزء A بنفس الطريقة، في عندنا شوية
+
+25
+00:02:21,720 --> 00:02:24,300
+ملاحظات على النظرية
+
+26
+00:02:31,050 --> 00:02:36,390
+نعطيلها رقم تسعة و ثلاثين، فالملاحظات 
+
+27
+00:02:36,390 --> 00:02:44,850
+في عندي تلت ملاحظات، الملاحظة الأولى أنه theorem
+
+28
+00:02:44,850 --> 00:02:47,910
+النظرية
+
+29
+00:02:47,910 --> 00:02:53,270
+السابقة، أعتقد أن هذا الرقم المفروض يكون ثلاثين
+
+30
+00:02:59,480 --> 00:03:15,440
+theorem 29 remains true، تبقى صحيحة، if condition if
+
+31
+00:03:15,440 --> 00:03:22,720
+condition star is replaced، إذا بدلنا الشرط star by
+
+32
+00:03:22,720 --> 00:03:34,060
+the weaker condition، by the weaker condition
+
+33
+00:03:34,060 --> 00:03:39,440
+اللي 
+
+34
+00:03:39,440 --> 00:03:46,080
+هو xn less than or equal yn، لكل
+
+35
+00:03:46,080 --> 00:03:55,540
+n أكبر من أو ساوي m، for some n natural number
+
+36
+00:03:58,530 --> 00:04:05,110
+يعني بدل ما Xn أصغر من أو ساوي Yn لكل الان، لكل
+
+37
+00:04:05,110 --> 00:04:09,910
+الأعداد الطبيعية N، فلنفرض أن يوجد M عدد طبيعي
+
+38
+00:04:09,910 --> 00:04:15,190
+نفرض أن يوجد some M عدد طبيعي بحيث أن المتباينة هذه
+
+39
+00:04:15,190 --> 00:04:21,590
+تتحقق لكل N أكبر من أو ساوي M، يعني مش شرط تتحقق
+
+40
+00:04:21,590 --> 00:04:26,270
+للأعداد الطبيعية اللي أصغر من M، فالنظرية برضه تبقى
+
+41
+00:04:26,270 --> 00:04:32,390
+صحيحة، ولو بدنا نبرهن النظرية اللي فاتت تحت الشرط
+
+42
+00:04:32,390 --> 00:04:37,150
+الأضعف، هذا الشرط أضعف من الشرط ال star، لكن برضه 
+
+43
+00:04:37,150 --> 00:04:44,970
+بيعطيني نفس النظرية، فال ..
+
+44
+00:04:44,970 --> 00:04:55,390
+ففي الحالة هذه، in fact، في حقيقة الأمر، in fact، in
+
+45
+00:04:55,390 --> 00:05:07,220
+the proofs، in the proofs of النظرية السابقة، take
+
+46
+00:05:07,220 --> 00:05:19,460
+the required، the required in to be that
+
+47
+00:05:19,460 --> 00:05:26,660
+corresponds، that corresponds
+
+48
+00:05:28,960 --> 00:05:34,420
+that corresponds to the given to
+
+49
+00:05:34,420 --> 00:05:38,880
+the given alpha or 
+
+50
+00:05:38,880 --> 00:05:45,360
+given beta to
+
+51
+00:05:45,360 --> 00:05:59,160
+be in عبارة عن ال maximum، the m و n of alpha أو n
+
+52
+00:05:59,160 --> 00:06:07,920
+بساوي ال maximum، الأكبر بين العدد الطبيعي m و n 
+
+53
+00:06:07,920 --> 00:06:17,920
+of beta، إذن
+
+54
+00:06:17,920 --> 00:06:22,480
+في البرهان مثلاً، هذه الجزء التالي برهاننا عادي هي
+
+55
+00:06:22,480 --> 00:06:29,810
+نقلت الكلام هذا صحيح، ويوجد capital N هتعتمد على
+
+56
+00:06:29,810 --> 00:06:34,910
+beta بحيث أن الكلام هذا يتحقق، الآن ال star ما قدرش
+
+57
+00:06:34,910 --> 00:06:38,850
+أ say bye star، هذه هتكون double star بدل ال star
+
+58
+00:06:38,850 --> 00:06:45,070
+فأنا سميها double star، فالآن
+
+59
+00:06:45,070 --> 00:06:51,590
+بأخد بعرف n، ال n هذه بعرفها على أنها الأكبر بين m
+
+60
+00:06:51,590 --> 00:06:59,720
+و n of beta، وبالتالي ال n هذه أكبر من أو ساوي M و
+
+61
+00:06:59,720 --> 00:07:05,920
+أكبر من أو ساوي N of beta، وبالتالي لما أجي أخد N
+
+62
+00:07:05,920 --> 00:07:10,640
+أكبر من أو ساوي capital N، بأضمن أن ال N تبعتي هذه
+
+63
+00:07:10,640 --> 00:07:17,820
+أكبر من أو ساوي M، وبالتالي XN أصغر من أو ساوي YN
+
+64
+00:07:17,820 --> 00:07:22,940
+وكذلك
+
+65
+00:07:22,940 --> 00:07:28,120
+ال N لما تكون ال N أكبر من أو ساوي N، الـ N هذه
+
+66
+00:07:28,120 --> 00:07:33,020
+فبأضمن أنها أكبر من أو ساوي N of beta، وبالتالي
+
+67
+00:07:33,020 --> 00:07:40,000
+الكلام هذا بيتحقق، ويعني هيك بيكون برهن الجزء B بال
+
+68
+00:07:40,000 --> 00:07:44,960
+.. باستخدام الشرط الأضعف double star، بالمثل طبعاً
+
+69
+00:07:44,960 --> 00:07:48,960
+ممكن نعطيه في حالة لما يكون بدي أبرهن الجزء A
+
+70
+00:07:48,960 --> 00:07:53,540
+فبأخد ال N المطلوبة هي ال maximum ل M وN of alpha
+
+71
+00:07:53,540 --> 00:08:00,680
+في برهان A تحت شرط star، هذه أول ملاحظة، الملاحظة
+
+72
+00:08:00,680 --> 00:08:15,360
+الثانية، الملاحظة 
+
+73
+00:08:15,360 --> 00:08:25,480
+الثانية، if condition star holds، إذا كان الشرط star
+
+74
+00:08:25,480 --> 00:08:27,660
+holds، then
+
+75
+00:08:37,130 --> 00:08:42,070
+النتيجة أن y in tends to infinity does not 
+
+76
+00:08:42,070 --> 00:08:46,650
+necessarily implies، أن x in tends to infinity
+
+77
+00:08:53,430 --> 00:09:00,270
+وكان limit ال yn بساوي infinity، فليس من الضروري أن
+
+78
+00:09:00,270 --> 00:09:06,090
+يكون limit xn بساوي infinity، وهي مثال يوضح ذلك، for
+
+79
+00:09:06,090 --> 00:09:13,550
+example، على سبيل المثال، consider، consider
+
+80
+00:09:13,550 --> 00:09:20,230
+ال sequence 1 على n، أصغر من أو بساوي n، لكل n في n
+
+81
+00:09:21,990 --> 00:09:27,890
+إذن هي، أنا عندي xn وهي عندي yn وهي xn أصغر من
+
+82
+00:09:27,890 --> 00:09:34,090
+يساوي yn، الشرط الصغير متحقق، لكن أنا عندي ال limit
+
+83
+00:09:34,090 --> 00:09:39,330
+لـ sequence yn، اللي الحد العام تبعها n، هذي بساوي
+
+84
+00:09:39,330 --> 00:09:51,440
+infinity، but ال limit ل xn اللي هي واحد على n، بساوي
+
+85
+00:09:51,440 --> 00:09:59,860
+صفر، لا تساوي infinity، الصفر لا يساوي infinity، okay
+
+86
+00:09:59,860 --> 00:10:06,440
+تمام، إذا الشرط star ما بيخلنيش أستنتج أنه limit x
+
+87
+00:10:06,440 --> 00:10:09,200
+in بساوي infinity، عندما limit y in بساوي
+
+88
+00:10:09,200 --> 00:10:19,280
+infinity، بالمثل، if condition star holds، إذا كان
+
+89
+00:10:19,280 --> 00:10:24,660
+الشرط الـ start متحقق، then x
+
+90
+00:10:24,660 --> 00:10:30,840
+in تقول إلى negative infinity، ليس بالضرورة بيؤدي
+
+91
+00:10:30,840 --> 00:10:36,620
+مش شرط يؤدي أن ال sequence y in تقول لـ negative
+
+92
+00:10:36,620 --> 00:10:44,360
+infinity، هذا مش شرط يكون صحيح، بنأ مثال على ذلك
+
+93
+00:10:44,360 --> 00:10:48,020
+ممكن نفس المثال بس
+
+94
+00:10:51,510 --> 00:10:57,370
+for example، بس نضرب في سالب، هي عندي negative n
+
+95
+00:10:57,370 --> 00:11:04,900
+أصغر من أو ساوي negative واحد على n، لكل n في n، هل
+
+96
+00:11:04,900 --> 00:11:09,780
+هذا كلام صح؟ أنا عندي واحد على n أصغر من أو ساوي n
+
+97
+00:11:09,780 --> 00:11:17,040
+لكل n، هذا صح، اضرب في سالب واحد، تناقص هاه؟ هي عندك 
+
+98
+00:11:17,040 --> 00:11:24,990
+xn بساوي سالب n، وهي عندنا yn بساوي negative واحد
+
+99
+00:11:24,990 --> 00:11:31,590
+على n، الآن أنا عندي limit xn اللي هو سالب n لما
+
+100
+00:11:31,590 --> 00:11:37,170
+طبعاً n تقول infinity بساوي negative infinity، but
+
+101
+00:11:37,170 --> 00:11:44,910
+لكن limit ال yn اللي هو واحد على n، ايش بتساوي؟
+
+102
+00:11:44,910 --> 00:11:49,950
+بساوي صفر، سالب واحد عفواً، سالب واحد على n، limit سالب
+
+103
+00:11:49,950 --> 00:11:56,630
+واحد على n بساوي صفر، وليست سالب infinity، okay
+
+104
+00:11:56,630 --> 00:12:02,430
+تمام؟ إذا النظرية ال comparison test لا يقبل .. لا
+
+105
+00:12:02,430 --> 00:12:07,010
+يقبل التأويل زي ما بيقولوا، بس النتائج تبعتها كما
+
+106
+00:12:07,010 --> 00:12:13,310
+هي في a و b، أي شيء آخر مش مظبوط، هو أمثلة بتوضح
+
+107
+00:12:13,310 --> 00:12:20,970
+الأشياء الأخرى، تمام؟ في كمان اختبار آخر زي هذا
+
+108
+00:12:20,970 --> 00:12:25,550
+بنسميه limit comparison
+
+109
+00:12:25,550 --> 00:12:34,250
+test، فال
+
+110
+00:12:34,250 --> 00:12:35,350
+.. نمسح
+
+111
+00:12:56,530 --> 00:13:11,090
+limit comparison test، خلّيني
+
+112
+00:13:11,090 --> 00:13:19,550
+آخد two sequences x in و y in، بـ sequences of 
+
+113
+00:13:19,550 --> 00:13:21,030
+positive real numbers
+
+114
+00:13:24,560 --> 00:13:28,760
+بالتالي سيكون الحدود
+
+115
+00:13:28,760 --> 00:13:33,660
+الموجبة، لكي يكونوا سالبة وبعضهم سالبة وبعضهم موجبة
+
+116
+00:13:33,660 --> 00:13:36,820
+بي
+
+117
+00:13:36,820 --> 00:13:41,240
+such that limit
+
+118
+00:13:41,240 --> 00:13:49,480
+لـ xn over yn، as n tends to infinity بساوي L، عدد
+
+119
+00:13:49,480 --> 00:13:50,200
+موجبة
+
+120
+00:13:55,720 --> 00:14:02,320
+بنسمي المعادلة add star، then
+
+121
+00:14:02,320 --> 00:14:12,380
+limit xn بساوي infinity، if and only if limit yn
+
+122
+00:14:12,380 --> 00:14:22,160
+بساوي infinity، إذا
+
+123
+00:14:22,160 --> 00:14:26,190
+هنا في عندي limit comparison test الذي يتم استخدامه
+
+124
+00:14:26,190 --> 00:14:28,110
+للسيقونسات، واحدة فقط من حدوث واحدة واحدة فقط من 
+
+125
+00:14:28,110 --> 00:14:29,250
+حدوث واحدة فقط من حدوث واحدة فقط من حدوث واحدة فقط
+
+126
+00:14:29,250 --> 00:14:33,210
+من حدوث واحدة فقط من حدوث واحدة فقط من حدوث
+
+127
+00:14:33,210 --> 00:14:35,150
+فقط من حدوث واحدة فقط من حدوث واحدة فقط من حدوث
+
+128
+00:14:35,150 --> 00:14:35,310
+واحدة فقط من حدوث واحدة فقط من حدوث واحدة فقط من
+
+129
+00:14:35,310 --> 00:14:37,070
+حدوث واحدة فقط من حدوث واحدة فقط من حدوث واحدة فقط
+
+130
+00:14:37,070 --> 00:14:43,510
+من حدوث واحدة فقط من حدوث واحدة فقط من حدوث واحدة
+
+131
+00:14:43,510 --> 00:14:51,010
+فقط من حدوث واحدة فقط من
+
+132
+00:14:51,010 --> 00:14:52,090
+حدوث واحدة، ف
+
+133
+00:14:59,920 --> 00:15:10,860
+let assume، ال أكبر من الصفر، satisfies
+
+134
+00:15:10,860 --> 00:15:15,300
+المعادلة
+
+135
+00:15:15,300 --> 00:15:21,390
+لسه نفرض أن في عدد حقيقي L وهو بحقق star، يعني هو
+
+136
+00:15:21,390 --> 00:15:30,890
+limit لـ ratio لـ xn على yn، تمام؟
+
+137
+00:15:30,890 --> 00:15:34,650
+take epsilon
+
+138
+00:15:34,650 --> 00:15:38,930
+بساوي
+
+139
+00:15:38,930 --> 00:15:46,920
+L على 2، Since L is positive، L over 2 is positive
+
+140
+00:15:46,920 --> 00:15:57,360
+لأن أنا جبت إبسلون which is positive، طيب since من
+
+
+
+141
+00:15:57,360 --> 00:16:08,300
+الفرض since the sequence XN over YN converges to Lوهي
+
+142
+00:16:08,300 --> 00:16:10,980
+عندي إبسلون أكبر من الصفر is given. إذا by
+
+143
+00:16:10,980 --> 00:16:15,400
+definition of convergence by epsilon capital N
+
+144
+00:16:15,400 --> 00:16:22,400
+definition لإبسلون هذه for this إبسلون, there exists
+
+145
+00:16:22,400 --> 00:16:29,600
+capital N يعتمد على إبسلون يعتمد
+
+146
+00:16:29,600 --> 00:16:34,780
+على الإبسلون اللي هي بتعتمد على العدد L natural
+
+147
+00:16:34,780 --> 00:16:41,620
+number بحيث أنه لكل N أكبر منه سوى capital N بيطلع
+
+148
+00:16:41,620 --> 00:16:48,220
+عندي absolute xn over yn negative L less than
+
+149
+00:16:48,220 --> 00:16:57,240
+إبسلون. هتبوت هيك؟ طيب الـ Y بساوي L over 2 خلينا
+
+150
+00:16:57,240 --> 00:17:02,560
+نشيل ال absolute value فبصير عندي Xn over Yn minus
+
+151
+00:17:02,560 --> 00:17:10,960
+L less than L over two bigger than negative L over اتنين و
+
+152
+00:17:10,960 --> 00:17:16,900
+هذا صحيح لكل N bigger than or equal N. اجمع L على 
+
+153
+00:17:16,900 --> 00:17:24,960
+كل الأطراف. إن أنا بطلع عندي xn over yn less than
+
+154
+00:17:24,960 --> 00:17:31,920
+three over two L bigger than L over two, and this
+
+155
+00:17:31,920 --> 00:17:36,620
+is true for every n bigger than or equal n. نسمي
+
+156
+00:17:36,620 --> 00:17:46,680
+المتباينة هذه double star now.
+
+157
+00:17:52,640 --> 00:18:02,400
+by double star, أنا عندي xn على yn أصغر من تلاتة ع
+
+158
+00:18:02,400 --> 00:18:09,020
+اتنين ال اللي هو الجزء هذا وهذا صحيح لكل n bigger
+
+159
+00:18:09,020 --> 00:18:15,820
+than or equal to n. بيقدي انه xn
+
+160
+00:18:24,520 --> 00:18:34,680
+في اتنين على التلاتة L less than YN وهذا صحيح لكل
+
+161
+00:18:34,680 --> 00:18:45,260
+N bigger than or equal N. تصبوت هيك صح؟ هذه
+
+162
+00:18:45,260 --> 00:18:49,720
+المتباينة بتقدر هذه هي نفس هذه
+
+163
+00:18:53,610 --> 00:19:05,290
+so as limit. احنا فرضنا... now now
+
+164
+00:19:05,290 --> 00:19:10,690
+أو .. أو so if 
+
+165
+00:19:12,930 --> 00:19:19,250
+limit xn بساوي infinity إذا كان limit xn بساوي
+
+166
+00:19:19,250 --> 00:19:23,490
+infinity و هذا ثابت موجب. this is positive constant
+
+167
+00:19:23,490 --> 00:19:31,650
+ف limit كل هذا برضه بساوي plus infinity و
+
+168
+00:19:31,650 --> 00:19:39,450
+بالتالي by comparison test then by comparison by
+
+169
+00:19:39,450 --> 00:19:40,170
+comparison
+
+170
+00:19:42,940 --> 00:19:47,480
+by comparison test. النظرية اللى فاتت مع الشرط
+
+171
+00:19:47,480 --> 00:19:51,640
+المخفف مع الشرط المخفف لأن في النظرية اللى فاتت
+
+172
+00:19:51,640 --> 00:19:56,940
+كان عندي xn أصغر من أو يسوى yn لكل n بعدين قلنا أن
+
+173
+00:19:56,940 --> 00:20:01,200
+هذا الشرط لو خففناه لكل n أكبر من أو يسوى عدد طبيعي
+
+174
+00:20:01,200 --> 00:20:06,440
+ما وليكن capital N هنا برضه بتظل صحيحة. فby
+
+175
+00:20:06,440 --> 00:20:12,620
+comparison test and limit ال sequence هذه بساوي
+
+176
+00:20:12,620 --> 00:20:20,700
+infinity إذا limit الأكبر limit yn بساوي infinity.
+
+177
+00:20:20,700 --> 00:20:31,220
+تمام؟ الآن بنثبت العكس نثبت الآن العكس. طيب
+
+178
+00:20:31,220 --> 00:20:32,380
+conversely
+
+179
+00:20:40,740 --> 00:20:51,080
+Conversely. Assume Assume المرهد أنه limit yn بساوي
+
+180
+00:20:51,080 --> 00:20:59,700
+infinity. من double star من double star لو أخدت
+
+181
+00:20:59,700 --> 00:21:06,520
+النص .. النص هذا من المتباينة النص الآخر. عندي أنا
+
+182
+00:21:06,520 --> 00:21:13,120
+L على 2 أصغر من XN على YN هذا صحيح for every N
+
+183
+00:21:13,120 --> 00:21:22,120
+أكبر من أو ساوية capital N. طيب هذا بيقدي ان ال L
+
+184
+00:21:22,120 --> 00:21:29,400
+over 2 في YN أصغر من XN لكل N bigger than or equal
+
+185
+00:21:29,400 --> 00:21:33,380
+to capital N. طيب
+
+186
+00:21:33,380 --> 00:21:34,900
+since
+
+187
+00:21:37,140 --> 00:21:44,200
+limit yn بساوي infinity بيقدي انه limit ثابت موجب
+
+188
+00:21:44,200 --> 00:21:51,060
+في yn لأن هذا بيقدي انه limit ثابت الا اتنين في yn
+
+189
+00:21:51,060 --> 00:21:57,760
+بساوي infinity. so by comparison
+
+190
+00:21:57,760 --> 00:22:01,700
+by comparison test
+
+191
+00:22:07,170 --> 00:22:11,210
+أنا عندي limit ال sequence لصغيرة infinity، إذا
+
+192
+00:22:11,210 --> 00:22:15,550
+limit ال sequence الأكبر بطلع infinity، إذا limit
+
+193
+00:22:15,550 --> 00:22:28,070
+xn equals infinity. وهذا بكمل البرهان، okay؟
+
+194
+00:22:28,070 --> 00:22:32,390
+تمام؟ إذا هذا بكمل ال limit comparison .. برهان ال
+
+195
+00:22:32,390 --> 00:22:36,780
+limit comparison test. طبعا ال test هذا و ال test
+
+196
+00:22:36,780 --> 00:22:40,100
+اللي جابله ال comparison test في عليهم هتجدوا فيه
+
+197
+00:22:40,100 --> 00:22:44,160
+بعض التمرين ممكن
+
+198
+00:22:44,160 --> 00:22:47,660
+تطبيقهم على بعض ال sequences موجودة في التمرين
+
+199
+00:22:47,660 --> 00:22:53,560
+فهنسيبكم طبعا تحلوا التمرين عشان تشوفوا كيف ممكن
+
+200
+00:22:53,560 --> 00:22:54,280
+تطبيقهم
+
+201
+00:22:58,500 --> 00:23:05,220
+باقي section واحد في ال chapter تلاتة
+
+202
+00:23:32,720 --> 00:23:37,660
+السيكشن الأخير سيكشن
+
+203
+00:23:37,660 --> 00:23:43,820
+تلاتة سبعة في شبكر 3 هذا هيكون أبراعا مقدمة
+
+204
+00:23:43,820 --> 00:23:47,860
+introduction to
+
+205
+00:23:47,860 --> 00:23:52,920
+infinite series
+
+206
+00:23:57,560 --> 00:24:02,380
+introduction to infinite series. مقدمة في
+
+207
+00:24:02,380 --> 00:24:06,940
+المتسلسلات اللانهائية
+
+208
+00:24:06,940 --> 00:24:19,460
+نعرف شو معناه متسلسلة لانهائية. let xn
+
+209
+00:24:19,460 --> 00:24:27,330
+contained in R be a sequence. sequence of real
+
+210
+00:24:27,330 --> 00:24:43,210
+numbers. sum
+
+211
+00:24:43,210 --> 00:24:47,010
+x1
+
+212
+00:24:47,010 --> 00:24:56,200
+plus x2 plus ..x3 plus و هكذا plus xn plus و هكذا
+
+213
+00:24:56,200 --> 00:25:03,400
+و ممكن نكتبه بالصورة using sigma notation نستخدم
+
+214
+00:25:03,400 --> 00:25:09,920
+رمز sigma. ممكن هذا نسميه summation from n equals
+
+215
+00:25:09,920 --> 00:25:17,280
+one to infinity إلى xn. فالصورة
+
+216
+00:25:17,280 --> 00:25:25,190
+المجموع هذاهذا expanded. هذا compact form of
+
+217
+00:25:25,190 --> 00:25:39,010
+summation is called an infinite series generated
+
+218
+00:25:39,010 --> 00:25:42,970
+by
+
+219
+00:25:46,320 --> 00:25:54,100
+متولدة من .. by the sequence x in. إذن
+
+220
+00:25:54,100 --> 00:26:00,280
+infinite series generated by the sequence x in. إذا
+
+221
+00:26:00,280 --> 00:26:04,680
+هذه عبارة عن infinite series بتسميها متولدة من الـ
+
+222
+00:26:04,680 --> 00:26:09,340
+sequence x in. طيب for every
+
+223
+00:26:12,430 --> 00:26:23,290
+for each n belong to N define خلينا نعرف S1 على
+
+224
+00:26:23,290 --> 00:26:37,950
+أنه X1. S2 على أنه S1 زاد X2 بساوي X1 زاد X2. S3
+
+225
+00:26:37,950 --> 00:26:50,000
+بساوي S2 زاد X3. يساوي X1 زايد X2 زايد X3 and
+
+226
+00:26:50,000 --> 00:26:58,380
+so on و هكذا. نعرف SN على انه SN negative one زايد
+
+227
+00:26:58,380 --> 00:27:07,060
+XN وطبعا ال SN negative one هيكون عبارة عن
+
+228
+00:27:07,060 --> 00:27:07,700
+summation
+
+229
+00:27:10,600 --> 00:27:19,140
+x1 زائد x2 زائد و هكذا إلى أخر حد xn-1. هذا عبارة
+
+230
+00:27:19,140 --> 00:27:25,620
+عن ايه؟ هذا عبارة عن s in negative one بنضيف لها xn
+
+231
+00:27:25,620 --> 00:27:34,340
+فهذا بيطلع بيساوي summation من k equals one to n
+
+232
+00:27:38,080 --> 00:27:43,700
+to for xk. إذا
+
+233
+00:27:43,700 --> 00:27:49,840
+sn هو مجموع الحدود من اول حد الى حد رقم n وهكذا
+
+234
+00:27:49,840 --> 00:27:55,960
+ممكن نستمر الى ملا نهاية and so on. الآن أنا كوّنت
+
+235
+00:27:55,960 --> 00:28:00,360
+sequence لاحظوا s1, s2, s3, sn هذا عبارة عن
+
+236
+00:28:00,360 --> 00:28:06,970
+sequence. ال sequence الجديدة هذه لها اسمو sequence
+
+237
+00:28:06,970 --> 00:28:12,210
+مهمة of partial sums. مظبوط؟ قعدت نسميها اذا طرست
+
+238
+00:28:12,210 --> 00:28:18,790
+تفاضل الف وفهمته الموضوع هذا هناك الموضوع ال
+
+239
+00:28:18,790 --> 00:28:23,110
+series قعدت نسميها the sequence of partial sums
+
+240
+00:28:23,110 --> 00:28:29,210
+إذا the sequence
+
+241
+00:28:30,940 --> 00:28:37,980
+SN from N equals one to infinity is called بنسميها
+
+242
+00:28:37,980 --> 00:28:51,180
+the sequence the sequence of partial sums
+
+243
+00:28:51,180 --> 00:29:03,040
+sequence of partial sums of the series اللي هي
+
+244
+00:29:03,040 --> 00:29:11,080
+sigma xn أو sigma من n بساعة واحد لانفينيتي. okay
+
+245
+00:29:11,080 --> 00:29:18,660
+الآن now if
+
+246
+00:29:18,660 --> 00:29:29,280
+the sequence sn converges, say
+
+247
+00:29:31,110 --> 00:29:42,090
+limit sn بالساوي عدد s ينتمي إلى r طبعا then
+
+248
+00:29:42,090 --> 00:29:51,390
+we say في الحالة هذه بنقول أنه the series اللي
+
+249
+00:29:51,390 --> 00:29:58,630
+هي summation xn from n equals one to infinity
+
+250
+00:29:58,630 --> 00:30:00,270
+converges
+
+251
+00:30:09,070 --> 00:30:18,290
+and its sum is summation from n equals one to
+
+252
+00:30:18,290 --> 00:30:23,530
+infinity ل x in. ال summation تبعها أو المجموعة
+
+253
+00:30:23,530 --> 00:30:28,450
+تبعها عبارة عن limit لل sequence of partial sums
+
+254
+00:30:28,450 --> 00:30:32,730
+اللي هو العدد S.
+
+255
+00:30:37,160 --> 00:30:43,180
+لو كانت ال sequence divergent
+
+256
+00:30:43,180 --> 00:30:50,740
+if the sequence is in diverges, we
+
+257
+00:30:50,740 --> 00:31:00,120
+say أنه ال series sigma
+
+258
+00:31:00,120 --> 00:31:05,880
+x in diverges
+
+259
+00:31:09,090 --> 00:31:13,410
+إذا ال convergence و ال divergence depends on the
+
+260
+00:31:13,410 --> 00:31:18,630
+divergence أو convergence of the infinite series
+
+261
+00:31:18,630 --> 00:31:23,910
+depends on the convergence or divergence of the
+
+262
+00:31:23,910 --> 00:31:30,690
+sequence of partial sums. مرتبط بيها ال sequence of
+
+263
+00:31:30,690 --> 00:31:34,350
+partial sums. convergent السيريز اللي تابع إليها
+
+264
+00:31:34,350 --> 00:31:38,360
+convergent. والعكس إذا كانت ال sequence of partial
+
+265
+00:31:38,360 --> 00:31:40,780
+sums divergent, ال series ال infinite series
+
+266
+00:31:40,780 --> 00:31:51,840
+التابعة إلى divergent. طيب
+
+267
+00:31:51,840 --> 00:31:58,180
+ناخد بعض الأمثلة طبعا
+
+268
+00:31:58,180 --> 00:32:01,720
+ال Sn هذا ال Sn
+
+269
+00:32:04,610 --> 00:32:15,810
+هذا بنسميه الانث partial sum. الانث partial sum
+
+270
+00:32:15,810 --> 00:32:25,690
+انث partial sum المجموع الجزئي أنوني. okay هو
+
+271
+00:32:25,690 --> 00:32:30,760
+الحد العام لل sequence و partial sums. إذا لما بدي
+
+272
+00:32:30,760 --> 00:32:34,760
+نخبر هل ال series convergent ولا divergent بجيب ال
+
+273
+00:32:34,760 --> 00:32:38,380
+sequence of partial sums وبجيب الحد العام لل
+
+274
+00:32:38,380 --> 00:32:41,780
+sequence of partial sums وبأفحص هل ال sequence هذي
+
+275
+00:32:41,780 --> 00:32:47,680
+convergent ولا divergent. ناخد
+
+276
+00:32:47,680 --> 00:32:48,620
+بعض الأمثلة
+
+277
+00:33:02,760 --> 00:33:14,560
+المثال الأول consider
+
+278
+00:33:14,560 --> 00:33:17,780
+sequence
+
+279
+00:33:17,780 --> 00:33:25,480
+R to N from N equals 0 to infinity. طبعا هذه
+
+280
+00:33:25,480 --> 00:33:33,650
+sequence of real numbers. Where R is a real number
+
+281
+00:33:33,650 --> 00:33:38,210
+which
+
+282
+00:33:38,210 --> 00:33:49,150
+generates هذه الsequence generates the geometric
+
+283
+00:33:49,150 --> 00:33:53,010
+.. the so-called geometric series .. geometric
+
+284
+00:33:53,010 --> 00:33:54,330
+series
+
+285
+00:33:57,050 --> 00:34:02,610
+اللي هي summation from n equals zero to infinity
+
+286
+00:34:02,610 --> 00:34:10,210
+from r to n okay إذا هي هذه الsequence of real
+
+287
+00:34:10,210 --> 00:34:15,650
+numbers بتولد infinite series أو generates this
+
+288
+00:34:15,650 --> 00:34:21,210
+infinite series اللي هي حدودها أول حد لما n بساوي
+
+289
+00:34:21,210 --> 00:34:33,620
+صفر واحد بعدين r بعدين r تربيعو R أس N و
+
+290
+00:34:33,620 --> 00:34:41,120
+هكذا ف such series is called geometric series هذه
+
+291
+00:34:41,120 --> 00:34:44,300
+الseries اللي على الصورة هذه بنسميها geometric
+
+292
+00:34:44,300 --> 00:34:49,820
+series الآن هذه الseries
+
+293
+00:34:58,170 --> 00:35:08,530
+this series واحد converges and
+
+294
+00:35:08,530 --> 00:35:15,910
+its sum اللي هو sigma from n equals zero to
+
+295
+00:35:15,910 --> 00:35:22,470
+infinity لRn بساوي واحد على واحد minus R إذا كان
+
+296
+00:35:22,470 --> 00:35:35,210
+absolute R أصغر من واحد and diverges and
+
+297
+00:35:35,210 --> 00:35:41,350
+اثنين diverges if
+
+298
+00:35:41,350 --> 00:35:48,830
+absolute R أكبر من أو يساوي واحد خلّينا
+
+299
+00:35:48,830 --> 00:35:50,050
+نثبت الجزء الأول
+
+300
+00:35:58,010 --> 00:36:04,110
+to prove one أنا
+
+301
+00:36:04,110 --> 00:36:12,410
+عندي ال SN بساوي سيجما
+
+302
+00:36:12,410 --> 00:36:21,050
+من K بساوي صفر إلى N ل R أس K اللي هو واحد زائد R
+
+303
+00:36:21,050 --> 00:36:34,640
+زائد R تربيع زائد R أس N وفي عندي .. في عندي ..
+
+304
+00:36:34,640 --> 00:36:37,780
+لو
+
+305
+00:36:37,780 --> 00:36:48,440
+ضربت SN في R فبضرب الطرف اليمين في R فبطلع R زائد R
+
+306
+00:36:48,440 --> 00:36:56,380
+تربيع زائد و هكذا زائد R أس N و آخر حد هيكون R أس N
+
+307
+00:36:56,380 --> 00:37:00,940
+زائد 1 تمام؟ الآن خلّينا نطرح ال subtract
+
+308
+00:37:05,370 --> 00:37:10,850
+subtract نطرح المعادلة اللي تحت من اللي فوق فبطلع عندي
+
+309
+00:37:10,850 --> 00:37:18,590
+SN في واحد minus R أخدت عامل مشترك SN ولمّا أطرح
+
+310
+00:37:18,590 --> 00:37:23,330
+هذا بروح مع هذا كل الحدود بتروح مع بعضها بظل عندي
+
+311
+00:37:23,330 --> 00:37:33,130
+واحد سالب R أس N زائد 1 تمام؟ ومن هنا إذا SN
+
+312
+00:37:36,050 --> 00:37:44,310
+بساوي واحد على واحد سالب R سالب R أس N زائد 1
+
+313
+00:37:44,310 --> 00:37:52,990
+على واحد سالب R ممكن
+
+314
+00:37:52,990 --> 00:37:59,070
+هذا نوديه على ناحية الثانية فبصير عندي هذا سالب
+
+315
+00:37:59,070 --> 00:38:01,350
+هذا بساوي
+
+316
+00:38:03,070 --> 00:38:08,930
+سالب R أس N زائد 1 على واحد سالب R الآن إذا
+
+317
+00:38:08,930 --> 00:38:13,590
+ناخد ال absolute value للطرفين SN سالب واحد على
+
+318
+00:38:13,590 --> 00:38:21,030
+واحد سالب R بيطلع بيساوي الكلام هذا وهذا أصغر من
+
+319
+00:38:21,030 --> 00:38:27,830
+أو يساوي absolute R أس N زائد 1 على absolute واحد
+
+320
+00:38:27,830 --> 00:38:30,870
+minus R تمام؟
+
+321
+00:38:34,940 --> 00:38:41,200
+إذا عندي أنا هاي واحد على absolute واحد سالب R ضرب
+
+322
+00:38:41,200 --> 00:38:51,360
+absolute R أس N زائد 1 الآن if absolute R أصغر
+
+323
+00:38:51,360 --> 00:39:00,870
+من واحد فهذا بيؤدي أن ال limit ل absolute R أس N زي
+
+324
+00:39:00,870 --> 00:39:05,790
+1 لما N تؤول ل infinity هذا بيساوي صفر أخذناها قبل
+
+325
+00:39:05,790 --> 00:39:10,430
+هيك وبالتالي
+
+326
+00:39:10,430 --> 00:39:14,950
+إذا ال ..
+
+327
+00:39:14,950 --> 00:39:18,290
+إذا أنا عندي ال absolute value هذه أكبر من أو يساوي
+
+328
+00:39:18,290 --> 00:39:24,270
+صفر و أصغر من أو يساوي ثابت موجب في هذه الsequence
+
+329
+00:39:24,270 --> 00:39:28,610
+هذه الsequence تؤول لـ 0 وهذه الـ sequence ثابتة
+
+330
+00:39:28,610 --> 00:39:32,330
+تؤول لـ 0 إذا by sandwich theorem
+
+331
+00:39:40,720 --> 00:39:47,760
+بتطلع عندي ال limit ل absolute SN سالب 1 على 1
+
+332
+00:39:47,760 --> 00:39:52,820
+minus R لما N تؤول ل infinity بساوي صفر وممكن
+
+333
+00:39:52,820 --> 00:39:58,600
+ندخل ال limit جوا فهذا بقدر أنه limit 1 على SN
+
+334
+00:39:58,600 --> 00:40:05,040
+عفوا limit SN لما N تؤول ل infinity بساوي 1 على 1
+
+335
+00:40:05,040 --> 00:40:11,560
+سالب R وبالتالي إذا الـ series sigma from N equal 0
+
+336
+00:40:11,560 --> 00:40:17,240
+to infinity لR أس N مجموعتها تطلع convergent
+
+337
+00:40:17,240 --> 00:40:23,180
+ومجموعها بساوي limit SN وهذا بساوي 1 على 1 minus R
+
+338
+00:40:24,110 --> 00:40:28,950
+إذن هذا بيثبت الجزء الأول الجزء الثاني ممكن إثباته
+
+339
+00:40:28,950 --> 00:40:34,190
+لو R بساوي واحد فبطلع عندي بجمع واحد على واحد عدد
+
+340
+00:40:34,190 --> 00:40:37,730
+لا نهائي من المرات وبالتالي ال sequence of partial
+
+341
+00:40:37,730 --> 00:40:40,170
+sums ممكن إثبات أنها unbounded وبالتالي not
+
+342
+00:40:40,170 --> 00:40:44,330
+convergent إذن الseries not convergent نفس الحاجة
+
+343
+00:40:44,330 --> 00:40:47,210
+لو كان absolute R أكبر من واحد فممكن إثبات أن ال
+
+344
+00:40:47,210 --> 00:40:50,550
+sequence of partial sums is divergent وبالتالي ال
+
+345
+00:40:50,550 --> 00:40:57,170
+series is divergent تمام؟ في أي سؤال؟ إذا بنوقف هنا
+
+346
+00:40:57,170 --> 00:41:02,830
+و بنكمل إن شاء الله الموضوع اللي جاي في المحاضرة
+
+347
+00:41:02,830 --> 00:41:04,630
+القادمة يوم السبت
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/XhLWrN2SkOQ_postprocess.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/XhLWrN2SkOQ_postprocess.srt
new file mode 100644
index 0000000000000000000000000000000000000000..0a1097ab4d6beab4aa79e4e70641be4882ca0e2b
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/XhLWrN2SkOQ_postprocess.srt
@@ -0,0 +1,1388 @@
+1
+00:00:20,890 --> 00:00:26,630
+أحنا هنكمل الموضوع لـ Properly Divergent Sequences
+
+2
+00:00:26,630 --> 00:00:31,690
+اللي بدأناه المحاضرة السابقة فشوفنا في المحاضرة
+
+3
+00:00:31,690 --> 00:00:36,530
+السابقة تعريف ما معنى أنه limit ل sequence xn
+
+4
+00:00:36,530 --> 00:00:41,450
+بساوي infinity وما معنى أنه limit ل sequence xn
+
+5
+00:00:41,450 --> 00:00:46,490
+بساوي negative infinityطبعاً الـ sequence بتكون
+
+6
+00:00:46,490 --> 00:00:49,470
+properly divergent إذا كانت ال limit تبعتها
+
+7
+00:00:49,470 --> 00:00:54,030
+بالساوي infinity أو سالب infinity في عندي
+
+8
+00:00:54,030 --> 00:00:58,550
+comparison test ل .. ل properly divergent
+
+9
+00:00:58,550 --> 00:01:01,790
+sequences هذا ال test بيقول لي لو في عندي two
+
+10
+00:01:01,790 --> 00:01:06,330
+sequences xn و yn two sequences of real numbers
+
+11
+00:01:06,330 --> 00:01:10,370
+بيحققوا الشرط star satisfy the condition star وهو
+
+12
+00:01:10,370 --> 00:01:15,400
+أن كل حدفي xn أصغر من أو ساوي الحد اللي بنظره في
+
+13
+00:01:15,400 --> 00:01:22,080
+ال sequence التانية yn هذا صحيح لكل n فإذا كانت ال
+
+14
+00:01:22,080 --> 00:01:26,140
+limit of the bigger sequence of the smaller
+
+15
+00:01:26,140 --> 00:01:30,240
+sequence is infinity then the limit of the bigger
+
+16
+00:01:30,240 --> 00:01:36,040
+sequence is infinity and if the limit of the big
+
+17
+00:01:36,040 --> 00:01:39,580
+the bigger sequence is negative infinity then the
+
+18
+00:01:39,580 --> 00:01:40,720
+limit of the smaller
+
+19
+00:01:50,070 --> 00:01:55,680
+الجزء بي هو نقاط نقاط نقاط نقاط نقاط نقاطan
+
+20
+00:01:55,680 --> 00:01:58,620
+application of the definition طبقنا التعريف
+
+21
+00:01:58,620 --> 00:02:03,740
+بالبرهان زي ما انتوا شايفينه برهان الجزء A similar
+
+22
+00:02:03,740 --> 00:02:09,840
+مشابه لجزء B فحنسيبوا تمرين لكم اتحاولوا يعني
+
+23
+00:02:09,840 --> 00:02:15,020
+اتبرهنوا بنفس الطريقة okay تمام فلو سمحتوا حاولوا
+
+24
+00:02:15,020 --> 00:02:21,720
+انكم اتبرهنوا الجزء A بنفس الطريقة في عندنا شوية
+
+25
+00:02:21,720 --> 00:02:24,300
+ملاحظات على النظرية
+
+26
+00:02:31,050 --> 00:02:36,390
+نعطيلها رقم تسعة و عشرين فالملاحظات
+
+27
+00:02:36,390 --> 00:02:44,850
+في عندي تلت ملاحظات الملاحظة الأولى انه theorem
+
+28
+00:02:44,850 --> 00:02:47,910
+النظرية
+
+29
+00:02:47,910 --> 00:02:53,270
+السابقة اعتقد ان هذا الرقم المفروض يكون تلاتين
+
+30
+00:02:59,480 --> 00:03:15,440
+theorem 29 remains true تبقى صحيحة if condition if
+
+31
+00:03:15,440 --> 00:03:22,720
+condition star is replaced إذا بدلنا الشرط star by
+
+32
+00:03:22,720 --> 00:03:34,060
+the weaker conditionby the weaker condition
+
+33
+00:03:34,060 --> 00:03:39,440
+اللي
+
+34
+00:03:39,440 --> 00:03:46,080
+هو xn less than or equal yn لكل
+
+35
+00:03:46,080 --> 00:03:55,540
+n أكبر من أو ساوي m for some n natural number
+
+36
+00:03:58,530 --> 00:04:05,110
+يعني بدل ما Xn أصغر من أو ساوي Yn لكل الان لكل
+
+37
+00:04:05,110 --> 00:04:09,910
+الأعداد الطبيعية N فلأ نفرض أن يوجد M عدد طبيعي
+
+38
+00:04:09,910 --> 00:04:15,190
+نفرض أن يوجد some M عدد طبيعي بحيث أن المتباين هذه
+
+39
+00:04:15,190 --> 00:04:21,590
+تتحقق لكل N أكبر من أو ساوي M يعني مش شرط تتحقق
+
+40
+00:04:21,590 --> 00:04:26,270
+للأعداد الطبيعية اللي أصغر من M فالنظرية برضه تبقى
+
+41
+00:04:26,270 --> 00:04:32,390
+صحيحةولو بدنا نبرهن النظرية اللى فاتت تحت الشرط
+
+42
+00:04:32,390 --> 00:04:37,150
+الأضعف هذا الشرط أضعف من الشرط ال star لكن برضه
+
+43
+00:04:37,150 --> 00:04:44,970
+بيعطينى نفس النظرية فال ..
+
+44
+00:04:44,970 --> 00:04:55,390
+ففي الحالة هذه in fact في حقيقة القمر in fact in
+
+45
+00:04:55,390 --> 00:05:07,220
+the proofsin the proofs of النظرية السابقة take
+
+46
+00:05:07,220 --> 00:05:19,460
+the required the required in to be that
+
+47
+00:05:19,460 --> 00:05:26,660
+corresponds that corresponds
+
+48
+00:05:28,960 --> 00:05:34,420
+that corresponds to the given to
+
+49
+00:05:34,420 --> 00:05:38,880
+the given alpha or
+
+50
+00:05:38,880 --> 00:05:45,360
+given beta to
+
+51
+00:05:45,360 --> 00:05:59,160
+be in ابارة عن ال maximum the m و n of alphaأو n
+
+52
+00:05:59,160 --> 00:06:07,920
+بالساوي ال maximum الأكبر بين العدد الطبيعي m و n
+
+53
+00:06:07,920 --> 00:06:17,920
+of beta إذن
+
+54
+00:06:17,920 --> 00:06:22,480
+في البرهان مثلا هذه الجزء التالي برهاننا عادي هي
+
+55
+00:06:22,480 --> 00:06:29,810
+نقلت الكلام هذا صحيح و يوجد capital Nهيعتمد على
+
+56
+00:06:29,810 --> 00:06:34,910
+beta بحيث ان الكلام هذا يتحقق الان ال star ماقدرش
+
+57
+00:06:34,910 --> 00:06:38,850
+ا say bye star هذه هتكون double star بدل ال star
+
+58
+00:06:38,850 --> 00:06:45,070
+فانا ساميها double star فالان
+
+59
+00:06:45,070 --> 00:06:51,590
+باخد بعرف n ال n هذه بعرفها على انها الأكبر بين m
+
+60
+00:06:51,590 --> 00:06:59,720
+و n of beta وبالتالي ال n هذهأكبر من أو ساوي M و
+
+61
+00:06:59,720 --> 00:07:05,920
+أكبر من أو ساوي N of beta وبالتالي لما أجي أخد N
+
+62
+00:07:05,920 --> 00:07:10,640
+أكبر من أو ساوي capital N بأضمن أن ال N تبعتي هذه
+
+63
+00:07:10,640 --> 00:07:17,820
+أكبر من أو ساوي M وبالتالي XN أصغر من أو ساوي YN
+
+64
+00:07:17,820 --> 00:07:22,940
+وكذلك
+
+65
+00:07:22,940 --> 00:07:28,120
+ال N لما تكون ال N أكبر من أو ساوي Nالـ N هذه
+
+66
+00:07:28,120 --> 00:07:33,020
+فبطمن أنها أكبر من أو ساوي N of beta وبالتالي
+
+67
+00:07:33,020 --> 00:07:40,000
+الكلام هذا بتحقق ويعني هيك بيكون برهن الجزء B بال
+
+68
+00:07:40,000 --> 00:07:44,960
+.. باستخدام الشرط الأضعف double star بالمثل طبعا
+
+69
+00:07:44,960 --> 00:07:48,960
+ممكن نعطيه في حالة لما يكون بدي أبرهن الجزء A
+
+70
+00:07:48,960 --> 00:07:53,540
+فباخد ال N المطلوبة هي ال maximum ل M وN of alpha
+
+71
+00:07:53,540 --> 00:08:00,680
+في برهان A تحت شرط starهذه أول ملاحظة الملاحظة
+
+72
+00:08:00,680 --> 00:08:15,360
+التانية الملاحظة
+
+73
+00:08:15,360 --> 00:08:25,480
+التانية if condition star holds إذا كان الشرط star
+
+74
+00:08:25,480 --> 00:08:27,660
+holds then
+
+75
+00:08:37,130 --> 00:08:42,070
+النتيجة ان y in tends to infinity does not
+
+76
+00:08:42,070 --> 00:08:46,650
+necessarily implies ان x in tends to infinity
+
+77
+00:08:53,430 --> 00:09:00,270
+وكان limit ال yn بساوي infinity فليس من الضروري ان
+
+78
+00:09:00,270 --> 00:09:06,090
+يكون limit xn بساوي infinity وهي مثال يوضح ذلك for
+
+79
+00:09:06,090 --> 00:09:13,550
+example على سبيل المثال consider consider
+
+80
+00:09:13,550 --> 00:09:20,230
+ال sequence 1 على n أصغر من أو بساوي n لكل n في n
+
+81
+00:09:21,990 --> 00:09:27,890
+إذن هى انا عندي xn وهى عندي yn وهى xn أصغر من
+
+82
+00:09:27,890 --> 00:09:34,090
+يساوي yn الشرط الصغير متحقق لكن أنا عندي ال limit
+
+83
+00:09:34,090 --> 00:09:39,330
+ل sequence yn اللى الحد العام تبعها n هدى بساوي
+
+84
+00:09:39,330 --> 00:09:51,440
+infinity but ال limit ل xn اللى هى واحد على nبساوي
+
+85
+00:09:51,440 --> 00:09:59,860
+سفر لا تساوي infinity السفر لا يساوي infinity okay
+
+86
+00:09:59,860 --> 00:10:06,440
+تمام إذا الشرط star ما بيخلنيش أستنتج أنه limit x
+
+87
+00:10:06,440 --> 00:10:09,200
+in بالساوية infinity عندما limit y in بالساوية
+
+88
+00:10:09,200 --> 00:10:19,280
+infinity بالمثل if condition star holdsإذا كان
+
+89
+00:10:19,280 --> 00:10:24,660
+الشرط الـ start متحقق then x
+
+90
+00:10:24,660 --> 00:10:30,840
+in تقول إلى negative infinity ليس بالضرورة بيؤدي
+
+91
+00:10:30,840 --> 00:10:36,620
+مش شرط يؤدي ان ال sequence y in تقول ل negative
+
+92
+00:10:36,620 --> 00:10:44,360
+infinity هذا مش شرط يكون صحيح بنا مثال على ذلك
+
+93
+00:10:44,360 --> 00:10:48,020
+ممكن نفس المثال بس
+
+94
+00:10:51,510 --> 00:10:57,370
+for example بس نضرب في سالب هي عندي negative n
+
+95
+00:10:57,370 --> 00:11:04,900
+أصغر من أو ساوي negative واحد على n لكل n في nهل
+
+96
+00:11:04,900 --> 00:11:09,780
+هذا كلام صح؟ انا عندي واحد على n أصغر من أو ساوي n
+
+97
+00:11:09,780 --> 00:11:17,040
+لكل n هذا صح اضرب في سالب واحد تناقص هاه؟ هاي عندك
+
+98
+00:11:17,040 --> 00:11:24,990
+xn بساوي سالب n وهي عندنا ynبساوي negative واحد
+
+99
+00:11:24,990 --> 00:11:31,590
+على n الان انا عندي limit xn اللي هو سالب n لما
+
+100
+00:11:31,590 --> 00:11:37,170
+طبعا n تقول infinity بساوي negative infinity but
+
+101
+00:11:37,170 --> 00:11:44,910
+لكن limit ال yn اللي هو واحد على n ايش بتساوي؟
+
+102
+00:11:44,910 --> 00:11:49,950
+ساوي سفر سالب واحد عفوا سالب واحد على n limit سالب
+
+103
+00:11:49,950 --> 00:11:56,630
+واحد على n بساوي سفرو ليست سالب infinity okay
+
+104
+00:11:56,630 --> 00:12:02,430
+تمام؟ إذا النظرية ال comparison test لا يقبل .. لا
+
+105
+00:12:02,430 --> 00:12:07,010
+يقبل التأويل زي ما بيقولوا بس النتائج تبعتها كما
+
+106
+00:12:07,010 --> 00:12:13,310
+هي في a و b أي شيء آخر مش مظبوطهو أمثلة بتوضح
+
+107
+00:12:13,310 --> 00:12:20,970
+الأشياء الأخرى تمام؟ في كمان اختبار أخر زي هذا
+
+108
+00:12:20,970 --> 00:12:25,550
+بنسميه limit comparison
+
+109
+00:12:25,550 --> 00:12:34,250
+test فال
+
+110
+00:12:34,250 --> 00:12:35,350
+.. نمسح
+
+111
+00:12:56,530 --> 00:13:11,090
+limit comparison test خلّيني
+
+112
+00:13:11,090 --> 00:13:19,550
+أاخد two sequences x in و y in بsequences of
+
+113
+00:13:19,550 --> 00:13:21,030
+positive real numbers
+
+114
+00:13:24,560 --> 00:13:28,760
+بالتالي سيكون الحدود
+
+115
+00:13:28,760 --> 00:13:33,660
+الموجبة لكي يكونوا سالبة وبعضهم سالبة وبعضهم موجبة
+
+116
+00:13:33,660 --> 00:13:36,820
+بي
+
+117
+00:13:36,820 --> 00:13:41,240
+such that limit
+
+118
+00:13:41,240 --> 00:13:49,480
+ل xn over yn as n tends to infinity بساوي L عدد
+
+119
+00:13:49,480 --> 00:13:50,200
+موجبة
+
+120
+00:13:55,720 --> 00:14:02,320
+بنسمي المعادلة add star then
+
+121
+00:14:02,320 --> 00:14:12,380
+limit xn بساوي infinity if and only if limit yn
+
+122
+00:14:12,380 --> 00:14:22,160
+بساوي infinity إذا
+
+123
+00:14:22,160 --> 00:14:26,190
+هنا في عندي limit comparison testالذي يتم استخدامه
+
+124
+00:14:26,190 --> 00:14:28,110
+لسيقونسات واحدة فقط من حدوث واحدة واحدة فقط من
+
+125
+00:14:28,110 --> 00:14:29,250
+حدوث واحدة فقط من حدوث واحدة فقط من حدوث واحدة فقط
+
+126
+00:14:29,250 --> 00:14:33,210
+من حدوث واحدة فقط من حدوث واحدة فقط من حدوث واحدة
+
+127
+00:14:33,210 --> 00:14:35,150
+فقط من حدوث واحدة فقط من حدوث واحدة فقط من حدوث
+
+128
+00:14:35,150 --> 00:14:35,310
+واحدة فقط من حدوث واحدة فقط من حدوث واحدة فقط من
+
+129
+00:14:35,310 --> 00:14:37,070
+حدوث واحدة فقط من حدوث واحدة فقط من حدوث واحدة فقط
+
+130
+00:14:37,070 --> 00:14:43,510
+من حدوث واحدة فقط من حدوث واحدة فقط من حدوث واحدة
+
+131
+00:14:43,510 --> 00:14:51,010
+فقط من حدوث واحدة فقط من
+
+132
+00:14:51,010 --> 00:14:52,090
+حدوث واحدة ف
+
+133
+00:14:59,920 --> 00:15:10,860
+let assume ال أكبر من السفر satisfies
+
+134
+00:15:10,860 --> 00:15:15,300
+المعادلة
+
+135
+00:15:15,300 --> 00:15:21,390
+لسه نفرض ان في عدد حقيقي الوهو بحقق star يعني هو
+
+136
+00:15:21,390 --> 00:15:30,890
+limit ل ratio ل xn على yn تمام؟
+
+137
+00:15:30,890 --> 00:15:34,650
+take epsilon
+
+138
+00:15:34,650 --> 00:15:38,930
+بساوي
+
+139
+00:15:38,930 --> 00:15:46,920
+L على 2Since L is positive L over 2 is positive
+
+140
+00:15:46,920 --> 00:15:57,360
+لأن أنا جبت إبسلون which is positive طيب since من
+
+141
+00:15:57,360 --> 00:16:08,300
+الفرض since by star XN over YN converges to Lوهي
+
+142
+00:16:08,300 --> 00:16:10,980
+عندي إبسلون أكبر من الصفر is given إذا by
+
+143
+00:16:10,980 --> 00:16:15,400
+definition of convergence by epsilon capital N
+
+144
+00:16:15,400 --> 00:16:22,400
+definition لإبسلون هذه for this إبسلونthere exist
+
+145
+00:16:22,400 --> 00:16:29,600
+capital N يعتمد على إبسلون يعتمد
+
+146
+00:16:29,600 --> 00:16:34,780
+على الإبسلون اللي هي بتعتمد على العدد L natural
+
+147
+00:16:34,780 --> 00:16:41,620
+number بحيث أنه لكل N أكبر منه سوى capital N بيطلع
+
+148
+00:16:41,620 --> 00:16:48,220
+عندي absolute xn over yn negative L less than
+
+149
+00:16:48,220 --> 00:16:57,240
+إبسلونهزبوت هيك؟ طيب الـ Y بسوي L over 2 خلّينا
+
+150
+00:16:57,240 --> 00:17:02,560
+نشيل ال absolute value فبصير اندي Xn over Yn minus
+
+151
+00:17:02,560 --> 00:17:10,960
+L less than L two bigger than negative L ع اتنين و
+
+152
+00:17:10,960 --> 00:17:16,900
+هذا صحيح لكل N bigger than or equal N اجمع L على
+
+153
+00:17:16,900 --> 00:17:24,960
+كل الأطرافإن أنا بطلع عندي xn over yn less than
+
+154
+00:17:24,960 --> 00:17:31,920
+three over two L bigger than L over two and this
+
+155
+00:17:31,920 --> 00:17:36,620
+is true for every n bigger than or equal n نسمي
+
+156
+00:17:36,620 --> 00:17:46,680
+المتباينة هذه double star now
+
+157
+00:17:52,640 --> 00:18:02,400
+by double star انا عندي xn على yn اصغر من تلاتة ع
+
+158
+00:18:02,400 --> 00:18:09,020
+اتنين ال اللي هو الجزء هذا وهذا صحيح لكل n bigger
+
+159
+00:18:09,020 --> 00:18:15,820
+than or equal to n بيقدي انه xn
+
+160
+00:18:24,520 --> 00:18:34,680
+في اتنين على التلاتة L less than Y N وهذا صحيح لكل
+
+161
+00:18:34,680 --> 00:18:45,260
+N bigger than or equal N تصبوت هيك صح؟ هذه
+
+162
+00:18:45,260 --> 00:18:49,720
+المتباينة بتقدر هذه هي نفس هذه
+
+163
+00:18:53,610 --> 00:19:05,290
+so as limit احنا فرض .. now now
+
+164
+00:19:05,290 --> 00:19:10,690
+او .. او so if
+
+165
+00:19:12,930 --> 00:19:19,250
+limit xn بساوي infinity إذا كان limit xn بساوي
+
+166
+00:19:19,250 --> 00:19:23,490
+infinity و هذا ثابت موجب this is positive constant
+
+167
+00:19:23,490 --> 00:19:31,650
+ف limit كل هذا برضه بساوي plus infinity و
+
+168
+00:19:31,650 --> 00:19:39,450
+بالتالي by comparison test then by comparison by
+
+169
+00:19:39,450 --> 00:19:40,170
+comparison
+
+170
+00:19:42,940 --> 00:19:47,480
+by comparison test النظرية اللى فاتت مع الشرط
+
+171
+00:19:47,480 --> 00:19:51,640
+المخفف مع الشرط المخفف لأن فى النظرية اللى فاتت
+
+172
+00:19:51,640 --> 00:19:56,940
+كان عندي xn أصغر من أو يسوى yn لكل n بعدين قلنا أن
+
+173
+00:19:56,940 --> 00:20:01,200
+هذا الشرط لو خففناه لكل n أكبر من أو سوى عدد طبيعى
+
+174
+00:20:01,200 --> 00:20:06,440
+ما وليكن capital N هنا برضه بتظل صحيحةف by
+
+175
+00:20:06,440 --> 00:20:12,620
+comparison test and limit ال sequence هذه بساوي
+
+176
+00:20:12,620 --> 00:20:20,700
+infinity إذا limit الأكبر limit yn بساوي infinity
+
+177
+00:20:20,700 --> 00:20:31,220
+تمام؟ الآن بنثبت العكس نثبت الآن العكس طيب
+
+178
+00:20:31,220 --> 00:20:32,380
+conversely
+
+179
+00:20:40,740 --> 00:20:51,080
+Conversely Assume Assume المرهد أنه limit yn بساوي
+
+180
+00:20:51,080 --> 00:20:59,700
+infinity من double star من double star لو أخدت
+
+181
+00:20:59,700 --> 00:21:06,520
+النص .. النص هذا من المتباينة النص الآخرفعندي انا
+
+182
+00:21:06,520 --> 00:21:13,120
+L على 2 أصغر من XN على YN هذا صحيح for every N
+
+183
+00:21:13,120 --> 00:21:22,120
+أكبر من أو ساوية capital N طيب هذا بيقدي ان ال L
+
+184
+00:21:22,120 --> 00:21:29,400
+over 2 في YN أصغر من XN لكل N bigger than or equal
+
+185
+00:21:29,400 --> 00:21:33,380
+to capital N طيب
+
+186
+00:21:33,380 --> 00:21:34,900
+since
+
+187
+00:21:37,140 --> 00:21:44,200
+limit yn بساوي infinity بيقدي انه limit ثابت موجب
+
+188
+00:21:44,200 --> 00:21:51,060
+في yn لأن هذا بيقدي انه limit ثابت الا اتنين في yn
+
+189
+00:21:51,060 --> 00:21:57,760
+بساوي infinity so by comparison
+
+190
+00:21:57,760 --> 00:22:01,700
+by comparison test
+
+191
+00:22:07,170 --> 00:22:11,210
+أنا عندي limit ال sequence لصغيرة infinity، إذا
+
+192
+00:22:11,210 --> 00:22:15,550
+limit ال sequence الأكبر بطلع infinity، إذا limit
+
+193
+00:22:15,550 --> 00:22:28,070
+xn equals infinity وهذا بكمل البرهان، okay؟
+
+194
+00:22:28,070 --> 00:22:32,390
+تمام؟ إذا هذا بكمل ال limit comparison .. برهان ال
+
+195
+00:22:32,390 --> 00:22:36,780
+limit comparison testطبعا ال test هذا و ال test
+
+196
+00:22:36,780 --> 00:22:40,100
+اللي جابله ال comparison test في عليهم هتجدوا فيه
+
+197
+00:22:40,100 --> 00:22:44,160
+بعض التمرين ممكن
+
+198
+00:22:44,160 --> 00:22:47,660
+تطبيقهم على بعض ال sequences موجودة في التمرين
+
+199
+00:22:47,660 --> 00:22:53,560
+فهنسيبكم طبعا تحلوا التمرين عشان تشوفوا كيف ممكن
+
+200
+00:22:53,560 --> 00:22:54,280
+تطبيقهم
+
+201
+00:22:58,500 --> 00:23:05,220
+باقي section واحد في ال chapter تلاتة
+
+202
+00:23:32,720 --> 00:23:37,660
+السيكشن الأخير سيكشن
+
+203
+00:23:37,660 --> 00:23:43,820
+تلاتة سبعة في شبكر 3 هذا هيكون أبراعا مقدمة
+
+204
+00:23:43,820 --> 00:23:47,860
+introduction to
+
+205
+00:23:47,860 --> 00:23:52,920
+infinite series
+
+206
+00:23:57,560 --> 00:24:02,380
+introduction to infinite series مقدمة في
+
+207
+00:24:02,380 --> 00:24:06,940
+المتسلسلات اللانهائية
+
+208
+00:24:06,940 --> 00:24:19,460
+نعرف شو معناه متسلسلة لانهائية let xn
+
+209
+00:24:19,460 --> 00:24:27,330
+contained in R be a sequencesequence of real
+
+210
+00:24:27,330 --> 00:24:43,210
+numbers sum
+
+211
+00:24:43,210 --> 00:24:47,010
+x1
+
+212
+00:24:47,010 --> 00:24:56,200
+plus x2 plus ..x3 plus و هكذا plus xn plus و هكذا
+
+213
+00:24:56,200 --> 00:25:03,400
+و ممكن نكتبه بالصورة using sigma notation نستخدم
+
+214
+00:25:03,400 --> 00:25:09,920
+رمز sigma ممكن هذا نسميه summation from n equals
+
+215
+00:25:09,920 --> 00:25:17,280
+one to infinity إلى xn فالصن
+
+216
+00:25:17,280 --> 00:25:25,190
+المجموع هذاهذا expanded هذا compact form of
+
+217
+00:25:25,190 --> 00:25:39,010
+summation is called an infinite series generated
+
+218
+00:25:39,010 --> 00:25:42,970
+by
+
+219
+00:25:46,320 --> 00:25:54,100
+متولدة من .. by الـ sequence x in إذن
+
+220
+00:25:54,100 --> 00:26:00,280
+infinite series generated by الـ sequence x in إذا
+
+221
+00:26:00,280 --> 00:26:04,680
+هذه عبارة عن infinite series بتسميها متولدة من الـ
+
+222
+00:26:04,680 --> 00:26:09,340
+sequence x in طيب for every
+
+223
+00:26:12,430 --> 00:26:23,290
+for each n belong to N define خلّيني أعرف S1 على
+
+224
+00:26:23,290 --> 00:26:37,950
+أنه X1 S2 على أنه S1 زاد X2 بساوي X1 زاد X2 S3
+
+225
+00:26:37,950 --> 00:26:50,000
+بساوي S2 زاد X3Y ساوي X1 زايد X2 زايد X3 and
+
+226
+00:26:50,000 --> 00:26:58,380
+so on و هكذا نعرف SN على انه SN negative one زايد
+
+227
+00:26:58,380 --> 00:27:07,060
+XN و طبعا ال SN negative one هيكون عبارة عن
+
+228
+00:27:07,060 --> 00:27:07,700
+summation
+
+229
+00:27:10,600 --> 00:27:19,140
+x1 زائد x2 زائد و هكذا إلى أخر حد xn-1 هذا عبارة
+
+230
+00:27:19,140 --> 00:27:25,620
+عن ايه هذا عبارة عن s in negative one بنضيف لها xn
+
+231
+00:27:25,620 --> 00:27:34,340
+فهذا بيطلع بيساوي summation من k equals one to n
+
+232
+00:27:38,080 --> 00:27:43,700
+to for xk اذا
+
+233
+00:27:43,700 --> 00:27:49,840
+sn هو مجموع الحدود من اول حد الى حد رقم n وهكذا
+
+234
+00:27:49,840 --> 00:27:55,960
+ممكن نستمر الى ملا نهاية and so on الان انا كوّنت
+
+235
+00:27:55,960 --> 00:28:00,360
+sequence لاحظوا s1, s2, s3, sn هذا عبارة عن
+
+236
+00:28:00,360 --> 00:28:06,970
+sequence ال sequence الجديدة هذه لها اسمو sequence
+
+237
+00:28:06,970 --> 00:28:12,210
+مهمة of partial sums مظبوط قعدت نسميها اذا طرست
+
+238
+00:28:12,210 --> 00:28:18,790
+تفاضل الف وفهمته الموضوع هذا هناك الموضوع ال
+
+239
+00:28:18,790 --> 00:28:23,110
+series قعدت نسميها the sequence of partial sums
+
+240
+00:28:23,110 --> 00:28:29,210
+اذا the sequence
+
+241
+00:28:30,940 --> 00:28:37,980
+SN from N equals one to infinity is called بنسميها
+
+242
+00:28:37,980 --> 00:28:51,180
+the sequence the sequence of partial sums
+
+243
+00:28:51,180 --> 00:29:03,040
+sequence of partial sums of the seriesاللي هي
+
+244
+00:29:03,040 --> 00:29:11,080
+sigma xn او sigma من n بساعة واحد لانفينيتي okay
+
+245
+00:29:11,080 --> 00:29:18,660
+الان now if
+
+246
+00:29:18,660 --> 00:29:29,280
+ال sequence sn converges say
+
+247
+00:29:31,110 --> 00:29:42,090
+limit sn بالساوي عدد s ينتمي إلى r طبعا then
+
+248
+00:29:42,090 --> 00:29:51,390
+we say في الحالة هذه بنقول أنه the series اللي
+
+249
+00:29:51,390 --> 00:29:58,630
+هي summation xn from n equals one to infinity
+
+250
+00:29:58,630 --> 00:30:00,270
+converges
+
+251
+00:30:09,070 --> 00:30:18,290
+and its sum is summation from n equals one to
+
+252
+00:30:18,290 --> 00:30:23,530
+infinity ل x in ال summation تبعها أو المجموعة
+
+253
+00:30:23,530 --> 00:30:28,450
+تبعها عبارة عن limit لل sequence of partial sums
+
+254
+00:30:28,450 --> 00:30:32,730
+اللي هو العدد S
+
+255
+00:30:37,160 --> 00:30:43,180
+لو كانت الـ sequence divergent
+
+256
+00:30:43,180 --> 00:30:50,740
+if الـ sequence is in diverges we
+
+257
+00:30:50,740 --> 00:31:00,120
+say أنه الـ series sigma
+
+258
+00:31:00,120 --> 00:31:05,880
+x in diverges
+
+259
+00:31:09,090 --> 00:31:13,410
+إذا ال convergence و ال divergence depends on the
+
+260
+00:31:13,410 --> 00:31:18,630
+divergence أو convergence of the infinite series
+
+261
+00:31:18,630 --> 00:31:23,910
+depends on the convergence or divergence of the
+
+262
+00:31:23,910 --> 00:31:30,690
+sequence of partial sums مرتبط بيها ال sequence of
+
+263
+00:31:30,690 --> 00:31:34,350
+partial sums convergent السيريز اللي تابع إليها
+
+264
+00:31:34,350 --> 00:31:38,360
+convergentو العكس إذا كانت ال sequence of partial
+
+265
+00:31:38,360 --> 00:31:40,780
+sums divergent ال series ال infinite series
+
+266
+00:31:40,780 --> 00:31:51,840
+التابعة إلى divergent طيب
+
+267
+00:31:51,840 --> 00:31:58,180
+ناخد بعض الأمثلة طبعا
+
+268
+00:31:58,180 --> 00:32:01,720
+ال Sn هذا ال Sn
+
+269
+00:32:04,610 --> 00:32:15,810
+هذا بنسميه الانث partial sum الانث partial sum
+
+270
+00:32:15,810 --> 00:32:25,690
+انث partial sum المجموع الجزئي أنوني okay هو
+
+271
+00:32:25,690 --> 00:32:30,760
+الحد العام لل sequence و partial sumsأذا لما بدى
+
+272
+00:32:30,760 --> 00:32:34,760
+نخبر هل ال series convergent ولا divergent بجيب ال
+
+273
+00:32:34,760 --> 00:32:38,380
+sequence of partial sums وبجيب الحد العام لل
+
+274
+00:32:38,380 --> 00:32:41,780
+sequence of partial sums وبأفحص هل ال sequence هذي
+
+275
+00:32:41,780 --> 00:32:47,680
+convergent ولا divergent ناخد
+
+276
+00:32:47,680 --> 00:32:48,620
+بعض الأمثلة
+
+277
+00:33:02,760 --> 00:33:14,560
+المثال الأول consider
+
+278
+00:33:14,560 --> 00:33:17,780
+sequence
+
+279
+00:33:17,780 --> 00:33:25,480
+R to N from N equals 0 to infinity طبعا هذه
+
+280
+00:33:25,480 --> 00:33:33,650
+sequence of real numbersWhere R is a real number
+
+281
+00:33:33,650 --> 00:33:38,210
+which
+
+282
+00:33:38,210 --> 00:33:49,150
+generates الsequence هذه generates the geometric
+
+283
+00:33:49,150 --> 00:33:53,010
+.. the so-called geometric series .. geometric
+
+284
+00:33:53,010 --> 00:33:54,330
+series
+
+285
+00:33:57,050 --> 00:34:02,610
+اللي هي summation from n equals zero to infinity
+
+286
+00:34:02,610 --> 00:34:10,210
+from r to n okay اذا هي ال sequence هذه of real
+
+287
+00:34:10,210 --> 00:34:15,650
+numbers بتولد infinite series او generates this
+
+288
+00:34:15,650 --> 00:34:21,210
+infinite series اللي هي حدودها اول حد لما n بساوي
+
+289
+00:34:21,210 --> 00:34:33,620
+سفر واحد بعدين r بعدين r تربيهو R أس N و
+
+290
+00:34:33,620 --> 00:34:41,120
+هكذا ف such series is called geometric series هذه
+
+291
+00:34:41,120 --> 00:34:44,300
+ال series اللي على الصورة هذه بنسميها geometric
+
+292
+00:34:44,300 --> 00:34:49,820
+series الآن ال series هذه
+
+293
+00:34:58,170 --> 00:35:08,530
+this series واحد converges and
+
+294
+00:35:08,530 --> 00:35:15,910
+its sum اللي هو sigma from n equals zero to
+
+295
+00:35:15,910 --> 00:35:22,470
+infinity لRn بساوي واحد على واحد minus R إذا كان
+
+296
+00:35:22,470 --> 00:35:35,210
+absolute R أصغر من واحدand diverges and
+
+297
+00:35:35,210 --> 00:35:41,350
+اتنين diverges if
+
+298
+00:35:41,350 --> 00:35:48,830
+absolute are أكبر من أو يساوي واحد خلّينا
+
+299
+00:35:48,830 --> 00:35:50,050
+نثبت الجزء الأول
+
+300
+00:35:58,010 --> 00:36:04,110
+to prove one أنا
+
+301
+00:36:04,110 --> 00:36:12,410
+عندي ال S N بساوي سيجما
+
+302
+00:36:12,410 --> 00:36:21,050
+من K بساوي سفر إلى N ل R أس K اللي هو واحد زائد R
+
+303
+00:36:21,050 --> 00:36:34,640
+زائد R تلبية زائدR Sn وفي عندي .. في عندي ..
+
+304
+00:36:34,640 --> 00:36:37,780
+لو
+
+305
+00:36:37,780 --> 00:36:48,440
+ضربت Sn في Rفادرب الطرف اليمين في R فبطلع R زاد R
+
+306
+00:36:48,440 --> 00:36:56,380
+تربيه زاد و هكذا زاد R أس N و آخر حد هيكون R أس N
+
+307
+00:36:56,380 --> 00:37:00,940
+زاد 1 تمام؟ الآن خلّينا نطرح ال subtract
+
+308
+00:37:05,370 --> 00:37:10,850
+subtract نطرح المعادلة لتحت من اللي فوق فبطلع عندي
+
+309
+00:37:10,850 --> 00:37:18,590
+sn في واحد minus r أخدت عامل مشترك sn ولمّا أطرح
+
+310
+00:37:18,590 --> 00:37:23,330
+هذا بروح مع هذا كل الهدوء بتروح مع بعضها بظل عندي
+
+311
+00:37:23,330 --> 00:37:33,130
+واحد سالب r to n زاد واحد تمام ومن هنا اذا sn
+
+312
+00:37:36,050 --> 00:37:44,310
+بساوي واحد على واحد سالب R سالب R to N زايد واحد
+
+313
+00:37:44,310 --> 00:37:52,990
+على واحد سالب R ممكن
+
+314
+00:37:52,990 --> 00:37:59,070
+هذا نوديه على ناحية التانية فبصير عندى هذا سالب
+
+315
+00:37:59,070 --> 00:38:01,350
+هذا بساوي
+
+316
+00:38:03,070 --> 00:38:08,930
+سالب R to N زياد واحد على واحد سالب R الآن إذا
+
+317
+00:38:08,930 --> 00:38:13,590
+ناخد ال absolute value للطرفين Sn سالب واحد على
+
+318
+00:38:13,590 --> 00:38:21,030
+واحد سالب R بيطلع بيساوي الكلام هذا وهذا أصغر من
+
+319
+00:38:21,030 --> 00:38:27,830
+أو ساوي absolute R أسن زياد واحد على absolute واحد
+
+320
+00:38:27,830 --> 00:38:30,870
+minus R تمام؟
+
+321
+00:38:34,940 --> 00:38:41,200
+أذا عندي أنا هاي واحد على absolute واحد سالب R ضرب
+
+322
+00:38:41,200 --> 00:38:51,360
+absolute R أسن زائد واحد الان if absolute R أصغر
+
+323
+00:38:51,360 --> 00:39:00,870
+من واحدفهذا بيؤدي ان ال limit ل absolute R to N زي
+
+324
+00:39:00,870 --> 00:39:05,790
+1 لما N تقول ل infinity هذا بيساوي سفر أخدناها قبل
+
+325
+00:39:05,790 --> 00:39:10,430
+هيك وبالتالي
+
+326
+00:39:10,430 --> 00:39:14,950
+اذا ال ..
+
+327
+00:39:14,950 --> 00:39:18,290
+اذا انا عند ال absolute value هذه أكبر من أو ساوي
+
+328
+00:39:18,290 --> 00:39:24,270
+سفر و أصغر من أو ساوي ثابت موجب في sequenceالـ
+
+329
+00:39:24,270 --> 00:39:28,610
+sequence هذه تقول لـ 0 وهذه الـ sequence ثابتة
+
+330
+00:39:28,610 --> 00:39:32,330
+تقول لـ 0 اذا by sandwich theorem
+
+331
+00:39:40,720 --> 00:39:47,760
+بتطلع عند ال limit ل absolute sn minus 1 على 1
+
+332
+00:39:47,760 --> 00:39:52,820
+minus r لما n تقول ل infinity بساوي سفر و ممكن
+
+333
+00:39:52,820 --> 00:39:58,600
+ندخل ال limit جوا فهذا بقدر انه limit 1 على sn
+
+334
+00:39:58,600 --> 00:40:05,040
+عفوا limit sn لما n تقول ل infinity بساوي 1 على 1
+
+335
+00:40:05,040 --> 00:40:11,560
+سالب rوبالتالي إذا الـ series sigma from N equal 0
+
+336
+00:40:11,560 --> 00:40:17,240
+to infinity لR to N المجموعة تبعها تطلع convergent
+
+337
+00:40:17,240 --> 00:40:23,180
+ومجموعة بساوي limit SN وهذا بساوي 1 على 1 minus R
+
+338
+00:40:24,110 --> 00:40:28,950
+إذن هذا بثبت الجزء الأول الجزء التاني ممكن اثباته
+
+339
+00:40:28,950 --> 00:40:34,190
+لو R بساوي واحد فبطل عندي بجمع واحد على واحد عدد
+
+340
+00:40:34,190 --> 00:40:37,730
+لا نهائي من المرات وبالتالي ال sequence of partial
+
+341
+00:40:37,730 --> 00:40:40,170
+sums ممكن اثبات أنها unbounded وبالتالي not
+
+342
+00:40:40,170 --> 00:40:44,330
+convergent إذن ال series not convergent نفس الحاجة
+
+343
+00:40:44,330 --> 00:40:47,210
+لو كان absolute R أكبر من واحد فممكن اثبات أن ال
+
+344
+00:40:47,210 --> 00:40:50,550
+sequence of partial sums is divergent وبالتالي ال
+
+345
+00:40:50,550 --> 00:40:57,170
+series is divergentتمام؟ في أي سؤال؟ إذا بنوقف هنا
+
+346
+00:40:57,170 --> 00:41:02,830
+و بنكمل ان شاء الله الموضوع اللي جاي في المحاضرة
+
+347
+00:41:02,830 --> 00:41:04,630
+القادمة يوم السبت
+
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/XhLWrN2SkOQ_raw.json b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/XhLWrN2SkOQ_raw.json
new file mode 100644
index 0000000000000000000000000000000000000000..a0263660a8545f6465140b8aa7a7d5777522ade4
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/XhLWrN2SkOQ_raw.json
@@ -0,0 +1 @@
+{"segments": [{"id": 1, "seek": 4285, "start": 20.89, "end": 42.85, "text": "أحنا هنكمل الموضوع لـ Properly Divergent Sequences اللي بدأناه المحاضرة السابقة فشوفنا في المحاضرة السابقة تعريف ما معنى أنه limit ل sequence xn بساوي infinity وما معنى أنه limit ل sequence xn بساوي negative infinity", "tokens": [10721, 5016, 8315, 8032, 1863, 24793, 1211, 9673, 2407, 11242, 45367, 5296, 39184, 27627, 356, 413, 1837, 6930, 46859, 2667, 13672, 1829, 47525, 10721, 8315, 3224, 9673, 5016, 46958, 25720, 21136, 16758, 28671, 6156, 8592, 38688, 8315, 8978, 9673, 5016, 46958, 25720, 21136, 16758, 28671, 37279, 16572, 5172, 19446, 20449, 1863, 7578, 14739, 3224, 4948, 5296, 8310, 2031, 77, 4724, 3794, 995, 45865, 13202, 4032, 15042, 20449, 1863, 7578, 14739, 3224, 4948, 5296, 8310, 2031, 77, 4724, 3794, 995, 45865, 3671, 13202], "avg_logprob": -0.17752259036144577, "compression_ratio": 1.7282608695652173, "no_speech_prob": 0.0, "words": [{"start": 20.89, "end": 21.89, "word": "أحنا", "probability": 0.6949055989583334}, {"start": 21.89, "end": 22.77, "word": " هنكمل", "probability": 0.9686279296875}, {"start": 22.77, "end": 23.75, "word": " الموضوع", "probability": 0.9283447265625}, {"start": 23.75, "end": 23.97, "word": " لـ", "probability": 0.71630859375}, {"start": 23.97, "end": 24.49, "word": " Properly", "probability": 0.724365234375}, {"start": 24.49, "end": 25.27, "word": " Divergent", "probability": 0.8753255208333334}, {"start": 25.27, "end": 26.63, "word": " Sequences", "probability": 0.9140625}, {"start": 26.63, "end": 27.03, "word": " اللي", "probability": 0.73193359375}, {"start": 27.03, "end": 28.57, "word": " بدأناه", "probability": 0.86083984375}, {"start": 28.57, "end": 29.15, "word": " المحاضرة", "probability": 0.927490234375}, {"start": 29.15, "end": 29.75, "word": " السابقة", "probability": 0.97314453125}, {"start": 29.75, "end": 31.11, "word": " فشوفنا", "probability": 0.8868408203125}, {"start": 31.11, "end": 31.23, "word": " في", "probability": 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37.51, "end": 39.25, "word": " وما", "probability": 0.734375}, {"start": 39.25, "end": 39.61, "word": " معنى", "probability": 0.9630533854166666}, {"start": 39.61, "end": 39.93, "word": " أنه", "probability": 0.938720703125}, {"start": 39.93, "end": 40.19, "word": " limit", "probability": 0.94873046875}, {"start": 40.19, "end": 40.41, "word": " ل", "probability": 0.95751953125}, {"start": 40.41, "end": 40.81, "word": " sequence", "probability": 0.78759765625}, {"start": 40.81, "end": 41.45, "word": " xn", "probability": 0.98681640625}, {"start": 41.45, "end": 41.89, "word": " بساوي", "probability": 0.9755859375}, {"start": 41.89, "end": 42.25, "word": " negative", "probability": 0.923828125}, {"start": 42.25, "end": 42.85, "word": " infinity", "probability": 0.91650390625}], "temperature": 1.0}, {"id": 2, "seek": 7139, "start": 45.09, "end": 71.39, "text": "طبعاً الـ sequence بتكون properly divergent إذا كانت ال limit تبعتها بالساوي infinity أو سالب infinity في عندي comparison test ل .. ل properly divergent sequences هذا ال test بيقول لي لو في عندي two sequences xn و yn two sequences of real numbers بيحققوا الشرط star satisfy the condition star وهو أن كل حد", "tokens": [9566, 3555, 3615, 995, 14111, 2423, 39184, 8310, 39894, 30544, 6108, 18558, 6930, 11933, 15730, 25961, 2655, 2423, 4948, 6055, 3555, 34268, 11296, 20666, 3794, 995, 45865, 13202, 34051, 8608, 6027, 3555, 13202, 8978, 18871, 16254, 9660, 1500, 5296, 4386, 5296, 6108, 18558, 6930, 22978, 23758, 2423, 1500, 4724, 1829, 39648, 32239, 45164, 8978, 18871, 16254, 732, 22978, 2031, 77, 4032, 17861, 732, 22978, 295, 957, 3547, 4724, 1829, 5016, 4587, 4587, 14407, 25124, 2288, 9566, 3543, 19319, 264, 4188, 3543, 37037, 2407, 14739, 28242, 11331, 3215], "avg_logprob": -0.27343749322674493, "compression_ratio": 1.596774193548387, "no_speech_prob": 0.0, "words": [{"start": 45.09, "end": 45.55, "word": "طبعاً", "probability": 0.66661376953125}, {"start": 45.55, "end": 45.71, "word": " الـ", "probability": 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two", "probability": 0.640625}, {"start": 64.85, "end": 65.41, "word": " sequences", "probability": 0.9697265625}, {"start": 65.41, "end": 65.59, "word": " of", "probability": 0.89990234375}, {"start": 65.59, "end": 65.81, "word": " real", "probability": 0.96728515625}, {"start": 65.81, "end": 66.33, "word": " numbers", "probability": 0.9052734375}, {"start": 66.33, "end": 67.71, "word": " بيحققوا", "probability": 0.94580078125}, {"start": 67.71, "end": 68.11, "word": " الشرط", "probability": 0.8483072916666666}, {"start": 68.11, "end": 68.53, "word": " star", "probability": 0.65478515625}, {"start": 68.53, "end": 68.95, "word": " satisfy", "probability": 0.386474609375}, {"start": 68.95, "end": 69.23, "word": " the", "probability": 0.91552734375}, {"start": 69.23, "end": 69.63, "word": " condition", "probability": 0.951171875}, {"start": 69.63, "end": 70.03, "word": " star", "probability": 0.8349609375}, {"start": 70.03, "end": 70.37, "word": " وهو", "probability": 0.7100830078125}, 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{"start": 436.34, "end": 436.52, "word": " أو", "probability": 0.99169921875}, {"start": 436.52, "end": 436.8, "word": " ساوي", "probability": 0.9794921875}, {"start": 436.8, "end": 437.82, "word": " YN", "probability": 0.92626953125}, {"start": 437.82, "end": 442.94, "word": " وكذلك", "probability": 0.9195556640625}, {"start": 442.94, "end": 443.22, "word": " ال", "probability": 0.720703125}, {"start": 443.22, "end": 443.48, "word": " N", "probability": 0.92431640625}, {"start": 443.48, "end": 443.74, "word": " لما", "probability": 0.947021484375}, {"start": 443.74, "end": 444.06, "word": " تكون", "probability": 0.99072265625}, {"start": 444.06, "end": 444.24, "word": " ال", "probability": 0.7626953125}, {"start": 444.24, "end": 444.54, "word": " N", "probability": 0.9423828125}, {"start": 444.54, "end": 445.14, "word": " أكبر", "probability": 0.98193359375}, {"start": 445.14, "end": 445.32, "word": " من", "probability": 0.9970703125}, {"start": 445.32, "end": 445.5, "word": " أو", "probability": 0.99072265625}, {"start": 445.5, "end": 445.96, "word": " ساوي", "probability": 0.9786783854166666}, {"start": 445.96, "end": 446.26, "word": " N", "probability": 0.8310546875}], "temperature": 1.0}, {"id": 16, "seek": 47666, "start": 447.34, "end": 476.66, "text": "الـ N هذه فبطمن أنها أكبر من أو ساوي N of beta وبالتالي الكلام هذا بتحقق ويعني هيك بيكون برهن الجزء B بال .. باستخدام الشرط الأضعف double star بالمثل طبعا ممكن نعطيه في حالة لما يكون بدي أبرهن الجزء A فباخد ال N المطلوبة هي ال maximum ل M وN of alpha في برهان A تحت شرط star", "tokens": [6027, 39184, 426, 29538, 6156, 3555, 9566, 27842, 14739, 11296, 5551, 4117, 26890, 9154, 34051, 8608, 995, 45865, 426, 295, 9861, 46599, 6027, 2655, 6027, 1829, 2423, 28820, 10943, 23758, 39894, 5016, 4587, 4587, 4032, 40228, 22653, 39896, 4117, 4724, 1829, 30544, 4724, 2288, 3224, 1863, 25724, 11622, 38207, 363, 20666, 4386, 4724, 995, 14851, 9778, 3215, 10943, 25124, 2288, 9566, 16247, 11242, 3615, 5172, 3834, 3543, 20666, 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464.96, "word": " طبعا", "probability": 0.993896484375}, {"start": 464.96, "end": 465.4, "word": " ممكن", "probability": 0.9794921875}, {"start": 465.4, "end": 466.46, "word": " نعطيه", "probability": 0.8708984375}, {"start": 466.46, "end": 466.74, "word": " في", "probability": 0.77587890625}, {"start": 466.74, "end": 467.08, "word": " حالة", "probability": 0.99267578125}, {"start": 467.08, "end": 467.34, "word": " لما", "probability": 0.8974609375}, {"start": 467.34, "end": 467.72, "word": " يكون", "probability": 0.7470703125}, {"start": 467.72, "end": 468.0, "word": " بدي", "probability": 0.64990234375}, {"start": 468.0, "end": 468.3, "word": " أبرهن", "probability": 0.88525390625}, {"start": 468.3, "end": 468.68, "word": " الجزء", "probability": 0.9736328125}, {"start": 468.68, "end": 468.96, "word": " A", "probability": 0.9580078125}, {"start": 468.96, "end": 469.62, "word": " فباخد", "probability": 0.947021484375}, {"start": 469.62, "end": 469.76, "word": " ال", "probability": 0.97265625}, {"start": 469.76, "end": 469.96, "word": " N", "probability": 0.63232421875}, {"start": 469.96, "end": 470.9, "word": " المطلوبة", "probability": 0.98486328125}, {"start": 470.9, "end": 471.44, "word": " هي", "probability": 0.875}, {"start": 471.44, "end": 471.6, "word": " ال", "probability": 0.422119140625}, {"start": 471.6, "end": 472.0, "word": " maximum", "probability": 0.97900390625}, {"start": 472.0, "end": 472.3, "word": " ل", "probability": 0.98046875}, {"start": 472.3, "end": 472.56, "word": " M", "probability": 0.79248046875}, {"start": 472.56, "end": 472.96, "word": " وN", "probability": 0.782958984375}, {"start": 472.96, "end": 473.12, "word": " of", "probability": 0.96630859375}, {"start": 473.12, "end": 473.54, "word": " alpha", "probability": 0.430419921875}, {"start": 473.54, "end": 474.26, "word": " في", "probability": 0.7490234375}, {"start": 474.26, "end": 474.82, "word": " برهان", "probability": 0.959716796875}, {"start": 474.82, "end": 475.24, "word": " A", "probability": 0.91064453125}, {"start": 475.24, "end": 475.82, "word": " تحت", "probability": 0.986572265625}, {"start": 475.82, "end": 476.28, "word": " شرط", "probability": 0.9010416666666666}, {"start": 476.28, "end": 476.66, "word": " star", "probability": 0.60205078125}], "temperature": 1.0}, {"id": 17, "seek": 50766, "start": 478.34, "end": 507.66, "text": "هذه أول ملاحظة الملاحظة التانية الملاحظة التانية if condition star holds إذا كان الشرط star holds then", "tokens": [3224, 24192, 5551, 12610, 3714, 15040, 5016, 19913, 3660, 9673, 15040, 5016, 19913, 3660, 16712, 7649, 10632, 9673, 15040, 5016, 19913, 3660, 16712, 7649, 10632, 498, 4188, 3543, 9190, 11933, 15730, 25961, 25124, 2288, 9566, 3543, 9190, 550], "avg_logprob": -0.1271033669129396, "compression_ratio": 1.5196078431372548, "no_speech_prob": 0.0, "words": [{"start": 478.34, "end": 478.68, "word": "هذه", "probability": 0.814208984375}, {"start": 478.68, "end": 478.92, "word": " أول", "probability": 0.8212890625}, {"start": 478.92, "end": 479.7, "word": " ملاحظة", "probability": 0.98134765625}, {"start": 479.7, "end": 480.68, "word": " الملاحظة", "probability": 0.95732421875}, {"start": 480.68, "end": 481.6, "word": " التانية", "probability": 0.9773763020833334}, {"start": 481.6, "end": 495.36, "word": " الملاحظة", "probability": 0.9798828125}, {"start": 495.36, "end": 496.46, "word": " التانية", "probability": 0.9928385416666666}, {"start": 496.46, "end": 498.28, "word": " if", "probability": 0.7685546875}, {"start": 498.28, "end": 499.18, "word": " condition", "probability": 0.921875}, {"start": 499.18, "end": 501.5, "word": " star", "probability": 0.748046875}, {"start": 501.5, "end": 502.18, "word": " holds", "probability": 0.74951171875}, {"start": 502.18, "end": 504.44, "word": " إذا", "probability": 0.669189453125}, {"start": 504.44, "end": 504.66, "word": " كان", "probability": 0.79296875}, {"start": 504.66, "end": 505.06, "word": " الشرط", "probability": 0.8548177083333334}, {"start": 505.06, "end": 505.48, "word": " star", "probability": 0.8271484375}, {"start": 505.48, "end": 505.98, "word": " holds", "probability": 0.775390625}, {"start": 505.98, "end": 507.66, "word": " then", "probability": 0.79833984375}], "temperature": 1.0}, {"id": 18, "seek": 52665, "start": 517.13, "end": 526.65, "text": "النتيجة ان y in tends to infinity does not necessarily implies ان x in tends to infinity", "tokens": [6027, 29399, 1829, 7435, 3660, 16472, 288, 294, 12258, 281, 13202, 775, 406, 4725, 18779, 16472, 2031, 294, 12258, 281, 13202], "avg_logprob": -0.4055397618900646, "compression_ratio": 1.1927710843373494, "no_speech_prob": 0.0, "words": [{"start": 517.13, "end": 518.05, "word": "النتيجة", "probability": 0.8849609375}, {"start": 518.05, "end": 518.27, "word": " ان", "probability": 0.48388671875}, {"start": 518.27, "end": 518.61, "word": " y", "probability": 0.16796875}, {"start": 518.61, "end": 519.05, "word": " in", "probability": 0.232177734375}, {"start": 519.05, "end": 519.63, "word": " tends", "probability": 0.50927734375}, {"start": 519.63, "end": 519.79, "word": " to", "probability": 0.89599609375}, {"start": 519.79, "end": 520.35, "word": " infinity", "probability": 0.87109375}, {"start": 520.35, "end": 521.75, "word": " does", "probability": 0.396240234375}, {"start": 521.75, "end": 522.07, "word": " not", "probability": 0.958984375}, {"start": 522.07, "end": 522.79, "word": " necessarily", "probability": 0.7431640625}, {"start": 522.79, "end": 523.79, "word": " implies", "probability": 0.8857421875}, {"start": 523.79, "end": 524.69, "word": " ان", "probability": 0.76171875}, {"start": 524.69, "end": 525.11, "word": " x", "probability": 0.90380859375}, {"start": 525.11, "end": 525.51, "word": " in", "probability": 0.9521484375}, {"start": 525.51, "end": 525.97, "word": " tends", "probability": 0.387939453125}, {"start": 525.97, "end": 526.11, "word": " to", "probability": 0.9091796875}, {"start": 526.11, "end": 526.65, "word": " infinity", "probability": 0.8916015625}], "temperature": 1.0}, {"id": 19, "seek": 56023, "start": 533.43, "end": 560.23, "text": "وكان limit ال yn بساوي infinity فليس من الضروري ان يكون limit xn بساوي infinity وهي مثال يوضح ذلك for example على سبيل المثال consider consider ال sequence 1 على n أصغر من أو بساوي n لكل n في n", "tokens": [2407, 41361, 4948, 2423, 288, 77, 4724, 3794, 995, 45865, 13202, 6156, 20292, 3794, 9154, 6024, 114, 2288, 13063, 1829, 16472, 7251, 30544, 4948, 2031, 77, 4724, 3794, 995, 45865, 13202, 37037, 1829, 50113, 6027, 7251, 2407, 11242, 5016, 29910, 23275, 337, 1365, 15844, 8608, 21292, 1211, 9673, 12984, 6027, 1949, 1949, 2423, 8310, 502, 15844, 297, 5551, 9381, 17082, 2288, 9154, 34051, 4724, 3794, 995, 45865, 297, 5296, 28820, 297, 8978, 297], "avg_logprob": -0.16986908662963557, "compression_ratio": 1.5706214689265536, "no_speech_prob": 0.0, "words": [{"start": 533.43, "end": 534.01, "word": "وكان", "probability": 0.79638671875}, {"start": 534.01, "end": 534.49, 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"probability": 0.9522705078125}, {"start": 542.35, "end": 542.89, "word": " infinity", "probability": 0.82958984375}, {"start": 542.89, "end": 543.69, "word": " وهي", "probability": 0.5765380859375}, {"start": 543.69, "end": 544.17, "word": " مثال", "probability": 0.967041015625}, {"start": 544.17, "end": 544.67, "word": " يوضح", "probability": 0.9853515625}, {"start": 544.67, "end": 545.19, "word": " ذلك", "probability": 0.994873046875}, {"start": 545.19, "end": 546.09, "word": " for", "probability": 0.69189453125}, {"start": 546.09, "end": 546.67, "word": " example", "probability": 0.9599609375}, {"start": 546.67, "end": 546.91, "word": " على", "probability": 0.83544921875}, {"start": 546.91, "end": 547.35, "word": " سبيل", "probability": 0.9851888020833334}, {"start": 547.35, "end": 548.03, "word": " المثال", "probability": 0.9866536458333334}, {"start": 548.03, "end": 550.49, "word": " consider", "probability": 0.9150390625}, {"start": 550.49, "end": 553.55, "word": " consider", "probability": 0.72802734375}, {"start": 553.55, "end": 553.71, "word": " ال", "probability": 0.48486328125}, {"start": 553.71, "end": 554.05, "word": " sequence", "probability": 0.97412109375}, {"start": 554.05, "end": 554.59, "word": " 1", "probability": 0.55029296875}, {"start": 554.59, "end": 554.97, "word": " على", "probability": 0.62744140625}, {"start": 554.97, "end": 555.39, "word": " n", "probability": 0.86328125}, {"start": 555.39, "end": 556.05, "word": " أصغر", "probability": 0.9461669921875}, {"start": 556.05, "end": 556.25, "word": " من", "probability": 0.9677734375}, {"start": 556.25, "end": 556.43, "word": " أو", "probability": 0.75439453125}, {"start": 556.43, "end": 556.99, "word": " بساوي", "probability": 0.895263671875}, {"start": 556.99, "end": 557.33, "word": " n", "probability": 0.744140625}, {"start": 557.33, "end": 558.99, "word": " لكل", "probability": 0.962890625}, {"start": 558.99, "end": 559.49, "word": " n", "probability": 0.87646484375}, {"start": 559.49, "end": 559.81, "word": " في", "probability": 0.98095703125}, {"start": 559.81, "end": 560.23, "word": " n", "probability": 0.80908203125}], "temperature": 1.0}, {"id": 20, "seek": 58961, "start": 561.99, "end": 589.61, "text": "إذن هى انا عندي xn وهى عندي yn وهى xn أصغر من يساوي yn الشرط الصغير متحقق لكن أنا عندي ال limit ل sequence yn اللى الحد العام تبعها n هدى بساوي infinity but ال limit ل xn اللى هى واحد على n", "tokens": [28814, 8848, 1863, 8032, 7578, 1975, 8315, 18871, 16254, 2031, 77, 37037, 7578, 18871, 16254, 17861, 37037, 7578, 2031, 77, 5551, 9381, 17082, 2288, 9154, 7251, 3794, 995, 45865, 17861, 25124, 2288, 9566, 2423, 9381, 17082, 13546, 44650, 5016, 4587, 4587, 44381, 41850, 18871, 16254, 2423, 4948, 5296, 8310, 17861, 13672, 7578, 21542, 3215, 18863, 10943, 6055, 3555, 3615, 11296, 297, 8032, 3215, 7578, 4724, 3794, 995, 45865, 13202, 457, 2423, 4948, 5296, 2031, 77, 13672, 7578, 8032, 7578, 36764, 24401, 15844, 297], "avg_logprob": -0.30189731397799086, 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من", "probability": 0.96533203125}, {"start": 567.89, "end": 568.41, "word": " يساوي", "probability": 0.5680389404296875}, {"start": 568.41, "end": 568.95, "word": " yn", "probability": 0.74267578125}, {"start": 568.95, "end": 569.99, "word": " الشرط", "probability": 0.9580078125}, {"start": 569.99, "end": 570.31, "word": " الصغير", "probability": 0.53790283203125}, {"start": 570.31, "end": 571.01, "word": " متحقق", "probability": 0.938720703125}, {"start": 571.01, "end": 572.89, "word": " لكن", "probability": 0.86767578125}, {"start": 572.89, "end": 573.13, "word": " أنا", "probability": 0.572265625}, {"start": 573.13, "end": 573.51, "word": " عندي", "probability": 0.95751953125}, {"start": 573.51, "end": 573.75, "word": " ال", "probability": 0.80810546875}, {"start": 573.75, "end": 574.09, "word": " limit", "probability": 0.93212890625}, {"start": 574.09, "end": 575.45, "word": " ل", "probability": 0.6201171875}, {"start": 575.45, "end": 576.07, "word": " sequence", "probability": 0.383544921875}, {"start": 576.07, "end": 576.83, "word": " yn", "probability": 0.84716796875}, {"start": 576.83, "end": 577.09, "word": " اللى", "probability": 0.869873046875}, {"start": 577.09, "end": 577.47, "word": " الحد", "probability": 0.889892578125}, {"start": 577.47, "end": 577.73, "word": " العام", "probability": 0.86669921875}, {"start": 577.73, "end": 578.21, "word": " تبعها", "probability": 0.81396484375}, {"start": 578.21, "end": 578.47, "word": " n", "probability": 0.7412109375}, {"start": 578.47, "end": 578.91, "word": " هدى", "probability": 0.6095377604166666}, {"start": 578.91, "end": 579.33, "word": " بساوي", "probability": 0.73388671875}, {"start": 579.33, "end": 579.87, "word": " infinity", "probability": 0.8828125}, {"start": 579.87, "end": 582.55, "word": " but", "probability": 0.7275390625}, {"start": 582.55, "end": 585.43, "word": " ال", "probability": 0.73291015625}, {"start": 585.43, "end": 585.81, "word": " limit", "probability": 0.97314453125}, {"start": 585.81, "end": 587.19, "word": " ل", "probability": 0.9658203125}, {"start": 587.19, "end": 587.95, "word": " xn", "probability": 0.833251953125}, {"start": 587.95, "end": 588.29, "word": " اللى", "probability": 0.985107421875}, {"start": 588.29, "end": 588.55, "word": " هى", "probability": 0.849609375}, {"start": 588.55, "end": 589.05, "word": " واحد", "probability": 0.88623046875}, {"start": 589.05, "end": 589.27, "word": " على", "probability": 0.6064453125}, {"start": 589.27, "end": 589.61, "word": " n", "probability": 0.845703125}], "temperature": 1.0}, {"id": 21, "seek": 61660, "start": 590.84, "end": 616.6, "text": "بساوي سفر لا تساوي infinity السفر لا يساوي infinity okay تمام إذا الشرط star ما بيخلنيش أستنتج أنه limit x in بالساوية infinity عندما limit y in بالساوية infinity بالمثل if condition star holds", "tokens": [3555, 3794, 995, 45865, 8608, 5172, 2288, 20193, 6055, 3794, 995, 45865, 13202, 21136, 5172, 2288, 20193, 7251, 3794, 995, 45865, 13202, 1392, 46811, 10943, 11933, 15730, 25124, 2288, 9566, 3543, 19446, 4724, 1829, 9778, 1211, 22653, 8592, 5551, 14851, 29399, 7435, 14739, 3224, 4948, 2031, 294, 20666, 3794, 995, 2407, 10632, 13202, 43242, 15042, 4948, 288, 294, 20666, 3794, 995, 2407, 10632, 13202, 20666, 2304, 12984, 1211, 498, 4188, 3543, 9190], "avg_logprob": -0.2564212394087282, "compression_ratio": 1.6890243902439024, "no_speech_prob": 0.0, "words": [{"start": 590.84, "end": 591.44, "word": "بساوي", "probability": 0.63128662109375}, {"start": 591.44, "end": 591.84, "word": " سفر", "probability": 0.7852376302083334}, {"start": 591.84, "end": 593.8, "word": " لا", "probability": 0.47216796875}, {"start": 593.8, "end": 594.44, "word": " تساوي", "probability": 0.9468994140625}, {"start": 594.44, "end": 594.92, "word": " infinity", "probability": 0.56494140625}, {"start": 594.92, "end": 596.52, "word": " السفر", "probability": 0.90869140625}, {"start": 596.52, "end": 596.68, "word": " لا", "probability": 0.95068359375}, {"start": 596.68, "end": 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643.82, "end": 644.36, "word": " ذلك", "probability": 0.993896484375}, {"start": 644.36, "end": 646.78, "word": " ممكن", "probability": 0.967529296875}, {"start": 646.78, "end": 647.1, "word": " نفس", "probability": 0.99609375}, {"start": 647.1, "end": 647.6, "word": " المثال", "probability": 0.9659830729166666}, {"start": 647.6, "end": 648.02, "word": " بس", "probability": 0.948486328125}], "temperature": 1.0}, {"id": 23, "seek": 66277, "start": 651.51, "end": 662.77, "text": "for example بس نضرب في سالب هي عندي negative n أصغر من أو ساوي negative واحد على n لكل n في n", "tokens": [2994, 1365, 4724, 3794, 8717, 11242, 25513, 8978, 8608, 6027, 3555, 39896, 18871, 16254, 3671, 297, 5551, 9381, 17082, 2288, 9154, 34051, 8608, 995, 45865, 3671, 36764, 24401, 15844, 297, 5296, 28820, 297, 8978, 297], "avg_logprob": -0.22504339615503946, "compression_ratio": 1.1637931034482758, "no_speech_prob": 0.0, "words": [{"start": 651.51, "end": 651.79, "word": "for", "probability": 0.286865234375}, {"start": 651.79, "end": 652.45, "word": " example", "probability": 0.970703125}, {"start": 652.45, "end": 653.87, "word": " بس", "probability": 0.782958984375}, {"start": 653.87, "end": 654.19, "word": " نضرب", "probability": 0.9703776041666666}, {"start": 654.19, "end": 654.35, "word": " في", "probability": 0.84716796875}, {"start": 654.35, "end": 655.17, "word": " سالب", "probability": 0.8681640625}, {"start": 655.17, "end": 655.91, "word": " هي", "probability": 0.199951171875}, {"start": 655.91, "end": 656.29, "word": " عندي", "probability": 0.764892578125}, {"start": 656.29, "end": 656.85, "word": " negative", "probability": 0.923828125}, {"start": 656.85, "end": 657.37, "word": " n", "probability": 0.609375}, {"start": 657.37, "end": 658.03, "word": " أصغر", "probability": 0.95166015625}, {"start": 658.03, "end": 658.17, "word": " من", "probability": 0.8955078125}, {"start": 658.17, "end": 658.33, "word": " أو", "probability": 0.389892578125}, {"start": 658.33, "end": 658.79, "word": " ساوي", "probability": 0.9054361979166666}, {"start": 658.79, "end": 659.17, "word": " negative", "probability": 0.8505859375}, {"start": 659.17, "end": 659.79, "word": " واحد", "probability": 0.942138671875}, {"start": 659.79, "end": 660.03, "word": " على", "probability": 0.73046875}, {"start": 660.03, "end": 660.41, "word": " n", "probability": 0.78466796875}, {"start": 660.41, "end": 661.67, "word": " لكل", "probability": 0.98486328125}, {"start": 661.67, "end": 662.11, "word": " n", "probability": 0.859375}, {"start": 662.11, "end": 662.39, "word": " في", "probability": 0.984375}, {"start": 662.39, "end": 662.77, "word": " n", "probability": 0.78466796875}], "temperature": 1.0}, {"id": 24, "seek": 68140, "start": 664.6, "end": 681.4, "text": "هل هذا كلام صح؟ انا عندي واحد على n أصغر من أو ساوي n لكل n هذا صح اضرب في سالب واحد تناقص هاه؟ هاي عندك xn بساوي سالب n وهي عندنا yn", "tokens": [3224, 1211, 23758, 28242, 10943, 20328, 5016, 22807, 1975, 8315, 18871, 16254, 36764, 24401, 15844, 297, 5551, 9381, 17082, 2288, 9154, 34051, 8608, 995, 45865, 297, 5296, 28820, 297, 23758, 20328, 5016, 1975, 11242, 25513, 8978, 8608, 6027, 3555, 36764, 24401, 6055, 8315, 4587, 9381, 8032, 40294, 22807, 8032, 47302, 43242, 4117, 2031, 77, 4724, 3794, 995, 45865, 8608, 6027, 3555, 297, 37037, 1829, 43242, 8315, 17861], "avg_logprob": -0.23851103226051612, "compression_ratio": 1.591549295774648, "no_speech_prob": 0.0, "words": [{"start": 664.6, "end": 664.9, "word": "هل", "probability": 0.9482421875}, {"start": 664.9, "end": 665.08, "word": " هذا", "probability": 0.837890625}, {"start": 665.08, "end": 665.44, "word": " كلام", "probability": 0.858642578125}, {"start": 665.44, "end": 666.86, "word": " صح؟", "probability": 0.9124348958333334}, {"start": 666.86, "end": 667.0, "word": " انا", "probability": 0.653076171875}, {"start": 667.0, "end": 667.36, "word": " عندي", "probability": 0.864990234375}, {"start": 667.36, "end": 667.82, "word": " واحد", "probability": 0.8779296875}, {"start": 667.82, "end": 668.02, "word": " على", "probability": 0.7138671875}, {"start": 668.02, "end": 668.34, "word": " n", "probability": 0.368896484375}, {"start": 668.34, "end": 668.88, "word": " أصغر", "probability": 0.7625732421875}, {"start": 668.88, "end": 668.98, "word": " من", "probability": 0.97802734375}, {"start": 668.98, "end": 669.14, "word": " أو", "probability": 0.55810546875}, {"start": 669.14, "end": 669.52, "word": " ساوي", "probability": 0.5887858072916666}, {"start": 669.52, "end": 669.78, "word": " n", "probability": 0.81982421875}, {"start": 669.78, "end": 670.2, "word": " لكل", "probability": 0.9697265625}, {"start": 670.2, "end": 670.48, "word": " n", "probability": 0.87451171875}, {"start": 670.48, "end": 670.5, "word": " هذا", "probability": 0.454345703125}, {"start": 670.5, "end": 672.42, "word": " صح", "probability": 0.991943359375}, {"start": 672.42, "end": 672.96, "word": " اضرب", "probability": 0.8811848958333334}, {"start": 672.96, "end": 673.12, "word": " في", "probability": 0.5927734375}, {"start": 673.12, "end": 673.46, "word": " سالب", "probability": 0.89453125}, {"start": 673.46, "end": 673.94, "word": " واحد", "probability": 0.985595703125}, {"start": 673.94, "end": 674.9, "word": " تناقص", "probability": 0.802490234375}, {"start": 674.9, "end": 675.72, "word": " هاه؟", "probability": 0.4392903645833333}, {"start": 675.72, "end": 676.62, "word": " هاي", "probability": 0.644287109375}, {"start": 676.62, "end": 677.04, "word": " عندك", "probability": 0.96728515625}, {"start": 677.04, "end": 677.7, "word": " xn", "probability": 0.73193359375}, {"start": 677.7, "end": 678.92, "word": " بساوي", "probability": 0.9259033203125}, {"start": 678.92, "end": 679.32, "word": " سالب", "probability": 0.9793294270833334}, {"start": 679.32, "end": 679.66, "word": " n", "probability": 0.90283203125}, {"start": 679.66, "end": 680.64, "word": " وهي", "probability": 0.812744140625}, {"start": 680.64, "end": 681.0, "word": " عندنا", "probability": 0.880615234375}, {"start": 681.0, "end": 681.4, "word": " yn", "probability": 0.90283203125}], "temperature": 1.0}, {"id": 25, "seek": 71131, "start": 683.69, "end": 711.31, "text": "بساوي negative واحد على n الان انا عندي limit xn اللي هو سالب n لما طبعا n تقول infinity بساوي negative infinity but لكن limit ال yn اللي هو واحد على n ايش بتساوي؟ ساوي سفر سالب واحد عفوا سالب واحد على n limit سالب واحد على n بساوي سفر", "tokens": [3555, 3794, 995, 45865, 3671, 36764, 24401, 15844, 297, 2423, 7649, 1975, 8315, 18871, 16254, 4948, 2031, 77, 13672, 1829, 31439, 8608, 6027, 3555, 297, 5296, 15042, 23032, 3555, 3615, 995, 297, 6055, 39648, 13202, 4724, 3794, 995, 45865, 3671, 13202, 457, 44381, 4948, 2423, 288, 77, 13672, 1829, 31439, 36764, 24401, 15844, 297, 1975, 1829, 8592, 39894, 3794, 995, 45865, 22807, 8608, 995, 45865, 8608, 5172, 2288, 8608, 6027, 3555, 36764, 24401, 6225, 5172, 14407, 8608, 6027, 3555, 36764, 24401, 15844, 297, 4948, 8608, 6027, 3555, 36764, 24401, 15844, 297, 4724, 3794, 995, 45865, 8608, 5172, 2288], "avg_logprob": -0.15072600829480876, "compression_ratio": 2.086705202312139, "no_speech_prob": 0.0, "words": [{"start": 683.69, "end": 684.19, "word": "بساوي", "probability": 0.852294921875}, {"start": 684.19, "end": 684.55, "word": " negative", "probability": 0.783203125}, {"start": 684.55, "end": 684.99, "word": " واحد", "probability": 0.9453125}, {"start": 684.99, "end": 685.17, "word": " على", "probability": 0.6142578125}, {"start": 685.17, "end": 685.39, "word": " n", "probability": 0.426025390625}, {"start": 685.39, "end": 686.39, "word": " الان", "probability": 0.6614990234375}, {"start": 686.39, "end": 686.61, "word": " انا", "probability": 0.84521484375}, {"start": 686.61, "end": 687.07, "word": " عندي", "probability": 0.89697265625}, {"start": 687.07, "end": 687.81, "word": " limit", "probability": 0.9873046875}, {"start": 687.81, "end": 689.35, "word": " xn", "probability": 0.778564453125}, {"start": 689.35, "end": 689.53, "word": " اللي", "probability": 0.87060546875}, {"start": 689.53, "end": 689.75, "word": " هو", "probability": 0.98974609375}, {"start": 689.75, "end": 690.25, "word": " سالب", "probability": 0.9620768229166666}, {"start": 690.25, "end": 690.53, "word": " n", "probability": 0.8212890625}, {"start": 690.53, "end": 691.59, "word": " لما", "probability": 0.940673828125}, {"start": 691.59, "end": 691.97, "word": " طبعا", "probability": 0.8994140625}, {"start": 691.97, "end": 692.25, "word": " n", "probability": 0.58251953125}, {"start": 692.25, "end": 692.65, "word": " تقول", "probability": 0.928466796875}, {"start": 692.65, "end": 693.31, "word": " infinity", "probability": 0.669921875}, {"start": 693.31, "end": 694.09, "word": " بساوي", "probability": 0.95458984375}, {"start": 694.09, "end": 694.51, "word": " negative", "probability": 0.97216796875}, {"start": 694.51, "end": 695.21, "word": " infinity", "probability": 0.91064453125}, {"start": 695.21, "end": 697.17, "word": " but", "probability": 0.7578125}, {"start": 697.17, "end": 697.93, "word": " لكن", "probability": 0.8076171875}, {"start": 697.93, "end": 698.49, "word": " limit", "probability": 0.98583984375}, {"start": 698.49, "end": 700.21, "word": " ال", "probability": 0.828125}, {"start": 700.21, "end": 700.93, "word": " yn", "probability": 0.683837890625}, {"start": 700.93, "end": 701.25, "word": " اللي", "probability": 0.9892578125}, {"start": 701.25, "end": 701.53, "word": " هو", "probability": 0.99169921875}, {"start": 701.53, "end": 702.09, "word": " واحد", "probability": 0.99072265625}, {"start": 702.09, "end": 702.27, "word": " على", "probability": 0.8779296875}, {"start": 702.27, "end": 702.65, "word": " n", "probability": 0.9296875}, {"start": 702.65, "end": 703.99, "word": " ايش", "probability": 0.85693359375}, {"start": 703.99, "end": 704.91, "word": " بتساوي؟", "probability": 0.89306640625}, {"start": 704.91, "end": 705.55, "word": " ساوي", "probability": 0.7461751302083334}, {"start": 705.55, "end": 705.99, "word": " سفر", "probability": 0.82958984375}, {"start": 705.99, "end": 706.87, "word": " سالب", "probability": 0.722900390625}, {"start": 706.87, "end": 707.05, "word": " واحد", "probability": 0.91162109375}, {"start": 707.05, "end": 707.33, "word": " عفوا", "probability": 0.8990885416666666}, {"start": 707.33, "end": 707.77, "word": " سالب", "probability": 0.8894856770833334}, {"start": 707.77, "end": 708.07, "word": " واحد", "probability": 0.99462890625}, {"start": 708.07, "end": 708.21, "word": " على", "probability": 0.79541015625}, {"start": 708.21, "end": 708.41, "word": " n", "probability": 0.89013671875}, {"start": 708.41, "end": 709.51, "word": " limit", "probability": 0.94482421875}, {"start": 709.51, "end": 709.95, "word": " سالب", "probability": 0.9601236979166666}, {"start": 709.95, "end": 710.27, "word": " واحد", "probability": 0.992431640625}, {"start": 710.27, "end": 710.41, "word": " على", "probability": 0.89453125}, {"start": 710.41, "end": 710.55, "word": " n", "probability": 0.9296875}, {"start": 710.55, "end": 710.93, "word": " بساوي", "probability": 0.9237060546875}, {"start": 710.93, "end": 711.31, "word": " سفر", "probability": 0.9894205729166666}], "temperature": 1.0}, {"id": 26, "seek": 73061, "start": 712.95, "end": 730.61, "text": "و ليست سالب infinity okay تمام؟ إذا النظرية ال comparison test لا يقبل .. لا يقبل التأويل زي ما بيقولوا بس النتائج تبعتها كما هي في a و b أي شيء آخر مش مظبوط", "tokens": [2407, 32239, 14851, 8608, 6027, 3555, 13202, 1392, 46811, 10943, 22807, 11933, 15730, 28239, 19913, 2288, 10632, 2423, 9660, 1500, 20193, 7251, 4587, 36150, 4386, 20193, 7251, 4587, 36150, 16712, 10721, 2407, 26895, 30767, 1829, 19446, 4724, 1829, 39648, 14407, 4724, 3794, 28239, 2655, 16373, 7435, 6055, 3555, 34268, 11296, 9122, 15042, 39896, 8978, 257, 4032, 272, 36632, 44049, 38207, 19753, 34740, 37893, 3714, 19913, 3555, 2407, 9566], "avg_logprob": -0.2250905814378158, "compression_ratio": 1.4180790960451977, "no_speech_prob": 0.0, "words": [{"start": 712.95, "end": 713.19, "word": "و", "probability": 0.74951171875}, {"start": 713.19, "end": 713.79, "word": " ليست", "probability": 0.711669921875}, {"start": 713.79, "end": 714.97, "word": " سالب", "probability": 0.6280924479166666}, {"start": 714.97, "end": 715.47, "word": " infinity", "probability": 0.7041015625}, {"start": 715.47, "end": 716.63, "word": " okay", "probability": 0.273193359375}, {"start": 716.63, "end": 717.43, "word": " تمام؟", "probability": 0.6568603515625}, {"start": 717.43, "end": 718.03, "word": " إذا", "probability": 0.64892578125}, {"start": 718.03, "end": 718.89, "word": " النظرية", "probability": 0.84075927734375}, {"start": 718.89, "end": 719.25, "word": " ال", "probability": 0.802734375}, {"start": 719.25, "end": 719.65, "word": " comparison", "probability": 0.8984375}, {"start": 719.65, "end": 720.19, "word": " test", "probability": 0.95458984375}, {"start": 720.19, "end": 721.13, "word": " لا", "probability": 0.81787109375}, {"start": 721.13, "end": 721.71, "word": " يقبل", "probability": 0.9778645833333334}, {"start": 721.71, "end": 722.17, "word": " ..", "probability": 0.60498046875}, {"start": 722.17, "end": 722.43, "word": " لا", "probability": 0.8212890625}, {"start": 722.43, "end": 722.81, "word": " يقبل", "probability": 0.9900716145833334}, {"start": 722.81, "end": 723.41, "word": " التأويل", "probability": 0.96142578125}, {"start": 723.41, "end": 723.65, "word": " زي", "probability": 0.944091796875}, {"start": 723.65, "end": 723.73, "word": " ما", "probability": 0.97607421875}, {"start": 723.73, "end": 724.47, "word": " بيقولوا", "probability": 0.853515625}, {"start": 724.47, "end": 725.43, "word": " بس", "probability": 0.9140625}, {"start": 725.43, "end": 726.21, "word": " النتائج", "probability": 0.887451171875}, {"start": 726.21, "end": 726.71, "word": " تبعتها", "probability": 0.76715087890625}, {"start": 726.71, "end": 727.01, "word": " كما", "probability": 0.928955078125}, {"start": 727.01, "end": 727.23, "word": " هي", "probability": 0.94384765625}, {"start": 727.23, "end": 727.43, "word": " في", "probability": 0.97412109375}, {"start": 727.43, "end": 727.59, "word": " a", "probability": 0.61865234375}, {"start": 727.59, "end": 727.71, "word": " و", "probability": 0.9521484375}, {"start": 727.71, "end": 727.89, "word": " b", "probability": 0.53271484375}, {"start": 727.89, "end": 728.15, "word": " أي", "probability": 0.37646484375}, {"start": 728.15, "end": 728.51, "word": " شيء", "probability": 0.86572265625}, {"start": 728.51, "end": 728.89, "word": " آخر", "probability": 0.770263671875}, {"start": 728.89, "end": 729.99, "word": " مش", "probability": 0.98828125}, {"start": 729.99, "end": 730.61, "word": " مظبوط", "probability": 0.97197265625}], "temperature": 1.0}, {"id": 27, "seek": 75535, "start": 732.05, "end": 755.35, "text": "هو أمثلة بتوضح الأشياء الأخرى تمام؟ في كمان اختبار أخر زي هذا بنسميه limit comparison test فال .. نمسح", 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{"id": 28, "seek": 80103, "start": 776.53, "end": 801.03, "text": "limit comparison test خلّيني أاخد two sequences x in و y in بsequences of positive real numbers", "tokens": [4197, 270, 9660, 1500, 16490, 1211, 11703, 9957, 1829, 5551, 47283, 3215, 732, 22978, 2031, 294, 4032, 288, 294, 4724, 11834, 2667, 295, 3353, 957, 3547], "avg_logprob": -0.29195602734883624, "compression_ratio": 1.0594059405940595, "no_speech_prob": 0.0, "words": [{"start": 776.53, "end": 777.77, "word": "limit", "probability": 0.6502685546875}, {"start": 777.77, "end": 779.01, "word": " comparison", "probability": 0.88525390625}, {"start": 779.01, "end": 779.65, "word": " test", "probability": 0.90380859375}, {"start": 779.65, "end": 791.09, "word": " خلّيني", "probability": 0.82080078125}, {"start": 791.09, "end": 791.59, "word": " أاخد", "probability": 0.644287109375}, {"start": 791.59, "end": 791.87, "word": " two", "probability": 0.75}, {"start": 791.87, "end": 792.55, "word": " sequences", "probability": 0.93896484375}, {"start": 792.55, "end": 793.03, "word": " x", "probability": 0.69091796875}, {"start": 793.03, "end": 793.41, "word": " in", "probability": 0.1802978515625}, {"start": 793.41, "end": 795.51, "word": " و", "probability": 0.8837890625}, {"start": 795.51, "end": 795.81, "word": " y", "probability": 0.8349609375}, {"start": 795.81, "end": 796.59, "word": " in", "probability": 0.9638671875}, {"start": 796.59, "end": 799.09, "word": " بsequences", "probability": 0.725830078125}, {"start": 799.09, "end": 799.55, "word": " of", "probability": 0.9248046875}, {"start": 799.55, "end": 800.01, "word": " positive", "probability": 0.9306640625}, {"start": 800.01, "end": 800.41, "word": " real", "probability": 0.97119140625}, {"start": 800.41, "end": 801.03, "word": " numbers", "probability": 0.89599609375}], "temperature": 1.0}, {"id": 29, "seek": 83020, "start": 804.56, "end": 830.2, "text": "بالتالي سيكون الحدود الموجبة لكي يكونوا سالبة وبعضهم سالبة وبعضهم موجبة بي such that limit ل xn over yn as n tends to infinity بساوي L عدد موجبة", "tokens": [3555, 6027, 2655, 6027, 1829, 8608, 1829, 30544, 21542, 3215, 23328, 9673, 29245, 49401, 5296, 4117, 1829, 7251, 30544, 14407, 8608, 6027, 49401, 46599, 3615, 11242, 16095, 8608, 6027, 49401, 46599, 3615, 11242, 16095, 3714, 29245, 49401, 4724, 1829, 1270, 300, 4948, 5296, 2031, 77, 670, 17861, 382, 297, 12258, 281, 13202, 4724, 3794, 995, 45865, 441, 6225, 3215, 3215, 3714, 29245, 3555, 3660], "avg_logprob": -0.48125, "compression_ratio": 1.4932432432432432, "no_speech_prob": 0.0, "words": [{"start": 804.56, "end": 804.88, "word": "بالتالي", "probability": 0.5478515625}, {"start": 804.88, "end": 805.34, "word": " سيكون", "probability": 0.5824381510416666}, {"start": 805.34, "end": 808.76, "word": " الحدود", "probability": 0.7108968098958334}, {"start": 808.76, "end": 809.26, "word": " الموجبة", "probability": 0.7732747395833334}, {"start": 809.26, "end": 809.78, "word": " لكي", "probability": 0.4790445963541667}, {"start": 809.78, "end": 809.98, "word": " يكونوا", "probability": 0.4297078450520833}, {"start": 809.98, "end": 810.88, "word": " سالبة", "probability": 0.51611328125}, {"start": 810.88, "end": 811.56, "word": " وبعضهم", "probability": 0.7420654296875}, {"start": 811.56, "end": 812.08, "word": " سالبة", "probability": 0.88720703125}, {"start": 812.08, "end": 813.16, "word": " وبعضهم", "probability": 0.9478759765625}, {"start": 813.16, "end": 813.66, "word": " موجبة", "probability": 0.94482421875}, {"start": 813.66, "end": 816.82, "word": " بي", "probability": 0.456787109375}, {"start": 816.82, "end": 817.24, "word": " such", "probability": 0.57177734375}, {"start": 817.24, "end": 817.82, "word": " that", "probability": 0.8935546875}, {"start": 817.82, "end": 821.24, "word": " limit", "probability": 0.414306640625}, {"start": 821.24, "end": 823.28, "word": " ل", "probability": 0.58447265625}, {"start": 823.28, "end": 823.96, "word": " xn", "probability": 0.28350830078125}, {"start": 823.96, "end": 824.42, "word": " over", "probability": 0.68212890625}, {"start": 824.42, "end": 825.24, "word": " yn", "probability": 0.94580078125}, {"start": 825.24, "end": 825.78, "word": " as", "probability": 0.80419921875}, {"start": 825.78, "end": 826.12, "word": " n", "probability": 0.87744140625}, {"start": 826.12, "end": 826.4, "word": " tends", "probability": 0.7275390625}, {"start": 826.4, "end": 826.52, "word": " to", "probability": 0.892578125}, {"start": 826.52, "end": 827.08, "word": " infinity", "probability": 0.88330078125}, {"start": 827.08, "end": 828.66, "word": " بساوي", "probability": 0.73602294921875}, {"start": 828.66, "end": 829.02, "word": " L", "probability": 0.3310546875}, {"start": 829.02, "end": 829.48, "word": " عدد", "probability": 0.9235026041666666}, {"start": 829.48, "end": 830.2, "word": " موجبة", "probability": 0.872314453125}], "temperature": 1.0}, {"id": 30, "seek": 86404, "start": 835.72, "end": 864.04, "text": "بنسمي المعادلة add star then limit xn بساوي infinity if and only if limit yn بساوي infinity إذا هنا في عندي limit comparison test", "tokens": [3555, 1863, 38251, 1829, 9673, 3615, 18513, 37977, 909, 3543, 550, 4948, 2031, 77, 4724, 3794, 995, 45865, 13202, 498, 293, 787, 498, 4948, 17861, 4724, 3794, 995, 45865, 13202, 11933, 15730, 34105, 8978, 18871, 16254, 4948, 9660, 1500], "avg_logprob": -0.2015624988824129, "compression_ratio": 1.3015873015873016, "no_speech_prob": 0.0, "words": [{"start": 835.7199999999999, "end": 836.8199999999999, "word": "بنسمي", "probability": 0.7830810546875}, {"start": 836.8199999999999, "end": 837.92, "word": " المعادلة", "probability": 0.9476318359375}, {"start": 837.92, "end": 838.18, "word": " add", "probability": 0.67626953125}, {"start": 838.18, "end": 838.64, "word": " star", "probability": 0.257080078125}, {"start": 838.64, "end": 842.32, "word": " then", "probability": 0.32958984375}, {"start": 842.32, "end": 844.06, "word": " limit", "probability": 0.9423828125}, {"start": 844.06, 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"probability": 0.9267578125}, {"start": 862.36, "end": 862.54, "word": " في", "probability": 0.63232421875}, {"start": 862.54, "end": 862.8, "word": " عندي", "probability": 0.92724609375}, {"start": 862.8, "end": 863.04, "word": " limit", "probability": 0.96484375}, {"start": 863.04, "end": 863.58, "word": " comparison", "probability": 0.94873046875}, {"start": 863.58, "end": 864.04, "word": " test", "probability": 0.955078125}], "temperature": 1.0}, {"id": 31, "seek": 89209, "start": 864.89, "end": 892.09, "text": "الذي يتم استخدامه لسيقونسات واحدة فقط من حدوث واحدة واحدة فقط من حدوث واحدة فقط من حدوث واحدة فقط من حدوث واحدة فقط من حدوث واحدة فقط من حدوث واحدة فقط من حدوث واحدة فقط من حدوث واحدة فقط من حدوث واحدة فقط من حدوث واحدة فقط من حدوث واحدة فقط من حدوث واحدة فقط من حدوث واحدة فقط من حدوث واحدة فقط من حدوث واحدة فقط من حدوث واحدة فقط من حدوث واحدة فقط من حدوث واحدة فقط من حدوث واحدة فقط من حدوث واحدة ف", "tokens": [6027, 8848, 1829, 7251, 39237, 44713, 9778, 3215, 10943, 3224, 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"probability": 0.988525390625}, {"start": 877.07, "end": 877.07, "word": " من", "probability": 0.98193359375}, {"start": 877.07, "end": 877.07, "word": " حدوث", "probability": 0.9620361328125}, {"start": 877.07, "end": 877.69, "word": " واحدة", "probability": 0.98681640625}, {"start": 877.69, "end": 878.03, "word": " فقط", "probability": 0.98779296875}, {"start": 878.03, "end": 878.11, "word": " من", "probability": 0.98193359375}, {"start": 878.11, "end": 878.11, "word": " حدوث", "probability": 0.965087890625}, {"start": 878.11, "end": 878.11, "word": " واحدة", "probability": 0.9853515625}, {"start": 878.11, "end": 878.11, "word": " فقط", "probability": 0.986083984375}, {"start": 878.11, "end": 880.07, "word": " من", "probability": 0.98193359375}, {"start": 880.07, "end": 882.85, "word": " حدوث", "probability": 0.9703369140625}, {"start": 882.85, "end": 883.51, "word": " واحدة", "probability": 0.9833984375}, {"start": 883.51, "end": 883.51, "word": " فقط", "probability": 0.98583984375}, {"start": 883.51, "end": 885.73, "word": " من", "probability": 0.98046875}, {"start": 885.73, "end": 885.95, "word": " حدوث", "probability": 0.9754638671875}, {"start": 885.95, "end": 886.61, "word": " واحدة", "probability": 0.9806315104166666}, {"start": 886.61, "end": 886.83, "word": " فقط", "probability": 0.983154296875}, {"start": 886.83, "end": 891.01, "word": " من", "probability": 0.97900390625}, {"start": 891.01, "end": 892.09, "word": " حدوث", "probability": 0.97705078125}, {"start": 892.09, "end": 892.09, "word": " واحدة", "probability": 0.9749348958333334}, {"start": 892.09, "end": 892.09, "word": " ف", "probability": 0.98486328125}], "temperature": 1.0}, {"id": 32, "seek": 91774, "start": 899.92, "end": 917.74, "text": "let assume ال أكبر من السفر satisfies المعادلة لسه نفرض ان في عدد حقيقي ال", "tokens": [2631, 6552, 2423, 5551, 4117, 26890, 9154, 21136, 5172, 2288, 44271, 9673, 3615, 18513, 37977, 5296, 3794, 3224, 8717, 5172, 43042, 16472, 8978, 6225, 3215, 3215, 11331, 38436, 38436, 2423], "avg_logprob": -0.22983870679332363, "compression_ratio": 1.16, "no_speech_prob": 0.0, "words": [{"start": 899.92, "end": 901.12, "word": "let", "probability": 0.62451171875}, {"start": 901.12, "end": 902.32, "word": " assume", "probability": 0.9033203125}, {"start": 902.32, "end": 905.26, "word": " ال", "probability": 0.62841796875}, {"start": 905.26, "end": 905.88, "word": " أكبر", "probability": 0.9226888020833334}, {"start": 905.88, "end": 906.12, "word": " من", "probability": 0.9951171875}, {"start": 906.12, "end": 906.72, "word": " السفر", "probability": 0.8116861979166666}, {"start": 906.72, "end": 910.86, "word": " satisfies", "probability": 0.677734375}, {"start": 910.86, "end": 915.3, "word": " المعادلة", "probability": 0.9676513671875}, {"start": 915.3, "end": 915.68, "word": " لسه", "probability": 0.6349283854166666}, {"start": 915.68, "end": 916.1, "word": " نفرض", "probability": 0.777587890625}, {"start": 916.1, "end": 916.24, "word": " ان", "probability": 0.283935546875}, {"start": 916.24, "end": 916.42, "word": " في", "probability": 0.78466796875}, {"start": 916.42, "end": 916.86, "word": " عدد", "probability": 0.93505859375}, {"start": 916.86, "end": 917.48, "word": " حقيقي", "probability": 0.9549153645833334}, {"start": 917.48, "end": 917.74, "word": " ال", "probability": 0.8408203125}], "temperature": 1.0}, {"id": 33, "seek": 94079, "start": 918.69, "end": 940.79, "text": "وهو بحقق star يعني هو limit ل ratio ل xn على yn تمام؟ take epsilon بساوي L على 2", "tokens": [2407, 3224, 2407, 4724, 5016, 4587, 4587, 3543, 37495, 22653, 31439, 4948, 5296, 8509, 5296, 2031, 77, 15844, 17861, 46811, 10943, 22807, 747, 17889, 4724, 3794, 995, 45865, 441, 15844, 568], "avg_logprob": -0.3100586012005806, "compression_ratio": 1.088235294117647, "no_speech_prob": 0.0, "words": [{"start": 918.69, "end": 919.45, "word": "وهو", "probability": 0.8693033854166666}, {"start": 919.45, "end": 920.49, "word": " بحقق", "probability": 0.8270263671875}, {"start": 920.49, "end": 920.91, "word": " star", "probability": 0.431884765625}, {"start": 920.91, "end": 921.17, "word": " يعني", "probability": 0.75048828125}, {"start": 921.17, "end": 921.39, "word": " هو", "probability": 0.9638671875}, {"start": 921.39, "end": 921.91, "word": " limit", "probability": 0.87451171875}, {"start": 921.91, "end": 923.21, "word": " ل", "probability": 0.5380859375}, {"start": 923.21, "end": 924.71, "word": " ratio", "probability": 0.78857421875}, {"start": 924.71, "end": 925.01, "word": " ل", "probability": 0.69677734375}, {"start": 925.01, "end": 925.53, "word": " xn", "probability": 0.54052734375}, {"start": 925.53, "end": 925.71, "word": " على", "probability": 0.736328125}, {"start": 925.71, "end": 926.13, "word": " yn", "probability": 0.9619140625}, {"start": 926.13, "end": 930.89, "word": " تمام؟", "probability": 0.6334635416666666}, {"start": 930.89, "end": 931.39, "word": " take", "probability": 0.7431640625}, {"start": 931.39, "end": 934.65, "word": " epsilon", "probability": 0.65576171875}, {"start": 934.65, "end": 938.93, "word": " بساوي", "probability": 0.9063720703125}, {"start": 938.93, "end": 940.35, "word": " L", "probability": 0.5205078125}, {"start": 940.35, "end": 940.57, "word": " على", "probability": 0.35791015625}, {"start": 940.57, "end": 940.79, "word": " 2", "probability": 0.81103515625}], "temperature": 1.0}, {"id": 34, "seek": 96586, "start": 943.48, "end": 965.86, "text": "Since L is positive L over 2 is positive لأن أنا جبت إبسلون which is positive طيب since من الفرض since by star XN over YN converges to L", "tokens": [50, 259, 384, 441, 307, 3353, 441, 670, 568, 307, 3353, 5296, 33456, 41850, 10874, 3555, 2655, 11933, 3555, 3794, 1211, 11536, 597, 307, 3353, 23032, 1829, 3555, 1670, 9154, 27188, 43042, 1670, 538, 3543, 1783, 45, 670, 398, 45, 9652, 2880, 281, 441], "avg_logprob": -0.3840277830759684, "compression_ratio": 1.2677165354330708, "no_speech_prob": 0.0, "words": [{"start": 943.48, "end": 944.12, "word": "Since", "probability": 0.545166015625}, {"start": 944.12, "end": 944.42, "word": " L", "probability": 0.74853515625}, {"start": 944.42, "end": 944.66, "word": " is", "probability": 0.78271484375}, {"start": 944.66, "end": 945.22, "word": " positive", "probability": 0.89208984375}, {"start": 945.22, "end": 945.68, "word": " L", "probability": 0.40478515625}, {"start": 945.68, "end": 946.0, "word": " over", "probability": 0.81005859375}, {"start": 946.0, "end": 946.28, "word": " 2", "probability": 0.70703125}, {"start": 946.28, "end": 946.46, "word": " is", "probability": 0.89892578125}, {"start": 946.46, "end": 946.92, "word": " positive", "probability": 0.94921875}, {"start": 946.92, "end": 947.12, "word": " لأن", "probability": 0.412353515625}, {"start": 947.12, "end": 947.42, "word": " أنا", "probability": 0.19921875}, {"start": 947.42, "end": 949.04, "word": " جبت", "probability": 0.845703125}, {"start": 949.04, "end": 949.62, "word": " إبسلون", "probability": 0.8116943359375}, {"start": 949.62, "end": 951.42, "word": " which", "probability": 0.485107421875}, {"start": 951.42, "end": 951.88, "word": " is", "probability": 0.966796875}, {"start": 951.88, "end": 953.62, "word": " positive", "probability": 0.94384765625}, {"start": 953.62, "end": 954.56, "word": " طيب", "probability": 0.7571614583333334}, {"start": 954.56, "end": 955.1, "word": " since", "probability": 0.327392578125}, {"start": 955.1, "end": 957.36, "word": " من", "probability": 0.927734375}, {"start": 957.36, "end": 957.9, "word": " الفرض", "probability": 0.9482421875}, {"start": 957.9, "end": 958.5, "word": " since", "probability": 0.58837890625}, {"start": 958.5, "end": 960.44, "word": " by", "probability": 0.64306640625}, {"start": 960.44, "end": 961.02, "word": " star", "probability": 0.83203125}, {"start": 961.02, "end": 961.94, "word": " XN", "probability": 0.51220703125}, {"start": 961.94, "end": 962.48, "word": " over", "probability": 0.916015625}, {"start": 962.48, "end": 963.62, "word": " YN", "probability": 0.875}, {"start": 963.62, "end": 964.98, "word": " converges", "probability": 0.964111328125}, {"start": 964.98, "end": 965.38, "word": " to", "probability": 0.98193359375}, {"start": 965.38, "end": 965.86, "word": " L", "probability": 0.9892578125}], "temperature": 1.0}, {"id": 35, "seek": 98030, "start": 967.88, "end": 980.3, "text": "وهي عندي إبسلون أكبر من الصفر is given إذا by definition of convergence by epsilon capital N definition لإبسلون هذه for this إبسلون", "tokens": [2407, 3224, 1829, 18871, 16254, 11933, 3555, 3794, 1211, 11536, 5551, 4117, 26890, 9154, 31767, 5172, 2288, 307, 2212, 11933, 15730, 538, 7123, 295, 32181, 538, 17889, 4238, 426, 7123, 5296, 28814, 3555, 3794, 1211, 11536, 29538, 337, 341, 11933, 3555, 3794, 1211, 11536], "avg_logprob": -0.2965277645323012, "compression_ratio": 1.2985074626865671, "no_speech_prob": 0.0, "words": [{"start": 967.88, "end": 968.3, "word": "وهي", "probability": 0.609375}, {"start": 968.3, "end": 968.62, "word": " عندي", "probability": 0.80908203125}, {"start": 968.62, "end": 969.1, "word": " إبسلون", "probability": 0.809423828125}, {"start": 969.1, "end": 969.48, "word": " أكبر", "probability": 0.9656575520833334}, {"start": 969.48, "end": 969.62, "word": " من", "probability": 0.9873046875}, {"start": 969.62, "end": 970.06, "word": " الصفر", "probability": 0.8333333333333334}, {"start": 970.06, "end": 970.18, "word": " is", "probability": 0.74755859375}, {"start": 970.18, "end": 970.46, "word": " given", "probability": 0.92236328125}, {"start": 970.46, "end": 970.74, "word": " إذا", "probability": 0.5506591796875}, {"start": 970.74, "end": 970.98, "word": " by", "probability": 0.87841796875}, {"start": 970.98, "end": 971.46, "word": " definition", "probability": 0.86181640625}, {"start": 971.46, "end": 971.96, "word": " of", "probability": 0.96826171875}, {"start": 971.96, "end": 973.58, "word": " convergence", "probability": 0.9765625}, {"start": 973.58, "end": 974.26, "word": " by", "probability": 0.5009765625}, {"start": 974.26, "end": 974.7, "word": " epsilon", "probability": 0.3623046875}, {"start": 974.7, "end": 975.08, "word": " capital", "probability": 0.7587890625}, {"start": 975.08, "end": 975.4, "word": " N", "probability": 0.8037109375}, {"start": 975.4, "end": 976.26, "word": " definition", "probability": 0.90380859375}, {"start": 976.26, "end": 978.32, "word": " لإبسلون", "probability": 0.9055989583333334}, {"start": 978.32, "end": 978.76, "word": " هذه", "probability": 0.404052734375}, {"start": 978.76, "end": 979.1, "word": " for", "probability": 0.78857421875}, {"start": 979.1, "end": 979.56, "word": " this", "probability": 0.96630859375}, {"start": 979.56, "end": 980.3, "word": " إبسلون", "probability": 0.827880859375}], "temperature": 1.0}, {"id": 36, "seek": 100886, "start": 981.5, "end": 1008.86, "text": "there exist capital N يعتمد على إبسلون يعتمد على الإبسلون اللي هي بتعتمد على العدد L natural number بحيث أنه لكل N أكبر منه سوى capital N بيطلع عندي absolute xn over yn negative L less than إبسلون", "tokens": [15456, 2514, 4238, 426, 7251, 34268, 2304, 3215, 15844, 11933, 3555, 3794, 1211, 11536, 7251, 34268, 2304, 3215, 15844, 33688, 3555, 3794, 1211, 11536, 13672, 1829, 39896, 39894, 34268, 2304, 3215, 15844, 18863, 3215, 3215, 441, 3303, 1230, 4724, 5016, 1829, 12984, 14739, 3224, 5296, 28820, 426, 5551, 4117, 26890, 9154, 3224, 8608, 2407, 7578, 4238, 426, 4724, 1829, 9566, 1211, 3615, 18871, 16254, 8236, 2031, 77, 670, 17861, 3671, 441, 1570, 813, 11933, 3555, 3794, 1211, 11536], "avg_logprob": -0.24327531758742996, "compression_ratio": 1.5524861878453038, "no_speech_prob": 0.0, "words": [{"start": 981.5, "end": 981.86, "word": "there", "probability": 0.285888671875}, {"start": 981.86, "end": 982.4, "word": " exist", "probability": 0.55224609375}, {"start": 982.4, "end": 982.94, "word": " capital", "probability": 0.64990234375}, {"start": 982.94, "end": 983.24, 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"probability": 0.638671875}, {"start": 993.22, "end": 994.78, "word": " natural", "probability": 0.62255859375}, {"start": 994.78, "end": 995.58, "word": " number", "probability": 0.982421875}, {"start": 995.58, "end": 996.24, "word": " بحيث", "probability": 0.9691162109375}, {"start": 996.24, "end": 996.6, "word": " أنه", "probability": 0.728515625}, {"start": 996.6, "end": 997.16, "word": " لكل", "probability": 0.9873046875}, {"start": 997.16, "end": 997.58, "word": " N", "probability": 0.87646484375}, {"start": 997.58, "end": 998.74, "word": " أكبر", "probability": 0.9728190104166666}, {"start": 998.74, "end": 999.1, "word": " منه", "probability": 0.748291015625}, {"start": 999.1, "end": 999.6, "word": " سوى", "probability": 0.7266438802083334}, {"start": 999.6, "end": 1000.08, "word": " capital", "probability": 0.626953125}, {"start": 1000.08, "end": 1000.48, "word": " N", "probability": 0.9384765625}, {"start": 1000.48, "end": 1001.62, "word": " بيطلع", "probability": 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Xn over Yn minus L less than L two bigger than negative L ع اتنين و هذا صحيح لكل N bigger than or equal N اجمع L على كل الأطراف", "tokens": [3224, 11622, 3555, 35473, 39896, 4117, 22807, 23032, 1829, 3555, 2423, 39184, 398, 4724, 3794, 45865, 441, 670, 568, 16490, 1211, 11703, 9957, 995, 8717, 8592, 26895, 2423, 8236, 2158, 6156, 3555, 9381, 13546, 16472, 16254, 1783, 77, 670, 398, 77, 3175, 441, 1570, 813, 441, 732, 3801, 813, 3671, 441, 6225, 1975, 2655, 1863, 9957, 4032, 23758, 20328, 5016, 1829, 5016, 5296, 28820, 426, 3801, 813, 420, 2681, 426, 1975, 7435, 2304, 3615, 441, 15844, 28242, 16247, 9566, 23557, 5172], "avg_logprob": -0.27667683362960815, "compression_ratio": 1.3142857142857143, "no_speech_prob": 0.0, "words": [{"start": 1010.22, "end": 1010.74, "word": "هزبوت", "probability": 0.669189453125}, {"start": 1010.74, "end": 1011.96, "word": " هيك؟", "probability": 0.79296875}, {"start": 1011.96, "end": 1012.66, "word": " طيب", "probability": 0.8235677083333334}, {"start": 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1019.72, "word": " اندي", "probability": 0.557861328125}, {"start": 1019.72, "end": 1020.28, "word": " Xn", "probability": 0.5662841796875}, {"start": 1020.28, "end": 1020.62, "word": " over", "probability": 0.90625}, {"start": 1020.62, "end": 1021.38, "word": " Yn", "probability": 0.6453857421875}, {"start": 1021.38, "end": 1022.56, "word": " minus", "probability": 0.74072265625}, {"start": 1022.56, "end": 1022.9, "word": " L", "probability": 0.8701171875}, {"start": 1022.9, "end": 1023.2, "word": " less", "probability": 0.84619140625}, {"start": 1023.2, "end": 1023.56, "word": " than", "probability": 0.95166015625}, {"start": 1023.56, "end": 1023.9, "word": " L", "probability": 0.9287109375}, {"start": 1023.9, "end": 1024.38, "word": " two", "probability": 0.416748046875}, {"start": 1024.38, "end": 1024.76, "word": " bigger", "probability": 0.9501953125}, {"start": 1024.76, "end": 1025.34, "word": " than", "probability": 0.94580078125}, {"start": 1025.34, "end": 1026.98, "word": " negative", "probability": 0.7451171875}, {"start": 1026.98, "end": 1027.74, "word": " L", "probability": 0.9560546875}, {"start": 1027.74, "end": 1027.92, "word": " ع", "probability": 0.48388671875}, {"start": 1027.92, "end": 1028.36, "word": " اتنين", "probability": 0.8043212890625}, {"start": 1028.36, "end": 1030.96, "word": " و", "probability": 0.454833984375}, {"start": 1030.96, "end": 1031.16, "word": " هذا", "probability": 0.74462890625}, {"start": 1031.16, "end": 1031.72, "word": " صحيح", "probability": 0.998046875}, {"start": 1031.72, "end": 1032.3, "word": " لكل", "probability": 0.979248046875}, {"start": 1032.3, "end": 1032.74, "word": " N", "probability": 0.437744140625}, {"start": 1032.74, "end": 1033.72, "word": " bigger", "probability": 0.9541015625}, {"start": 1033.72, "end": 1034.06, "word": " than", "probability": 0.96044921875}, {"start": 1034.06, "end": 1034.18, "word": " or", "probability": 0.8701171875}, {"start": 1034.18, "end": 1034.42, "word": " equal", "probability": 0.9111328125}, {"start": 1034.42, "end": 1034.86, "word": " N", "probability": 0.50439453125}, {"start": 1034.86, "end": 1036.4, "word": " اجمع", "probability": 0.90576171875}, {"start": 1036.4, "end": 1036.72, "word": " L", "probability": 0.724609375}, {"start": 1036.72, "end": 1036.9, "word": " على", "probability": 0.83203125}, {"start": 1036.9, "end": 1037.2, "word": " كل", "probability": 0.9912109375}, {"start": 1037.2, "end": 1037.92, "word": " الأطراف", "probability": 0.96484375}], "temperature": 1.0}, {"id": 38, "seek": 106668, "start": 1039.08, "end": 1066.68, "text": "إن أنا بطلع عندي xn over yn less than three over two L bigger than L over two and this is true for every n bigger than or equal n نسمي المتباينة هذه double star now", "tokens": [28814, 1863, 41850, 4724, 9566, 1211, 3615, 18871, 16254, 2031, 77, 670, 17861, 1570, 813, 1045, 670, 732, 441, 3801, 813, 441, 670, 732, 293, 341, 307, 2074, 337, 633, 297, 3801, 813, 420, 2681, 297, 8717, 38251, 1829, 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"end": 1046.66, "word": " over", "probability": 0.93212890625}, {"start": 1046.66, "end": 1047.12, "word": " two", "probability": 0.95947265625}, {"start": 1047.12, "end": 1047.5, "word": " L", "probability": 0.25634765625}, {"start": 1047.5, "end": 1047.88, "word": " bigger", "probability": 0.87255859375}, {"start": 1047.88, "end": 1048.4, "word": " than", "probability": 0.94580078125}, {"start": 1048.4, "end": 1048.84, "word": " L", "probability": 0.92431640625}, {"start": 1048.84, "end": 1049.26, "word": " over", "probability": 0.9140625}, {"start": 1049.26, "end": 1049.7, "word": " two", "probability": 0.9375}, {"start": 1049.7, "end": 1051.68, "word": " and", "probability": 0.64306640625}, {"start": 1051.68, "end": 1051.92, "word": " this", "probability": 0.9541015625}, {"start": 1051.92, "end": 1052.06, "word": " is", "probability": 0.88037109375}, {"start": 1052.06, "end": 1052.38, "word": " true", "probability": 0.96826171875}, {"start": 1052.38, "end": 1052.74, "word": " for", "probability": 0.97216796875}, {"start": 1052.74, "end": 1053.12, "word": " every", "probability": 0.8193359375}, {"start": 1053.12, "end": 1053.34, "word": " n", "probability": 0.69775390625}, {"start": 1053.34, "end": 1053.62, "word": " bigger", "probability": 0.9404296875}, {"start": 1053.62, "end": 1053.96, "word": " than", "probability": 0.9375}, {"start": 1053.96, "end": 1054.14, "word": " or", "probability": 0.9111328125}, {"start": 1054.14, "end": 1054.38, "word": " equal", "probability": 0.91015625}, {"start": 1054.38, "end": 1055.26, "word": " n", "probability": 0.49755859375}, {"start": 1055.26, "end": 1056.62, "word": " نسمي", "probability": 0.86083984375}, {"start": 1056.62, "end": 1057.78, "word": " المتباينة", "probability": 0.9383138020833334}, {"start": 1057.78, "end": 1058.5, "word": " هذه", "probability": 0.62841796875}, {"start": 1058.5, "end": 1060.16, "word": " double", "probability": 0.64794921875}, {"start": 1060.16, "end": 1060.7, "word": " star", "probability": 0.7451171875}, {"start": 1060.7, "end": 1066.68, "word": " now", "probability": 0.634765625}], "temperature": 1.0}, {"id": 39, "seek": 109582, "start": 1072.64, "end": 1095.82, "text": "by double star انا عندي xn على yn اصغر من تلاتة ع اتنين ال اللي هو الجزء هذا وهذا صحيح لكل n bigger than or equal to n بيقدي انه xn", "tokens": [2322, 3834, 3543, 1975, 8315, 18871, 16254, 2031, 77, 15844, 17861, 1975, 9381, 17082, 2288, 9154, 6055, 1211, 9307, 3660, 6225, 1975, 2655, 1863, 9957, 2423, 13672, 1829, 31439, 25724, 11622, 38207, 23758, 37037, 15730, 20328, 5016, 1829, 5016, 5296, 28820, 297, 3801, 813, 420, 2681, 281, 297, 4724, 1829, 4587, 16254, 16472, 3224, 2031, 77], "avg_logprob": -0.23670503758547598, "compression_ratio": 1.269736842105263, "no_speech_prob": 1.7881393432617188e-07, "words": [{"start": 1072.64, "end": 1073.18, "word": "by", "probability": 0.474365234375}, {"start": 1073.18, "end": 1073.84, "word": " double", "probability": 0.81396484375}, {"start": 1073.84, "end": 1074.38, "word": " star", "probability": 0.91162109375}, {"start": 1074.38, "end": 1077.04, "word": " انا", "probability": 0.6728515625}, {"start": 1077.04, "end": 1077.48, "word": " عندي", "probability": 0.799072265625}, {"start": 1077.48, "end": 1078.48, "word": " xn", "probability": 0.640869140625}, {"start": 1078.48, "end": 1080.04, "word": " على", "probability": 0.775390625}, {"start": 1080.04, "end": 1080.56, "word": " yn", "probability": 0.90771484375}, {"start": 1080.56, "end": 1081.42, "word": " اصغر", "probability": 0.885498046875}, {"start": 1081.42, "end": 1081.74, "word": " من", "probability": 0.99072265625}, {"start": 1081.74, "end": 1082.28, "word": " تلاتة", "probability": 0.8896484375}, {"start": 1082.28, "end": 1082.4, "word": " ع", "probability": 0.7021484375}, {"start": 1082.4, "end": 1082.8, "word": " اتنين", "probability": 0.832763671875}, {"start": 1082.8, "end": 1083.1, "word": " ال", "probability": 0.392333984375}, {"start": 1083.1, "end": 1084.08, "word": " اللي", "probability": 0.671875}, {"start": 1084.08, "end": 1084.28, "word": " هو", "probability": 0.97998046875}, {"start": 1084.28, "end": 1084.68, "word": " الجزء", "probability": 0.9752604166666666}, {"start": 1084.68, "end": 1085.08, "word": " هذا", "probability": 0.8271484375}, {"start": 1085.08, "end": 1087.36, "word": " وهذا", "probability": 0.769287109375}, {"start": 1087.36, "end": 1087.88, "word": " صحيح", "probability": 0.9970703125}, {"start": 1087.88, "end": 1088.4, "word": " لكل", "probability": 0.981689453125}, {"start": 1088.4, "end": 1088.74, "word": " n", "probability": 0.74267578125}, {"start": 1088.74, "end": 1089.02, "word": " bigger", "probability": 0.91357421875}, {"start": 1089.02, "end": 1089.34, "word": " than", "probability": 0.9482421875}, {"start": 1089.34, "end": 1089.5, "word": " or", "probability": 0.91552734375}, {"start": 1089.5, "end": 1089.66, "word": " equal", "probability": 0.9052734375}, {"start": 1089.66, "end": 1089.74, "word": " to", "probability": 0.38818359375}, {"start": 1089.74, "end": 1090.06, "word": " n", "probability": 0.87451171875}, {"start": 1090.06, "end": 1091.98, "word": " بيقدي", "probability": 0.70147705078125}, {"start": 1091.98, "end": 1093.96, "word": " انه", "probability": 0.674072265625}, {"start": 1093.96, "end": 1095.82, "word": " xn", "probability": 0.910888671875}], "temperature": 1.0}, {"id": 40, "seek": 112972, "start": 1104.52, "end": 1129.72, "text": "في اتنين على التلاتة L less than Y N وهذا صحيح لكل N bigger than or equal N تصبوت هيك صح؟ هذه المتباينة بتقدر هذه هي نفس هذه", "tokens": [41185, 1975, 2655, 1863, 9957, 15844, 16712, 1211, 9307, 3660, 441, 1570, 813, 398, 426, 37037, 15730, 20328, 5016, 1829, 5016, 5296, 28820, 426, 3801, 813, 420, 2681, 426, 6055, 9381, 3555, 35473, 39896, 4117, 20328, 5016, 22807, 29538, 9673, 2655, 3555, 995, 9957, 3660, 39894, 28543, 2288, 29538, 39896, 8717, 36178, 29538], "avg_logprob": -0.2660590222588292, "compression_ratio": 1.3172413793103448, "no_speech_prob": 0.0, "words": [{"start": 1104.52, "end": 1104.88, "word": "في", "probability": 0.7705078125}, {"start": 1104.88, "end": 1105.98, "word": " اتنين", "probability": 0.9376220703125}, {"start": 1105.98, "end": 1108.12, "word": " على", "probability": 0.7080078125}, {"start": 1108.12, "end": 1109.06, "word": " التلاتة", "probability": 0.9464111328125}, {"start": 1109.06, "end": 1109.44, "word": " L", "probability": 0.39892578125}, {"start": 1109.44, "end": 1110.72, "word": " less", "probability": 0.77001953125}, {"start": 1110.72, "end": 1111.14, "word": " than", "probability": 0.93017578125}, {"start": 1111.14, "end": 1111.62, "word": " Y", "probability": 0.41015625}, {"start": 1111.62, "end": 1111.98, "word": " N", "probability": 0.29541015625}, {"start": 1111.98, "end": 1113.6, "word": " وهذا", "probability": 0.79345703125}, {"start": 1113.6, "end": 1114.12, "word": " صحيح", "probability": 0.9962158203125}, {"start": 1114.12, "end": 1114.68, "word": " لكل", "probability": 0.969482421875}, {"start": 1114.68, "end": 1114.98, "word": " N", "probability": 0.86083984375}, {"start": 1114.98, "end": 1115.28, "word": " bigger", "probability": 0.93701171875}, {"start": 1115.28, "end": 1115.66, "word": " than", "probability": 0.96337890625}, {"start": 1115.66, "end": 1115.82, "word": " or", "probability": 0.78857421875}, {"start": 1115.82, "end": 1116.12, "word": " equal", "probability": 0.92333984375}, {"start": 1116.12, "end": 1118.86, "word": " N", "probability": 0.52685546875}, {"start": 1118.86, "end": 1120.6, "word": " تصبوت", "probability": 0.539306640625}, {"start": 1120.6, "end": 1120.88, "word": " هيك", "probability": 0.8212890625}, {"start": 1120.88, "end": 1121.84, "word": " صح؟", "probability": 0.7111002604166666}, {"start": 1121.84, "end": 1125.26, "word": " هذه", "probability": 0.23583984375}, {"start": 1125.26, "end": 1125.98, "word": " المتباينة", "probability": 0.8841145833333334}, {"start": 1125.98, "end": 1126.4, "word": " بتقدر", "probability": 0.71337890625}, {"start": 1126.4, "end": 1126.78, "word": " هذه", "probability": 0.68408203125}, {"start": 1126.78, "end": 1129.08, "word": " هي", "probability": 0.7509765625}, {"start": 1129.08, "end": 1129.36, "word": " نفس", "probability": 0.9990234375}, {"start": 1129.36, "end": 1129.72, "word": " هذه", "probability": 0.86767578125}], "temperature": 1.0}, {"id": 41, "seek": 115069, "start": 1133.61, "end": 1150.69, "text": "so as limit احنا فرض .. now now او .. او so if", "tokens": [539, 382, 4948, 1975, 5016, 8315, 6156, 43042, 4386, 586, 586, 1975, 2407, 4386, 1975, 2407, 370, 498], "avg_logprob": -0.31517268795716136, "compression_ratio": 1.0, "no_speech_prob": 0.0, "words": [{"start": 1133.61, "end": 1134.21, "word": "so", "probability": 0.501953125}, {"start": 1134.21, "end": 1135.57, "word": " as", "probability": 0.9345703125}, {"start": 1135.57, "end": 1138.01, "word": " limit", "probability": 0.92236328125}, {"start": 1138.01, "end": 1139.05, "word": " احنا", "probability": 0.83837890625}, {"start": 1139.05, "end": 1139.37, "word": " فرض", "probability": 0.87841796875}, {"start": 1139.37, "end": 1139.53, "word": " ..", "probability": 0.138671875}, {"start": 1139.53, "end": 1140.43, "word": " now", "probability": 0.861328125}, {"start": 1140.43, "end": 1145.29, "word": " now", "probability": 0.71142578125}, {"start": 1145.29, "end": 1147.17, "word": " او", "probability": 0.811767578125}, {"start": 1147.17, "end": 1147.25, "word": " ..", "probability": 0.432861328125}, {"start": 1147.25, "end": 1148.61, "word": " او", "probability": 0.937255859375}, {"start": 1148.61, "end": 1149.05, "word": " so", "probability": 0.8642578125}, {"start": 1149.05, "end": 1150.69, "word": " if", "probability": 0.94580078125}], "temperature": 1.0}, {"id": 42, "seek": 118017, "start": 1152.93, "end": 1180.17, "text": "limit xn بساوي infinity إذا كان limit xn بساوي infinity و هذا ثابت موجب this is positive constant ف limit كل هذا برضه بساوي plus infinity و بالتالي by comparison test then by comparison by comparison", "tokens": [4197, 270, 2031, 77, 4724, 3794, 995, 45865, 13202, 11933, 15730, 25961, 4948, 2031, 77, 4724, 3794, 995, 45865, 13202, 4032, 23758, 38637, 16758, 2655, 3714, 29245, 3555, 341, 307, 3353, 5754, 6156, 4948, 28242, 23758, 4724, 43042, 3224, 4724, 3794, 995, 45865, 1804, 13202, 4032, 20666, 2655, 6027, 1829, 538, 9660, 1500, 550, 538, 9660, 538, 9660], "avg_logprob": -0.1557203364574303, "compression_ratio": 1.6556291390728477, "no_speech_prob": 0.0, "words": [{"start": 1152.93, "end": 1153.99, "word": "limit", "probability": 0.84912109375}, {"start": 1153.99, "end": 1155.05, "word": " xn", "probability": 0.58935546875}, {"start": 1155.05, "end": 1156.07, "word": " بساوي", "probability": 0.6398162841796875}, {"start": 1156.07, "end": 1156.61, "word": " infinity", "probability": 0.76123046875}, {"start": 1156.61, "end": 1157.87, "word": " إذا", "probability": 0.687744140625}, {"start": 1157.87, "end": 1158.13, 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1163.49, "word": " constant", "probability": 0.96728515625}, {"start": 1163.49, "end": 1164.65, "word": " ف", "probability": 0.953125}, {"start": 1164.65, "end": 1165.01, "word": " limit", "probability": 0.7158203125}, {"start": 1165.01, "end": 1165.37, "word": " كل", "probability": 0.9716796875}, {"start": 1165.37, "end": 1165.79, "word": " هذا", "probability": 0.96484375}, {"start": 1165.79, "end": 1166.29, "word": " برضه", "probability": 0.97802734375}, {"start": 1166.29, "end": 1166.77, "word": " بساوي", "probability": 0.975341796875}, {"start": 1166.77, "end": 1167.13, "word": " plus", "probability": 0.943359375}, {"start": 1167.13, "end": 1167.69, "word": " infinity", "probability": 0.90771484375}, {"start": 1167.69, "end": 1171.65, "word": " و", "probability": 0.52734375}, {"start": 1171.65, "end": 1172.53, "word": " بالتالي", "probability": 0.8995361328125}, {"start": 1172.53, "end": 1172.73, "word": " by", "probability": 0.9375}, {"start": 1172.73, "end": 1173.33, "word": " comparison", "probability": 0.9365234375}, {"start": 1173.33, "end": 1173.71, "word": " test", "probability": 0.95703125}, {"start": 1173.71, "end": 1174.09, "word": " then", "probability": 0.646484375}, {"start": 1174.09, "end": 1175.55, "word": " by", "probability": 0.939453125}, {"start": 1175.55, "end": 1176.51, "word": " comparison", "probability": 0.908203125}, {"start": 1176.51, "end": 1179.45, "word": " by", "probability": 0.81201171875}, {"start": 1179.45, "end": 1180.17, "word": " comparison", "probability": 0.9169921875}], "temperature": 1.0}, {"id": 43, "seek": 120472, "start": 1182.94, "end": 1204.72, "text": "by comparison test النظرية اللى فاتت مع الشرط المخفف مع الشرط المخفف لأن فى النظرية اللى فاتت كان عندي xn أصغر من أو يسوى yn لكل n بعدين قلنا أن هذا الشرط لو خففناه لكل n أكبر من أو سوى عدد طبيعى ما وليكن capital N هنا برضه بتظل صحيحة", "tokens": [2322, 9660, 1500, 28239, 19913, 2288, 10632, 13672, 7578, 6156, 9307, 2655, 20449, 25124, 2288, 9566, 9673, 9778, 5172, 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"word": " النظرية", "probability": 0.84747314453125}, {"start": 1185.0, "end": 1185.18, "word": " اللى", "probability": 0.745849609375}, {"start": 1185.18, "end": 1185.76, "word": " فاتت", "probability": 0.99267578125}, {"start": 1185.76, "end": 1186.98, "word": " مع", "probability": 0.9794921875}, {"start": 1186.98, "end": 1187.48, "word": " الشرط", "probability": 0.98095703125}, {"start": 1187.48, "end": 1188.2, "word": " المخفف", "probability": 0.987060546875}, {"start": 1188.2, "end": 1189.52, "word": " مع", "probability": 0.4990234375}, {"start": 1189.52, "end": 1189.96, "word": " الشرط", "probability": 0.9884440104166666}, {"start": 1189.96, "end": 1190.5, "word": " المخفف", "probability": 0.9930419921875}, {"start": 1190.5, "end": 1190.68, "word": " لأن", "probability": 0.71826171875}, {"start": 1190.68, "end": 1190.82, "word": " فى", "probability": 0.76513671875}, {"start": 1190.82, "end": 1191.16, "word": " النظرية", "probability": 0.8929443359375}, {"start": 1191.16, "end": 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بعدين", "probability": 0.6494140625}, {"start": 1196.54, "end": 1196.8, "word": " قلنا", "probability": 0.735107421875}, {"start": 1196.8, "end": 1196.94, "word": " أن", "probability": 0.28955078125}, {"start": 1196.94, "end": 1197.22, "word": " هذا", "probability": 0.85546875}, {"start": 1197.22, "end": 1197.66, "word": " الشرط", "probability": 0.9820963541666666}, {"start": 1197.66, "end": 1197.78, "word": " لو", "probability": 0.96826171875}, {"start": 1197.78, "end": 1198.44, "word": " خففناه", "probability": 0.96650390625}, {"start": 1198.44, "end": 1198.82, "word": " لكل", "probability": 0.99169921875}, {"start": 1198.82, "end": 1199.1, "word": " n", "probability": 0.7919921875}, {"start": 1199.1, "end": 1199.88, "word": " أكبر", "probability": 0.9615885416666666}, {"start": 1199.88, "end": 1200.02, "word": " من", "probability": 0.9619140625}, {"start": 1200.02, "end": 1200.1, "word": " أو", "probability": 0.8876953125}, {"start": 1200.1, "end": 1200.44, "word": " سوى", "probability": 0.908203125}, {"start": 1200.44, "end": 1200.7, "word": " عدد", "probability": 0.9801432291666666}, {"start": 1200.7, "end": 1201.2, "word": " طبيعى", "probability": 0.9547119140625}, {"start": 1201.2, "end": 1201.46, "word": " ما", "probability": 0.681640625}, {"start": 1201.46, "end": 1202.6, "word": " وليكن", "probability": 0.7347005208333334}, {"start": 1202.6, "end": 1203.0, "word": " capital", "probability": 0.60400390625}, {"start": 1203.0, "end": 1203.26, "word": " N", "probability": 0.76171875}, {"start": 1203.26, "end": 1203.46, "word": " هنا", "probability": 0.9736328125}, {"start": 1203.46, "end": 1203.86, "word": " برضه", "probability": 0.94091796875}, {"start": 1203.86, "end": 1204.2, "word": " بتظل", "probability": 0.6466471354166666}, {"start": 1204.2, "end": 1204.72, "word": " صحيحة", "probability": 0.92314453125}], "temperature": 1.0}, {"id": 44, "seek": 123238, "start": 1205.72, "end": 1232.38, "text": "ف by comparison test and limit ال sequence هذه بساوي infinity إذا limit الأكبر limit yn بساوي infinity تمام؟ الآن بنثبت العكس نثبت الآن العكس طيب conversely", "tokens": [5172, 538, 9660, 1500, 293, 4948, 2423, 8310, 29538, 4724, 3794, 995, 45865, 13202, 11933, 15730, 4948, 16247, 4117, 26890, 4948, 17861, 4724, 3794, 995, 45865, 13202, 46811, 10943, 22807, 6024, 48506, 44945, 12984, 3555, 2655, 18863, 4117, 3794, 8717, 12984, 3555, 2655, 6024, 48506, 18863, 4117, 3794, 23032, 1829, 3555, 2615, 736], "avg_logprob": -0.263599529310509, "compression_ratio": 1.469387755102041, "no_speech_prob": 0.0, "words": [{"start": 1205.72, "end": 1206.0, "word": "ف", "probability": 0.5556640625}, {"start": 1206.0, "end": 1206.44, "word": " by", "probability": 0.40869140625}, {"start": 1206.44, "end": 1207.06, "word": " comparison", "probability": 0.900390625}, {"start": 1207.06, "end": 1208.4, "word": " test", "probability": 0.92333984375}, {"start": 1208.4, "end": 1209.82, "word": " and", "probability": 0.2266845703125}, {"start": 1209.82, "end": 1210.3, "word": " limit", "probability": 0.95947265625}, {"start": 1210.3, "end": 1211.4, "word": " ال", "probability": 0.63818359375}, {"start": 1211.4, "end": 1211.84, "word": " sequence", "probability": 0.810546875}, {"start": 1211.84, "end": 1212.2, "word": " هذه", "probability": 0.4912109375}, {"start": 1212.2, "end": 1212.62, "word": " بساوي", "probability": 0.8248291015625}, {"start": 1212.62, "end": 1213.16, "word": " infinity", "probability": 0.818359375}, {"start": 1213.16, "end": 1213.84, "word": " إذا", "probability": 0.52996826171875}, {"start": 1213.84, "end": 1214.18, "word": " limit", "probability": 0.64013671875}, {"start": 1214.18, "end": 1214.88, "word": " الأكبر", "probability": 0.97705078125}, {"start": 1214.88, "end": 1217.18, "word": " limit", "probability": 0.73486328125}, {"start": 1217.18, "end": 1219.14, "word": " yn", "probability": 0.495849609375}, {"start": 1219.14, "end": 1220.06, "word": " بساوي", "probability": 0.929931640625}, {"start": 1220.06, "end": 1220.7, "word": " infinity", "probability": 0.84716796875}, {"start": 1220.7, "end": 1223.26, "word": " تمام؟", "probability": 0.7381998697916666}, {"start": 1223.26, "end": 1223.76, "word": " الآن", "probability": 0.625244140625}, {"start": 1223.76, "end": 1224.34, "word": " بنثبت", "probability": 0.830078125}, {"start": 1224.34, "end": 1225.52, "word": " العكس", "probability": 0.9923502604166666}, {"start": 1225.52, "end": 1227.92, "word": " نثبت", "probability": 0.887939453125}, {"start": 1227.92, "end": 1228.3, "word": " الآن", "probability": 0.85498046875}, {"start": 1228.3, "end": 1229.22, "word": " العكس", "probability": 0.9978841145833334}, {"start": 1229.22, "end": 1231.22, "word": " طيب", "probability": 0.90087890625}, {"start": 1231.22, "end": 1232.38, "word": " conversely", "probability": 0.8935546875}], "temperature": 1.0}, {"id": 45, "seek": 126402, "start": 1240.74, "end": 1264.02, "text": "Conversely Assume Assume المرهد أنه limit yn بساوي infinity من double star من double star لو أخدت النص .. النص هذا من المتباينة النص الآخر", "tokens": [9838, 4308, 356, 6281, 2540, 6281, 2540, 9673, 2288, 3224, 3215, 14739, 3224, 4948, 17861, 4724, 3794, 995, 45865, 13202, 9154, 3834, 3543, 9154, 3834, 3543, 45164, 5551, 9778, 3215, 2655, 28239, 9381, 4386, 28239, 9381, 23758, 9154, 9673, 2655, 3555, 995, 9957, 3660, 28239, 9381, 6024, 95, 34740], "avg_logprob": -0.2425000011920929, "compression_ratio": 1.3402777777777777, "no_speech_prob": 0.0, "words": [{"start": 1240.7399999999998, "end": 1242.1399999999999, "word": "Conversely", "probability": 0.5782063802083334}, {"start": 1242.1399999999999, "end": 1243.54, "word": " Assume", "probability": 0.707763671875}, {"start": 1243.54, "end": 1245.98, "word": " Assume", "probability": 0.6297607421875}, {"start": 1245.98, "end": 1247.12, "word": " المرهد", "probability": 0.9090576171875}, {"start": 1247.12, "end": 1247.48, "word": " أنه", "probability": 0.593017578125}, {"start": 1247.48, "end": 1247.84, "word": " limit", "probability": 0.7548828125}, {"start": 1247.84, "end": 1250.16, "word": " yn", "probability": 0.251953125}, {"start": 1250.16, "end": 1251.08, "word": " بساوي", "probability": 0.845947265625}, {"start": 1251.08, "end": 1251.8, "word": " infinity", "probability": 0.77294921875}, {"start": 1251.8, "end": 1254.26, "word": " من", "probability": 0.962890625}, {"start": 1254.26, "end": 1255.62, "word": " double", "probability": 0.6435546875}, {"start": 1255.62, "end": 1256.26, "word": " star", "probability": 0.8720703125}, {"start": 1256.26, "end": 1257.12, "word": " من", "probability": 0.83056640625}, {"start": 1257.12, "end": 1257.5, "word": " double", "probability": 0.92138671875}, {"start": 1257.5, "end": 1258.04, "word": " star", "probability": 0.966796875}, {"start": 1258.04, "end": 1259.28, "word": " لو", "probability": 0.916015625}, {"start": 1259.28, "end": 1259.7, "word": " أخدت", "probability": 0.9742431640625}, {"start": 1259.7, "end": 1260.46, "word": " النص", "probability": 0.798828125}, {"start": 1260.46, "end": 1260.58, "word": " ..", "probability": 0.420654296875}, {"start": 1260.58, "end": 1261.04, "word": " النص", "probability": 0.934326171875}, {"start": 1261.04, "end": 1261.38, "word": " هذا", "probability": 0.3486328125}, {"start": 1261.38, "end": 1261.58, "word": " من", "probability": 0.98193359375}, {"start": 1261.58, "end": 1262.58, "word": " المتباينة", "probability": 0.9525553385416666}, {"start": 1262.58, "end": 1263.52, "word": " النص", "probability": 0.905517578125}, {"start": 1263.52, "end": 1264.02, "word": " الآخر", "probability": 0.9274088541666666}], "temperature": 1.0}, {"id": 46, "seek": 129490, "start": 1265.44, "end": 1294.9, "text": "فعندي انا L على 2 أصغر من XN على YN هذا صحيح for every N أكبر من أو ساوية capital N طيب هذا بيقدي ان ال L over 2 في YN أصغر من XN لكل N bigger than or equal to capital N طيب since", "tokens": [5172, 3615, 1863, 16254, 1975, 8315, 441, 15844, 568, 5551, 9381, 17082, 2288, 9154, 1783, 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البرهان،", "probability": 0.84482421875}, {"start": 1342.03, "end": 1348.07, "word": " okay؟", "probability": 0.759765625}, {"start": 1348.07, "end": 1349.07, "word": " تمام؟", "probability": 0.761474609375}, {"start": 1349.07, "end": 1349.63, "word": " إذا", "probability": 0.695556640625}, {"start": 1349.63, "end": 1349.89, "word": " هذا", "probability": 0.75146484375}, {"start": 1349.89, "end": 1350.51, "word": " بكمل", "probability": 0.9876302083333334}, {"start": 1350.51, "end": 1350.65, "word": " ال", "probability": 0.65966796875}, {"start": 1350.65, "end": 1350.89, "word": " limit", "probability": 0.767578125}, {"start": 1350.89, "end": 1351.47, "word": " comparison", "probability": 0.9189453125}, {"start": 1351.47, "end": 1351.61, "word": " ..", "probability": 0.53369140625}, {"start": 1351.61, "end": 1352.03, "word": " برهان", "probability": 0.9544677734375}, {"start": 1352.03, "end": 1352.39, "word": " ال", "probability": 0.490966796875}, {"start": 1352.39, "end": 1352.63, 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{"start": 1357.4, "end": 1357.78, "word": " comparison", "probability": 0.96044921875}, {"start": 1357.78, "end": 1358.16, "word": " test", "probability": 0.927734375}, {"start": 1358.16, "end": 1358.34, "word": " في", "probability": 0.779296875}, {"start": 1358.34, "end": 1358.76, "word": " عليهم", "probability": 0.93603515625}, {"start": 1358.76, "end": 1359.9, "word": " هتجدوا", "probability": 0.79130859375}, {"start": 1359.9, "end": 1360.1, "word": " فيه", "probability": 0.82470703125}, {"start": 1360.1, "end": 1360.32, "word": " بعض", "probability": 0.9462890625}, {"start": 1360.32, "end": 1360.94, "word": " التمرين", "probability": 0.8899739583333334}, {"start": 1360.94, "end": 1364.16, "word": " ممكن", "probability": 0.98193359375}, {"start": 1364.16, "end": 1364.76, "word": " تطبيقهم", "probability": 0.98623046875}, {"start": 1364.76, "end": 1364.96, "word": " على", "probability": 0.8193359375}, {"start": 1364.96, "end": 1365.3, "word": " بعض", "probability": 0.990966796875}, {"start": 1365.3, "end": 1365.72, "word": " ال", "probability": 0.982421875}, {"start": 1365.72, "end": 1366.36, "word": " sequences", "probability": 0.96630859375}, {"start": 1366.36, "end": 1366.92, "word": " موجودة", "probability": 0.9847412109375}, {"start": 1366.92, "end": 1367.04, "word": " في", "probability": 0.8984375}, {"start": 1367.04, "end": 1367.66, "word": " التمرين", "probability": 0.9931640625}, {"start": 1367.66, "end": 1369.8, "word": " فهنسيبكم", "probability": 0.8739885602678571}, {"start": 1369.8, "end": 1370.12, "word": " طبعا", "probability": 0.984130859375}, {"start": 1370.12, "end": 1370.56, "word": " تحلوا", "probability": 0.868408203125}, {"start": 1370.56, "end": 1371.16, "word": " التمرين", "probability": 0.9952799479166666}, {"start": 1371.16, "end": 1372.22, "word": " عشان", "probability": 0.9899088541666666}, {"start": 1372.22, "end": 1372.88, "word": " تشوفوا", "probability": 0.99658203125}, {"start": 1372.88, "end": 1373.16, "word": " كيف", "probability": 0.97216796875}, {"start": 1373.16, "end": 1373.56, "word": " ممكن", "probability": 0.990478515625}, {"start": 1373.56, "end": 1374.28, "word": " تطبيقهم", "probability": 0.99130859375}], "temperature": 1.0}, {"id": 50, "seek": 138522, "start": 1378.5, "end": 1385.22, "text": "باقي section واحد في ال chapter تلاتة", "tokens": [3555, 995, 38436, 3541, 36764, 24401, 8978, 2423, 7187, 6055, 1211, 9307, 3660], "avg_logprob": -0.18066406675747462, "compression_ratio": 0.8852459016393442, "no_speech_prob": 0.0, "words": [{"start": 1378.5, "end": 1379.32, "word": "باقي", "probability": 0.8665364583333334}, {"start": 1379.32, "end": 1379.94, "word": " section", "probability": 0.64453125}, {"start": 1379.94, "end": 1380.84, "word": " واحد", "probability": 0.9638671875}, {"start": 1380.84, "end": 1381.06, "word": " في", "probability": 0.9267578125}, {"start": 1381.06, "end": 1381.68, "word": " ال", "probability": 0.41357421875}, {"start": 1381.68, "end": 1383.28, "word": " chapter", "probability": 0.90576171875}, {"start": 1383.28, "end": 1385.22, "word": " تلاتة", "probability": 0.909423828125}], "temperature": 1.0}, {"id": 51, "seek": 143292, "start": 1412.72, "end": 1432.92, "text": "السيكشن الأخير سيكشن تلاتة سبعة في شبكر 3 هذا هيكون أبراعا مقدمة introduction to infinite series", "tokens": [6027, 3794, 1829, 4117, 8592, 1863, 16247, 9778, 13546, 8608, 1829, 4117, 8592, 1863, 6055, 1211, 9307, 3660, 8608, 3555, 27884, 8978, 13412, 3555, 4117, 2288, 805, 23758, 8032, 1829, 30544, 5551, 3555, 23557, 3615, 995, 3714, 28543, 46007, 9339, 281, 13785, 2638], "avg_logprob": -0.3433948972008445, "compression_ratio": 1.2131147540983607, "no_speech_prob": 0.0, "words": [{"start": 1412.72, "end": 1413.64, "word": "السيكشن", "probability": 0.6215413411458334}, {"start": 1413.64, "end": 1414.22, "word": " الأخير", "probability": 0.8566080729166666}, {"start": 1414.22, "end": 1417.66, "word": " سيكشن", "probability": 0.859765625}, {"start": 1417.66, "end": 1418.16, "word": " تلاتة", "probability": 0.888916015625}, {"start": 1418.16, "end": 1418.66, "word": " سبعة", "probability": 0.91650390625}, {"start": 1418.66, "end": 1419.06, "word": " في", "probability": 0.65185546875}, {"start": 1419.06, "end": 1419.74, "word": " شبكر", "probability": 0.52008056640625}, {"start": 1419.74, "end": 1420.08, "word": " 3", "probability": 0.411865234375}, {"start": 1420.08, "end": 1421.34, "word": " هذا", "probability": 0.5087890625}, {"start": 1421.34, "end": 1422.34, "word": " هيكون", "probability": 0.6173502604166666}, {"start": 1422.34, "end": 1423.12, "word": " أبراعا", "probability": 0.7595703125}, {"start": 1423.12, "end": 1423.82, "word": " مقدمة", "probability": 0.9622395833333334}, {"start": 1423.82, "end": 1424.7, "word": " introduction", "probability": 0.83740234375}, {"start": 1424.7, "end": 1427.86, "word": " to", "probability": 0.93212890625}, {"start": 1427.86, "end": 1428.6, "word": " infinite", "probability": 0.86376953125}, {"start": 1428.6, 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1440.36, "word": " series", "probability": 0.90185546875}, {"start": 1440.36, "end": 1441.96, "word": " مقدمة", "probability": 0.8011881510416666}, {"start": 1441.96, "end": 1442.38, "word": " في", "probability": 0.82568359375}, {"start": 1442.38, "end": 1443.86, "word": " المتسلسلات", "probability": 0.9664481026785714}, {"start": 1443.86, "end": 1446.94, "word": " اللانهائية", "probability": 0.891796875}, {"start": 1446.94, "end": 1449.42, "word": " نعرف", "probability": 0.8484700520833334}, {"start": 1449.42, "end": 1449.66, "word": " شو", "probability": 0.822021484375}, {"start": 1449.66, "end": 1450.96, "word": " معناه", "probability": 0.8357747395833334}, {"start": 1450.96, "end": 1451.94, "word": " متسلسلة", "probability": 0.91171875}, {"start": 1451.94, "end": 1452.86, "word": " لانهائية", "probability": 0.93310546875}, {"start": 1452.86, "end": 1455.26, "word": " let", "probability": 0.290771484375}, {"start": 1455.26, "end": 1459.46, "word": " xn", "probability": 0.41064453125}, {"start": 1459.46, "end": 1460.88, "word": " contained", "probability": 0.7451171875}, {"start": 1460.88, "end": 1461.16, "word": " in", "probability": 0.9658203125}, {"start": 1461.16, "end": 1461.6, "word": " R", "probability": 0.84716796875}, {"start": 1461.6, "end": 1463.3, "word": " be", "probability": 0.6416015625}, {"start": 1463.3, "end": 1463.92, "word": " a", "probability": 0.95361328125}, {"start": 1463.92, "end": 1464.56, "word": " sequence", "probability": 0.978515625}], "temperature": 1.0}, {"id": 53, "seek": 148937, "start": 1466.13, "end": 1489.37, "text": "sequence of real numbers sum x1 plus x2 plus ..", "tokens": [11834, 655, 295, 957, 3547, 2408, 2031, 16, 1804, 2031, 17, 1804, 4386], "avg_logprob": -0.49944196002823965, "compression_ratio": 0.94, "no_speech_prob": 0.0, "words": [{"start": 1466.13, "end": 1466.89, "word": "sequence", "probability": 0.6378173828125}, {"start": 1466.89, "end": 1467.13, "word": " of", "probability": 0.94384765625}, {"start": 1467.13, "end": 1467.33, "word": " real", "probability": 0.97119140625}, {"start": 1467.33, "end": 1467.95, "word": " numbers", "probability": 0.859375}, {"start": 1467.95, "end": 1483.21, "word": " sum", "probability": 0.0980224609375}, {"start": 1483.21, "end": 1487.01, "word": " x1", "probability": 0.468505859375}, {"start": 1487.01, "end": 1487.51, "word": " plus", "probability": 0.142578125}, {"start": 1487.51, "end": 1488.33, "word": " x2", "probability": 0.974365234375}, {"start": 1488.33, "end": 1488.91, "word": " plus", "probability": 0.96240234375}, {"start": 1488.91, "end": 1489.37, "word": " ..", "probability": 0.343505859375}], "temperature": 1.0}, {"id": 54, "seek": 151834, "start": 1490.4, "end": 1518.34, "text": "x3 plus و هكذا plus xn plus و هكذا و ممكن نكتبه بالصورة using sigma notation نستخدم رمز sigma ممكن هذا نسميه summation from n equals one to infinity إلى xn فالصن المجموع هذا", "tokens": [87, 18, 1804, 4032, 8032, 4117, 15730, 1804, 2031, 77, 1804, 4032, 8032, 4117, 15730, 4032, 3714, 43020, 8717, 4117, 2655, 3555, 3224, 20666, 9381, 13063, 3660, 1228, 12771, 24657, 8717, 14851, 9778, 40448, 12602, 2304, 11622, 12771, 3714, 43020, 23758, 8717, 38251, 1829, 3224, 28811, 490, 297, 6915, 472, 281, 13202, 30731, 2031, 77, 6156, 6027, 9381, 1863, 9673, 7435, 2304, 45367, 23758], "avg_logprob": -0.16502403846153846, "compression_ratio": 1.4226190476190477, "no_speech_prob": 0.0, "words": [{"start": 1490.4, "end": 1491.24, "word": "x3", "probability": 0.5057373046875}, {"start": 1491.24, "end": 1491.76, "word": " plus", "probability": 0.1259765625}, {"start": 1491.76, "end": 1491.98, "word": " و", "probability": 0.50439453125}, {"start": 1491.98, "end": 1492.54, "word": " هكذا", "probability": 0.7638346354166666}, {"start": 1492.54, "end": 1494.16, "word": " plus", "probability": 0.7177734375}, {"start": 1494.16, "end": 1494.96, "word": " xn", "probability": 0.906005859375}, {"start": 1494.96, "end": 1495.42, "word": " plus", "probability": 0.951171875}, {"start": 1495.42, "end": 1495.64, "word": " و", "probability": 0.9443359375}, {"start": 1495.64, "end": 1496.2, "word": " هكذا", "probability": 0.966796875}, {"start": 1496.2, "end": 1497.4, "word": " و", "probability": 0.76806640625}, {"start": 1497.4, "end": 1497.7, "word": " ممكن", "probability": 0.951416015625}, {"start": 1497.7, "end": 1498.28, "word": " نكتبه", "probability": 0.9671875}, {"start": 1498.28, "end": 1499.08, "word": " بالصورة", "probability": 0.9730224609375}, {"start": 1499.08, "end": 1499.68, "word": " using", "probability": 0.876953125}, {"start": 1499.68, "end": 1500.16, "word": " sigma", "probability": 0.6875}, {"start": 1500.16, "end": 1500.84, "word": " notation", "probability": 0.98095703125}, {"start": 1500.84, "end": 1503.4, "word": " نستخدم", "probability": 0.814208984375}, {"start": 1503.4, "end": 1503.88, "word": " رمز", "probability": 0.9939778645833334}, {"start": 1503.88, "end": 1504.38, "word": " sigma", "probability": 0.7939453125}, {"start": 1504.38, "end": 1505.8, "word": " ممكن", "probability": 0.845947265625}, {"start": 1505.8, "end": 1506.04, "word": " هذا", "probability": 0.72265625}, {"start": 1506.04, "end": 1506.78, "word": " نسميه", "probability": 0.9439697265625}, {"start": 1506.78, "end": 1507.46, "word": " summation", "probability": 0.74072265625}, {"start": 1507.46, "end": 1508.88, "word": " from", "probability": 0.892578125}, {"start": 1508.88, "end": 1509.32, "word": " n", "probability": 0.91650390625}, {"start": 1509.32, "end": 1509.92, "word": " equals", "probability": 0.84130859375}, {"start": 1509.92, "end": 1510.26, "word": " one", "probability": 0.84765625}, {"start": 1510.26, "end": 1510.48, "word": " to", "probability": 0.80126953125}, {"start": 1510.48, "end": 1511.0, "word": " infinity", "probability": 0.892578125}, {"start": 1511.0, "end": 1511.26, "word": " إلى", "probability": 0.814453125}, {"start": 1511.26, "end": 1511.94, "word": " xn", "probability": 0.950927734375}, {"start": 1511.94, "end": 1517.28, "word": " فالصن", "probability": 0.748046875}, {"start": 1517.28, "end": 1517.92, "word": " المجموع", "probability": 0.8973388671875}, {"start": 1517.92, "end": 1518.34, "word": " هذا", "probability": 0.8701171875}], "temperature": 1.0}, {"id": 55, "seek": 154297, "start": 1520.55, "end": 1542.97, "text": "هذا expanded هذا compact form of summation is called an infinite series generated by", "tokens": [3224, 15730, 14342, 23758, 14679, 1254, 295, 28811, 307, 1219, 364, 13785, 2638, 10833, 538], "avg_logprob": -0.3986816480755806, "compression_ratio": 1.0588235294117647, "no_speech_prob": 0.0, "words": [{"start": 1520.55, "end": 1520.93, "word": "هذا", "probability": 0.6181640625}, {"start": 1520.93, "end": 1521.59, "word": " expanded", "probability": 0.186767578125}, {"start": 1521.59, "end": 1522.83, "word": " هذا", "probability": 0.5517578125}, {"start": 1522.83, "end": 1523.53, "word": " compact", "probability": 0.8232421875}, {"start": 1523.53, "end": 1524.89, "word": " form", "probability": 0.775390625}, {"start": 1524.89, "end": 1525.19, "word": " of", "probability": 0.94384765625}, {"start": 1525.19, "end": 1525.73, "word": " summation", "probability": 0.84912109375}, {"start": 1525.73, "end": 1527.15, "word": " is", "probability": 0.67041015625}, {"start": 1527.15, "end": 1528.75, "word": " called", "probability": 0.92138671875}, {"start": 1528.75, "end": 1531.15, "word": " an", "probability": 0.5078125}, {"start": 1531.15, "end": 1531.67, "word": " infinite", "probability": 0.96044921875}, {"start": 1531.67, "end": 1532.45, "word": " series", "probability": 0.92041015625}, {"start": 1532.45, "end": 1539.01, "word": " generated", "probability": 0.464111328125}, {"start": 1539.01, "end": 1542.97, "word": " by", "probability": 0.802734375}], "temperature": 1.0}, {"id": 56, "seek": 156934, "start": 1546.32, "end": 1569.34, "text": "متولدة من .. by الـ sequence x in إذن infinite series generated by الـ sequence x in إذا هذه عبارة عن infinite series بتسميها متولدة من الـ sequence x in طيب for every", "tokens": [2304, 2655, 12610, 41891, 9154, 4386, 538, 2423, 39184, 8310, 2031, 294, 11933, 8848, 1863, 13785, 2638, 10833, 538, 2423, 39184, 8310, 2031, 294, 11933, 15730, 29538, 6225, 3555, 9640, 3660, 18871, 13785, 2638, 39894, 38251, 1829, 11296, 44650, 12610, 41891, 9154, 2423, 39184, 8310, 2031, 294, 23032, 1829, 3555, 337, 633], "avg_logprob": -0.2796285444835447, "compression_ratio": 1.6390977443609023, "no_speech_prob": 0.0, "words": [{"start": 1546.32, "end": 1547.2, "word": "متولدة", "probability": 0.81231689453125}, {"start": 1547.2, "end": 1547.66, "word": " من", "probability": 0.97802734375}, {"start": 1547.66, "end": 1547.78, "word": " ..", "probability": 0.200439453125}, {"start": 1547.78, "end": 1548.32, "word": " by", "probability": 0.73828125}, {"start": 1548.32, "end": 1549.14, "word": " الـ", "probability": 0.502685546875}, {"start": 1549.14, "end": 1549.58, "word": " sequence", "probability": 0.84521484375}, {"start": 1549.58, "end": 1550.0, "word": " x", "probability": 0.67041015625}, {"start": 1550.0, "end": 1550.3, "word": " in", "probability": 0.289794921875}, {"start": 1550.3, "end": 1554.1, "word": " إذن", "probability": 0.4612630208333333}, {"start": 1554.1, "end": 1554.48, "word": " infinite", "probability": 0.7802734375}, {"start": 1554.48, "end": 1555.24, "word": " series", "probability": 0.92041015625}, {"start": 1555.24, "end": 1556.66, "word": " generated", "probability": 0.77294921875}, {"start": 1556.66, "end": 1558.06, "word": " by", "probability": 0.974609375}, {"start": 1558.06, "end": 1559.26, "word": " الـ", "probability": 0.625732421875}, {"start": 1559.26, "end": 1559.6, "word": " sequence", "probability": 0.97998046875}, {"start": 1559.6, "end": 1559.9, "word": " x", "probability": 0.9619140625}, {"start": 1559.9, "end": 1560.1, "word": " in", "probability": 0.86279296875}, {"start": 1560.1, "end": 1560.28, "word": " إذا", "probability": 0.5767822265625}, {"start": 1560.28, "end": 1560.52, "word": " هذه", "probability": 0.6611328125}, {"start": 1560.52, "end": 1560.88, "word": " عبارة", "probability": 0.9400634765625}, {"start": 1560.88, "end": 1561.02, "word": " عن", "probability": 0.9921875}, {"start": 1561.02, "end": 1561.3, "word": " infinite", "probability": 0.9677734375}, {"start": 1561.3, "end": 1561.76, "word": " series", "probability": 0.93115234375}, {"start": 1561.76, "end": 1562.4, "word": " بتسميها", "probability": 0.80474853515625}, {"start": 1562.4, "end": 1563.6, "word": " متولدة", "probability": 0.97021484375}, {"start": 1563.6, "end": 1564.44, "word": " من", "probability": 0.9921875}, {"start": 1564.44, "end": 1564.68, "word": " الـ", "probability": 0.845703125}, {"start": 1564.68, "end": 1565.0, "word": " sequence", "probability": 0.984375}, {"start": 1565.0, "end": 1565.32, "word": " x", "probability": 0.9765625}, {"start": 1565.32, "end": 1565.64, "word": " in", "probability": 0.92919921875}, {"start": 1565.64, "end": 1567.28, "word": " طيب", "probability": 0.9791666666666666}, {"start": 1567.28, "end": 1568.9, "word": " for", "probability": 0.8857421875}, {"start": 1568.9, "end": 1569.34, "word": " every", "probability": 0.8447265625}], "temperature": 1.0}, {"id": 57, "seek": 160051, "start": 1572.43, "end": 1600.51, "text": "for each n belong to N define خلّيني أعرف S1 على أنه X1 S2 على أنه S1 زاد X2 بساوي X1 زاد X2 S3 بساوي S2 زاد X3", "tokens": [2994, 1184, 297, 5784, 281, 426, 6964, 16490, 1211, 11703, 9957, 1829, 5551, 3615, 28480, 318, 16, 15844, 14739, 3224, 1783, 16, 318, 17, 15844, 14739, 3224, 318, 16, 30767, 18513, 1783, 17, 4724, 3794, 995, 45865, 1783, 16, 30767, 18513, 1783, 17, 318, 18, 4724, 3794, 995, 45865, 318, 17, 30767, 18513, 1783, 18], "avg_logprob": -0.17508370668760367, "compression_ratio": 1.3103448275862069, "no_speech_prob": 0.0, "words": [{"start": 1572.43, "end": 1572.97, "word": "for", "probability": 0.7705078125}, {"start": 1572.97, "end": 1573.59, "word": " each", "probability": 0.947265625}, {"start": 1573.59, "end": 1575.21, "word": " n", "probability": 0.73193359375}, {"start": 1575.21, "end": 1575.67, "word": " belong", "probability": 0.501953125}, {"start": 1575.67, "end": 1575.95, "word": " to", "probability": 0.9853515625}, {"start": 1575.95, "end": 1576.41, "word": " N", "probability": 0.58056640625}, {"start": 1576.41, "end": 1578.67, "word": " define", "probability": 0.84326171875}, {"start": 1578.67, "end": 1580.37, "word": " خلّيني", "probability": 0.696728515625}, {"start": 1580.37, "end": 1581.01, "word": " أعرف", "probability": 0.7152506510416666}, {"start": 1581.01, "end": 1582.93, "word": " S1", "probability": 0.771728515625}, {"start": 1582.93, "end": 1583.29, "word": " على", "probability": 0.64404296875}, {"start": 1583.29, "end": 1583.85, "word": " أنه", "probability": 0.6904296875}, {"start": 1583.85, "end": 1584.71, "word": " X1", "probability": 0.901123046875}, {"start": 1584.71, "end": 1586.89, "word": " S2", "probability": 0.926513671875}, {"start": 1586.89, "end": 1587.17, "word": " على", "probability": 0.94970703125}, {"start": 1587.17, "end": 1588.51, "word": " أنه", "probability": 0.954345703125}, {"start": 1588.51, "end": 1590.29, "word": " S1", "probability": 0.977294921875}, {"start": 1590.29, "end": 1590.81, "word": " زاد", "probability": 0.62451171875}, {"start": 1590.81, "end": 1591.69, "word": " X2", "probability": 0.995361328125}, {"start": 1591.69, "end": 1592.63, "word": " بساوي", "probability": 0.8148193359375}, {"start": 1592.63, "end": 1593.35, "word": " X1", "probability": 0.966552734375}, {"start": 1593.35, "end": 1593.99, "word": " زاد", "probability": 0.943115234375}, {"start": 1593.99, "end": 1595.53, "word": " X2", "probability": 0.9970703125}, {"start": 1595.53, "end": 1597.95, "word": " S3", "probability": 0.958251953125}, {"start": 1597.95, "end": 1598.65, "word": " بساوي", "probability": 0.9365234375}, {"start": 1598.65, "end": 1599.33, "word": " S2", "probability": 0.977783203125}, {"start": 1599.33, "end": 1599.87, "word": " زاد", "probability": 0.967041015625}, {"start": 1599.87, "end": 1600.51, "word": " X3", "probability": 0.997314453125}], "temperature": 1.0}, {"id": 58, "seek": 162770, "start": 1602.3, "end": 1627.7, "text": "Y ساوي X1 زايد X2 زايد X3 and so on و هكذا نعرف SN على انه SN negative one زايد XN و طبعا ال SN negative one هيكون عبارة عن summation", "tokens": [56, 8608, 995, 45865, 1783, 16, 30767, 995, 25708, 1783, 17, 30767, 995, 25708, 1783, 18, 293, 370, 322, 4032, 8032, 4117, 15730, 8717, 3615, 28480, 13955, 15844, 16472, 3224, 13955, 3671, 472, 30767, 995, 25708, 1783, 45, 4032, 23032, 3555, 3615, 995, 2423, 13955, 3671, 472, 39896, 30544, 6225, 3555, 9640, 3660, 18871, 28811], "avg_logprob": -0.2420479870800461, "compression_ratio": 1.326086956521739, "no_speech_prob": 0.0, "words": [{"start": 1602.3, "end": 1602.66, "word": "Y", "probability": 0.13720703125}, {"start": 1602.66, "end": 1603.24, "word": " ساوي", "probability": 0.5895309448242188}, {"start": 1603.24, "end": 1603.8, "word": " X1", "probability": 0.71875}, {"start": 1603.8, "end": 1604.26, "word": " زايد", "probability": 0.6979166666666666}, {"start": 1604.26, "end": 1604.98, "word": " X2", "probability": 0.9814453125}, {"start": 1604.98, "end": 1605.68, "word": " زايد", "probability": 0.9720052083333334}, {"start": 1605.68, "end": 1606.54, "word": " X3", "probability": 0.993896484375}, {"start": 1606.54, "end": 1610.0, "word": " and", "probability": 0.426025390625}, {"start": 1610.0, "end": 1610.34, "word": " so", "probability": 0.83251953125}, {"start": 1610.34, "end": 1610.68, "word": " on", "probability": 0.939453125}, {"start": 1610.68, "end": 1610.82, "word": " و", "probability": 0.7890625}, {"start": 1610.82, "end": 1611.42, "word": " هكذا", "probability": 0.8870442708333334}, {"start": 1611.42, "end": 1612.74, "word": " نعرف", "probability": 0.7586263020833334}, {"start": 1612.74, "end": 1613.58, "word": " SN", "probability": 0.41259765625}, {"start": 1613.58, "end": 1614.12, "word": " على", "probability": 0.857421875}, {"start": 1614.12, "end": 1614.72, "word": " انه", "probability": 0.6600341796875}, {"start": 1614.72, "end": 1616.4, "word": " SN", "probability": 0.61328125}, {"start": 1616.4, "end": 1617.12, "word": " negative", "probability": 0.57568359375}, {"start": 1617.12, "end": 1617.62, "word": " one", "probability": 0.830078125}, {"start": 1617.62, "end": 1618.38, "word": " زايد", "probability": 0.9739583333333334}, {"start": 1618.38, "end": 1619.24, "word": " XN", "probability": 0.932861328125}, {"start": 1619.24, "end": 1621.02, "word": " و", "probability": 0.75439453125}, {"start": 1621.02, "end": 1621.42, "word": " طبعا", "probability": 0.958984375}, {"start": 1621.42, "end": 1622.46, "word": " ال", "probability": 0.87841796875}, {"start": 1622.46, "end": 1623.04, "word": " SN", "probability": 0.65234375}, {"start": 1623.04, "end": 1624.1, "word": " negative", "probability": 0.5556640625}, {"start": 1624.1, "end": 1624.9, "word": " one", "probability": 0.9580078125}, {"start": 1624.9, "end": 1626.38, "word": " هيكون", "probability": 0.85009765625}, {"start": 1626.38, "end": 1626.84, "word": " عبارة", "probability": 0.87451171875}, {"start": 1626.84, "end": 1627.06, "word": " عن", "probability": 0.9951171875}, {"start": 1627.06, "end": 1627.7, "word": " summation", "probability": 0.8486328125}], "temperature": 1.0}, {"id": 59, "seek": 165434, "start": 1630.6, "end": 1654.34, "text": "x1 زائد x2 زائد و هكذا إلى أخر حد xn-1 هذا عبارة عن ايه هذا عبارة عن s in negative one بنضيف لها xn فهذا بيطلع بيساوي summation من k equals one to n", "tokens": [87, 16, 30767, 16373, 3215, 2031, 17, 30767, 16373, 3215, 4032, 8032, 4117, 15730, 30731, 5551, 34740, 11331, 3215, 2031, 77, 12, 16, 23758, 6225, 3555, 9640, 3660, 18871, 1975, 1829, 3224, 23758, 6225, 3555, 9640, 3660, 18871, 262, 294, 3671, 472, 44945, 11242, 33911, 5296, 11296, 2031, 77, 6156, 3224, 15730, 4724, 1829, 9566, 1211, 3615, 4724, 1829, 3794, 995, 45865, 28811, 9154, 350, 6915, 472, 281, 297], "avg_logprob": -0.2720982044935226, "compression_ratio": 1.391025641025641, "no_speech_prob": 5.960464477539063e-08, "words": [{"start": 1630.6, "end": 1631.32, "word": "x1", "probability": 0.373321533203125}, {"start": 1631.32, "end": 1631.84, "word": " زائد", "probability": 0.487548828125}, {"start": 1631.84, "end": 1632.74, "word": " x2", "probability": 0.6834716796875}, {"start": 1632.74, "end": 1633.44, "word": " زائد", "probability": 0.9588216145833334}, {"start": 1633.44, "end": 1633.56, "word": " و", "probability": 0.5673828125}, {"start": 1633.56, "end": 1634.2, "word": " هكذا", "probability": 0.7649739583333334}, {"start": 1634.2, "end": 1635.14, "word": " إلى", "probability": 0.6728515625}, {"start": 1635.14, "end": 1635.52, "word": " أخر", "probability": 0.7022705078125}, {"start": 1635.52, "end": 1635.84, "word": " حد", "probability": 0.994140625}, {"start": 1635.84, "end": 1636.52, "word": " xn", "probability": 0.71142578125}, {"start": 1636.52, "end": 1637.36, "word": "-1", "probability": 0.60040283203125}, {"start": 1637.36, "end": 1638.72, "word": " هذا", "probability": 0.63720703125}, {"start": 1638.72, "end": 1639.14, "word": " عبارة", "probability": 0.9923095703125}, {"start": 1639.14, "end": 1639.36, "word": " عن", "probability": 0.9912109375}, {"start": 1639.36, "end": 1639.68, "word": " ايه", "probability": 0.746337890625}, {"start": 1639.68, "end": 1640.46, "word": " هذا", "probability": 0.61181640625}, {"start": 1640.46, "end": 1640.78, "word": " عبارة", "probability": 0.9951171875}, {"start": 1640.78, "end": 1640.94, "word": " عن", "probability": 0.9931640625}, {"start": 1640.94, "end": 1641.28, "word": " s", "probability": 0.50830078125}, {"start": 1641.28, "end": 1642.1, "word": " in", "probability": 0.1923828125}, {"start": 1642.1, "end": 1642.46, "word": " negative", "probability": 0.80419921875}, {"start": 1642.46, "end": 1642.84, "word": " one", "probability": 0.810546875}, {"start": 1642.84, "end": 1644.64, "word": " بنضيف", "probability": 0.8284505208333334}, {"start": 1644.64, "end": 1644.96, "word": " لها", "probability": 0.7423095703125}, {"start": 1644.96, "end": 1645.62, "word": " xn", "probability": 0.94580078125}, {"start": 1645.62, "end": 1647.96, "word": " فهذا", "probability": 0.9658203125}, {"start": 1647.96, "end": 1648.4, "word": " بيطلع", "probability": 0.860546875}, {"start": 1648.4, "end": 1649.26, "word": " بيساوي", "probability": 0.857421875}, {"start": 1649.26, "end": 1650.24, "word": " summation", "probability": 0.80517578125}, {"start": 1650.24, "end": 1652.12, "word": " من", "probability": 0.9619140625}, {"start": 1652.12, "end": 1652.58, "word": " k", "probability": 0.78759765625}, {"start": 1652.58, "end": 1653.38, "word": " equals", "probability": 0.64453125}, {"start": 1653.38, "end": 1653.72, "word": " one", "probability": 0.82666015625}, {"start": 1653.72, "end": 1653.98, "word": " to", "probability": 0.9423828125}, {"start": 1653.98, "end": 1654.34, "word": " n", "probability": 0.90966796875}], "temperature": 1.0}, {"id": 60, "seek": 168394, "start": 1658.08, "end": 1683.94, "text": "to for xk اذا sn هو مجموع الحدود من اول حد الى حد رقم n وهكذا ممكن نستمر الى ملا نهاية and so on الان انا كوّنت sequence لاحظوا s1, s2, s3, sn هذا عبارة عن sequence ال sequence الجديدة هذه لها اسم", "tokens": [1353, 337, 2031, 74, 1975, 15730, 2406, 31439, 3714, 7435, 2304, 45367, 21542, 3215, 23328, 9154, 1975, 12610, 11331, 3215, 2423, 7578, 11331, 3215, 12602, 4587, 2304, 297, 37037, 4117, 15730, 3714, 43020, 8717, 14851, 29973, 2423, 7578, 3714, 15040, 8717, 11296, 10632, 293, 370, 322, 2423, 7649, 1975, 8315, 9122, 2407, 11703, 29399, 8310, 20193, 5016, 19913, 14407, 262, 16, 11, 262, 17, 11, 262, 18, 11, 2406, 23758, 6225, 3555, 9640, 3660, 18871, 8310, 2423, 8310, 25724, 16254, 41891, 29538, 5296, 11296, 24525, 2304], "avg_logprob": -0.26867817119620313, "compression_ratio": 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الى", "probability": 0.715576171875}, {"start": 1667.28, "end": 1667.62, "word": " حد", "probability": 0.87646484375}, {"start": 1667.62, "end": 1668.02, "word": " رقم", "probability": 0.98291015625}, {"start": 1668.02, "end": 1668.32, "word": " n", "probability": 0.744140625}, {"start": 1668.32, "end": 1669.84, "word": " وهكذا", "probability": 0.82568359375}, {"start": 1669.84, "end": 1670.12, "word": " ممكن", "probability": 0.83447265625}, {"start": 1670.12, "end": 1670.68, "word": " نستمر", "probability": 0.9845377604166666}, {"start": 1670.68, "end": 1671.0, "word": " الى", "probability": 0.94775390625}, {"start": 1671.0, "end": 1671.32, "word": " ملا", "probability": 0.6982421875}, {"start": 1671.32, "end": 1671.7, "word": " نهاية", "probability": 0.83349609375}, {"start": 1671.7, "end": 1671.92, "word": " and", "probability": 0.81689453125}, {"start": 1671.92, "end": 1672.12, "word": " so", "probability": 0.8974609375}, {"start": 1672.12, "end": 1672.64, "word": " on", 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عن", "probability": 0.98828125}, {"start": 1680.36, "end": 1680.92, "word": " sequence", "probability": 0.9619140625}, {"start": 1680.92, "end": 1682.16, "word": " ال", "probability": 0.53369140625}, {"start": 1682.16, "end": 1682.58, "word": " sequence", "probability": 0.93408203125}, {"start": 1682.58, "end": 1683.08, "word": " الجديدة", "probability": 0.9148763020833334}, {"start": 1683.08, "end": 1683.32, "word": " هذه", "probability": 0.463623046875}, {"start": 1683.32, "end": 1683.54, "word": " لها", "probability": 0.60302734375}, {"start": 1683.54, "end": 1683.94, "word": " اسم", "probability": 0.904541015625}], "temperature": 1.0}, {"id": 61, "seek": 170921, "start": 1685.15, "end": 1709.21, "text": "و sequence مهمة of partial sums مظبوط قعدت نسميها اذا طرست تفاضل الف وفهمته الموضوع هذا هناك الموضوع ال series قعدت نسميها the sequence of partial sums اذا the sequence", "tokens": [2407, 8310, 3714, 16095, 3660, 295, 14641, 34499, 3714, 19913, 3555, 2407, 9566, 12174, 22488, 2655, 8717, 38251, 1829, 11296, 1975, 15730, 23032, 2288, 14851, 6055, 5172, 46958, 1211, 27188, 4032, 5172, 16095, 47395, 9673, 2407, 11242, 45367, 23758, 34105, 4117, 9673, 2407, 11242, 45367, 2423, 2638, 12174, 22488, 2655, 8717, 38251, 1829, 11296, 264, 8310, 295, 14641, 34499, 1975, 15730, 264, 8310], "avg_logprob": -0.2805175641551614, "compression_ratio": 1.6333333333333333, "no_speech_prob": 0.0, "words": [{"start": 1685.15, "end": 1685.55, "word": "و", "probability": 0.46484375}, {"start": 1685.55, "end": 1686.97, "word": " sequence", "probability": 0.740234375}, {"start": 1686.97, "end": 1687.99, "word": " مهمة", "probability": 0.9708658854166666}, {"start": 1687.99, "end": 1688.19, "word": " of", "probability": 0.517578125}, {"start": 1688.19, "end": 1688.59, "word": " partial", "probability": 0.93017578125}, {"start": 1688.59, "end": 1688.99, "word": " sums", "probability": 0.951171875}, {"start": 1688.99, "end": 1690.69, "word": " مظبوط", "probability": 0.8494140625}, {"start": 1690.69, "end": 1691.01, "word": " قعدت", "probability": 0.4569498697916667}, {"start": 1691.01, "end": 1691.63, "word": " نسميها", "probability": 0.84539794921875}, {"start": 1691.63, "end": 1691.79, "word": " اذا", "probability": 0.66064453125}, {"start": 1691.79, "end": 1692.21, "word": " طرست", "probability": 0.74755859375}, {"start": 1692.21, "end": 1692.71, "word": " تفاضل", "probability": 0.54840087890625}, {"start": 1692.71, "end": 1693.01, "word": " الف", "probability": 0.457763671875}, {"start": 1693.01, "end": 1694.57, "word": " وفهمته", "probability": 0.771240234375}, {"start": 1694.57, "end": 1696.51, "word": " الموضوع", "probability": 0.9974365234375}, {"start": 1696.51, "end": 1696.89, "word": " هذا", "probability": 0.76611328125}, {"start": 1696.89, "end": 1697.55, "word": " هناك", "probability": 0.996337890625}, {"start": 1697.55, "end": 1698.61, "word": " الموضوع", "probability": 0.907958984375}, {"start": 1698.61, "end": 1698.79, "word": " ال", "probability": 0.90869140625}, {"start": 1698.79, "end": 1699.25, "word": " series", "probability": 0.931640625}, {"start": 1699.25, "end": 1700.33, "word": " قعدت", "probability": 0.7369791666666666}, {"start": 1700.33, "end": 1700.95, "word": " نسميها", "probability": 0.9932861328125}, {"start": 1700.95, "end": 1701.13, "word": " the", "probability": 0.71044921875}, {"start": 1701.13, "end": 1701.85, "word": " sequence", "probability": 0.97900390625}, {"start": 1701.85, "end": 1702.15, "word": " of", "probability": 0.9833984375}, {"start": 1702.15, "end": 1702.59, "word": " partial", "probability": 0.96044921875}, {"start": 1702.59, "end": 1703.11, "word": " sums", "probability": 0.9462890625}, {"start": 1703.11, "end": 1705.83, "word": " اذا", "probability": 0.83251953125}, {"start": 1705.83, "end": 1706.57, "word": " the", "probability": 0.54541015625}, {"start": 1706.57, "end": 1709.21, "word": " sequence", "probability": 0.99169921875}], "temperature": 1.0}, {"id": 62, "seek": 174030, "start": 1710.94, "end": 1740.3, "text": "SN from N equals one to infinity is called بنسميها the sequence the sequence of partial sums sequence of partial sums of the series", "tokens": [32481, 490, 426, 6915, 472, 281, 13202, 307, 1219, 44945, 38251, 1829, 11296, 264, 8310, 264, 8310, 295, 14641, 34499, 8310, 295, 14641, 34499, 295, 264, 2638], "avg_logprob": -0.33816964605024885, "compression_ratio": 1.3529411764705883, "no_speech_prob": 0.0, "words": [{"start": 1710.94, "end": 1711.94, "word": "SN", "probability": 0.09149169921875}, {"start": 1711.94, "end": 1712.46, "word": " from", "probability": 0.2919921875}, {"start": 1712.46, "end": 1712.82, "word": " N", "probability": 0.7470703125}, {"start": 1712.82, "end": 1713.3, "word": " equals", "probability": 0.378662109375}, {"start": 1713.3, "end": 1713.56, "word": " one", "probability": 0.6474609375}, {"start": 1713.56, "end": 1713.76, "word": " to", "probability": 0.9267578125}, {"start": 1713.76, "end": 1714.26, "word": " infinity", "probability": 0.86083984375}, {"start": 1714.26, "end": 1715.58, "word": " is", "probability": 0.5}, {"start": 1715.58, "end": 1716.3, "word": " called", "probability": 0.91748046875}, {"start": 1716.3, "end": 1717.98, "word": " بنسميها", "probability": 0.82666015625}, {"start": 1717.98, "end": 1719.86, "word": " the", "probability": 0.32470703125}, {"start": 1719.86, "end": 1720.8, "word": " sequence", "probability": 0.97216796875}, {"start": 1720.8, "end": 1723.72, "word": " the", "probability": 0.25732421875}, {"start": 1723.72, "end": 1724.68, "word": " sequence", "probability": 0.9716796875}, {"start": 1724.68, "end": 1726.14, "word": " of", "probability": 0.974609375}, {"start": 1726.14, "end": 1727.66, "word": " partial", "probability": 0.9345703125}, {"start": 1727.66, "end": 1731.18, "word": " sums", "probability": 0.927734375}, {"start": 1731.18, "end": 1733.24, "word": " sequence", "probability": 0.94287109375}, {"start": 1733.24, "end": 1734.02, "word": " of", "probability": 0.978515625}, {"start": 1734.02, "end": 1734.64, "word": " partial", "probability": 0.94677734375}, {"start": 1734.64, "end": 1735.3, "word": " sums", "probability": 0.93896484375}, {"start": 1735.3, "end": 1736.88, "word": " of", "probability": 0.96728515625}, {"start": 1736.88, "end": 1739.5, "word": " the", "probability": 0.9130859375}, {"start": 1739.5, "end": 1740.3, "word": " series", "probability": 0.9130859375}], "temperature": 1.0}, {"id": 63, "seek": 176928, "start": 1742.34, "end": 1769.28, "text": "اللي هي sigma xn او sigma من n بساعة واحد لانفينيتي okay الان now if ال sequence sn converges say", "tokens": [6027, 20292, 39896, 12771, 2031, 77, 1975, 2407, 12771, 9154, 297, 4724, 3794, 995, 27884, 36764, 24401, 5296, 7649, 5172, 9957, 1829, 31371, 1392, 2423, 7649, 586, 498, 2423, 8310, 2406, 9652, 2880, 584], "avg_logprob": -0.3506696479661124, "compression_ratio": 1.1293103448275863, "no_speech_prob": 0.0, "words": [{"start": 1742.34, "end": 1742.78, "word": "اللي", "probability": 0.73388671875}, {"start": 1742.78, "end": 1743.04, "word": " هي", "probability": 0.90478515625}, {"start": 1743.04, "end": 1743.52, "word": " sigma", "probability": 0.465087890625}, {"start": 1743.52, "end": 1744.52, "word": " xn", "probability": 0.574951171875}, {"start": 1744.52, "end": 1746.34, "word": " او", "probability": 0.712646484375}, {"start": 1746.34, "end": 1746.7, "word": " sigma", "probability": 0.7626953125}, {"start": 1746.7, "end": 1747.44, "word": " من", "probability": 0.94921875}, {"start": 1747.44, "end": 1747.78, "word": " n", "probability": 0.69775390625}, {"start": 1747.78, "end": 1748.2, "word": " بساعة", "probability": 0.756103515625}, {"start": 1748.2, "end": 1748.7, "word": " واحد", "probability": 0.941650390625}, {"start": 1748.7, "end": 1750.08, "word": " لانفينيتي", "probability": 0.7154134114583334}, {"start": 1750.08, "end": 1751.08, "word": " okay", "probability": 0.4248046875}, {"start": 1751.08, "end": 1753.2, "word": " الان", "probability": 0.67578125}, {"start": 1753.2, "end": 1754.32, "word": " now", "probability": 0.403564453125}, {"start": 1754.32, "end": 1758.66, "word": " if", "probability": 0.83837890625}, {"start": 1758.66, "end": 1761.22, "word": " ال", "probability": 0.638671875}, {"start": 1761.22, "end": 1761.84, "word": " sequence", "probability": 0.95361328125}, {"start": 1761.84, "end": 1762.94, "word": " sn", "probability": 0.448486328125}, {"start": 1762.94, "end": 1765.42, "word": " converges", "probability": 0.863525390625}, {"start": 1765.42, "end": 1769.28, "word": " say", "probability": 0.84375}], "temperature": 1.0}, {"id": 64, "seek": 180027, "start": 1771.11, "end": 1800.27, "text": "limit sn بالساوي عدد s ينتمي إلى r طبعا then we say في الحالة هذه بنقول أنه the series اللي هي summation xn from n equals one to infinity converges", "tokens": [4197, 270, 2406, 20666, 3794, 995, 45865, 6225, 3215, 3215, 262, 7251, 29399, 2304, 1829, 30731, 367, 23032, 3555, 3615, 995, 550, 321, 584, 8978, 21542, 6027, 3660, 29538, 44945, 39648, 14739, 3224, 264, 2638, 13672, 1829, 39896, 28811, 2031, 77, 490, 297, 6915, 472, 281, 13202, 9652, 2880], "avg_logprob": -0.24187500149011612, "compression_ratio": 1.2435897435897436, "no_speech_prob": 0.0, "words": [{"start": 1771.1100000000001, "end": 1772.23, "word": "limit", "probability": 0.757080078125}, {"start": 1772.23, "end": 1773.35, "word": " sn", "probability": 0.2998046875}, {"start": 1773.35, "end": 1774.13, "word": " بالساوي", "probability": 0.5901718139648438}, {"start": 1774.13, "end": 1774.61, "word": " عدد", "probability": 0.9500325520833334}, {"start": 1774.61, "end": 1775.09, "word": " s", "probability": 0.5751953125}, {"start": 1775.09, "end": 1776.39, "word": " ينتمي", "probability": 0.83624267578125}, {"start": 1776.39, "end": 1776.53, "word": " إلى", "probability": 0.5888671875}, {"start": 1776.53, "end": 1776.79, "word": " r", "probability": 0.424072265625}, {"start": 1776.79, "end": 1777.31, "word": " طبعا", "probability": 0.9744873046875}, {"start": 1777.31, "end": 1782.09, "word": " then", "probability": 0.55859375}, {"start": 1782.09, "end": 1782.65, "word": " we", "probability": 0.91748046875}, {"start": 1782.65, "end": 1783.39, "word": " say", "probability": 0.9677734375}, {"start": 1783.39, "end": 1784.47, "word": " في", "probability": 0.7353515625}, {"start": 1784.47, "end": 1784.81, "word": " الحالة", "probability": 0.9498697916666666}, {"start": 1784.81, "end": 1784.99, "word": " هذه", "probability": 0.7080078125}, {"start": 1784.99, "end": 1785.45, "word": " بنقول", "probability": 0.8203125}, {"start": 1785.45, "end": 1786.75, "word": " أنه", "probability": 0.5953369140625}, {"start": 1786.75, "end": 1787.05, "word": " the", "probability": 0.64453125}, {"start": 1787.05, "end": 1787.81, "word": " series", "probability": 0.923828125}, {"start": 1787.81, "end": 1791.39, "word": " اللي", "probability": 0.760498046875}, {"start": 1791.39, "end": 1791.83, "word": " هي", "probability": 0.94921875}, {"start": 1791.83, "end": 1793.85, "word": " summation", "probability": 0.86572265625}, {"start": 1793.85, "end": 1795.67, "word": " xn", "probability": 0.77685546875}, {"start": 1795.67, "end": 1795.99, "word": " from", "probability": 0.76416015625}, {"start": 1795.99, "end": 1796.37, "word": " n", "probability": 0.916015625}, {"start": 1796.37, "end": 1797.61, "word": " equals", "probability": 0.82373046875}, {"start": 1797.61, "end": 1797.89, "word": " one", "probability": 0.8466796875}, {"start": 1797.89, "end": 1798.07, "word": " to", "probability": 0.95654296875}, {"start": 1798.07, "end": 1798.63, "word": " infinity", "probability": 0.90380859375}, {"start": 1798.63, "end": 1800.27, "word": " converges", "probability": 0.892578125}], "temperature": 1.0}, {"id": 65, "seek": 183273, "start": 1809.07, "end": 1832.73, "text": "and its sum is summation from n equals one to infinity ل x in ال summation تبعها أو المجموعة تبعها عبارة عن limit لل sequence of partial sums اللي هو العدد S", "tokens": [474, 1080, 2408, 307, 28811, 490, 297, 6915, 472, 281, 13202, 5296, 2031, 294, 2423, 28811, 6055, 3555, 3615, 11296, 34051, 9673, 7435, 2304, 2407, 27884, 6055, 3555, 3615, 11296, 6225, 3555, 9640, 3660, 18871, 4948, 24976, 8310, 295, 14641, 34499, 13672, 1829, 31439, 18863, 3215, 3215, 318], "avg_logprob": -0.22863520408163265, "compression_ratio": 1.342281879194631, "no_speech_prob": 0.0, "words": [{"start": 1809.0700000000002, "end": 1810.1100000000001, "word": "and", "probability": 0.10943603515625}, {"start": 1810.1100000000001, "end": 1811.15, "word": " its", "probability": 0.59228515625}, {"start": 1811.15, "end": 1812.53, "word": " sum", "probability": 0.92041015625}, {"start": 1812.53, "end": 1814.49, "word": " is", "probability": 0.93310546875}, {"start": 1814.49, "end": 1816.35, "word": " summation", "probability": 0.77392578125}, {"start": 1816.35, "end": 1817.01, "word": " from", "probability": 0.8642578125}, {"start": 1817.01, "end": 1817.35, "word": " n", "probability": 0.763671875}, {"start": 1817.35, "end": 1817.85, "word": " equals", "probability": 0.488037109375}, {"start": 1817.85, "end": 1818.11, "word": " one", "probability": 0.8271484375}, {"start": 1818.11, "end": 1818.29, "word": " to", "probability": 0.9453125}, {"start": 1818.29, "end": 1818.81, "word": " infinity", "probability": 0.87109375}, {"start": 1818.81, "end": 1819.99, "word": " ل", "probability": 0.74609375}, {"start": 1819.99, "end": 1820.37, "word": " x", "probability": 0.272216796875}, {"start": 1820.37, "end": 1820.77, "word": " in", "probability": 0.27587890625}, {"start": 1820.77, "end": 1821.29, "word": " ال", "probability": 0.381103515625}, {"start": 1821.29, "end": 1821.73, "word": " summation", "probability": 0.74072265625}, {"start": 1821.73, "end": 1822.41, "word": " تبعها", "probability": 0.9278564453125}, {"start": 1822.41, "end": 1822.95, "word": " أو", "probability": 0.7236328125}, {"start": 1822.95, "end": 1823.53, "word": " المجموعة", "probability": 0.97568359375}, {"start": 1823.53, "end": 1824.03, "word": " تبعها", "probability": 0.9754638671875}, {"start": 1824.03, "end": 1824.49, "word": " عبارة", "probability": 0.9476318359375}, {"start": 1824.49, "end": 1824.77, "word": " عن", "probability": 0.998046875}, {"start": 1824.77, "end": 1825.31, "word": " limit", "probability": 0.646484375}, {"start": 1825.31, "end": 1826.51, "word": " لل", "probability": 0.453857421875}, {"start": 1826.51, "end": 1826.95, "word": " sequence", "probability": 0.94140625}, {"start": 1826.95, "end": 1827.35, "word": " of", "probability": 0.97412109375}, {"start": 1827.35, "end": 1827.87, "word": " partial", "probability": 0.94091796875}, {"start": 1827.87, "end": 1828.45, "word": " sums", "probability": 0.95751953125}, {"start": 1828.45, "end": 1830.89, "word": " اللي", "probability": 0.95849609375}, {"start": 1830.89, "end": 1831.95, "word": " هو", "probability": 0.921875}, {"start": 1831.95, "end": 1832.45, "word": " العدد", "probability": 0.9583333333333334}, {"start": 1832.45, "end": 1832.73, "word": " S", "probability": 0.5}], "temperature": 1.0}, {"id": 66, "seek": 186588, "start": 1837.16, "end": 1865.88, "text": "لو كانت الـ sequence divergent if الـ sequence is in diverges we say أنه الـ series sigma x in diverges", "tokens": [1211, 2407, 25961, 2655, 2423, 39184, 8310, 18558, 6930, 498, 2423, 39184, 8310, 307, 294, 18558, 2880, 321, 584, 14739, 3224, 2423, 39184, 2638, 12771, 2031, 294, 18558, 2880], "avg_logprob": -0.3981770714124044, "compression_ratio": 1.3595505617977528, "no_speech_prob": 0.0, "words": [{"start": 1837.16, "end": 1837.54, "word": "لو", "probability": 0.85498046875}, {"start": 1837.54, "end": 1837.98, "word": " كانت", "probability": 0.990234375}, {"start": 1837.98, "end": 1838.08, "word": " الـ", "probability": 0.795166015625}, {"start": 1838.08, "end": 1838.66, "word": " sequence", "probability": 0.76806640625}, {"start": 1838.66, "end": 1843.18, "word": " divergent", "probability": 0.6280517578125}, {"start": 1843.18, "end": 1844.94, "word": " if", "probability": 0.354736328125}, {"start": 1844.94, "end": 1845.62, "word": " الـ", "probability": 0.64111328125}, {"start": 1845.62, "end": 1846.06, "word": " sequence", "probability": 0.9208984375}, {"start": 1846.06, "end": 1846.36, "word": " is", "probability": 0.349365234375}, {"start": 1846.36, "end": 1846.7, "word": " in", "probability": 0.88916015625}, {"start": 1846.7, "end": 1847.62, "word": " diverges", "probability": 0.6141357421875}, {"start": 1847.62, "end": 1850.74, "word": " we", "probability": 0.58154296875}, {"start": 1850.74, "end": 1851.32, "word": " say", "probability": 0.9609375}, {"start": 1851.32, "end": 1854.02, "word": " أنه", "probability": 0.4552001953125}, {"start": 1854.02, "end": 1854.2, "word": " الـ", "probability": 0.666748046875}, {"start": 1854.2, "end": 1854.66, "word": " series", "probability": 0.861328125}, {"start": 1854.66, "end": 1860.12, "word": " sigma", "probability": 0.255859375}, {"start": 1860.12, "end": 1862.74, "word": " x", "probability": 0.7236328125}, {"start": 1862.74, "end": 1863.24, "word": " in", "probability": 0.84912109375}, {"start": 1863.24, "end": 1865.88, "word": " diverges", "probability": 0.822021484375}], "temperature": 1.0}, {"id": 67, "seek": 189511, "start": 1869.09, "end": 1895.11, "text": "إذا ال convergence و ال divergence depends on the divergence أو convergence of the infinite series depends on the convergence or divergence of the sequence of partial sums مرتبط بيها ال sequence of partial sums convergent السيريز اللي تابع إليها convergent", "tokens": [28814, 15730, 2423, 32181, 4032, 2423, 47387, 5946, 322, 264, 47387, 34051, 32181, 295, 264, 13785, 2638, 5946, 322, 264, 32181, 420, 47387, 295, 264, 8310, 295, 14641, 34499, 3714, 43500, 3555, 9566, 4724, 1829, 11296, 2423, 8310, 295, 14641, 34499, 9652, 6930, 21136, 13546, 1829, 11622, 13672, 1829, 6055, 16758, 3615, 11933, 20292, 11296, 9652, 6930], "avg_logprob": -0.23612608477987093, "compression_ratio": 1.879746835443038, "no_speech_prob": 0.0, "words": [{"start": 1869.09, "end": 1869.43, "word": "إذا", "probability": 0.615478515625}, {"start": 1869.43, "end": 1869.61, "word": " ال", "probability": 0.93798828125}, {"start": 1869.61, "end": 1870.17, "word": " convergence", "probability": 0.6689453125}, {"start": 1870.17, "end": 1870.53, "word": " و", "probability": 0.71923828125}, {"start": 1870.53, "end": 1870.61, "word": " ال", "probability": 0.7294921875}, {"start": 1870.61, "end": 1871.15, "word": " divergence", "probability": 0.9951171875}, {"start": 1871.15, "end": 1872.75, "word": " depends", "probability": 0.446533203125}, {"start": 1872.75, "end": 1873.23, "word": " on", "probability": 0.92578125}, {"start": 1873.23, "end": 1873.41, "word": " the", "probability": 0.525390625}, {"start": 1873.41, "end": 1875.63, "word": " divergence", "probability": 0.17041015625}, {"start": 1875.63, "end": 1876.11, "word": " أو", "probability": 0.4208984375}, {"start": 1876.11, "end": 1876.67, "word": " convergence", "probability": 0.93994140625}, {"start": 1876.67, "end": 1877.63, "word": " of", "probability": 0.908203125}, {"start": 1877.63, "end": 1877.83, "word": " the", "probability": 0.87109375}, {"start": 1877.83, "end": 1878.11, "word": " infinite", "probability": 0.85693359375}, {"start": 1878.11, "end": 1878.63, "word": " series", "probability": 0.8984375}, {"start": 1878.63, "end": 1879.51, "word": " depends", "probability": 0.70849609375}, {"start": 1879.51, "end": 1880.09, "word": " on", "probability": 0.95556640625}, {"start": 1880.09, "end": 1880.31, "word": " the", "probability": 0.8583984375}, {"start": 1880.31, "end": 1880.83, "word": " convergence", "probability": 0.94482421875}, {"start": 1880.83, "end": 1881.27, "word": " or", "probability": 0.77197265625}, {"start": 1881.27, "end": 1881.79, "word": " divergence", "probability": 0.95361328125}, {"start": 1881.79, "end": 1883.03, "word": " of", "probability": 0.97314453125}, {"start": 1883.03, "end": 1883.91, "word": " the", "probability": 0.83935546875}, {"start": 1883.91, "end": 1885.65, "word": " sequence", "probability": 0.95263671875}, {"start": 1885.65, "end": 1885.83, "word": " of", "probability": 0.7626953125}, {"start": 1885.83, "end": 1886.25, "word": " partial", "probability": 0.93359375}, {"start": 1886.25, "end": 1886.73, "word": " sums", "probability": 0.96875}, {"start": 1886.73, "end": 1888.99, "word": " مرتبط", "probability": 0.8701171875}, {"start": 1888.99, "end": 1889.31, "word": " بيها", "probability": 0.95263671875}, {"start": 1889.31, "end": 1890.11, "word": " ال", "probability": 0.6572265625}, {"start": 1890.11, "end": 1890.39, "word": " sequence", "probability": 0.94775390625}, {"start": 1890.39, "end": 1890.69, "word": " of", "probability": 0.94921875}, {"start": 1890.69, "end": 1890.99, "word": " partial", "probability": 0.92041015625}, {"start": 1890.99, "end": 1891.31, "word": " sums", "probability": 0.95849609375}, {"start": 1891.31, "end": 1891.93, "word": " convergent", "probability": 0.766357421875}, {"start": 1891.93, "end": 1892.59, "word": " السيريز", "probability": 0.78045654296875}, {"start": 1892.59, "end": 1893.67, "word": " اللي", "probability": 0.8466796875}, {"start": 1893.67, "end": 1894.01, "word": " تابع", "probability": 0.9716796875}, {"start": 1894.01, "end": 1894.35, "word": " إليها", "probability": 0.7615559895833334}, {"start": 1894.35, "end": 1895.11, "word": " convergent", "probability": 0.793212890625}], "temperature": 1.0}, {"id": 68, "seek": 192172, "start": 1896.34, "end": 1921.72, "text": "و العكس إذا كانت ال sequence of partial sums divergent ال series ال infinite series التابعة إلى divergent طيب ناخد بعض الأمثلة طبعا ال Sn هذا ال Sn", "tokens": [2407, 18863, 4117, 3794, 11933, 15730, 25961, 2655, 2423, 8310, 295, 14641, 34499, 18558, 6930, 2423, 2638, 2423, 13785, 2638, 16712, 16758, 27884, 30731, 18558, 6930, 23032, 1829, 3555, 8717, 47283, 3215, 45030, 11242, 16247, 2304, 12984, 37977, 23032, 3555, 3615, 995, 2423, 9264, 23758, 2423, 9264], "avg_logprob": -0.2503255183498065, "compression_ratio": 1.4166666666666667, "no_speech_prob": 0.0, "words": [{"start": 1896.34, "end": 1896.54, "word": "و", "probability": 0.67431640625}, {"start": 1896.54, "end": 1896.88, "word": " العكس", "probability": 0.8328450520833334}, {"start": 1896.88, "end": 1897.06, "word": " إذا", "probability": 0.76904296875}, {"start": 1897.06, "end": 1897.4, "word": " كانت", "probability": 0.9873046875}, {"start": 1897.4, "end": 1897.5, "word": " ال", "probability": 0.6123046875}, {"start": 1897.5, "end": 1897.7, "word": " sequence", "probability": 0.88330078125}, {"start": 1897.7, "end": 1898.06, "word": " of", "probability": 0.93603515625}, {"start": 1898.06, "end": 1898.36, "word": " partial", "probability": 0.93310546875}, {"start": 1898.36, "end": 1898.72, "word": " sums", "probability": 0.97607421875}, {"start": 1898.72, "end": 1899.4, "word": " divergent", "probability": 0.6175537109375}, {"start": 1899.4, "end": 1899.62, "word": " ال", "probability": 0.42138671875}, {"start": 1899.62, "end": 1899.98, "word": " series", "probability": 0.7890625}, {"start": 1899.98, "end": 1900.16, "word": " ال", "probability": 0.86962890625}, {"start": 1900.16, "end": 1900.38, "word": " infinite", "probability": 0.6513671875}, {"start": 1900.38, "end": 1900.78, "word": " series", "probability": 0.8720703125}, {"start": 1900.78, "end": 1901.26, "word": " التابعة", "probability": 0.904296875}, {"start": 1901.26, "end": 1901.52, "word": " إلى", "probability": 0.3076171875}, {"start": 1901.52, "end": 1903.14, "word": " divergent", "probability": 0.786865234375}, {"start": 1903.14, "end": 1911.84, "word": " طيب", "probability": 0.6883138020833334}, {"start": 1911.84, "end": 1912.92, "word": " ناخد", "probability": 0.9602864583333334}, {"start": 1912.92, "end": 1913.24, "word": " بعض", "probability": 0.975830078125}, {"start": 1913.24, "end": 1913.88, "word": " الأمثلة", "probability": 0.9820556640625}, {"start": 1913.88, "end": 1918.18, "word": " طبعا", "probability": 0.85845947265625}, {"start": 1918.18, "end": 1918.38, "word": " ال", "probability": 0.9599609375}, {"start": 1918.38, "end": 1918.98, "word": " Sn", "probability": 0.29443359375}, {"start": 1918.98, "end": 1919.44, "word": " هذا", "probability": 0.7236328125}, {"start": 1919.44, "end": 1921.08, "word": " ال", "probability": 0.89453125}, {"start": 1921.08, "end": 1921.72, "word": " Sn", "probability": 0.802734375}], "temperature": 1.0}, {"id": 69, "seek": 194829, "start": 1924.61, "end": 1948.29, "text": "هذا بنسميه الانث partial sum الانث partial sum انث partial sum المجموع الجزئي أنوني okay هو الحد العام لل sequence و partial sums", "tokens": [3224, 15730, 44945, 38251, 1829, 3224, 2423, 7649, 12984, 14641, 2408, 2423, 7649, 12984, 14641, 2408, 16472, 12984, 14641, 2408, 9673, 7435, 2304, 45367, 25724, 11622, 19986, 1829, 14739, 11536, 1829, 1392, 31439, 21542, 3215, 18863, 10943, 24976, 8310, 4032, 14641, 34499], "avg_logprob": -0.26507993492969245, "compression_ratio": 1.525, "no_speech_prob": 0.0, "words": [{"start": 1924.61, "end": 1924.97, "word": "هذا", "probability": 0.823486328125}, {"start": 1924.97, "end": 1925.83, "word": " بنسميه", "probability": 0.7672119140625}, {"start": 1925.83, "end": 1926.39, "word": " الانث", "probability": 0.6005452473958334}, {"start": 1926.39, "end": 1926.89, "word": " partial", "probability": 0.80712890625}, {"start": 1926.89, "end": 1927.43, "word": " sum", "probability": 0.9453125}, {"start": 1927.43, "end": 1929.37, "word": " الانث", "probability": 0.8601888020833334}, {"start": 1929.37, "end": 1930.61, "word": " partial", "probability": 0.8828125}, {"start": 1930.61, "end": 1935.81, "word": " sum", "probability": 0.7392578125}, {"start": 1935.81, "end": 1936.53, "word": " انث", "probability": 0.591796875}, {"start": 1936.53, "end": 1937.05, "word": " partial", "probability": 0.88525390625}, {"start": 1937.05, "end": 1937.71, "word": " sum", "probability": 0.95068359375}, {"start": 1937.71, "end": 1939.19, "word": " المجموع", "probability": 0.900146484375}, {"start": 1939.19, "end": 1939.89, "word": " الجزئي", "probability": 0.763671875}, {"start": 1939.89, "end": 1940.75, "word": " أنوني", "probability": 0.81591796875}, {"start": 1940.75, "end": 1942.41, "word": " okay", "probability": 0.496337890625}, {"start": 1942.41, "end": 1945.69, "word": " هو", "probability": 0.62255859375}, {"start": 1945.69, "end": 1946.13, "word": " الحد", "probability": 0.99072265625}, {"start": 1946.13, "end": 1946.55, "word": " العام", "probability": 0.986083984375}, {"start": 1946.55, "end": 1946.75, "word": " لل", "probability": 0.8515625}, {"start": 1946.75, "end": 1947.15, "word": " sequence", "probability": 0.9072265625}, {"start": 1947.15, "end": 1947.53, "word": " و", "probability": 0.6943359375}, {"start": 1947.53, "end": 1947.85, "word": " partial", "probability": 0.947265625}, {"start": 1947.85, "end": 1948.29, "word": " sums", "probability": 0.7685546875}], "temperature": 1.0}, {"id": 70, "seek": 196862, "start": 1949.5, "end": 1968.62, "text": "أذا لما بدى نخبر هل ال series convergent ولا divergent بجيب ال sequence of partial sums وبجيب الحد العام لل sequence of partial sums وبأفحص هل ال sequence هذي convergent ولا divergent ناخد بعض الأمثلة", "tokens": [10721, 15730, 5296, 15042, 47525, 7578, 8717, 9778, 26890, 8032, 1211, 2423, 2638, 9652, 6930, 49429, 18558, 6930, 4724, 7435, 1829, 3555, 2423, 8310, 295, 14641, 34499, 46599, 7435, 1829, 3555, 21542, 3215, 18863, 10943, 24976, 8310, 295, 14641, 34499, 46599, 10721, 5172, 5016, 9381, 8032, 1211, 2423, 8310, 8032, 8848, 1829, 9652, 6930, 49429, 18558, 6930, 8717, 47283, 3215, 45030, 11242, 16247, 2304, 12984, 37977], "avg_logprob": -0.2621268799055868, "compression_ratio": 1.732484076433121, "no_speech_prob": 1.1920928955078125e-07, "words": [{"start": 1949.5, "end": 1950.24, "word": "أذا", "probability": 0.44976806640625}, {"start": 1950.24, "end": 1950.44, "word": " لما", "probability": 0.7091064453125}, {"start": 1950.44, "end": 1950.76, "word": " بدى", "probability": 0.3966064453125}, {"start": 1950.76, "end": 1951.32, "word": " نخبر", "probability": 0.7205403645833334}, {"start": 1951.32, "end": 1951.78, "word": " هل", "probability": 0.92236328125}, {"start": 1951.78, "end": 1951.88, "word": " ال", "probability": 0.9306640625}, {"start": 1951.88, "end": 1952.1, "word": " series", "probability": 0.70361328125}, {"start": 1952.1, "end": 1952.64, "word": " convergent", "probability": 0.6898193359375}, {"start": 1952.64, "end": 1952.8, "word": " ولا", "probability": 0.55517578125}, {"start": 1952.8, "end": 1953.52, "word": " divergent", "probability": 0.6627197265625}, {"start": 1953.52, "end": 1954.62, "word": " بجيب", "probability": 0.92529296875}, {"start": 1954.62, "end": 1954.76, "word": " ال", "probability": 0.8544921875}, {"start": 1954.76, "end": 1955.06, "word": " sequence", "probability": 0.888671875}, {"start": 1955.06, "end": 1955.3, "word": " of", "probability": 0.87255859375}, {"start": 1955.3, "end": 1955.68, "word": " partial", "probability": 0.947265625}, {"start": 1955.68, "end": 1956.04, "word": " sums", "probability": 0.96484375}, {"start": 1956.04, "end": 1956.48, "word": " وبجيب", "probability": 0.8253173828125}, {"start": 1956.48, "end": 1956.86, "word": " الحد", "probability": 0.98193359375}, {"start": 1956.86, "end": 1957.28, "word": " العام", "probability": 0.988037109375}, {"start": 1957.28, "end": 1958.38, "word": " لل", "probability": 0.8134765625}, {"start": 1958.38, "end": 1958.82, "word": " sequence", "probability": 0.93408203125}, {"start": 1958.82, "end": 1959.02, "word": " of", "probability": 0.93310546875}, {"start": 1959.02, "end": 1959.36, "word": " partial", "probability": 0.94873046875}, {"start": 1959.36, "end": 1959.7, "word": " sums", "probability": 0.90673828125}, {"start": 1959.7, "end": 1960.42, "word": " وبأفحص", "probability": 0.77119140625}, {"start": 1960.42, "end": 1960.64, "word": " هل", "probability": 0.966796875}, {"start": 1960.64, "end": 1960.76, "word": " ال", "probability": 0.9189453125}, {"start": 1960.76, "end": 1961.16, "word": " sequence", "probability": 0.98779296875}, {"start": 1961.16, "end": 1961.78, "word": " هذي", "probability": 0.564697265625}, {"start": 1961.78, "end": 1962.82, "word": " convergent", "probability": 0.891845703125}, {"start": 1962.82, "end": 1963.06, "word": " ولا", "probability": 0.89599609375}, {"start": 1963.06, "end": 1963.8, "word": " divergent", "probability": 0.898681640625}, {"start": 1963.8, "end": 1967.68, "word": " ناخد", "probability": 0.672119140625}, {"start": 1967.68, "end": 1967.96, "word": " بعض", "probability": 0.964111328125}, {"start": 1967.96, "end": 1968.62, "word": " الأمثلة", "probability": 0.98828125}], "temperature": 1.0}, {"id": 71, "seek": 200702, "start": 1982.76, "end": 2007.02, "text": "المثال الأول consider sequence R to N from N equals 0 to infinity طبعا هذه sequence of real numbers", "tokens": [45340, 12984, 6027, 16247, 12610, 1949, 8310, 497, 281, 426, 490, 426, 6915, 1958, 281, 13202, 23032, 3555, 3615, 995, 29538, 8310, 295, 957, 3547], "avg_logprob": -0.27073317995438206, "compression_ratio": 1.0636363636363637, "no_speech_prob": 0.0, "words": [{"start": 1982.76, "end": 1983.4, "word": "المثال", "probability": 0.9402669270833334}, {"start": 1983.4, "end": 1983.98, "word": " الأول", "probability": 0.94287109375}, {"start": 1983.98, "end": 1994.56, "word": " consider", "probability": 0.448974609375}, {"start": 1994.56, "end": 1997.78, "word": " sequence", "probability": 0.51123046875}, {"start": 1997.78, "end": 2000.06, "word": " R", "probability": 0.278076171875}, {"start": 2000.06, "end": 2000.34, "word": " to", "probability": 0.611328125}, {"start": 2000.34, "end": 2000.7, "word": " N", "probability": 0.97802734375}, {"start": 2000.7, "end": 2001.16, "word": " from", "probability": 0.826171875}, {"start": 2001.16, "end": 2001.58, "word": " N", "probability": 0.94287109375}, {"start": 2001.58, "end": 2002.06, "word": " equals", "probability": 0.266845703125}, {"start": 2002.06, "end": 2002.5, "word": " 0", "probability": 0.5029296875}, {"start": 2002.5, "end": 2002.7, "word": " to", "probability": 0.9765625}, {"start": 2002.7, "end": 2003.32, "word": " infinity", "probability": 0.7880859375}, {"start": 2003.32, "end": 2005.28, "word": " طبعا", "probability": 0.9146728515625}, {"start": 2005.28, "end": 2005.48, "word": " هذه", "probability": 0.47998046875}, {"start": 2005.48, "end": 2005.88, "word": " sequence", "probability": 0.74609375}, {"start": 2005.88, "end": 2006.2, "word": " of", "probability": 0.88525390625}, {"start": 2006.2, "end": 2006.46, "word": " real", "probability": 0.92822265625}, {"start": 2006.46, "end": 2007.02, "word": " numbers", "probability": 0.837890625}], "temperature": 1.0}, {"id": 72, "seek": 203433, "start": 2009.15, "end": 2034.33, "text": "Where R is a real number which generates الsequence هذه generates the geometric .. the so-called geometric series .. geometric series", "tokens": [14436, 497, 307, 257, 957, 1230, 597, 23815, 2423, 11834, 655, 29538, 23815, 264, 33246, 4386, 264, 370, 12, 11880, 33246, 2638, 4386, 33246, 2638], "avg_logprob": -0.5054086446762085, "compression_ratio": 1.3663366336633664, "no_speech_prob": 0.0, "words": [{"start": 2009.15, "end": 2009.69, "word": "Where", "probability": 0.08111572265625}, {"start": 2009.69, "end": 2010.67, "word": " R", "probability": 0.60205078125}, {"start": 2010.67, "end": 2012.07, "word": " is", "probability": 0.78466796875}, {"start": 2012.07, "end": 2012.85, "word": " a", "probability": 0.95654296875}, {"start": 2012.85, "end": 2013.15, "word": " real", "probability": 0.93896484375}, {"start": 2013.15, "end": 2013.65, "word": " number", "probability": 0.970703125}, {"start": 2013.65, "end": 2018.21, "word": " which", "probability": 0.470703125}, {"start": 2018.21, "end": 2020.77, "word": " generates", "probability": 0.93359375}, {"start": 2020.77, "end": 2023.13, "word": " الsequence", "probability": 0.5599365234375}, {"start": 2023.13, "end": 2023.45, "word": " هذه", "probability": 0.61865234375}, {"start": 2023.45, "end": 2025.03, "word": " generates", "probability": 0.182861328125}, {"start": 2025.03, "end": 2026.87, "word": " the", "probability": 0.70849609375}, {"start": 2026.87, "end": 2029.15, "word": " geometric", "probability": 0.8427734375}, {"start": 2029.15, "end": 2029.63, "word": " ..", "probability": 0.321044921875}, {"start": 2029.63, "end": 2029.79, "word": " the", "probability": 0.56298828125}, {"start": 2029.79, "end": 2030.01, "word": " so", "probability": 0.94189453125}, {"start": 2030.01, "end": 2030.45, "word": "-called", "probability": 0.6717529296875}, {"start": 2030.45, "end": 2031.01, "word": " geometric", "probability": 0.939453125}, {"start": 2031.01, "end": 2031.77, "word": " series", "probability": 0.9169921875}, {"start": 2031.77, "end": 2032.61, "word": " ..", "probability": 0.5556640625}, {"start": 2032.61, "end": 2033.01, "word": " geometric", "probability": 0.85791015625}, {"start": 2033.01, "end": 2034.33, "word": " series", "probability": 0.91357421875}], "temperature": 1.0}, {"id": 73, "seek": 206527, "start": 2037.05, "end": 2065.27, "text": "اللي هي summation from n equals zero to infinity from r to n okay اذا هي ال sequence هذه of real numbers بتولد infinite series او generates this infinite series اللي هي حدودها اول حد لما n بساوي سفر واحد بعدين r بعدين r تربيه", "tokens": [6027, 20292, 39896, 28811, 490, 297, 6915, 4018, 281, 13202, 490, 367, 281, 297, 1392, 1975, 15730, 39896, 2423, 8310, 29538, 295, 957, 3547, 39894, 12610, 3215, 13785, 2638, 1975, 2407, 23815, 341, 13785, 2638, 13672, 1829, 39896, 11331, 3215, 23328, 11296, 1975, 12610, 11331, 3215, 5296, 15042, 297, 4724, 3794, 995, 45865, 8608, 5172, 2288, 36764, 24401, 39182, 9957, 367, 39182, 9957, 367, 6055, 2288, 21292, 3224], "avg_logprob": -0.3211050776467807, "compression_ratio": 1.544502617801047, "no_speech_prob": 0.0, "words": [{"start": 2037.05, "end": 2037.47, "word": "اللي", "probability": 0.569091796875}, {"start": 2037.47, "end": 2037.85, "word": " هي", "probability": 0.89453125}, {"start": 2037.85, "end": 2038.71, "word": " summation", "probability": 0.6640625}, {"start": 2038.71, "end": 2040.77, "word": " from", "probability": 0.70263671875}, {"start": 2040.77, "end": 2041.09, "word": " n", "probability": 0.62451171875}, {"start": 2041.09, "end": 2041.47, "word": " equals", "probability": 0.370361328125}, {"start": 2041.47, "end": 2041.83, "word": " zero", "probability": 0.478759765625}, {"start": 2041.83, "end": 2042.03, "word": " to", "probability": 0.9658203125}, {"start": 2042.03, "end": 2042.61, "word": " infinity", "probability": 0.8779296875}, {"start": 2042.61, "end": 2043.87, "word": " from", "probability": 0.302734375}, {"start": 2043.87, "end": 2044.25, "word": " r", "probability": 0.3994140625}, {"start": 2044.25, "end": 2044.49, "word": " to", "probability": 0.85693359375}, {"start": 2044.49, "end": 2044.79, "word": " n", "probability": 0.92578125}, {"start": 2044.79, "end": 2045.55, "word": " okay", "probability": 0.1973876953125}, {"start": 2045.55, "end": 2048.15, "word": " اذا", "probability": 0.5694580078125}, {"start": 2048.15, "end": 2048.49, "word": " هي", "probability": 0.58056640625}, {"start": 2048.49, "end": 2048.73, "word": " ال", "probability": 0.83837890625}, {"start": 2048.73, "end": 2049.17, "word": " sequence", "probability": 0.78662109375}, {"start": 2049.17, "end": 2049.63, "word": " هذه", "probability": 0.469970703125}, {"start": 2049.63, "end": 2049.97, "word": " of", "probability": 0.505859375}, {"start": 2049.97, "end": 2050.21, "word": " real", "probability": 0.9619140625}, {"start": 2050.21, "end": 2050.73, "word": " numbers", "probability": 0.8212890625}, {"start": 2050.73, "end": 2052.71, "word": " بتولد", "probability": 0.95947265625}, {"start": 2052.71, "end": 2053.31, "word": " infinite", "probability": 0.94970703125}, {"start": 2053.31, "end": 2053.77, "word": " series", "probability": 0.92041015625}, {"start": 2053.77, "end": 2054.03, "word": " او", "probability": 0.812744140625}, {"start": 2054.03, "end": 2054.61, "word": " generates", "probability": 0.7509765625}, {"start": 2054.61, "end": 2055.65, "word": " this", "probability": 0.87744140625}, {"start": 2055.65, "end": 2056.05, "word": " infinite", "probability": 0.9501953125}, {"start": 2056.05, "end": 2056.75, "word": " series", "probability": 0.8974609375}, {"start": 2056.75, "end": 2059.11, "word": " اللي", "probability": 0.9326171875}, {"start": 2059.11, "end": 2059.35, "word": " هي", "probability": 0.9091796875}, {"start": 2059.35, "end": 2059.87, "word": " حدودها", "probability": 0.950927734375}, {"start": 2059.87, "end": 2060.11, "word": " اول", "probability": 0.833984375}, {"start": 2060.11, "end": 2060.37, "word": " حد", "probability": 0.970703125}, {"start": 2060.37, "end": 2060.63, "word": " لما", "probability": 0.6826171875}, {"start": 2060.63, "end": 2060.81, "word": " n", "probability": 0.55224609375}, {"start": 2060.81, "end": 2061.21, "word": " بساوي", "probability": 0.649658203125}, {"start": 2061.21, "end": 2061.61, "word": " سفر", "probability": 0.8170572916666666}, {"start": 2061.61, "end": 2062.11, "word": " واحد", "probability": 0.9833984375}, {"start": 2062.11, "end": 2063.35, "word": " بعدين", "probability": 0.807861328125}, {"start": 2063.35, "end": 2063.79, "word": " r", "probability": 0.6767578125}, {"start": 2063.79, "end": 2064.35, "word": " بعدين", "probability": 0.81640625}, {"start": 2064.35, "end": 2064.65, "word": " r", "probability": 0.67333984375}, {"start": 2064.65, "end": 2065.27, "word": " تربيه", "probability": 0.59765625}], "temperature": 1.0}, {"id": 74, "seek": 208982, "start": 2066.58, "end": 2089.82, "text": "و R أس N و هكذا ف such series is called geometric series هذه ال series اللي على الصورة هذه بنسميها geometric series الآن ال series هذه", "tokens": [2407, 497, 5551, 3794, 426, 4032, 8032, 4117, 15730, 6156, 1270, 2638, 307, 1219, 33246, 2638, 29538, 2423, 2638, 13672, 1829, 15844, 31767, 13063, 3660, 29538, 44945, 38251, 1829, 11296, 33246, 2638, 6024, 48506, 2423, 2638, 29538], "avg_logprob": -0.3595805999479796, "compression_ratio": 1.40625, "no_speech_prob": 0.0, "words": [{"start": 2066.58, "end": 2066.92, "word": "و", "probability": 0.489990234375}, {"start": 2066.92, "end": 2067.5, "word": " R", "probability": 0.29541015625}, {"start": 2067.5, "end": 2068.12, "word": " أس", "probability": 0.4912109375}, {"start": 2068.12, "end": 2069.92, "word": " N", "probability": 0.174560546875}, {"start": 2069.92, "end": 2073.62, "word": " و", "probability": 0.64208984375}, {"start": 2073.62, "end": 2074.86, "word": " هكذا", "probability": 0.90087890625}, {"start": 2074.86, "end": 2075.8, "word": " ف", "probability": 0.8671875}, {"start": 2075.8, "end": 2076.52, "word": " such", "probability": 0.38134765625}, {"start": 2076.52, "end": 2078.5, "word": " series", "probability": 0.60302734375}, {"start": 2078.5, "end": 2079.7, "word": " is", "probability": 0.90625}, {"start": 2079.7, "end": 2080.06, "word": " called", "probability": 0.8876953125}, {"start": 2080.06, "end": 2080.52, "word": " geometric", "probability": 0.84716796875}, {"start": 2080.52, "end": 2080.94, "word": " series", "probability": 0.91357421875}, {"start": 2080.94, "end": 2081.12, "word": " هذه", "probability": 0.34228515625}, {"start": 2081.12, "end": 2081.54, "word": " ال", "probability": 0.80517578125}, {"start": 2081.54, "end": 2081.82, "word": " series", "probability": 0.88671875}, {"start": 2081.82, "end": 2082.02, "word": " اللي", "probability": 0.630615234375}, {"start": 2082.02, "end": 2082.12, "word": " على", "probability": 0.6796875}, {"start": 2082.12, "end": 2082.48, "word": " الصورة", "probability": 0.97900390625}, {"start": 2082.48, "end": 2082.88, "word": " هذه", "probability": 0.52294921875}, {"start": 2082.88, "end": 2083.88, "word": " بنسميها", "probability": 0.867919921875}, {"start": 2083.88, "end": 2084.3, "word": " geometric", "probability": 0.841796875}, {"start": 2084.3, "end": 2085.06, "word": " series", "probability": 0.92529296875}, {"start": 2085.06, "end": 2086.62, "word": " الآن", "probability": 0.680419921875}, {"start": 2086.62, "end": 2088.56, "word": " ال", "probability": 0.9423828125}, {"start": 2088.56, "end": 2089.26, "word": " series", "probability": 0.9658203125}, {"start": 2089.26, "end": 2089.82, "word": " هذه", "probability": 0.8681640625}], "temperature": 1.0}, {"id": 75, "seek": 212609, "start": 2098.17, "end": 2126.09, "text": "this series واحد converges and its sum اللي هو sigma from n equals zero to infinity لRn بساوي واحد على واحد minus R إذا كان absolute R أصغر من واحد", "tokens": [11176, 2638, 36764, 24401, 9652, 2880, 293, 1080, 2408, 13672, 1829, 31439, 12771, 490, 297, 6915, 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واحد", "probability": 0.98388671875}, {"start": 2119.83, "end": 2120.33, "word": " minus", "probability": 0.96875}, {"start": 2120.33, "end": 2120.79, "word": " R", "probability": 0.9228515625}, {"start": 2120.79, "end": 2121.95, "word": " إذا", "probability": 0.8134765625}, {"start": 2121.95, "end": 2122.47, "word": " كان", "probability": 0.99658203125}, {"start": 2122.47, "end": 2123.91, "word": " absolute", "probability": 0.85595703125}, {"start": 2123.91, "end": 2124.81, "word": " R", "probability": 0.904296875}, {"start": 2124.81, "end": 2125.43, "word": " أصغر", "probability": 0.9881591796875}, {"start": 2125.43, "end": 2125.63, "word": " من", "probability": 0.99755859375}, {"start": 2125.63, "end": 2126.09, "word": " واحد", "probability": 0.975830078125}], "temperature": 1.0}, {"id": 76, "seek": 215005, "start": 2128.09, "end": 2150.05, "text": "and diverges and اتنين diverges if absolute are أكبر من أو يساوي واحد خلّينا نثبت الجزء الأول", "tokens": [474, 18558, 2880, 293, 1975, 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{"start": 2142.87, "end": 2143.53, "word": " أكبر", "probability": 0.8307291666666666}, {"start": 2143.53, "end": 2143.85, "word": " من", "probability": 0.978515625}, {"start": 2143.85, "end": 2144.09, "word": " أو", "probability": 0.888671875}, {"start": 2144.09, "end": 2144.67, "word": " يساوي", "probability": 0.857177734375}, {"start": 2144.67, "end": 2145.47, "word": " واحد", "probability": 0.978759765625}, {"start": 2145.47, "end": 2148.83, "word": " خلّينا", "probability": 0.648779296875}, {"start": 2148.83, "end": 2149.23, "word": " نثبت", "probability": 0.98828125}, {"start": 2149.23, "end": 2149.57, "word": " الجزء", "probability": 0.94482421875}, {"start": 2149.57, "end": 2150.05, "word": " الأول", "probability": 0.960693359375}], "temperature": 1.0}, {"id": 77, "seek": 218325, "start": 2158.01, "end": 2183.25, "text": "to prove one أنا عندي ال S N بساوي سيجما من K بساوي سفر إلى N ل R أس K اللي هو واحد زائد R زائد R تلبية زائد", "tokens": [1353, 7081, 472, 41850, 18871, 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K", "probability": 0.92578125}, {"start": 2176.35, "end": 2179.27, "word": " اللي", "probability": 0.947509765625}, {"start": 2179.27, "end": 2179.53, "word": " هو", "probability": 0.9921875}, {"start": 2179.53, "end": 2180.01, "word": " واحد", "probability": 0.935791015625}, {"start": 2180.01, "end": 2180.57, "word": " زائد", "probability": 0.9098307291666666}, {"start": 2180.57, "end": 2181.05, "word": " R", "probability": 0.96630859375}, {"start": 2181.05, "end": 2181.57, "word": " زائد", "probability": 0.9431966145833334}, {"start": 2181.57, "end": 2181.95, "word": " R", "probability": 0.984375}, {"start": 2181.95, "end": 2182.57, "word": " تلبية", "probability": 0.64093017578125}, {"start": 2182.57, "end": 2183.25, "word": " زائد", "probability": 0.97705078125}], "temperature": 1.0}, {"id": 78, "seek": 220098, "start": 2185.34, "end": 2200.98, "text": "R Sn وفي عندي .. في عندي .. لو ضربت Sn في R", "tokens": [49, 9264, 4032, 41185, 18871, 16254, 4386, 8978, 18871, 16254, 4386, 45164, 48812, 25513, 2655, 9264, 8978, 497], "avg_logprob": -0.44490131265238714, "compression_ratio": 1.2307692307692308, "no_speech_prob": 0.0, "words": [{"start": 2185.34, "end": 2185.92, "word": "R", "probability": 0.368408203125}, {"start": 2185.92, "end": 2186.52, "word": " Sn", "probability": 0.125732421875}, {"start": 2186.52, "end": 2187.36, "word": " وفي", "probability": 0.4676513671875}, {"start": 2187.36, "end": 2188.5, "word": " عندي", "probability": 0.758056640625}, {"start": 2188.5, "end": 2188.5, "word": " ..", "probability": 0.420654296875}, {"start": 2188.5, "end": 2190.64, "word": " في", "probability": 0.6162109375}, {"start": 2190.64, "end": 2191.32, "word": " عندي", "probability": 0.97802734375}, {"start": 2191.32, "end": 2194.64, "word": " ..", "probability": 0.46923828125}, {"start": 2194.64, "end": 2197.78, "word": " لو", "probability": 0.96142578125}, {"start": 2197.78, "end": 2198.42, "word": " ضربت", "probability": 0.970703125}, {"start": 2198.42, "end": 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هكذا", "probability": 0.8541666666666666}, {"start": 2210.6, "end": 2212.04, "word": " زاد", "probability": 0.7685546875}, {"start": 2212.04, "end": 2212.48, "word": " R", "probability": 0.91015625}, {"start": 2212.48, "end": 2212.82, "word": " أس", "probability": 0.663330078125}, {"start": 2212.82, "end": 2213.1, "word": " N", "probability": 0.85986328125}, {"start": 2213.1, "end": 2213.3, "word": " و", "probability": 0.76708984375}, {"start": 2213.3, "end": 2213.62, "word": " آخر", "probability": 0.83349609375}, {"start": 2213.62, "end": 2213.92, "word": " حد", "probability": 0.967041015625}, {"start": 2213.92, "end": 2214.5, "word": " هيكون", "probability": 0.844482421875}, {"start": 2214.5, "end": 2215.06, "word": " R", "probability": 0.9541015625}, {"start": 2215.06, "end": 2216.16, "word": " أس", "probability": 0.956298828125}, {"start": 2216.16, "end": 2216.38, "word": " N", "probability": 0.9619140625}, {"start": 2216.38, "end": 2216.78, "word": " زاد", "probability": 0.943359375}, {"start": 2216.78, "end": 2217.16, "word": " 1", "probability": 0.55322265625}, {"start": 2217.16, "end": 2218.94, "word": " تمام؟", "probability": 0.6971028645833334}, {"start": 2218.94, "end": 2219.38, "word": " الآن", "probability": 0.7056884765625}, {"start": 2219.38, "end": 2219.92, "word": " خلّينا", "probability": 0.76162109375}, {"start": 2219.92, "end": 2220.42, "word": " نطرح", "probability": 0.9775390625}, {"start": 2220.42, "end": 2220.58, "word": " ال", "probability": 0.4990234375}, {"start": 2220.58, "end": 2220.94, "word": " subtract", "probability": 0.3662109375}], "temperature": 1.0}, {"id": 80, "seek": 225313, "start": 2225.37, "end": 2253.13, "text": "subtract نطرح المعادلة لتحت من اللي فوق فبطلع عندي sn في واحد minus r أخدت عامل مشترك sn ولمّا أطرح هذا بروح مع هذا كل الهدوء بتروح مع بعضها بظل عندي واحد سالب r to n زاد واحد تمام ومن هنا اذا sn", "tokens": [30131, 83, 1897, 8717, 9566, 2288, 5016, 9673, 3615, 18513, 37977, 5296, 2655, 33753, 9154, 13672, 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"end": 2229.25, "word": " من", "probability": 0.91748046875}, {"start": 2229.25, "end": 2229.49, "word": " اللي", "probability": 0.738037109375}, {"start": 2229.49, "end": 2229.87, "word": " فوق", "probability": 0.849853515625}, {"start": 2229.87, "end": 2230.39, "word": " فبطلع", "probability": 0.930859375}, {"start": 2230.39, "end": 2230.85, "word": " عندي", "probability": 0.728271484375}, {"start": 2230.85, "end": 2232.69, "word": " sn", "probability": 0.36669921875}, {"start": 2232.69, "end": 2233.79, "word": " في", "probability": 0.6171875}, {"start": 2233.79, "end": 2234.29, "word": " واحد", "probability": 0.7281494140625}, {"start": 2234.29, "end": 2234.79, "word": " minus", "probability": 0.94873046875}, {"start": 2234.79, "end": 2235.07, "word": " r", "probability": 0.51904296875}, {"start": 2235.07, "end": 2235.45, "word": " أخدت", "probability": 0.7796630859375}, {"start": 2235.45, "end": 2235.67, "word": " عامل", "probability": 0.83837890625}, {"start": 2235.67, "end": 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"probability": 0.8564453125}, {"start": 2264.67, "end": 2266.25, "word": " واحد", "probability": 0.987548828125}, {"start": 2266.25, "end": 2266.79, "word": " سالب", "probability": 0.9842122395833334}, {"start": 2266.79, "end": 2267.17, "word": " R", "probability": 0.9677734375}, {"start": 2267.17, "end": 2272.99, "word": " ممكن", "probability": 0.965087890625}, {"start": 2272.99, "end": 2273.27, "word": " هذا", "probability": 0.7431640625}, {"start": 2273.27, "end": 2273.77, "word": " نوديه", "probability": 0.79144287109375}, {"start": 2273.77, "end": 2273.87, "word": " على", "probability": 0.5029296875}, {"start": 2273.87, "end": 2274.25, "word": " ناحية", "probability": 0.7843424479166666}, {"start": 2274.25, "end": 2274.83, "word": " التانية", "probability": 0.9736328125}, {"start": 2274.83, "end": 2276.97, "word": " فبصير", "probability": 0.92919921875}, {"start": 2276.97, "end": 2277.97, "word": " عندى", "probability": 0.66015625}, {"start": 2277.97, "end": 2278.51, "word": " 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0.58868408203125}, {"start": 2370.91, "end": 2371.17, "word": " by", "probability": 0.8046875}, {"start": 2371.17, "end": 2371.67, "word": " sandwich", "probability": 0.9423828125}, {"start": 2371.67, "end": 2372.33, "word": " theorem", "probability": 0.89013671875}], "temperature": 1.0}, {"id": 86, "seek": 240582, "start": 2380.72, "end": 2405.82, "text": "بتطلع عند ال limit ل absolute sn minus 1 على 1 minus r لما n تقول ل infinity بساوي سفر و ممكن ندخل ال limit جوا فهذا بقدر انه limit 1 على sn عفوا limit sn لما n تقول ل infinity بساوي 1 على 1 سالب r", "tokens": [3555, 2655, 9566, 1211, 3615, 43242, 2423, 4948, 5296, 8236, 2406, 3175, 502, 15844, 502, 3175, 367, 5296, 15042, 297, 6055, 4587, 12610, 5296, 13202, 4724, 3794, 995, 45865, 8608, 5172, 2288, 4032, 3714, 43020, 8717, 3215, 9778, 1211, 2423, 4948, 10874, 14407, 6156, 3224, 15730, 4724, 28543, 2288, 16472, 3224, 4948, 502, 15844, 2406, 6225, 5172, 14407, 4948, 2406, 5296, 15042, 297, 6055, 4587, 12610, 5296, 13202, 4724, 3794, 995, 45865, 502, 15844, 502, 8608, 6027, 3555, 367], "avg_logprob": -0.3253906279802322, "compression_ratio": 1.650887573964497, "no_speech_prob": 1.7881393432617188e-07, "words": [{"start": 2380.72, "end": 2381.56, "word": "بتطلع", "probability": 0.481103515625}, {"start": 2381.56, "end": 2381.8, "word": " عند", "probability": 0.9130859375}, {"start": 2381.8, "end": 2381.96, "word": " ال", "probability": 0.60009765625}, {"start": 2381.96, "end": 2382.32, "word": " limit", "probability": 0.73046875}, {"start": 2382.32, "end": 2383.38, "word": " ل", "probability": 0.8603515625}, {"start": 2383.38, "end": 2384.04, "word": " absolute", "probability": 0.478759765625}, {"start": 2384.04, "end": 2384.88, "word": " sn", "probability": 0.2265625}, {"start": 2384.88, "end": 2385.5, "word": " minus", "probability": 0.259765625}, {"start": 2385.5, "end": 2386.3, "word": " 1", "probability": 0.64501953125}, {"start": 2386.3, "end": 2387.36, "word": " على", "probability": 0.5654296875}, {"start": 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"word": " ندخل", "probability": 0.9720458984375}, {"start": 2393.32, "end": 2393.42, "word": " ال", "probability": 0.83447265625}, {"start": 2393.42, "end": 2393.64, "word": " limit", "probability": 0.9697265625}, {"start": 2393.64, "end": 2394.1, "word": " جوا", "probability": 0.6962890625}, {"start": 2394.1, "end": 2395.26, "word": " فهذا", "probability": 0.8326822916666666}, {"start": 2395.26, "end": 2395.66, "word": " بقدر", "probability": 0.5862630208333334}, {"start": 2395.66, "end": 2395.9, "word": " انه", "probability": 0.629638671875}, {"start": 2395.9, "end": 2396.22, "word": " limit", "probability": 0.97119140625}, {"start": 2396.22, "end": 2397.5, "word": " 1", "probability": 0.410400390625}, {"start": 2397.5, "end": 2397.86, "word": " على", "probability": 0.73046875}, {"start": 2397.86, "end": 2398.6, "word": " sn", "probability": 0.6904296875}, {"start": 2398.6, "end": 2399.5, "word": " عفوا", "probability": 0.8087565104166666}, {"start": 2399.5, "end": 2399.8, "word": " limit", "probability": 0.90087890625}, {"start": 2399.8, "end": 2400.58, "word": " sn", "probability": 0.857421875}, {"start": 2400.58, "end": 2401.96, "word": " لما", "probability": 0.963623046875}, {"start": 2401.96, "end": 2402.22, "word": " n", "probability": 0.96044921875}, {"start": 2402.22, "end": 2402.6, "word": " تقول", "probability": 0.9039713541666666}, {"start": 2402.6, "end": 2402.72, "word": " ل", "probability": 0.966796875}, {"start": 2402.72, "end": 2403.24, "word": " infinity", "probability": 0.8720703125}, {"start": 2403.24, "end": 2403.96, "word": " بساوي", "probability": 0.9664306640625}, {"start": 2403.96, "end": 2404.34, "word": " 1", "probability": 0.900390625}, {"start": 2404.34, "end": 2404.66, "word": " على", "probability": 0.84033203125}, {"start": 2404.66, "end": 2405.04, "word": " 1", "probability": 0.9345703125}, {"start": 2405.04, "end": 2405.56, "word": " سالب", "probability": 0.7744140625}, {"start": 2405.56, "end": 2405.82, "word": " r", "probability": 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0.95654296875}, {"start": 2442.11, "end": 2442.37, "word": " not", "probability": 0.8310546875}, {"start": 2442.37, "end": 2442.99, "word": " convergent", "probability": 0.96630859375}, {"start": 2442.99, "end": 2443.29, "word": " نفس", "probability": 0.96875}, {"start": 2443.29, "end": 2444.33, "word": " الحاجة", "probability": 0.99267578125}, {"start": 2444.33, "end": 2444.53, "word": " لو", "probability": 0.89306640625}, {"start": 2444.53, "end": 2444.77, "word": " كان", "probability": 0.97900390625}, {"start": 2444.77, "end": 2445.07, "word": " absolute", "probability": 0.83154296875}, {"start": 2445.07, "end": 2445.39, "word": " R", "probability": 0.88134765625}, {"start": 2445.39, "end": 2445.73, "word": " أكبر", "probability": 0.9609375}, {"start": 2445.73, "end": 2445.87, "word": " من", "probability": 0.99267578125}, {"start": 2445.87, "end": 2446.23, "word": " واحد", "probability": 0.954833984375}, {"start": 2446.23, "end": 2446.61, "word": " فممكن", "probability": 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0.962890625}, {"start": 2450.91, "end": 2451.37, "word": " is", "probability": 0.92822265625}, {"start": 2451.37, "end": 2452.21, "word": " divergent", "probability": 0.938720703125}], "temperature": 1.0}, {"id": 89, "seek": 246463, "start": 2453.37, "end": 2464.63, "text": "تمام؟ في أي سؤال؟ إذا بنوقف هنا و بنكمل ان شاء الله الموضوع اللي جاي في المحاضرة القادمة يوم السبت", "tokens": [39237, 10943, 22807, 8978, 36632, 8608, 33604, 6027, 22807, 11933, 15730, 44945, 30543, 5172, 34105, 4032, 44945, 24793, 1211, 16472, 13412, 16606, 21984, 9673, 2407, 11242, 45367, 13672, 1829, 10874, 47302, 8978, 9673, 5016, 46958, 25720, 25062, 18513, 46007, 7251, 20498, 21136, 3555, 2655], "avg_logprob": -0.12769097884496053, "compression_ratio": 1.475, "no_speech_prob": 0.0, "words": [{"start": 2453.37, "end": 2454.47, "word": "تمام؟", "probability": 0.9134114583333334}, {"start": 2454.47, "end": 2454.63, "word": " في", "probability": 0.88916015625}, {"start": 2454.63, "end": 2454.77, "word": " أي", "probability": 0.52978515625}, {"start": 2454.77, "end": 2455.47, "word": " سؤال؟", "probability": 0.91943359375}, {"start": 2455.47, "end": 2456.09, "word": " إذا", "probability": 0.69287109375}, {"start": 2456.09, "end": 2456.79, "word": " بنوقف", "probability": 0.857421875}, {"start": 2456.79, "end": 2457.17, "word": " هنا", "probability": 0.998046875}, {"start": 2457.17, "end": 2457.91, "word": " و", "probability": 0.86474609375}, {"start": 2457.91, "end": 2459.97, "word": " بنكمل", "probability": 0.8759765625}, {"start": 2459.97, "end": 2460.11, "word": " ان", "probability": 0.78564453125}, {"start": 2460.11, "end": 2460.27, "word": " شاء", "probability": 0.98388671875}, {"start": 2460.27, "end": 2460.37, "word": " الله", "probability": 0.95947265625}, {"start": 2460.37, "end": 2460.83, "word": " الموضوع", "probability": 0.9967041015625}, {"start": 2460.83, "end": 2460.99, "word": " اللي", "probability": 0.72412109375}, {"start": 2460.99, "end": 2461.27, "word": " جاي", "probability": 0.91259765625}, {"start": 2461.27, "end": 2462.17, "word": " في", "probability": 0.484375}, {"start": 2462.17, "end": 2462.83, "word": " المحاضرة", "probability": 0.9840087890625}, {"start": 2462.83, "end": 2463.43, "word": " القادمة", "probability": 0.9908854166666666}, {"start": 2463.43, "end": 2463.99, "word": " يوم", "probability": 0.982666015625}, {"start": 2463.99, "end": 2464.63, "word": " السبت", "probability": 0.8990885416666666}], "temperature": 1.0}], "language": "ar", "language_probability": 1.0, "duration": 2465.6805, "duration_after_vad": 2206.1874999999936}
\ No newline at end of file
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/XjWoXKhuE-o.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/XjWoXKhuE-o.srt
new file mode 100644
index 0000000000000000000000000000000000000000..e3bb7e5a2eb32e2afa72bcde9d90eb5f1a3b9bab
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/XjWoXKhuE-o.srt
@@ -0,0 +1,1767 @@
+1
+00:00:21,090 --> 00:00:26,570
+إذن في المحاضرة هذه إن شاء الله هنحل بعض التمارين
+
+2
+00:00:26,570 --> 00:00:35,250
+للـ homework اللي تابع لـ section ثلاثة واحد وثلاثة 
+
+3
+00:00:35,250 --> 00:00:44,390
+اثنين فأزملتكم سألوا عن ال .. نحن نحل السؤال 13
+
+4
+00:00:44,390 --> 00:00:46,030
+section ثلاثة واحد
+
+5
+00:00:49,430 --> 00:01:15,310
+نكتب السؤال على اللوحة section
+
+6
+00:01:15,310 --> 00:01:25,990
+السؤال 13 section ثلاثة واحد أنا 
+
+7
+00:01:25,990 --> 00:01:31,150
+عندي b is real number أكبر من صفر أصغر من واحد
+
+8
+00:01:31,150 --> 00:01:36,010
+وبينا 
+
+9
+00:01:36,010 --> 00:01:40,990
+نثبت show أن الـ limit
+
+10
+00:01:44,880 --> 00:01:53,020
+للـ sequence اللي الحد العام تبعها n في b to n لما 
+
+11
+00:01:53,020 --> 00:02:03,560
+n تؤول إلى infinity يساوي صفر والكتاب جايب لكم
+
+12
+00:02:03,560 --> 00:02:08,840
+use الـ binomial theorem كما في مثال 3-1-11 الجزء
+
+13
+00:02:08,840 --> 00:02:16,010
+develop حاولتم تستخدموا نفس أسلوب البرهان تبع
+
+14
+00:02:16,010 --> 00:02:20,350
+المثال اللي استخدمنا فيه الـ binomial theorem
+
+15
+00:02:20,350 --> 00:02:28,550
+فهتصلوا للنتيجة فهي البرهان نشوف
+
+16
+00:02:28,550 --> 00:02:32,170
+كيف نستخدم الـ binomial theorem في الوصول إلى 
+
+17
+00:02:32,170 --> 00:02:40,300
+المطلوب أنا عندي من الفرض صفر أصغر من b أصغر من
+
+18
+00:02:40,300 --> 00:02:48,300
+واحد هذا يؤدي أن واحد على b أكبر من واحد
+
+19
+00:02:48,300 --> 00:02:55,500
+وبالتالي هذا يؤدي أن واحد على b سالب واحد أكبر
+
+20
+00:02:55,500 --> 00:03:08,230
+من صفر إذا نأخذ let let a خليني أعرف عدد a على أنه 
+
+21
+00:03:08,230 --> 00:03:13,850
+العدد الموجب واحد على b سالب واحد طبعا هذا عدد
+
+22
+00:03:13,850 --> 00:03:20,110
+موجب حسب ما شفنا وهذا
+
+23
+00:03:20,110 --> 00:03:28,210
+يؤدي أن العدد لو حليت المعادلة هذه في b فهيطلع
+
+24
+00:03:28,210 --> 00:03:38,250
+b يساوي واحد على واحد زائد الـ a وبالتالي 
+
+25
+00:03:38,250 --> 00:03:49,150
+so by الـ binomial باستخدام
+
+26
+00:03:49,150 --> 00:03:59,120
+الـ binomial theorem أنا عندي واحد زائد a الكل أس n
+
+27
+00:03:59,120 --> 00:04:09,300
+يساوي واحد زائد n في a زائد نصف n في n سالب واحد
+
+28
+00:04:09,300 --> 00:04:17,800
+في a تربيع زائد وهكذا تمام
+
+29
+00:04:17,800 --> 00:04:24,890
+إلى آخر حد طبعا هيكون a to n هذا بالضبط زي ما عملنا 
+
+30
+00:04:24,890 --> 00:04:31,950
+في مثال ثلاثة وبالتالي
+
+31
+00:04:31,950 --> 00:04:42,090
+هذا يؤدي من هنا هذا
+
+32
+00:04:42,090 --> 00:04:51,620
+المجموعة بيطلع أكبر من أو يساوي نصف n في n سالب
+
+33
+00:04:51,620 --> 00:05:00,000
+واحد في a تربيع يعني أنا أخذت بس الحد الثالث من
+
+34
+00:05:00,000 --> 00:05:05,120
+المجموعة دي المجموعة طبعا مجموعة أعداد موجبة كلها 
+
+35
+00:05:05,120 --> 00:05:10,180
+فالمجموعة دي بالتأكيد أكبر من أو يساوي الحد الثالث
+
+36
+00:05:10,180 --> 00:05:14,740
+في a تربيع هذا صحيح مافيش مشكلة تمام
+
+37
+00:05:17,950 --> 00:05:33,450
+وبالتالي إذا n في b أس n إيش بيساوي؟ بيساوي n على
+
+38
+00:05:33,450 --> 00:05:43,480
+واحد زائد a الكل أس n صح؟ هذه b فـ b أس n يساوي واحد
+
+39
+00:05:43,480 --> 00:05:50,900
+على واحد زائد a to n وأضرب في n فبيصير هيك طيب
+
+40
+00:05:50,900 --> 00:05:58,560
+من هنا مقلوب واحد زائد a الكل أس n هيطلع أصغر من أو
+
+41
+00:05:58,560 --> 00:06:09,580
+يساوي مقلوب العدد هذا إذا هذا أصغر من أو يساوي n
+
+42
+00:06:14,820 --> 00:06:20,480
+على n في 
+
+43
+00:06:20,480 --> 00:06:28,380
+n سالب واحد في .. في n سالب واحد في a تربيع على
+
+44
+00:06:28,380 --> 00:06:38,980
+اثنين وفي عندنا كمان n العكس
+
+45
+00:06:38,980 --> 00:06:39,600
+العكس
+
+46
+00:06:46,180 --> 00:06:55,020
+هي عندي n ومقلوب هذا بيطلع اثنين n في n سالب واحد
+
+47
+00:06:55,020 --> 00:07:01,760
+في a تربيع تمام؟ إذا هذا إيجى من هنا الآن بختصر الـ
+
+48
+00:07:01,760 --> 00:07:12,620
+n مع الـ n فهدا بيطلع اثنين على n سالب واحد في a
+
+49
+00:07:12,620 --> 00:07:14,040
+تربيع تمام؟
+
+50
+00:07:16,500 --> 00:07:23,140
+الآن هذا الكلام صحيح لكل n أكبر من واحد طبعا ممنوع
+
+51
+00:07:23,140 --> 00:07:27,100
+نأخذ n يساوي واحد لأن في الحالة هذه بيصير في قسمة
+
+52
+00:07:27,100 --> 00:07:31,780
+على صفر لأن لكل الأعداد الطبيعية n أكبر من واحد n
+
+53
+00:07:31,780 --> 00:07:36,980
+في b to n بيطلع أصغر من أو يساوي اثنين على n سالب
+
+54
+00:07:36,980 --> 00:07:44,420
+واحد في a تربيع الآن تعالوا نثبت أن الـ limit للـ
+
+55
+00:07:44,420 --> 00:07:46,240
+sequence هذه يساوي صفر
+
+56
+00:07:50,790 --> 00:07:57,390
+هنستخدم تعريف epsilon capital N لـ limit إذن let 
+
+57
+00:07:57,390 --> 00:08:02,090
+epsilon let 
+
+58
+00:08:02,090 --> 00:08:12,830
+epsilon أكبر من صفر be given
+
+59
+00:08:12,830 --> 00:08:19,210
+Archimedean property by Archimedean property حسب
+
+60
+00:08:19,210 --> 00:08:25,750
+خاصية أرخميدس يوجد نقدر نلاقي عدد طبيعي capital
+
+61
+00:08:25,750 --> 00:08:34,530
+N ينتمي إلى N يعتمد طبعا على إبسلون بحيث أن مقلوب 
+
+62
+00:08:34,530 --> 00:08:41,590
+capital N أصغر من a تربيع في إبسلون على اثنين
+
+63
+00:08:49,130 --> 00:08:54,230
+الـ a تربيع عدد موجب إبسلون على اثنين عدد موجب إذا هذا 
+
+64
+00:08:54,230 --> 00:09:00,830
+عدد موجب الـ Archimedean property بتقول لأي عدد 
+
+65
+00:09:00,830 --> 00:09:05,450
+موجب زي هذا بقدر ألاقي عدد طبيعي capital N مقلوبه
+
+66
+00:09:05,450 --> 00:09:09,250
+وأصغر من العدد الموجب وبالتالي capital N هذا زي ما
+
+67
+00:09:09,250 --> 00:09:13,510
+أنتم شايفين مرتبط بإبسلون بالمتباينة هذه وبالتالي 
+
+68
+00:09:13,510 --> 00:09:18,970
+capital N هذا depends أو يعتمد على إبسلون okay إذا
+
+69
+00:09:18,970 --> 00:09:22,510
+هذا من الـ Archimedean Property طب ليش أنا اخترت
+
+70
+00:09:22,510 --> 00:09:29,930
+هذا العدد عشان نخلي المسافة بين xn و 0 أصغر من إبسلون
+
+71
+00:09:29,930 --> 00:09:38,290
+فركبناها أو ركبناها عشان نصل لإيه الهدف هذا تعالوا 
+
+72
+00:09:38,290 --> 00:09:46,950
+نشوف إذا hence وبالتالي hence بناء على ذلك لو أخذت 
+
+73
+00:09:46,950 --> 00:09:57,950
+n أكبر من capital N هذا يؤدي أن n سالب واحد أكبر
+
+74
+00:09:57,950 --> 00:10:08,270
+من أو يساوي capital N وهذا يؤدي أن absolute n في b
+
+75
+00:10:08,270 --> 00:10:16,690
+to n سالب صفر إيش هذا بيساوي؟ بيساوي n في b to n لأن
+
+76
+00:10:16,690 --> 00:10:26,710
+هذا عدد موجب ومن هنا من هنا n في b to n أصغر من أو
+
+77
+00:10:26,710 --> 00:10:33,690
+يساوي اثنين على n
+
+78
+00:10:33,690 --> 00:10:40,750
+سالب واحد في a تربيع وهذا 
+
+79
+00:10:40,750 --> 00:10:42,390
+أصغر من أو يساوي
+
+80
+00:10:50,580 --> 00:11:00,000
+هذا أصغر من أو يساوي واحد على capital N في اثنين
+
+81
+00:11:00,000 --> 00:11:11,660
+على a تربيع يعني
+
+82
+00:11:11,660 --> 00:11:19,140
+أنا من هنا من واحد على n سالب واحد مقلوب n سالب
+
+83
+00:11:19,140 --> 00:11:27,000
+واحد هيطلع أعظم أو يساوي مقلوب capital N وهذا
+
+84
+00:11:27,000 --> 00:11:33,400
+عبارة عن واحد على n سالب واحد اثنين على a تربيع
+
+85
+00:11:36,970 --> 00:11:42,010
+فمقلوب n سالب واحد أصغر من أو يساوي مقلوب capital N
+
+86
+00:11:42,010 --> 00:11:51,630
+في اثنين على a تربيع تمام؟ شفتم من أين أتيت؟
+
+87
+00:11:51,630 --> 00:11:58,690
+طيب أنا من هنا من هنا واحد مقلوب capital N أصغر من
+
+88
+00:11:58,690 --> 00:12:09,470
+a تربيع في إبسلون على اثنين ضربت اثنين على a تربيع
+
+89
+00:12:09,470 --> 00:12:13,630
+إذا شوفتم ليه أخذت n هنا a تربيع في إبسلون على 
+
+90
+00:12:13,630 --> 00:12:19,210
+اثنين عشان أختصر a تربيع مع a تربيع واثنين مع
+
+91
+00:12:19,210 --> 00:12:26,870
+اثنين ويبقى إبسلون إذا
+
+92
+00:12:26,870 --> 00:12:35,170
+ماذا أثبتنا؟ أثبتنا أن لأي given إبسلون عدد موجب
+
+93
+00:12:35,980 --> 00:12:42,520
+يوجد capital N تعتمد على epsilon بحيث لكل n أكبر
+
+94
+00:12:42,520 --> 00:12:48,260
+من capital N طلع عندي المسافة بين الحد العام للـ
+
+95
+00:12:48,260 --> 00:12:51,900
+sequence اللي هو n في b to n والـ limit المنشودة اللي 
+
+96
+00:12:51,900 --> 00:12:57,860
+هي صفر المسافة بينهم طلعت أصغر من epsilon إذا حسب 
+
+97
+00:12:57,860 --> 00:13:03,100
+تعريف epsilon capital N للـ limit هذا معناه أن الـ
+
+98
+00:13:03,100 --> 00:13:06,720
+limit بما أن هذا صحيح لأي epsilon، epsilon was
+
+99
+00:13:06,720 --> 00:13:11,540
+arbitrary إذاً هيك ممكن أثبتنا إن limit n في b to
+
+100
+00:13:11,540 --> 00:13:16,760
+n as n tends to infinity يساوي صفر وهو المطلوب
+
+101
+00:13:16,760 --> 00:13:22,640
+okay تمام؟ إذاً
+
+102
+00:13:22,640 --> 00:13:27,950
+هنا استخدمنا الـ binomial theorem ساعدتني في الوصول
+
+103
+00:13:27,950 --> 00:13:33,590
+للمتباينة هذه والوصول للمتباينة هذه اللي احنا 
+
+104
+00:13:33,590 --> 00:13:42,730
+استخدمناها في البرهان سهلة البرهان تمام بفهم 
+
+105
+00:13:42,730 --> 00:13:45,970
+الخطوة هذه أقول أن الـ limit يعني آخذ الـ limit 
+
+106
+00:13:45,970 --> 00:13:49,750
+للتربيع أقول أن واحد على n ناقص الواحد ماهي close
+
+107
+00:13:49,750 --> 00:13:55,840
+to zero إذا الـ limit المقدار من أين المتباينة؟ هذه؟
+
+108
+00:13:55,840 --> 00:14:02,160
+بنفع آه بنفع يعني أنت عندك هنا ممكن واحد يستخدم الـ
+
+109
+00:14:02,160 --> 00:14:08,980
+sandwich أو الـ squeeze theorem فبدل ما نستخدم
+
+110
+00:14:08,980 --> 00:14:15,680
+تعريف epsilon capital N نيجي نقول أن الآن أنا 
+
+111
+00:14:15,680 --> 00:14:25,150
+عندي هذه n في b to n طلعت أصغر من أو يساوي اثنين على
+
+112
+00:14:25,150 --> 00:14:31,450
+n سالب واحد في a تربيع وطبعا بالتأكيد هذا أكبر من 
+
+113
+00:14:31,450 --> 00:14:35,390
+أو يساوي صفر لأن الـ n عدد موجب والـ b to n عدد موجب
+
+114
+00:14:35,390 --> 00:14:43,530
+وهذا صحيح لكل n أكبر من واحد الآن هذا عبارة عن 
+
+115
+00:14:43,530 --> 00:14:47,410
+sequence هي الحد العام تبعها لما n تؤول إلى infinity
+
+116
+00:14:47,410 --> 00:14:52,230
+مقلوب n سالب واحد تؤول إلى infinity وبالتالي مقلوبها 
+
+117
+00:14:52,230 --> 00:14:55,990
+تؤول إلى infinity في ثابت موجب اثنين على a تربيع
+
+118
+00:14:55,990 --> 00:15:01,570
+عفوا لما n تؤول إلى infinity المقام بيروح لـ infinity
+
+119
+00:15:01,570 --> 00:15:07,110
+وبالتالي مقلوبه وبيروح لـ صفر تمام؟
+
+120
+00:15:16,990 --> 00:15:22,190
+إذن هذه الـ sequence تؤول إلى 0 نهايتها 0 وهذه الـ
+
+121
+00:15:22,190 --> 00:15:26,970
+constant sequence 0 نهايتها 0 إذن by squeeze 
+
+122
+00:15:26,970 --> 00:15:30,410
+theorem limit الـ sequence هذه يساوي 0 وبلاش 
+
+123
+00:15:30,410 --> 00:15:35,350
+نستخدم تعريف epsilon capital N لكن هذا السؤال في 
+
+124
+00:15:35,350 --> 00:15:39,750
+section 3-1 ما كناش واخدين الـ squeeze theorem فلازم
+
+125
+00:15:39,750 --> 00:15:43,770
+نحلها على طريقة باستخدام الـ definition لكن لو 
+
+126
+00:15:43,770 --> 00:15:48,910
+في الامتحان وممكن ما تفرقش أنت متعلم الـ definition 
+
+127
+00:15:48,910 --> 00:15:52,630
+ومتلم الـ squeeze theorem واستخدم أي طريقة 
+
+128
+00:15:52,630 --> 00:15:58,330
+تعجبك okay تمام في
+
+129
+00:15:58,330 --> 00:16:01,010
+أسئلة ثانية في حد عنده أي سؤال ثاني في section 
+
+130
+00:16:01,010 --> 00:16:07,890
+ثلاثة واحد وثلاثة اثنين تفضلي في أي section ثلاثة
+
+131
+00:16:07,890 --> 00:16:10,510
+واحد طيب ماشي الحال
+
+132
+00:16:50,410 --> 00:17:09,310
+السؤال عشرة section ثلاثة واحد السؤال هذا بيقول if
+
+133
+00:17:09,310 --> 00:17:20,060
+limit sequence xn يساوي x والـ x هذا أكبر من
+
+134
+00:17:20,060 --> 00:17:24,880
+الصفر then
+
+135
+00:17:24,880 --> 00:17:29,340
+then
+
+136
+00:17:29,340 --> 00:17:36,780
+there exist يوجد capital N عدد طبيعي أو capital M
+
+137
+00:17:36,780 --> 00:17:48,170
+natural number عدد طبيعي such that xn أكبر من الصفر
+
+138
+00:17:48,170 --> 00:18:08,950
+لكل n أكبر من أو يساوي m لت
+
+139
+00:18:08,950 --> 00:18:13,250
+y أكبر من الصفر be given
+
+140
+00:18:17,620 --> 00:18:23,600
+خذ أي إبسلون أكبر من الصفر إذن 
+
+141
+00:18:23,600 --> 00:18:30,900
+إبسيلون على اتنين برضه بيطلع عدد موجب طيب
+
+142
+00:18:30,900 --> 00:18:38,880
+احنا فرضنا ان limit xn بيساوي x إذن since xn 
+
+143
+00:18:38,880 --> 00:18:44,960
+converges to x وهي إبسيلون على اتنين عدد أكبر من 
+
+144
+00:18:44,960 --> 00:18:54,480
+الصفر إذا يوجد M عدد طبيعي يعتمد على 
+
+145
+00:18:54,480 --> 00:18:58,840
+إبسيلون عدد
+
+146
+00:18:58,840 --> 00:19:05,140
+طبيعي بحيث أنه لكل N أكبر من أو يساوي M
+
+147
+00:19:05,140 --> 00:19:33,800
+تطلع المسافة من xn إلى x أصغر من إبسيلون على اتنين طيب
+
+148
+00:19:33,800 --> 00:19:34,980
+أنا ال epsilon هذا
+
+149
+00:19:37,520 --> 00:19:44,200
+ممكن آخده أنا عندي من الفرض x أكبر من 0 فممكن آخد
+
+150
+00:19:44,200 --> 00:19:49,640
+ال epsilon هذا بيساوي x بيساوي
+
+151
+00:19:49,640 --> 00:19:56,480
+x أنا
+
+152
+00:19:56,480 --> 00:20:04,400
+ممكن آخد ال epsilon بيساوي x أو حتى x على 2 أو x على 2
+
+153
+00:20:04,400 --> 00:20:10,380
+هذا بالتأكيد الإبسيلون هذا هو عدد موجب اعتبره هو 
+
+154
+00:20:10,380 --> 00:20:15,660
+given وبالتالي
+
+155
+00:20:15,660 --> 00:20:20,580
+أنا أخذت الآن إبسيلون = x عدد موجب إذا x على اتنين عدد
+
+156
+00:20:20,580 --> 00:20:26,270
+موجب وأخذت إبسيلون عبارة عن x على اتنين فاعتبر هذا given
+
+157
+00:20:26,270 --> 00:20:31,070
+إبسيلون إبسيلون معطى مُسبقا فحسب التعريف بما أن x
+
+158
+00:20:31,070 --> 00:20:34,390
+in converges to x إذا يوجد عدد طبيعي يعتمد على
+
+159
+00:20:34,390 --> 00:20:38,710
+إبسيلون بحيث لكل n أكبر من أو يساوي M
+
+160
+00:20:38,710 --> 00:20:45,730
+المسافة هذه أصغر من إبسيلون الآن عوض عن إبسيلون
+
+161
+00:20:45,730 --> 00:20:54,490
+بيساوى x على 2 فهذا يؤدي الآن فك ال absolute value
+
+162
+00:20:54,490 --> 00:21:03,070
+فبيطلع عندي xn - x أصغر من x على 2 وأكبر من -x
+
+163
+00:21:03,070 --> 00:21:08,570
+على 2، مظبوط؟
+
+164
+00:21:08,570 --> 00:21:15,370
+طيب
+
+165
+00:21:15,370 --> 00:21:17,790
+لو أخذت هذا الجزء من المتباينة
+
+166
+00:21:20,790 --> 00:21:28,770
+فبيصير عندي xn أكبر من وادي x على الناحية التالية
+
+167
+00:21:28,770 --> 00:21:38,050
+أكبر من x - x على 2 وبالتالي
+
+168
+00:21:38,050 --> 00:21:46,710
+إذا أنا عندي هي xn أكبر من x على 2 وهذا أكبر من 
+
+169
+00:21:46,710 --> 00:21:57,210
+الصفر تمام؟ وهذا صحيح إذا طلع عندي xn أكبر من الصفر
+
+170
+00:21:57,210 --> 00:22:07,170
+وهذا صحيح لكل n أكبر من أو يساوي M وهو
+
+171
+00:22:07,170 --> 00:22:12,630
+المطلوب تمام إذا هنا استخدمنا تعريف epsilon M
+
+172
+00:22:12,630 --> 00:22:19,690
+وهنا استنتجنا إن لازم xn يطلع أكبر من الصفر لكل
+
+173
+00:22:19,690 --> 00:22:32,210
+n أكبر من أو يساوي M تمام واضح البرهان طيب
+
+174
+00:22:32,210 --> 00:22:34,110
+في أي أسئلة تانية؟
+
+175
+00:22:37,830 --> 00:22:48,330
+section ثلاثة اثنين مين
+
+176
+00:22:48,330 --> 00:22:54,390
+عنده سؤال أي سؤال في أي section ثلاثة اثنين ثلاثة
+
+177
+00:22:54,390 --> 00:23:03,070
+اثنين سبعة عشر
+
+178
+00:23:03,070 --> 00:23:05,150
+section ثلاثة اثنين
+
+179
+00:23:40,180 --> 00:23:44,200
+أنا في عندي هنا sequence of positive real numbers
+
+180
+00:23:44,200 --> 00:23:55,680
+إذا xn حدودها موجبة بقى لكل n such
+
+181
+00:23:55,680 --> 00:24:00,560
+that limit ل
+
+182
+00:24:00,560 --> 00:24:11,550
+xn زائد واحد على xn لما n تؤول إلى infinity بيساوي عددًا
+
+183
+00:24:11,550 --> 00:24:20,550
+أكبر من واحد والمقلوب show اثبت في الحالة هذه أن
+
+184
+00:24:20,550 --> 00:24:25,750
+ال sequence
+
+185
+00:24:25,750 --> 00:24:30,170
+xn is
+
+186
+00:24:30,170 --> 00:24:34,350
+unbounded is not bounded
+
+187
+00:24:38,480 --> 00:24:46,100
+and hence not
+
+188
+00:24:46,100 --> 00:24:53,460
+convergent لأن لو كانت convergent بتطلع bounded
+
+189
+00:25:13,370 --> 00:25:17,190
+يعني من الشرط هذا ممكن نثبت أن ال sequence
+
+190
+00:25:17,190 --> 00:25:21,290
+increasing متزايدة
+
+191
+00:26:05,950 --> 00:26:08,750
+أه ..
+
+192
+00:26:31,500 --> 00:26:38,240
+ممكن نعمل برهان بالـ ... بالتناقض نفترض
+
+193
+00:26:38,240 --> 00:26:48,640
+أنها bounded وممكن نصل لتناقض من تعريف الـ ... هنا
+
+194
+00:26:48,640 --> 00:26:56,680
+ال sequence هذه of quotient convergent لعدد L أكبر
+
+195
+00:26:56,680 --> 00:27:00,220
+من واحد ممكن باستخدامه
+
+196
+00:27:02,850 --> 00:27:14,250
+باستخدام تعريف ال convergence زائد أو
+
+197
+00:27:14,250 --> 00:27:18,390
+ممكن من الفرض هذا نثبت أنه ال sequence unbounded
+
+198
+00:27:18,390 --> 00:27:22,870
+أو ممكن بالتناقض إما باستخدام تعريف epsilon
+
+199
+00:27:22,870 --> 00:27:29,600
+N من ال convergence هذانعمل برهان بالتناقض
+
+200
+00:27:29,600 --> 00:27:35,560
+لنصل إلى حاجة يعني تتناقض مع الفرض اللي هنا على أي 
+
+201
+00:27:35,560 --> 00:27:40,540
+حال أنا هأسيب في حد يحل السؤال هذا طيب أنا هأسيبكم
+
+202
+00:27:40,540 --> 00:27:45,320
+تفكروا فيه وتقرؤوا برهان شوفوا برهان أنا في
+
+203
+00:27:45,320 --> 00:27:49,380
+البرهان النظرية هذه اللي كنت قلت لكم اقرؤوا
+
+204
+00:27:49,380 --> 00:27:54,650
+فحاولوا إنكم تستفيدوا من البرهان تبع النظرية اللي
+
+205
+00:27:54,650 --> 00:27:57,930
+كانت بتقول إن لو كانت ال limit هذه بيساوي L أصغر من
+
+206
+00:27:57,930 --> 00:28:03,370
+واحد فبتطلع ال sequence convergent للصفر فإقرأوا
+
+207
+00:28:03,370 --> 00:28:08,710
+البرهان تبع النظرية هذه وشوفوا كيف يعني النظرية
+
+208
+00:28:08,710 --> 00:28:12,750
+هذه أثبتت وشوفوا لو كان ال L أكبر من واحد كيف
+
+209
+00:28:12,750 --> 00:28:17,450
+بيطلع البرهان إيش اللي بيخلي البرهان هذا يبطل صحيح
+
+210
+00:28:18,870 --> 00:28:23,230
+أه فعيدوا قراءته وحاولكم تحلوه وإذا ما حلتوهوش
+
+211
+00:28:23,230 --> 00:28:27,290
+يعني المرة الجاية ممكن نحله مع بعض أه ماشي الحال
+
+212
+00:28:27,290 --> 00:28:30,470
+فإقرأوا
+
+213
+00:28:30,470 --> 00:28:35,150
+برهان النظرية اللي سيبنا قلنا لكم برهانها موجود
+
+214
+00:28:35,150 --> 00:28:38,030
+في الكتاب وبدي إنكم تقرأوا تفهموا هل قرأتوا 
+
+215
+00:28:38,030 --> 00:28:45,010
+البرهان؟ حاولوا تقرأوا إيه حاولوا تتعملوا إيه تشوفوا
+
+216
+00:28:45,010 --> 00:28:50,070
+وين في البرهان الـ L أكبر من واحد بتخلي البرهان
+
+217
+00:28:50,070 --> 00:28:55,050
+يبطل صح وين المشكلة وشوفوا 
+
+218
+00:28:55,050 --> 00:28:58,210
+إذا كانوا تقدروا تحلو ولا لأ إذا أنا هأسيبكم 
+
+219
+00:28:58,210 --> 00:29:02,610
+تفكروا فيه مرة تانية وتحاولوا تحلوه إذا ما عرفتووش
+
+220
+00:29:02,610 --> 00:29:09,110
+ممكن نحله مرة تانية أو في المرة القادمة نعم مين
+
+221
+00:29:09,110 --> 00:29:13,190
+اللي بتحكي هذه ما حدش لو سمحت تحكي إلا غير ترفع
+
+222
+00:29:13,190 --> 00:29:18,790
+يدها الأول وبعدين تكلم طيب إذا هذا السؤال 
+
+223
+00:29:18,790 --> 00:29:22,510
+هنسيبكم يتفكروا فيه مرة تانية في أي أسئلة تانية
+
+224
+00:29:22,510 --> 00:29:26,710
+section ثلاثة اثنين أو ثلاثة واحد
+
+225
+00:29:45,050 --> 00:29:50,450
+في حد عندها سؤال في نفس
+
+226
+00:29:50,450 --> 00:29:55,770
+ال section نعم فالقاعدة ما أعطينا sequence إنه احنا
+
+227
+00:29:55,770 --> 00:29:59,390
+نشوف إذا هي تتقارب ولا تتباعد استخدمت ال ratio test
+
+228
+00:29:59,390 --> 00:30:04,310
+نعم طلعت ال limit بتساوي واحد واحنا الشرط إن تكون
+
+229
+00:30:04,310 --> 00:30:09,790
+ال limit أقل من واحد صح فالقاعدة هذه بتطلع تطلع ال
+
+230
+00:30:09,790 --> 00:30:12,430
+limit ل sequence لو معطينيها تساوي صفر
+
+231
+00:30:15,730 --> 00:30:21,110
+لأ لازم يكون أصغر من واحد ما بتساويش الواحد معناته
+
+232
+00:30:21,110 --> 00:30:26,150
+ال test بيفشل لأ هي تساوي واحد إذا بالتساوي واحد
+
+233
+00:30:26,150 --> 00:30:33,430
+ارجعي لهي تمرين 16 بقول إذا كانت ال limit بالتساوي
+
+234
+00:30:33,430 --> 00:30:38,710
+واحد فممكن 
+
+235
+00:30:38,710 --> 00:30:41,650
+تكون ال sequence convergent أو divergent يعني هذا
+
+236
+00:30:41,650 --> 00:30:46,920
+ال test ال ratio test بيفشل هي في سؤال 16 هتجيب
+
+237
+00:30:46,920 --> 00:30:52,480
+بمثالين أول شيء إذا كانت ال limit هذه بالتساوي واحد 
+
+238
+00:30:52,480 --> 00:30:59,740
+فهتجيب بمثالين ال limit تبع ال quotient تبع كل 
+
+239
+00:30:59,740 --> 00:31:03,220
+واحدة بالتساوي واحد لكن واحدة convergent واحدة
+
+240
+00:31:03,220 --> 00:31:08,140
+divergent وبالتالي ال test هذا بيفشل إذا كانت ال L
+
+241
+00:31:08,140 --> 00:31:12,420
+بالتساوي واحد أما لو كانت ال L أصغر من واحد فال
+
+242
+00:31:12,420 --> 00:31:16,400
+sequence xn بتطلع convergent للصفر إذا كان ال L 
+
+243
+00:31:16,400 --> 00:31:21,740
+أكبر من 1 فال sequence بتطلع divergent okay تمام
+
+244
+00:31:21,740 --> 00:31:30,340
+هذا هو ال ratio test فهل جبت أمثلة؟ كويس ممتاز طيب
+
+245
+00:31:30,340 --> 00:31:36,220
+إيش دخل دي؟ دي معناته بدك تستخدم طريقة تانية غير
+
+246
+00:31:36,220 --> 00:31:43,260
+ال ratio test صحيح لأن حسب سؤال 16 ال test بيفشل
+
+247
+00:31:43,260 --> 00:31:48,320
+إذا كانت limit ال ratio ال ratio test بيفشل إذا
+
+248
+00:31:48,320 --> 00:31:53,020
+كانت limit لل ratio بتساوي واحد وبالتالي بدك تبحث
+
+249
+00:31:53,020 --> 00:31:54,300
+عن طريقة تانية
+
+250
+00:32:12,840 --> 00:32:31,940
+طيب في أسئلة تانية في
+
+251
+00:32:31,940 --> 00:32:35,300
+section ثلاثة واحد وثلاثة اثنين في عندكم أي سؤال
+
+252
+00:32:35,300 --> 00:32:42,490
+ما فيش أسئلة لسه مش دارسين مش محاضرين كانت واحدة بس
+
+253
+00:32:42,490 --> 00:32:51,430
+للدراسة وهم اللي بيسألوا الأسئلة والباقي مستمع طيب
+
+254
+00:32:51,430 --> 00:32:54,930
+بتحبوا نرجع لأسئلة chapter اثنين في أسئلة في
+
+255
+00:32:54,930 --> 00:33:01,130
+chapter اثنين إذا
+
+256
+00:33:01,130 --> 00:33:10,470
+في عندكم أسئلة في section اثنين
+
+257
+00:33:10,470 --> 00:33:11,010
+أربعة
+
+258
+00:33:26,250 --> 00:33:35,710
+السؤال هذا يعني في الكتاب أعطيكم hint كيف
+
+259
+00:33:35,710 --> 00:33:41,790
+يعني تحلوه موجود في نهاية الكتاب فحاولوا تقرأوا إيه
+
+260
+00:33:41,790 --> 00:33:46,530
+تقرا ال hint هذا وتستفيدي منه وتشوفي يعني هذا
+
+261
+00:33:46,530 --> 00:33:54,780
+أكيد هساعدك في حل السؤال شفتيه قبل هيك؟ طيب طلعي
+
+262
+00:33:54,780 --> 00:33:59,360
+خلف الكتاب فيه hint أو إرشادات لبعض التمارين
+
+263
+00:33:59,360 --> 00:34:06,680
+بيعطيكي يعني طريقة مقتضبة للحل أو بيحط رجلك على طريق 
+
+264
+00:34:06,680 --> 00:34:12,840
+الحل فحاولي تقرأي إيه وتستفيدي منه وإذا فهمتي
+
+265
+00:34:12,840 --> 00:34:19,640
+الإرشاد هذا ممكن تحلي السؤال أنتِ وزميلاتك تطلعوا 
+
+266
+00:34:19,640 --> 00:34:23,580
+على الإرشادات هذه تبعت التمرين أو بعض الحلول 
+
+267
+00:34:23,580 --> 00:34:28,240
+المختصرة وحاولوا تستفيدوا منها وتفصلوها وتكتبوا
+
+268
+00:34:28,240 --> 00:34:35,340
+الحل بطريقة واضحة وكاملة فهأسيبكم
+
+269
+00:34:35,340 --> 00:34:42,440
+تقرؤوا الإرشاد وتحاولوا تستفيدوا منه أي أسئلة
+
+270
+00:34:42,440 --> 00:34:49,980
+تانية في section 2 4 2 3 2 2 إن واحد الجزء اللي
+
+271
+00:34:49,980 --> 00:34:56,460
+داخل الامتحان، في عندكم أي سؤال فيه؟ منين في عندها
+
+272
+00:34:56,460 --> 00:35:00,260
+سؤال؟
+
+273
+00:35:00,260 --> 00:35:07,020
+في أسئلة كتير حلوة ومهمة ويا بدوا إنكم مش مدرسين
+
+274
+00:35:07,020 --> 00:35:08,680
+ولا حتى مستعدين للامتحان
+
+275
+00:35:16,700 --> 00:35:20,800
+في أي أسئلة في chapter 2 أو chapter 3 الجزء الداخل
+
+276
+00:35:20,800 --> 00:35:21,960
+في الامتحان
+
+277
+00:36:04,610 --> 00:36:11,090
+فيش أسئلة؟ طيب
+
+278
+00:36:11,090 --> 00:36:15,390
+أنا هأحل لكم يعني كمان سؤالين واحد من section ثلاثة 
+
+279
+00:36:15,390 --> 00:36:21,070
+واحد وواحد من ثلاثة اثنين
+
+280
+00:36:21,070 --> 00:36:28,670
+خليني 
+
+281
+00:36:28,670 --> 00:36:29,830
+أحل السؤال
+
+282
+00:36:46,350 --> 00:36:58,770
+يعني مثلا يعني
+
+283
+00:36:58,770 --> 00:37:04,090
+مثلا السؤال الخامسة
+
+284
+00:37:04,090 --> 00:37:10,530
+السؤال
+
+285
+00:37:10,530 --> 00:37:16,320
+الخامسة الفرع دي section تلاتة واحد use definition
+
+286
+00:37:16,320 --> 00:37:25,660
+use definition of limit to
+
+287
+00:37:25,660 --> 00:37:33,880
+establish أنه
+
+288
+00:37:33,880 --> 00:37:37,800
+ال limit لإن
+
+289
+00:37:37,800 --> 00:37:44,970
+تربية سالب واحد على اتنين انتر بيه زائد تلاتة ال
+
+290
+00:37:44,970 --> 00:37:52,850
+sequence اللي حد العم تبعها الكاسر هذا بيساوي نص و
+
+291
+00:37:52,850 --> 00:37:56,410
+بيثبت ان ال sequence هذي convergence و نهايتها نص
+
+292
+00:37:56,410 --> 00:38:00,390
+بيستخدم ال definition ماهو ال definition المقصود
+
+293
+00:38:00,390 --> 00:38:06,700
+في هنا اللي هو تعريف epsilon capital N لل limit أو
+
+294
+00:38:06,700 --> 00:38:21,360
+للنهاية تعريف epsilon capital N طيب أنا
+
+295
+00:38:21,360 --> 00:38:27,300
+في النهاية في نهاية المطاف تعريف epsilon capital N
+
+296
+00:38:31,470 --> 00:38:36,710
+عايزني أثبت أن المسافة بين xn اللي هو انتر بيها
+
+297
+00:38:36,710 --> 00:38:42,510
+سالب واحد على اتنين انتر بيها زائد تلاتة سالب نص
+
+298
+00:38:42,510 --> 00:38:47,270
+بدنا هذا يكون أصغر من أي given epsilon عدد موجب
+
+299
+00:38:47,270 --> 00:38:53,950
+لكل n أكبر من أو يساوي capital N حيث capital N عدد
+
+300
+00:38:53,950 --> 00:39:00,410
+طبيعي هنجيبه ويعتمد على ال epsilon فنشوف مع بعض هذا
+
+301
+00:39:00,410 --> 00:39:07,410
+إيه من الآخر طيب إذا هنا solution إذا بقول أنا
+
+302
+00:39:07,410 --> 00:39:12,490
+عايز في النهاية absolute value سالب واحد على
+
+303
+00:39:12,490 --> 00:39:17,970
+اتنين انتر بيه زائد تلاتة سالب نص بسأل نفسي متى
+
+304
+00:39:17,970 --> 00:39:24,570
+هذا بيكون أصغر من أي epsilon موجب هذا بكافئ
+
+305
+00:39:27,220 --> 00:39:34,160
+الـ absolute value بين واحد المقامات هي اتنين في
+
+306
+00:39:34,160 --> 00:39:40,260
+اتنين انتر بيه زائد تلاتة و بيصير عندنا اتنين
+
+307
+00:39:40,260 --> 00:39:46,720
+انتر بيه سالب اتنين تضرب هذا في اتنين سالب اتنين
+
+308
+00:39:46,720 --> 00:39:53,680
+انتر بيه موجبة بتلاتة لان هذا المقدار اللي فوق
+
+309
+00:39:53,680 --> 00:39:55,580
+بيبقى اصغر من epsilon
+
+310
+00:39:58,630 --> 00:40:02,730
+طيب أنا عندي اتنين in تربيع و هاي سالب اتنين in
+
+311
+00:40:02,730 --> 00:40:07,450
+تربيع بيروحوا مع بعض و عندي سالب اتنين و السالب
+
+312
+00:40:07,450 --> 00:40:10,630
+تلاتة بطلع خمسة يعني دلوقتي بصير absolute سالب
+
+313
+00:40:10,630 --> 00:40:16,330
+خمسة على اتنين في
+
+314
+00:40:16,330 --> 00:40:19,230
+اتنين in تربيع زائد تلاتة
+
+315
+00:40:24,890 --> 00:40:31,830
+بدي هذا يكون أصغر من epsilon طيب
+
+316
+00:40:31,830 --> 00:40:38,990
+هاد عبارة عن خمسة هاد
+
+317
+00:40:38,990 --> 00:40:48,570
+عبارة عن خمسة على اتنين اتنين انتر بيه زي
+
+318
+00:40:48,570 --> 00:40:49,370
+التلاتة
+
+319
+00:40:52,780 --> 00:41:02,080
+متى بيكون هذا أصغر من epsilon هذا
+
+320
+00:41:02,080 --> 00:41:09,220
+بكافئ هذا
+
+321
+00:41:09,220 --> 00:41:15,900
+بكافئ ان اقول واحد متى بيكون واحد على اتنين انتر
+
+322
+00:41:15,900 --> 00:41:30,390
+بيه زائد تلاتة أصغر من اتنين على خمسة إبسلون طيب
+
+323
+00:41:30,390 --> 00:41:35,550
+إذا
+
+324
+00:41:35,550 --> 00:41:42,470
+أنا ممكن أستخدم ال Archimedean property إذا هنا
+
+325
+00:41:42,470 --> 00:41:49,690
+let epsilon أكبر من الصفر
+
+326
+00:41:51,720 --> 00:41:57,880
+نبدأ بـ epsilon أكبر من الصفر تعريف epsilon capital N
+
+327
+00:41:57,880 --> 00:42:02,160
+بيقول ابدا بـ epsilon أكبر من الصفر و جيب capital N
+
+328
+00:42:03,440 --> 00:42:07,880
+بحيث أن المسافة بين XN و X أصغر من epsilon لكل N
+
+329
+00:42:07,880 --> 00:42:15,440
+أكبر من أو يساوي capital N بحيث أن المسافة بين XN
+
+330
+00:42:15,440 --> 00:42:17,600
+بحيث أن المسافة بين XN و X أصغر من epsilon لكل N
+
+331
+00:42:17,600 --> 00:42:17,940
+أكبر من أو يساوي capital N بحيث أن المسافة بين XN
+
+332
+00:42:17,940 --> 00:42:20,660
+و X أصغر من epsilon لكل N أكبر من أو يساوي capital N
+
+333
+00:42:20,660 --> 00:42:21,360
+بحيث أن المسافة بين XN و X أصغر من epsilon لكل N
+
+334
+00:42:21,360 --> 00:42:23,640
+أكبر من أو يساوي capital N بحيث أن المسافة بين XN
+
+335
+00:42:23,640 --> 00:42:28,200
+و X أصغر
+
+336
+00:42:28,200 --> 00:42:35,040
+من epsilon لكل N أكبر من أو يساوي capital N بحيث
+
+337
+00:42:35,040 --> 00:42:43,620
+choose it choose طبعا
+
+338
+00:42:43,620 --> 00:42:51,500
+by Archimedean property capital
+
+339
+00:42:51,500 --> 00:43:01,200
+N عدد طبيعي بحيث انه واحد على اتنين في capital N
+
+340
+00:43:01,200 --> 00:43:07,820
+تربيع زائد تلاتة أصغر من اتنين على خمسة epsilon
+
+341
+00:43:07,820 --> 00:43:20,180
+ممكن
+
+342
+00:43:20,180 --> 00:43:26,070
+ألاقي capital N عدد طبيعي ممكن 2 في n تربيع زائد
+
+343
+00:43:26,070 --> 00:43:32,170
+تلاتة طبعا تلاتة مش epsilon واحد على اتنين n
+
+344
+00:43:32,170 --> 00:43:41,290
+تربيع زائد تلاتة اصغر من اتنين على خمسة epsilon الان
+
+345
+00:43:41,290 --> 00:43:46,770
+اذا لو اخدت small n اكبر من او يساوي ال capital N هذا
+
+346
+00:43:46,770 --> 00:44:00,110
+بيقدي انه واحد على اتنين n تربيع زائد تلاتة او بلاش
+
+347
+00:44:00,110 --> 00:44:09,230
+absolute اه بيقدي ان absolute طيب
+
+348
+00:44:09,230 --> 00:44:16,750
+هذا بيقدي ان الكلام هذا اصغر من أو يساوي واحد على
+
+349
+00:44:16,750 --> 00:44:25,510
+اتنين capital N تربيع زائد تلاتة وبالتالي هذا
+
+350
+00:44:25,510 --> 00:44:31,390
+بيقدي ان ال absolute value ل n تربيع سالب واحد على
+
+351
+00:44:31,390 --> 00:44:42,670
+اتنين n تربيع زائد تلاتة سالب نص طلع هذا
+
+352
+00:45:09,580 --> 00:45:16,580
+خمسة على اتنين
+
+353
+00:45:16,580 --> 00:45:20,140
+في اتنين n تربيع زائد التلاتة
+
+354
+00:45:28,400 --> 00:45:34,680
+وهذا هيطلع أصغر من أو يساوي خمسة على اتنين في اتنين
+
+355
+00:45:34,680 --> 00:45:42,000
+capital N تربيع زائد تلاتة ومن هنا هذا أصغر من
+
+356
+00:45:42,000 --> 00:45:47,320
+خمسة
+
+357
+00:45:47,320 --> 00:45:54,280
+على اتنين ضرب اتنين على خمسة في epsilon اللي هو
+
+358
+00:45:54,280 --> 00:45:55,160
+بيطلع epsilon
+
+359
+00:45:59,840 --> 00:46:03,560
+إذن هذه لأي epsilon أكبر من صفر لجيت فيه capital N
+
+360
+00:46:03,560 --> 00:46:08,200
+مرتبطة لـ capital N هي في epsilon depends on epsilon
+
+361
+00:46:08,200 --> 00:46:12,280
+بتعتمد على epsilon بحيث لكل n أكبر من أو يساوي
+
+362
+00:46:12,280 --> 00:46:17,920
+capital N طلع absolute xn minus x أصغر من epsilon
+
+363
+00:46:19,350 --> 00:46:24,350
+طبعا إذا هذا حسب تعريف by definition of epsilon
+
+364
+00:46:24,350 --> 00:46:29,770
+capital N of limit بطلع عندي limit n تربيع سالب
+
+365
+00:46:29,770 --> 00:46:34,750
+واحد على اتنين n تربيع زائد تلاتة لما n تؤول
+
+366
+00:46:34,750 --> 00:46:37,830
+infinity بساوي نص
+
+367
+00:46:44,620 --> 00:46:48,560
+بالمثل ممكن نحل باقي التمرين اللي هي الفروع A وB
+
+368
+00:46:48,560 --> 00:46:54,940
+و C باستخدام التعريف فحاولوا تتدربوا على التمرين
+
+369
+00:46:54,940 --> 00:47:02,700
+هذه و تحلوا أسئلة زيها في حد عنده أي سؤال تاني في
+
+370
+00:47:02,700 --> 00:47:07,260
+هذا ال section طيب
+
+371
+00:47:07,260 --> 00:47:12,220
+نحل كمان سؤال في section تلاتة اتنين
+
+372
+00:47:27,570 --> 00:47:34,750
+في انكم أي سؤال في section تلاتة اتنين اخر
+
+373
+00:47:34,750 --> 00:47:35,250
+سؤال
+
+374
+00:47:57,080 --> 00:48:03,480
+هي سؤال واحد وعشرين section تلاتة
+
+375
+00:48:03,480 --> 00:48:13,760
+اتنين suppose
+
+376
+00:48:13,760 --> 00:48:24,980
+افترضي ان ال sequence xn converge إلى x و ال
+
+377
+00:48:24,980 --> 00:48:33,200
+sequence yn و yn is such that is a sequence
+
+378
+00:48:33,200 --> 00:48:40,900
+such that for any epsilon for
+
+379
+00:48:40,900 --> 00:48:46,240
+any epsilon أكبر من الصفر يوجد
+
+380
+00:48:46,240 --> 00:48:53,780
+m بحيث يوجد عدد m such that
+
+381
+00:48:56,580 --> 00:49:06,460
+absolute xn minus yn أصغر من epsilon لكل N أكبر من
+
+382
+00:49:06,460 --> 00:49:14,260
+أو يساوي capital N فالسؤال
+
+383
+00:49:14,260 --> 00:49:19,060
+does it
+
+384
+00:49:19,060 --> 00:49:22,820
+follow هل
+
+385
+00:49:22,820 --> 00:49:34,030
+ينتج من ذلك هل ال sequence yn تطلع
+
+386
+00:49:34,030 --> 00:49:44,210
+convergent فنشوف
+
+387
+00:49:44,210 --> 00:49:44,930
+مع بعض
+
+388
+00:49:53,440 --> 00:49:59,260
+كمان مرة اندي two sequences واحدة xn واحدة yn
+
+389
+00:49:59,260 --> 00:50:04,280
+ال sequence xn مُعطى انها convergent to some x
+
+390
+00:50:04,280 --> 00:50:08,880
+إلى عدد ما x ال limit تبقى تاكس و ال sequence yn
+
+391
+00:50:08,880 --> 00:50:14,600
+بتحقق الشرط هذا وهو
+
+392
+00:50:14,600 --> 00:50:19,600
+انه لأي epsilon أكبر من صفر في عدد طبيعي حتى هذا
+
+393
+00:50:19,600 --> 00:50:27,790
+عدد طبيعي المفروض يكون بنسميه capital N بحيث انه لكل
+
+394
+00:50:27,790 --> 00:50:31,810
+n أكبر من أو يساوي capital N المسافة بين xn و yn
+
+395
+00:50:31,810 --> 00:50:35,510
+أصغر من epsilon هل هذا بيقدم ال sequence yn
+
+396
+00:50:35,510 --> 00:50:40,870
+convergent؟ هنشوف الآن أن فعلا تطلع ال sequence yn
+
+397
+00:50:40,870 --> 00:50:46,130
+convergent ونهايتها هي نفس نهاية ال sequence xn
+
+398
+00:50:46,130 --> 00:50:51,270
+لأن هنا الإجابة yes
+
+399
+00:50:53,550 --> 00:51:01,270
+and yn converge to x لكن
+
+400
+00:51:01,270 --> 00:51:07,570
+هذا بيده برهان اذا
+
+401
+00:51:07,570 --> 00:51:11,370
+to see this
+
+402
+00:51:11,370 --> 00:51:16,610
+نبدأ
+
+403
+00:51:16,610 --> 00:51:18,610
+بـ epsilon أكبر من الصفر
+
+404
+00:51:36,810 --> 00:51:44,450
+let by hypothesis من الفرض من
+
+405
+00:51:44,450 --> 00:51:50,820
+الفرض من ال hypothesis أنا عندي absolute xn minus
+
+406
+00:51:50,820 --> 00:51:54,860
+yn أصغر
+
+407
+00:51:54,860 --> 00:52:03,700
+من epsilon أكبر من أو يساوي صفر وهذا صحيح لكل n أكبر
+
+408
+00:52:03,700 --> 00:52:10,440
+من أو يساوي capital M وهذا
+
+409
+00:52:10,440 --> 00:52:15,380
+الكلام صحيح لكل epsilon أكبر من الصفر
+
+410
+00:52:24,820 --> 00:52:36,980
+فمن هنا فمن
+
+411
+00:52:36,980 --> 00:52:45,680
+هنا بهدف بيقدي ان ال limit ل xn minus yn لما n
+
+412
+00:52:45,680 --> 00:52:49,420
+تؤول infinity بساوي صفر
+
+413
+00:52:54,150 --> 00:52:58,570
+مش شرط هذا أنا
+
+414
+00:52:58,570 --> 00:53:03,950
+عندي ال ..
+
+415
+00:53:03,950 --> 00:53:08,010
+ما معناه ان limit ال sequence هذه بساوة صفر؟ معناه
+
+416
+00:53:08,010 --> 00:53:16,620
+لأي epsilon أكبر من الصفر يوجد capital M عدد طبيعي
+
+417
+00:53:16,620 --> 00:53:21,840
+يعتمد على epsilon بحيث أنه لكل n أكبر من أو يساوي
+
+418
+00:53:21,840 --> 00:53:28,860
+capital N هذا بيقدي أن absolute xn minus yn minus
+
+419
+00:53:28,860 --> 00:53:34,700
+الصفر أصغر من epsilon هي معنى ان limit ال sequence
+
+420
+00:53:34,700 --> 00:53:40,740
+للفرق بساوي صفر ايش معنى هذا لأي epsilon أكبر من 
+
+421
+00:53:40,740 --> 00:53:46,660
+سفر يوجد  M يعتمد على N عدد طبيعي يعتمد على
+
+422
+00:53:46,660 --> 00:53:51,020
+ال epsilon بحيث لكل N أكبر من أو يساوي M
+
+423
+00:53:51,020 --> 00:53:55,540
+المسافة بين الحد العام لل sequence و limit اللي هي 
+
+424
+00:53:55,540 --> 00:54:00,140
+سفر أصغر من epsilon هذا الكلام هي متحقق هنا هي
+
+425
+00:54:00,140 --> 00:54:04,850
+متحققة تمام؟ إذا هذا بنحصل عليه وبالتالي limit xn
+
+426
+00:54:04,850 --> 00:54:14,070
+minus yn بساوي سفر ومنها الآن أنا عندي ال yn ممكن
+
+427
+00:54:14,070 --> 00:54:20,870
+كتبتها على صورة yn 
+
+428
+00:54:20,870 --> 00:54:32,610
+سالب xn موجب xn وهذا بيساوي سالب Xn سالب Yn زائد Xn
+
+429
+00:54:32,610 --> 00:54:40,630
+تمام؟ إذا ال limit ل Yn as n tends to infinity
+
+430
+00:54:40,630 --> 00:54:49,110
+بيساوي limit الطرف اليمين ف limit Xn سالب Yn 
+
+431
+00:54:49,110 --> 00:54:56,410
+مضروبة في سالب واحد بيطلع برا ال limit زائد limit
+
+432
+00:54:56,410 --> 00:55:03,770
+xn لما n تقول لإنفينيتي وهنا 
+
+433
+00:55:03,770 --> 00:55:08,770
+لسه احنا مثبتين هذا عبارة عن سالب limit sequence
+
+434
+00:55:08,770 --> 00:55:16,570
+xn minus yn بالساوية سفر، سالب واحد في سفر 
+
+435
+00:55:19,990 --> 00:55:26,850
+زائد limit xn اللي هي x تمام اذا limit ال sequence 
+
+436
+00:55:26,850 --> 00:55:32,370
+yn تطلع بالساوي x اذا 
+
+437
+00:55:32,370 --> 00:55:37,210
+هنا اثبتنا ان ال sequence yn تطلع convergent وال
+
+438
+00:55:37,210 --> 00:55:44,210
+limit تبعتها بالساوي x تمام البرهان هنا اعتمد على 
+
+439
+00:55:44,210 --> 00:55:49,890
+انه من الفرض انا عندي المثال لأي epsilon هذا الفرض
+
+440
+00:55:49,890 --> 00:55:57,390
+معناه ان limit ال sequence xn minus yn بالساوي
+
+441
+00:55:57,390 --> 00:56:04,290
+سفر وهذا اللي ساعدنا في الحل وهذا ناتج هي من تعريف 
+
+442
+00:56:04,290 --> 00:56:09,190
+epsilon N لل limit هذا هو البرهان
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/XjWoXKhuE-o_raw.json b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/XjWoXKhuE-o_raw.json
new file mode 100644
index 0000000000000000000000000000000000000000..ad7bc6c37a9ffba4760eb49c39cc4fbb9494457b
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/XjWoXKhuE-o_raw.json
@@ -0,0 +1 @@
+{"segments": [{"id": 1, "seek": 4603, "start": 21.09, "end": 46.03, "text": "إذن في المحاضرة هذه ان شاء الله هنحل بعض التمرين لل homework اللي تابع ل section تلاتة واحد و تلاتة اتنين فأزملتكم تسأل عن ال .. نحن نحل السؤال 13 section تلاتة واحد", "tokens": [28814, 8848, 1863, 8978, 9673, 5016, 46958, 25720, 29538, 16472, 13412, 16606, 21984, 8032, 1863, 5016, 1211, 45030, 11242, 16712, 29973, 9957, 24976, 14578, 13672, 1829, 6055, 16758, 3615, 5296, 3541, 6055, 1211, 9307, 3660, 36764, 24401, 4032, 6055, 1211, 9307, 3660, 1975, 2655, 1863, 9957, 6156, 10721, 11622, 42213, 2655, 24793, 6055, 3794, 10721, 1211, 18871, 2423, 4386, 8717, 5016, 1863, 8717, 5016, 1211, 21136, 33604, 6027, 3705, 3541, 6055, 1211, 9307, 3660, 36764, 24401], "avg_logprob": -0.212459414036243, "compression_ratio": 1.6287425149700598, "no_speech_prob": 0.0, "words": [{"start": 21.09, "end": 21.55, "word": "إذن", "probability": 0.6546223958333334}, {"start": 21.55, "end": 22.03, "word": " في", "probability": 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"end": 152.95, "text": "فلو حاولتوا تستخدموا نفس أسلوب البرهان تبع المثال اللي استخدمنا فيه ال binomial theorem فهتصلوا للنتيجة فهي البرهان نشوف كيف نستخدم ال binomial theorem في الوصول إلى المطلوب", "tokens": [5172, 1211, 2407, 11331, 995, 12610, 2655, 14407, 6055, 14851, 9778, 40448, 14407, 8717, 36178, 5551, 3794, 1211, 37746, 2423, 26890, 3224, 7649, 6055, 3555, 3615, 9673, 12984, 6027, 13672, 1829, 44713, 9778, 40448, 8315, 8978, 3224, 2423, 5171, 47429, 20904, 6156, 3224, 2655, 36520, 14407, 24976, 29399, 1829, 7435, 3660, 6156, 3224, 1829, 2423, 26890, 3224, 7649, 8717, 8592, 38688, 9122, 33911, 8717, 14851, 9778, 40448, 2423, 5171, 47429, 20904, 8978, 2423, 2407, 9381, 12610, 30731, 9673, 9566, 1211, 37746], "avg_logprob": -0.17606707862237606, "compression_ratio": 1.7, "no_speech_prob": 0.0, "words": [{"start": 130.41, "end": 131.11, "word": "فلو", "probability": 0.77587890625}, {"start": 131.11, "end": 131.87, "word": " حاولتوا", "probability": 0.83486328125}, {"start": 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وهذا صحيح لكل n أكبر من أو ساوي capital M وهو المطلوب", "tokens": [39237, 10943, 22807, 37037, 15730, 20328, 5016, 1829, 5016, 11933, 15730, 23032, 1211, 3615, 18871, 16254, 2031, 77, 5551, 4117, 26890, 9154, 21136, 5172, 2288, 37037, 15730, 20328, 5016, 1829, 5016, 5296, 28820, 297, 5551, 4117, 26890, 9154, 34051, 8608, 995, 45865, 4238, 376, 37037, 2407, 9673, 9566, 1211, 37746], "avg_logprob": -0.2826286881577735, "compression_ratio": 1.3495934959349594, "no_speech_prob": 0.0, "words": [{"start": 1309.67, "end": 1310.73, "word": "تمام؟", "probability": 0.5128072102864584}, {"start": 1310.73, "end": 1311.75, "word": " وهذا", "probability": 0.715576171875}, {"start": 1311.75, "end": 1312.27, "word": " صحيح", "probability": 0.9931640625}, {"start": 1312.27, "end": 1312.51, "word": " إذا", "probability": 0.5570068359375}, {"start": 1312.51, "end": 1312.95, "word": " طلع", "probability": 0.9254557291666666}, {"start": 1312.95, "end": 1313.59, "word": " عندي", "probability": 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0.9072265625}, {"start": 1326.05, "end": 1326.55, "word": " capital", "probability": 0.474853515625}, {"start": 1326.55, "end": 1326.85, "word": " M", "probability": 0.8798828125}, {"start": 1326.85, "end": 1327.17, "word": " وهو", "probability": 0.77294921875}, {"start": 1327.17, "end": 1327.65, "word": " المطلوب", "probability": 0.9849853515625}], "temperature": 1.0}, {"id": 47, "seek": 135411, "start": 1329.63, "end": 1354.11, "text": "تمام إذا هنا استخدمنا تعريف epsilon capital M وهنا استنتجنا إن لازم xn يطلع أكبر من السفر لكل M أكبر من أو يساوي capital M تمام واضح البرهان طيب في أي أسئلة تانية؟", "tokens": [2655, 15042, 2304, 11933, 15730, 34105, 44713, 9778, 40448, 8315, 37279, 16572, 5172, 17889, 4238, 376, 37037, 8315, 44713, 29399, 7435, 8315, 36145, 5296, 31377, 2304, 2031, 77, 7251, 9566, 1211, 3615, 5551, 4117, 26890, 9154, 21136, 5172, 2288, 5296, 28820, 376, 5551, 4117, 26890, 9154, 34051, 7251, 3794, 995, 45865, 4238, 376, 46811, 10943, 4032, 46958, 5016, 2423, 26890, 3224, 7649, 23032, 1829, 3555, 8978, 36632, 5551, 3794, 19986, 37977, 6055, 7649, 10632, 22807], "avg_logprob": -0.26521382755354833, "compression_ratio": 1.5314285714285714, "no_speech_prob": 0.0, "words": [{"start": 1329.63, "end": 1330.07, "word": "تمام", "probability": 0.47802734375}, {"start": 1330.07, "end": 1330.33, "word": " إذا", "probability": 0.4835205078125}, {"start": 1330.33, "end": 1330.55, "word": " هنا", "probability": 0.837890625}, {"start": 1330.55, "end": 1331.17, "word": " استخدمنا", "probability": 0.898681640625}, {"start": 1331.17, "end": 1331.65, "word": " تعريف", "probability": 0.9895833333333334}, {"start": 1331.65, "end": 1332.13, "word": " epsilon", "probability": 0.402587890625}, {"start": 1332.13, "end": 1332.63, "word": " capital", "probability": 0.35009765625}, {"start": 1332.63, "end": 1333.01, "word": " M", "probability": 0.935546875}, {"start": 1333.01, "end": 1334.65, "word": " وهنا", "probability": 0.631591796875}, {"start": 1334.65, "end": 1335.31, "word": " استنتجنا", "probability": 0.8463134765625}, {"start": 1335.31, "end": 1335.47, "word": " إن", "probability": 0.6279296875}, {"start": 1335.47, "end": 1335.95, "word": " لازم", "probability": 0.8463541666666666}, {"start": 1335.95, "end": 1336.59, "word": " xn", "probability": 0.3741455078125}, {"start": 1336.59, "end": 1337.07, "word": " يطلع", "probability": 0.9598388671875}, {"start": 1337.07, "end": 1337.47, "word": " أكبر", "probability": 0.9557291666666666}, {"start": 1337.47, "end": 1337.65, "word": " من", "probability": 0.99462890625}, {"start": 1337.65, "end": 1338.23, "word": " السفر", "probability": 0.7254231770833334}, {"start": 1338.23, "end": 1339.69, "word": " لكل", "probability": 0.978759765625}, {"start": 1339.69, "end": 1340.03, "word": " M", "probability": 0.85107421875}, {"start": 1340.03, "end": 1340.59, "word": " أكبر", "probability": 0.9720052083333334}, {"start": 1340.59, "end": 1340.81, "word": " من", "probability": 0.94775390625}, {"start": 1340.81, "end": 1341.01, "word": " أو", "probability": 0.93017578125}, {"start": 1341.01, "end": 1341.37, "word": " يساوي", "probability": 0.74786376953125}, {"start": 1341.37, "end": 1341.83, "word": " capital", "probability": 0.394775390625}, {"start": 1341.83, "end": 1342.15, "word": " M", "probability": 0.97607421875}, {"start": 1342.15, "end": 1343.09, "word": " تمام", "probability": 0.786376953125}, {"start": 1343.09, "end": 1345.57, "word": " واضح", "probability": 0.9049479166666666}, {"start": 1345.57, "end": 1346.55, "word": " البرهان", "probability": 0.85302734375}, {"start": 1346.55, "end": 1352.21, "word": " طيب", "probability": 0.92822265625}, {"start": 1352.21, "end": 1352.45, "word": " في", "probability": 0.8779296875}, {"start": 1352.45, "end": 1352.75, "word": " أي", "probability": 0.77880859375}, {"start": 1352.75, "end": 1353.27, "word": " أسئلة", "probability": 0.97265625}, {"start": 1353.27, "end": 1354.11, "word": " تانية؟", "probability": 0.9307861328125}], "temperature": 1.0}, {"id": 48, "seek": 138515, "start": 1357.83, "end": 1385.15, "text": "section تلاتة اتنين مين عنده سؤال اي سؤال في اي section تلاتة اتنين تلاتة اتنين سبعتاش section تلاتة اتنين", "tokens": [11963, 6055, 1211, 9307, 3660, 1975, 2655, 1863, 9957, 3714, 9957, 43242, 3224, 8608, 33604, 6027, 1975, 1829, 8608, 33604, 6027, 8978, 1975, 1829, 3541, 6055, 1211, 9307, 3660, 1975, 2655, 1863, 9957, 6055, 1211, 9307, 3660, 1975, 2655, 1863, 9957, 8608, 3555, 34268, 33599, 3541, 6055, 1211, 9307, 3660, 1975, 2655, 1863, 9957], "avg_logprob": -0.22017044858499007, "compression_ratio": 1.9222222222222223, "no_speech_prob": 0.0, "words": [{"start": 1357.83, "end": 1358.65, "word": "section", "probability": 0.59814453125}, {"start": 1358.65, "end": 1359.69, "word": " تلاتة", "probability": 0.9002685546875}, {"start": 1359.69, "end": 1360.15, "word": " اتنين", "probability": 0.974853515625}, {"start": 1360.15, "end": 1368.33, "word": " مين", "probability": 0.553466796875}, {"start": 1368.33, "end": 1368.67, "word": " عنده", "probability": 0.6326904296875}, {"start": 1368.67, "end": 1369.07, "word": " سؤال", "probability": 0.94921875}, {"start": 1369.07, "end": 1370.27, "word": " اي", "probability": 0.542816162109375}, {"start": 1370.27, "end": 1370.65, "word": " سؤال", "probability": 0.9910481770833334}, {"start": 1370.65, "end": 1371.35, "word": " في", "probability": 0.64208984375}, {"start": 1371.35, "end": 1372.15, "word": " اي", "probability": 0.826171875}, {"start": 1372.15, "end": 1372.57, "word": " section", "probability": 0.9140625}, {"start": 1372.57, "end": 1373.37, "word": " تلاتة", "probability": 0.828125}, {"start": 1373.37, "end": 1373.79, "word": " اتنين", "probability": 0.9906005859375}, {"start": 1373.79, "end": 1374.39, "word": " تلاتة", "probability": 0.81390380859375}, {"start": 1374.39, "end": 1376.35, "word": " اتنين", "probability": 0.9912109375}, {"start": 1376.35, "end": 1383.07, "word": " سبعتاش", "probability": 0.815185546875}, {"start": 1383.07, "end": 1383.65, "word": " section", "probability": 0.2489013671875}, {"start": 1383.65, "end": 1384.61, "word": " تلاتة", "probability": 0.977783203125}, {"start": 1384.61, "end": 1385.15, "word": " اتنين", "probability": 0.99365234375}], "temperature": 1.0}, {"id": 49, "seek": 144822, "start": 1420.18, "end": 1448.22, "text": "أنا في عندي هنا sequence of positive real numbers إذا xn حدود عموجة بقى لكل n such that limit ل xn زاد واحد على xn لما n تقول infinity", "tokens": [10721, 8315, 8978, 18871, 16254, 34105, 8310, 295, 3353, 957, 3547, 11933, 15730, 2031, 77, 11331, 3215, 23328, 6225, 2304, 29245, 3660, 4724, 4587, 7578, 5296, 28820, 297, 1270, 300, 4948, 5296, 2031, 77, 30767, 18513, 36764, 24401, 15844, 2031, 77, 5296, 15042, 297, 6055, 39648, 13202], "avg_logprob": -0.27099610430498916, "compression_ratio": 1.2380952380952381, "no_speech_prob": 0.0, "words": [{"start": 1420.18, "end": 1420.56, "word": "أنا", "probability": 0.6328125}, {"start": 1420.56, "end": 1420.7, "word": " في", "probability": 0.75927734375}, {"start": 1420.7, "end": 1421.1, "word": " عندي", "probability": 0.6865234375}, {"start": 1421.1, "end": 1421.36, "word": " هنا", "probability": 0.74609375}, {"start": 1421.36, "end": 1421.86, "word": " sequence", "probability": 0.8291015625}, {"start": 1421.86, "end": 1422.46, "word": " of", "probability": 0.9375}, {"start": 1422.46, "end": 1423.12, "word": " positive", "probability": 0.85888671875}, {"start": 1423.12, "end": 1423.6, "word": " real", "probability": 0.962890625}, {"start": 1423.6, "end": 1424.2, "word": " numbers", "probability": 0.83642578125}, {"start": 1424.2, "end": 1426.42, "word": " إذا", "probability": 0.4354248046875}, {"start": 1426.42, "end": 1427.4, "word": " xn", "probability": 0.59130859375}, {"start": 1427.4, "end": 1428.84, "word": " حدود", "probability": 0.9728190104166666}, {"start": 1428.84, "end": 1429.56, "word": " عموجة", "probability": 0.8731689453125}, {"start": 1429.56, "end": 1431.22, "word": " بقى", "probability": 0.7252604166666666}, {"start": 1431.22, "end": 1432.04, "word": " لكل", "probability": 0.9873046875}, {"start": 1432.04, "end": 1432.58, "word": " n", "probability": 0.77587890625}, {"start": 1432.58, "end": 1435.68, "word": " such", "probability": 0.728515625}, {"start": 1435.68, "end": 1436.14, "word": " that", "probability": 0.9541015625}, {"start": 1436.14, "end": 1437.04, "word": " limit", "probability": 0.85791015625}, {"start": 1437.04, "end": 1440.56, "word": " ل", "probability": 0.8515625}, {"start": 1440.56, "end": 1441.88, "word": " xn", "probability": 0.774169921875}, {"start": 1441.88, "end": 1442.6, "word": " زاد", "probability": 0.59619140625}, {"start": 1442.6, "end": 1443.16, "word": " واحد", "probability": 0.84619140625}, {"start": 1443.16, "end": 1443.44, "word": " على", "probability": 0.7080078125}, {"start": 1443.44, "end": 1444.48, "word": " xn", "probability": 0.9755859375}, {"start": 1444.48, "end": 1446.38, "word": " لما", "probability": 0.931884765625}, {"start": 1446.38, "end": 1446.8, "word": " n", "probability": 0.78955078125}, {"start": 1446.8, "end": 1447.52, "word": " تقول", "probability": 0.7978515625}, {"start": 1447.52, "end": 1448.22, "word": " infinity", "probability": 0.228759765625}], "temperature": 1.0}, {"id": 50, "seek": 147435, "start": 1449.75, "end": 1474.35, "text": "بساوي عدد ال أكبر من واحد و المقلوب show اثبت في الحالة هذه ان ال sequence xm is unbounded is not bounded", "tokens": [3555, 3794, 995, 45865, 6225, 3215, 3215, 2423, 5551, 4117, 26890, 9154, 36764, 24401, 4032, 9673, 4587, 1211, 37746, 855, 1975, 12984, 3555, 2655, 8978, 21542, 6027, 3660, 29538, 16472, 2423, 8310, 2031, 76, 307, 517, 18767, 292, 307, 406, 37498], "avg_logprob": -0.28031992770376657, "compression_ratio": 1.2258064516129032, "no_speech_prob": 0.0, "words": [{"start": 1449.75, "end": 1450.59, "word": "بساوي", "probability": 0.7938232421875}, {"start": 1450.59, "end": 1451.17, "word": " عدد", "probability": 0.9016927083333334}, {"start": 1451.17, "end": 1451.55, "word": " ال", "probability": 0.26171875}, {"start": 1451.55, "end": 1452.25, "word": " أكبر", "probability": 0.9326171875}, {"start": 1452.25, "end": 1452.45, "word": " من", "probability": 0.990234375}, {"start": 1452.45, "end": 1453.01, "word": " واحد", "probability": 0.95751953125}, {"start": 1453.01, "end": 1455.39, "word": " و", "probability": 0.369140625}, {"start": 1455.39, "end": 1456.33, "word": " المقلوب", "probability": 0.843994140625}, {"start": 1456.33, "end": 1456.97, "word": " show", "probability": 0.833984375}, {"start": 1456.97, "end": 1459.25, "word": " اثبت", "probability": 0.6942901611328125}, {"start": 1459.25, "end": 1459.41, "word": " في", "probability": 0.8935546875}, {"start": 1459.41, "end": 1459.85, "word": " الحالة", "probability": 0.96728515625}, {"start": 1459.85, "end": 1460.23, "word": " هذه", "probability": 0.78564453125}, {"start": 1460.23, "end": 1460.55, "word": " ان", "probability": 0.7431640625}, {"start": 1460.55, "end": 1461.53, "word": " ال", "probability": 0.297119140625}, {"start": 1461.53, "end": 1465.75, "word": " sequence", "probability": 0.85888671875}, {"start": 1465.75, "end": 1466.49, "word": " xm", "probability": 0.7138671875}, {"start": 1466.49, "end": 1470.17, "word": " is", "probability": 0.84619140625}, {"start": 1470.17, "end": 1471.53, "word": " unbounded", "probability": 0.8479817708333334}, {"start": 1471.53, "end": 1471.89, "word": " is", "probability": 0.48681640625}, {"start": 1471.89, "end": 1472.31, "word": " not", "probability": 0.9619140625}, {"start": 1472.31, "end": 1474.35, "word": " bounded", "probability": 0.923828125}], "temperature": 1.0}, {"id": 51, "seek": 149346, "start": 1478.48, "end": 1493.46, "text": "and hence not convergent لأن لو كانت convergent تطلع bounded", "tokens": [474, 16678, 406, 9652, 6930, 5296, 33456, 45164, 25961, 2655, 9652, 6930, 6055, 9566, 1211, 3615, 37498], "avg_logprob": -0.2884114483992259, "compression_ratio": 1.042857142857143, "no_speech_prob": 0.0, "words": [{"start": 1478.48, "end": 1478.98, "word": "and", "probability": 0.347412109375}, {"start": 1478.98, "end": 1479.5, "word": " hence", "probability": 0.837890625}, {"start": 1479.5, "end": 1486.1, "word": " not", "probability": 0.9208984375}, {"start": 1486.1, "end": 1487.26, "word": " convergent", "probability": 0.93212890625}, {"start": 1487.26, "end": 1489.58, "word": " لأن", "probability": 0.603515625}, {"start": 1489.58, "end": 1489.76, "word": " لو", "probability": 0.865234375}, {"start": 1489.76, "end": 1490.14, "word": " كانت", "probability": 0.974365234375}, {"start": 1490.14, "end": 1490.94, "word": " convergent", "probability": 0.8798828125}, {"start": 1490.94, "end": 1492.06, "word": " تطلع", "probability": 0.931640625}, {"start": 1492.06, "end": 1493.46, "word": " bounded", "probability": 0.2095947265625}], "temperature": 1.0}, {"id": 52, "seek": 152129, "start": 1513.37, "end": 1521.29, "text": "يعني من الشرط هذا ممكن تباطم الـ sequence increasing متزايدة", "tokens": [40228, 22653, 9154, 25124, 2288, 9566, 23758, 3714, 43020, 6055, 3555, 995, 9566, 2304, 2423, 39184, 8310, 5662, 44650, 11622, 995, 25708, 3660], "avg_logprob": -0.3723958445092042, "compression_ratio": 1.0333333333333334, "no_speech_prob": 0.0, "words": [{"start": 1513.37, "end": 1513.91, "word": "يعني", "probability": 0.6400146484375}, {"start": 1513.91, "end": 1514.31, "word": " من", "probability": 0.919921875}, {"start": 1514.31, "end": 1515.23, "word": " الشرط", "probability": 0.9580078125}, {"start": 1515.23, "end": 1515.59, "word": " هذا", "probability": 0.6767578125}, {"start": 1515.59, "end": 1516.01, "word": " ممكن", "probability": 0.85400390625}, {"start": 1516.01, "end": 1516.59, "word": " تباطم", "probability": 0.561572265625}, {"start": 1516.59, "end": 1516.79, "word": " الـ", "probability": 0.813232421875}, {"start": 1516.79, "end": 1517.19, "word": " sequence", "probability": 0.45068359375}, {"start": 1517.19, "end": 1518.01, "word": " increasing", "probability": 0.3505859375}, {"start": 1518.01, "end": 1521.29, "word": " متزايدة", "probability": 0.846728515625}], "temperature": 1.0}, {"id": 53, "seek": 156875, "start": 1565.95, "end": 1568.75, "text": "أه ..", "tokens": [10721, 3224, 4386], "avg_logprob": -1.37109375, "compression_ratio": 0.4666666666666667, "no_speech_prob": 2.6702880859375e-05, "words": [{"start": 1565.9499999999998, "end": 1567.35, "word": "أه", "probability": 0.15814208984375}, {"start": 1567.35, "end": 1568.75, "word": " ..", "probability": 0.4775390625}], "temperature": 1.0}, {"id": 54, "seek": 162022, "start": 1591.5, "end": 1620.22, "text": "ممكن نعمل برهان بال .. بالتناقض افرم انها bounded وممكن نصل لتناقض من تعريف ال .. هنا ال sequence هذه of quotient convergent لعدد L أكبر من واحد ممكن باستخدامه", "tokens": [2304, 43020, 8717, 25957, 1211, 4724, 2288, 3224, 7649, 20666, 4386, 20666, 2655, 8315, 4587, 11242, 1975, 5172, 2288, 2304, 16472, 11296, 37498, 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باستخدام التعريف فحاولوا تتدربوا على التمرين هادى و تحلوا أسئلة زيها فى حد عنده أي سؤال تانى فى هذا ال section طيب نحل كمان سؤال فى section تلاتة اتنين", "tokens": [3555, 45340, 12984, 1211, 3714, 43020, 8717, 5016, 1211, 4724, 995, 4587, 7578, 16712, 29973, 9957, 13672, 7578, 8032, 7578, 27188, 32887, 3615, 316, 4032, 33, 4032, 34, 4724, 995, 14851, 9778, 3215, 10943, 16712, 3615, 16572, 5172, 6156, 5016, 995, 12610, 14407, 6055, 2655, 3215, 25513, 14407, 15844, 16712, 29973, 9957, 8032, 18513, 7578, 4032, 6055, 5016, 1211, 14407, 5551, 3794, 19986, 37977, 30767, 1829, 11296, 6156, 7578, 11331, 3215, 43242, 3224, 36632, 8608, 33604, 6027, 6055, 7649, 7578, 6156, 7578, 23758, 2423, 3541, 23032, 1829, 3555, 8717, 5016, 1211, 9122, 2304, 7649, 8608, 33604, 6027, 6156, 7578, 3541, 6055, 1211, 9307, 3660, 1975, 2655, 1863, 9957], "avg_logprob": -0.1346760277354389, "compression_ratio": 1.7241379310344827, "no_speech_prob": 0.0, "words": [{"start": 2804.62, "end": 2805.18, "word": "بالمثل", 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such that for any epsilon for any epsilon أكبر من السفر يوجد m بحيث يوجد عدد m such that", "tokens": [474, 17861, 307, 1270, 300, 307, 257, 8310, 1270, 300, 337, 604, 17889, 337, 604, 17889, 5551, 4117, 26890, 9154, 21136, 5172, 2288, 7251, 29245, 3215, 275, 4724, 5016, 1829, 12984, 7251, 29245, 3215, 6225, 3215, 3215, 275, 1270, 300], "avg_logprob": -0.17816310975609756, "compression_ratio": 1.3333333333333333, "no_speech_prob": 0.0, "words": [{"start": 2907.58, "end": 2908.16, "word": "and", "probability": 0.253173828125}, {"start": 2908.16, "end": 2908.96, "word": " yn", "probability": 0.28125}, {"start": 2908.96, "end": 2911.04, "word": " is", "probability": 0.85888671875}, {"start": 2911.04, "end": 2911.68, "word": " such", "probability": 0.927734375}, {"start": 2911.68, "end": 2912.16, "word": " that", "probability": 0.94873046875}, {"start": 2912.16, "end": 2912.44, "word": " is", "probability": 0.63134765625}, {"start": 2912.44, "end": 2912.62, "word": " a", "probability": 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تطلع ال sequence yn convergent ونهايتها هي نفس نهاية ال sequence xn لأن هنا الإجابة yes", "tokens": [3555, 1863, 8592, 2288, 2423, 1230, 4724, 5016, 1829, 12984, 16472, 3224, 5296, 28820, 297, 5551, 4117, 26890, 9154, 19446, 7251, 14851, 2407, 7578, 4238, 426, 9673, 3794, 31845, 3660, 49374, 2031, 77, 4032, 2534, 5551, 9381, 17082, 2288, 9154, 8717, 36178, 11296, 8032, 1211, 23758, 4724, 1829, 28543, 2304, 2423, 8310, 17861, 9652, 6930, 22807, 8032, 1863, 8592, 38688, 6024, 48506, 14739, 6156, 3615, 15040, 6055, 9566, 1211, 3615, 2423, 8310, 17861, 9652, 6930, 4032, 1863, 11296, 36081, 11296, 39896, 8717, 36178, 8717, 11296, 10632, 2423, 8310, 2031, 77, 5296, 33456, 34105, 33688, 7435, 16758, 3660, 2086], "avg_logprob": -0.2627840872966882, "compression_ratio": 1.6238532110091743, "no_speech_prob": 1.1920928955078125e-07, "words": [{"start": 3022.13, "end": 3022.83, "word": "بنشر", "probability": 0.460540771484375}, {"start": 3022.83, "end": 3022.97, "word": " ال", "probability": 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"probability": 0.6284993489583334}, {"start": 3061.95, "end": 3062.49, "word": " برهان", "probability": 0.9490966796875}, {"start": 3062.49, "end": 3067.57, "word": " اذا", "probability": 0.5267333984375}, {"start": 3067.57, "end": 3067.97, "word": " to", "probability": 0.7216796875}, {"start": 3067.97, "end": 3068.31, "word": " see", "probability": 0.93798828125}, {"start": 3068.31, "end": 3071.37, "word": " this", "probability": 0.9599609375}, {"start": 3071.37, "end": 3076.61, "word": " نبدأ", "probability": 0.880859375}, {"start": 3076.61, "end": 3077.29, "word": " بإبسلون", "probability": 0.8448893229166666}, {"start": 3077.29, "end": 3077.77, "word": " أكبر", "probability": 0.921875}, {"start": 3077.77, "end": 3077.97, "word": " من", "probability": 0.9921875}, {"start": 3077.97, "end": 3078.61, "word": " السفر", "probability": 0.8245442708333334}], "temperature": 1.0}, {"id": 105, "seek": 310589, "start": 3096.81, "end": 3105.89, "text": "let by hypothesis من الفرض من الفرض من ال hypothesis", "tokens": [2631, 538, 17291, 9154, 27188, 43042, 9154, 27188, 43042, 9154, 2423, 17291], "avg_logprob": -0.2593148946762085, "compression_ratio": 1.5217391304347827, "no_speech_prob": 0.0, "words": [{"start": 3096.81, "end": 3097.89, "word": "let", "probability": 0.56884765625}, {"start": 3097.89, "end": 3098.97, "word": " by", "probability": 0.82421875}, {"start": 3098.97, "end": 3099.73, "word": " hypothesis", "probability": 0.304443359375}, {"start": 3099.73, "end": 3100.15, "word": " من", "probability": 0.9453125}, {"start": 3100.15, "end": 3100.71, "word": " الفرض", "probability": 0.841552734375}, {"start": 3100.71, "end": 3104.45, "word": " من", "probability": 0.8212890625}, {"start": 3104.45, "end": 3104.97, "word": " الفرض", "probability": 0.87939453125}, {"start": 3104.97, "end": 3105.23, "word": " من", "probability": 0.9443359375}, {"start": 3105.23, "end": 3105.37, "word": " ال", "probability": 0.83349609375}, {"start": 3105.37, "end": 3105.89, "word": " hypothesis", "probability": 0.76416015625}], "temperature": 1.0}, {"id": 106, "seek": 313538, "start": 3107.12, "end": 3135.38, "text": "أنا عندي absolute xn minus yn أصغر من إبسلون أكبر من أو ساوي سفر وهذا صحيح لكل n أكبر من أو ساوي capital M وهذا الكلام صحيح لكل إبسلون أكبر من السفر", "tokens": [10721, 8315, 18871, 16254, 8236, 2031, 77, 3175, 17861, 5551, 9381, 17082, 2288, 9154, 11933, 3555, 3794, 1211, 11536, 5551, 4117, 26890, 9154, 34051, 8608, 995, 45865, 8608, 5172, 2288, 37037, 15730, 20328, 5016, 1829, 5016, 5296, 28820, 297, 5551, 4117, 26890, 9154, 34051, 8608, 995, 45865, 4238, 376, 37037, 15730, 2423, 28820, 10943, 20328, 5016, 1829, 5016, 5296, 28820, 11933, 3555, 3794, 1211, 11536, 5551, 4117, 26890, 9154, 21136, 5172, 2288], "avg_logprob": -0.13366865785154577, "compression_ratio": 1.6950354609929077, "no_speech_prob": 0.0, "words": [{"start": 3107.12, "end": 3107.44, "word": "أنا", "probability": 0.7919921875}, {"start": 3107.44, "end": 3107.86, "word": " عندي", 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"word": " minus", "probability": 0.40576171875}, {"start": 3162.96, "end": 3163.66, "word": " yn", "probability": 0.9560546875}, {"start": 3163.66, "end": 3165.38, "word": " لما", "probability": 0.850341796875}, {"start": 3165.38, "end": 3165.68, "word": " n", "probability": 0.6435546875}, {"start": 3165.68, "end": 3166.14, "word": " تقول", "probability": 0.74365234375}, {"start": 3166.14, "end": 3166.98, "word": " infinity", "probability": 0.445068359375}, {"start": 3166.98, "end": 3168.94, "word": " بساوي", "probability": 0.724609375}, {"start": 3168.94, "end": 3169.42, "word": " سفر", "probability": 0.91064453125}], "temperature": 1.0}, {"id": 108, "seek": 319385, "start": 3174.15, "end": 3193.85, "text": "مش شرط هذا انا عندي ال .. ما معناه ان limit ال sequence هذه بساوة سفر؟ معناه لأي epsilon أكبر من السفر يوجد capital M", "tokens": [2304, 8592, 13412, 2288, 9566, 23758, 1975, 8315, 18871, 16254, 2423, 4386, 19446, 20449, 8315, 3224, 16472, 4948, 2423, 8310, 29538, 4724, 3794, 995, 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"probability": 0.99658203125}, {"start": 3191.69, "end": 3192.23, "word": " السفر", "probability": 0.788330078125}, {"start": 3192.23, "end": 3192.83, "word": " يوجد", "probability": 0.9698893229166666}, {"start": 3192.83, "end": 3193.35, "word": " capital", "probability": 0.79931640625}, {"start": 3193.35, "end": 3193.85, "word": " M", "probability": 0.93017578125}], "temperature": 1.0}, {"id": 109, "seek": 321096, "start": 3195.26, "end": 3210.96, "text": "عدد طبيعي يعتمد على إبسلن بحيث أنه لكل n أكبر من أو ساوي capital N هذا بيقدي أن absolute xn minus yn minus الصفر أصغر من إبسلن", "tokens": [22488, 3215, 23032, 21292, 3615, 1829, 7251, 34268, 2304, 3215, 15844, 11933, 3555, 3794, 1211, 1863, 4724, 5016, 1829, 12984, 14739, 3224, 5296, 28820, 297, 5551, 4117, 26890, 9154, 34051, 8608, 995, 45865, 4238, 426, 23758, 4724, 1829, 4587, 16254, 14739, 8236, 2031, 77, 3175, 17861, 3175, 31767, 5172, 2288, 5551, 9381, 17082, 2288, 9154, 11933, 3555, 3794, 1211, 1863], "avg_logprob": 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0.9384765625}, {"start": 3201.18, "end": 3201.32, "word": " أو", "probability": 0.92724609375}, {"start": 3201.32, "end": 3201.84, "word": " ساوي", "probability": 0.9197591145833334}, {"start": 3201.84, "end": 3202.34, "word": " capital", "probability": 0.45361328125}, {"start": 3202.34, "end": 3202.68, "word": " N", "probability": 0.8408203125}, {"start": 3202.68, "end": 3203.5, "word": " هذا", "probability": 0.68798828125}, {"start": 3203.5, "end": 3204.02, "word": " بيقدي", "probability": 0.677734375}, {"start": 3204.02, "end": 3204.22, "word": " أن", "probability": 0.55517578125}, {"start": 3204.22, "end": 3204.86, "word": " absolute", "probability": 0.317138671875}, {"start": 3204.86, "end": 3205.9, "word": " xn", "probability": 0.63671875}, {"start": 3205.9, "end": 3206.66, "word": " minus", "probability": 0.61474609375}, {"start": 3206.66, "end": 3208.02, "word": " yn", "probability": 0.9296875}, {"start": 3208.02, "end": 3208.86, "word": " minus", "probability": 0.95361328125}, {"start": 3208.86, "end": 3209.58, "word": " الصفر", "probability": 0.7859700520833334}, {"start": 3209.58, "end": 3210.18, "word": " أصغر", "probability": 0.9774169921875}, {"start": 3210.18, "end": 3210.34, "word": " من", "probability": 0.98779296875}, {"start": 3210.34, "end": 3210.96, "word": " إبسلن", "probability": 0.96943359375}], "temperature": 1.0}, {"id": 110, "seek": 324078, "start": 3212.68, "end": 3240.78, "text": "هي معنى ان limit ال sequence للفرق بساوي سفر ايش معنى هذا لأي epsilon أكبر من سفر يوجد capital M يعتمد على N عدد طبيعي يعتمد على ال epsilon بحيث لكل N أكبر من أو ساوي capital N المسافة بين الحد العام لل sequence و limit اللي هي سفر أصغر من epsilon هذا الكلام هى متحقق هنا هى متحقق", "tokens": [3224, 1829, 20449, 1863, 7578, 16472, 4948, 2423, 8310, 24976, 5172, 2288, 4587, 4724, 3794, 995, 45865, 8608, 5172, 2288, 1975, 1829, 8592, 20449, 1863, 7578, 23758, 5296, 10721, 1829, 17889, 5551, 4117, 26890, 9154, 8608, 5172, 2288, 7251, 29245, 3215, 4238, 376, 7251, 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"probability": 0.8963216145833334}, {"start": 3221.18, "end": 3221.66, "word": " يوجد", "probability": 0.9856770833333334}, {"start": 3221.66, "end": 3222.08, "word": " capital", "probability": 0.67626953125}, {"start": 3222.08, "end": 3222.48, "word": " M", "probability": 0.90771484375}, {"start": 3222.48, "end": 3224.04, "word": " يعتمد", "probability": 0.9473876953125}, {"start": 3224.04, "end": 3224.22, "word": " على", "probability": 0.88720703125}, {"start": 3224.22, "end": 3224.46, "word": " N", "probability": 0.76806640625}, {"start": 3224.46, "end": 3224.78, "word": " عدد", "probability": 0.7823893229166666}, {"start": 3224.78, "end": 3225.38, "word": " طبيعي", "probability": 0.9697265625}, {"start": 3225.38, "end": 3226.5, "word": " يعتمد", "probability": 0.9261474609375}, {"start": 3226.5, "end": 3226.66, "word": " على", "probability": 0.91796875}, {"start": 3226.66, "end": 3226.84, "word": " ال", "probability": 0.9189453125}, {"start": 3226.84, "end": 3227.22, "word": " epsilon", "probability": 0.5029296875}, {"start": 3227.22, "end": 3228.66, "word": " بحيث", "probability": 0.975830078125}, {"start": 3228.66, "end": 3229.1, "word": " لكل", "probability": 0.97802734375}, {"start": 3229.1, "end": 3229.32, "word": " N", "probability": 0.84228515625}, {"start": 3229.32, "end": 3229.7, "word": " أكبر", "probability": 0.9791666666666666}, {"start": 3229.7, "end": 3229.84, "word": " من", "probability": 0.896484375}, {"start": 3229.84, "end": 3229.96, "word": " أو", "probability": 0.1578369140625}, {"start": 3229.96, "end": 3230.3, "word": " ساوي", "probability": 0.84765625}, {"start": 3230.3, "end": 3230.72, "word": " capital", "probability": 0.68310546875}, {"start": 3230.72, "end": 3231.02, "word": " N", "probability": 0.9404296875}, {"start": 3231.02, "end": 3232.16, "word": " المسافة", "probability": 0.982177734375}, {"start": 3232.16, "end": 3232.44, "word": " بين", "probability": 0.9775390625}, {"start": 3232.44, "end": 3232.9, "word": " الحد", "probability": 0.964111328125}, {"start": 3232.9, "end": 3233.22, "word": " العام", "probability": 0.9833984375}, {"start": 3233.22, "end": 3233.38, "word": " لل", "probability": 0.9248046875}, {"start": 3233.38, "end": 3233.94, "word": " sequence", "probability": 0.96484375}, {"start": 3233.94, "end": 3234.2, "word": " و", "probability": 0.76318359375}, {"start": 3234.2, "end": 3235.2, "word": " limit", "probability": 0.73779296875}, {"start": 3235.2, "end": 3235.4, "word": " اللي", "probability": 0.6705322265625}, {"start": 3235.4, "end": 3235.54, "word": " هي", "probability": 0.845703125}, {"start": 3235.54, "end": 3235.82, "word": " سفر", "probability": 0.7926432291666666}, {"start": 3235.82, "end": 3236.1, "word": " أصغر", "probability": 0.7784423828125}, {"start": 3236.1, "end": 3236.24, "word": " من", "probability": 0.98974609375}, {"start": 3236.24, "end": 3236.56, "word": " epsilon", "probability": 0.689453125}, {"start": 3236.56, "end": 3237.68, "word": " هذا", "probability": 0.20703125}, {"start": 3237.68, "end": 3238.44, "word": " الكلام", "probability": 0.92529296875}, {"start": 3238.44, "end": 3238.66, "word": " هى", "probability": 0.6192626953125}, {"start": 3238.66, "end": 3239.3, "word": " متحقق", "probability": 0.9136962890625}, {"start": 3239.3, "end": 3239.56, "word": " هنا", "probability": 0.97314453125}, {"start": 3239.56, "end": 3240.14, "word": " هى", "probability": 0.7412109375}, {"start": 3240.14, "end": 3240.78, "word": " متحقق", "probability": 0.9874267578125}], "temperature": 1.0}, {"id": 111, "seek": 326395, "start": 3241.43, "end": 3263.95, "text": "تامام؟ إذا هذا بنحصل عليه وبالتالي limit xn minus yn بساوي سفر ومنها الآن أنا عندي ال yn ممكن كتبتها على صورة yn سالب xn موجب xn", "tokens": [2655, 10943, 10943, 22807, 11933, 15730, 23758, 44945, 5016, 36520, 47356, 46599, 6027, 2655, 6027, 1829, 4948, 2031, 77, 3175, 17861, 4724, 3794, 995, 45865, 8608, 5172, 2288, 4032, 27842, 11296, 6024, 48506, 41850, 18871, 16254, 2423, 288, 77, 3714, 43020, 9122, 2655, 3555, 2655, 11296, 15844, 20328, 13063, 3660, 288, 77, 8608, 6027, 3555, 2031, 77, 3714, 29245, 3555, 2031, 77], "avg_logprob": -0.31349205781543066, "compression_ratio": 1.395973154362416, "no_speech_prob": 0.0, "words": [{"start": 3241.43, "end": 3242.27, "word": "تامام؟", "probability": 0.334197998046875}, {"start": 3242.27, "end": 3242.47, "word": " إذا", "probability": 0.718017578125}, {"start": 3242.47, "end": 3242.77, "word": " هذا", "probability": 0.378662109375}, {"start": 3242.77, "end": 3243.21, "word": " بنحصل", "probability": 0.808837890625}, {"start": 3243.21, "end": 3243.51, "word": " عليه", "probability": 0.98291015625}, {"start": 3243.51, "end": 3244.03, "word": " وبالتالي", "probability": 0.8455078125}, {"start": 3244.03, "end": 3244.33, "word": " limit", "probability": 0.81103515625}, {"start": 3244.33, "end": 3244.85, "word": " xn", "probability": 0.5406494140625}, {"start": 3244.85, "end": 3245.27, "word": " minus", "probability": 0.373291015625}, {"start": 3245.27, "end": 3245.69, "word": " yn", "probability": 0.9208984375}, {"start": 3245.69, "end": 3246.19, "word": " بساوي", "probability": 0.77978515625}, {"start": 3246.19, "end": 3247.17, "word": " سفر", "probability": 0.8816731770833334}, {"start": 3247.17, "end": 3248.39, "word": " ومنها", "probability": 0.8914388020833334}, {"start": 3248.39, "end": 3250.31, "word": " الآن", "probability": 0.6182861328125}, {"start": 3250.31, "end": 3250.55, "word": " أنا", "probability": 0.837890625}, {"start": 3250.55, "end": 3250.99, "word": " عندي", "probability": 0.720703125}, {"start": 3250.99, "end": 3251.41, "word": " ال", "probability": 0.9814453125}, {"start": 3251.41, "end": 3252.23, "word": " yn", "probability": 0.598876953125}, {"start": 3252.23, "end": 3254.07, "word": " ممكن", "probability": 0.96240234375}, {"start": 3254.07, "end": 3254.67, "word": " كتبتها", "probability": 0.84814453125}, {"start": 3254.67, "end": 3254.83, "word": " على", "probability": 0.9267578125}, {"start": 3254.83, "end": 3256.45, "word": " صورة", "probability": 0.9713541666666666}, {"start": 3256.45, "end": 3260.87, "word": " yn", "probability": 0.6253662109375}, {"start": 3260.87, "end": 3261.61, "word": " سالب", "probability": 0.7815755208333334}, {"start": 3261.61, "end": 3262.47, "word": " xn", "probability": 0.955810546875}, {"start": 3262.47, "end": 3263.29, "word": " موجب", "probability": 0.845703125}, {"start": 3263.29, "end": 3263.95, "word": " xn", "probability": 0.9560546875}], "temperature": 1.0}, {"id": 112, "seek": 329259, "start": 3265.31, "end": 3292.59, "text": "وهذا بيساوي سالب Xn سالب Yn زاد Xn تمام؟ إذا ال limit ل Yn as n tends to infinity بيساوي limit الطرف اليمين ف limit Xn سالب Yn مضروبة في سالب واحد بيطلع برا ال limit", "tokens": [2407, 3224, 15730, 4724, 1829, 3794, 995, 45865, 8608, 6027, 3555, 1783, 77, 8608, 6027, 3555, 398, 77, 30767, 18513, 1783, 77, 46811, 10943, 22807, 11933, 15730, 2423, 4948, 5296, 398, 77, 382, 297, 12258, 281, 13202, 4724, 1829, 3794, 995, 45865, 4948, 41950, 28480, 2423, 32640, 9957, 6156, 4948, 1783, 77, 8608, 6027, 3555, 398, 77, 3714, 11242, 32887, 49401, 8978, 8608, 6027, 3555, 36764, 24401, 4724, 1829, 9566, 1211, 3615, 4724, 23557, 2423, 4948], "avg_logprob": -0.17573052567320985, "compression_ratio": 1.6333333333333333, "no_speech_prob": 0.0, "words": [{"start": 3265.31, "end": 3265.85, "word": "وهذا", "probability": 0.7262369791666666}, {"start": 3265.85, "end": 3266.57, "word": " بيساوي", "probability": 0.80732421875}, {"start": 3266.57, "end": 3267.41, "word": " سالب", "probability": 0.7337239583333334}, {"start": 3267.41, "end": 3269.37, "word": " Xn", "probability": 0.3472900390625}, {"start": 3269.37, "end": 3270.09, "word": " سالب", "probability": 0.9117838541666666}, {"start": 3270.09, "end": 3271.03, "word": " Yn", "probability": 0.82177734375}, {"start": 3271.03, "end": 3271.89, "word": " زاد", "probability": 0.636962890625}, {"start": 3271.89, "end": 3272.61, "word": " Xn", "probability": 0.973876953125}, {"start": 3272.61, "end": 3274.05, "word": " تمام؟", "probability": 0.720458984375}, {"start": 3274.05, "end": 3275.59, "word": " إذا", "probability": 0.688232421875}, {"start": 3275.59, "end": 3275.81, "word": " ال", "probability": 0.75341796875}, {"start": 3275.81, "end": 3276.21, "word": " limit", "probability": 0.853515625}, {"start": 3276.21, "end": 3277.79, "word": " ل", "probability": 0.82177734375}, {"start": 3277.79, "end": 3278.63, "word": " Yn", "probability": 0.828125}, {"start": 3278.63, "end": 3279.19, "word": " as", "probability": 0.8544921875}, {"start": 3279.19, "end": 3279.61, "word": " n", "probability": 0.6787109375}, {"start": 3279.61, "end": 3279.91, "word": " tends", "probability": 0.74169921875}, {"start": 3279.91, "end": 3280.09, "word": " to", "probability": 0.96435546875}, {"start": 3280.09, "end": 3280.63, "word": " infinity", "probability": 0.87255859375}, {"start": 3280.63, "end": 3281.57, "word": " بيساوي", "probability": 0.8919921875}, {"start": 3281.57, "end": 3282.69, "word": " limit", "probability": 0.970703125}, {"start": 3282.69, "end": 3283.17, "word": " الطرف", "probability": 0.972900390625}, {"start": 3283.17, "end": 3283.79, "word": " اليمين", "probability": 0.8629557291666666}, {"start": 3283.79, "end": 3285.11, "word": " ف", "probability": 0.86279296875}, {"start": 3285.11, "end": 3285.65, "word": " limit", "probability": 0.8916015625}, {"start": 3285.65, "end": 3287.67, "word": " Xn", "probability": 0.973388671875}, {"start": 3287.67, "end": 3288.41, "word": " سالب", "probability": 0.9822591145833334}, {"start": 3288.41, "end": 3289.11, "word": " Yn", "probability": 0.95263671875}, {"start": 3289.11, "end": 3289.65, "word": " مضروبة", "probability": 0.970703125}, {"start": 3289.65, "end": 3289.79, "word": " في", "probability": 0.88232421875}, {"start": 3289.79, "end": 3290.15, "word": " سالب", "probability": 0.9694010416666666}, {"start": 3290.15, "end": 3290.63, "word": " واحد", "probability": 0.9365234375}, {"start": 3290.63, "end": 3291.75, "word": " بيطلع", "probability": 0.8646484375}, {"start": 3291.75, "end": 3292.13, "word": " برا", "probability": 0.805419921875}, {"start": 3292.13, "end": 3292.27, "word": " ال", "probability": 0.939453125}, {"start": 3292.27, "end": 3292.59, "word": " limit", "probability": 0.97314453125}], "temperature": 1.0}, {"id": 113, "seek": 331657, "start": 3295.23, "end": 3316.57, "text": "زائد limit xn لما n تقول لإنفينيتي وهنا لسه احنا مثبتين هذا عبارة عن سالب limit sequence xn minus yn بالساوية سفر، سالد واحد في سفر", "tokens": [11622, 16373, 3215, 4948, 2031, 77, 5296, 15042, 297, 6055, 39648, 5296, 28814, 1863, 5172, 9957, 1829, 31371, 37037, 8315, 5296, 3794, 3224, 1975, 5016, 8315, 50113, 3555, 2655, 9957, 23758, 6225, 3555, 9640, 3660, 18871, 8608, 6027, 3555, 4948, 8310, 2031, 77, 3175, 17861, 20666, 3794, 995, 2407, 10632, 8608, 5172, 2288, 12399, 8608, 6027, 3215, 36764, 24401, 8978, 8608, 5172, 2288], "avg_logprob": -0.31079100351780653, "compression_ratio": 1.38, "no_speech_prob": 0.0, "words": [{"start": 3295.23, "end": 3295.99, "word": "زائد", "probability": 0.5196533203125}, {"start": 3295.99, "end": 3296.41, "word": " limit", "probability": 0.91845703125}, {"start": 3296.41, "end": 3298.67, "word": " xn", "probability": 0.54443359375}, {"start": 3298.67, "end": 3299.07, "word": " لما", "probability": 0.5548095703125}, {"start": 3299.07, "end": 3299.35, "word": " n", "probability": 0.5263671875}, {"start": 3299.35, "end": 3299.85, "word": " تقول", "probability": 0.849853515625}, {"start": 3299.85, "end": 3300.69, "word": " لإنفينيتي", "probability": 0.6565638950892857}, {"start": 3300.69, "end": 3303.77, "word": " وهنا", "probability": 0.5377197265625}, {"start": 3303.77, "end": 3304.11, "word": " لسه", "probability": 0.8098958333333334}, {"start": 3304.11, "end": 3304.33, "word": " احنا", "probability": 0.892578125}, {"start": 3304.33, "end": 3304.91, "word": " مثبتين", "probability": 0.81298828125}, {"start": 3304.91, "end": 3305.07, "word": " هذا", "probability": 0.2744140625}, {"start": 3305.07, "end": 3305.45, "word": " عبارة", "probability": 0.9866943359375}, {"start": 3305.45, "end": 3305.73, "word": " عن", "probability": 0.99560546875}, {"start": 3305.73, "end": 3306.41, "word": " سالب", "probability": 0.9148763020833334}, {"start": 3306.41, "end": 3307.65, "word": " limit", "probability": 0.9521484375}, {"start": 3307.65, "end": 3308.77, "word": " sequence", "probability": 0.908203125}, {"start": 3308.77, "end": 3309.31, "word": " xn", "probability": 0.907958984375}, {"start": 3309.31, "end": 3309.71, "word": " minus", "probability": 0.8291015625}, {"start": 3309.71, "end": 3310.39, "word": " yn", "probability": 0.8427734375}, {"start": 3310.39, "end": 3311.81, "word": " بالساوية", "probability": 0.718017578125}, {"start": 3311.81, "end": 3312.33, "word": " سفر،", "probability": 0.71075439453125}, {"start": 3312.33, "end": 3313.63, "word": " سالد", "probability": 0.7132161458333334}, {"start": 3313.63, "end": 3315.71, "word": " واحد", "probability": 0.966552734375}, {"start": 3315.71, "end": 3316.01, "word": " في", "probability": 0.9765625}, {"start": 3316.01, "end": 3316.57, "word": " سفر", "probability": 0.9765625}], "temperature": 1.0}, {"id": 114, "seek": 334009, "start": 3319.99, "end": 3340.09, "text": "زاد limit xn اللي هي x تمام اذا limit ال sequence yn تطلع بالساوي x اذا هنا اثبتنا ان ال sequence yn تطلع convergent وال limit تبعتها بالساوي x تمام", "tokens": [11622, 18513, 4948, 2031, 77, 13672, 1829, 39896, 2031, 46811, 10943, 1975, 15730, 4948, 2423, 8310, 17861, 6055, 9566, 1211, 3615, 20666, 3794, 995, 45865, 2031, 1975, 15730, 34105, 1975, 12984, 3555, 2655, 8315, 16472, 2423, 8310, 17861, 6055, 9566, 1211, 3615, 9652, 6930, 16070, 4948, 6055, 3555, 34268, 11296, 20666, 3794, 995, 45865, 2031, 46811, 10943], "avg_logprob": -0.24339977857367745, "compression_ratio": 1.6194029850746268, "no_speech_prob": 5.960464477539063e-08, "words": [{"start": 3319.99, "end": 3320.79, "word": "زاد", "probability": 0.624755859375}, {"start": 3320.79, "end": 3321.27, "word": " limit", "probability": 0.76513671875}, {"start": 3321.27, "end": 3322.05, "word": " xn", "probability": 0.736572265625}, {"start": 3322.05, "end": 3322.25, "word": " اللي", "probability": 0.49627685546875}, {"start": 3322.25, "end": 3322.49, "word": " هي", "probability": 0.900390625}, {"start": 3322.49, "end": 3322.97, "word": " x", "probability": 0.783203125}, {"start": 3322.97, "end": 3324.31, "word": " تمام", "probability": 0.6832275390625}, {"start": 3324.31, "end": 3325.45, "word": " اذا", "probability": 0.58685302734375}, {"start": 3325.45, "end": 3326.23, "word": " limit", "probability": 0.8046875}, {"start": 3326.23, "end": 3326.43, "word": " ال", "probability": 0.57568359375}, {"start": 3326.43, "end": 3326.85, "word": " sequence", "probability": 0.97119140625}, {"start": 3326.85, "end": 3327.35, "word": " yn", "probability": 0.77001953125}, {"start": 3327.35, "end": 3328.01, "word": " تطلع", "probability": 0.92919921875}, {"start": 3328.01, "end": 3328.63, "word": " بالساوي", "probability": 0.835693359375}, {"start": 3328.63, "end": 3329.07, "word": " x", "probability": 0.91845703125}, {"start": 3329.07, "end": 3332.37, "word": " اذا", "probability": 0.775146484375}, {"start": 3332.37, "end": 3332.67, "word": " هنا", "probability": 0.372802734375}, {"start": 3332.67, "end": 3333.19, "word": " اثبتنا", "probability": 0.89658203125}, {"start": 3333.19, "end": 3333.37, "word": " ان", "probability": 0.93310546875}, {"start": 3333.37, "end": 3333.49, "word": " ال", "probability": 0.8828125}, {"start": 3333.49, "end": 3334.03, "word": " sequence", "probability": 0.98193359375}, {"start": 3334.03, "end": 3334.99, "word": " yn", "probability": 0.970703125}, {"start": 3334.99, "end": 3335.45, "word": " تطلع", "probability": 0.9400634765625}, {"start": 3335.45, "end": 3336.27, "word": " convergent", "probability": 0.738037109375}, {"start": 3336.27, "end": 3337.21, "word": " وال", "probability": 0.29345703125}, {"start": 3337.21, "end": 3337.47, "word": " limit", "probability": 0.90966796875}, {"start": 3337.47, "end": 3338.07, "word": " تبعتها", "probability": 0.8646240234375}, {"start": 3338.07, "end": 3338.91, "word": " بالساوي", "probability": 0.956787109375}, {"start": 3338.91, "end": 3339.41, "word": " x", "probability": 0.94482421875}, {"start": 3339.41, "end": 3340.09, "word": " تمام", "probability": 0.978271484375}], "temperature": 1.0}, {"id": 115, "seek": 336919, "start": 3342.29, "end": 3369.19, "text": "البرهان هنا اعتمد على انه من الفرض انا عندي المثال لأي epsilon هذا الفرض معناه ان limit ال sequence x in minus y in بالساوي سفر وهذا اللي ساعدنا في الحل وهذا ناتج هي من تعريف epsilon capital N لل limit هذا هو البرهان", "tokens": [6027, 26890, 3224, 7649, 34105, 1975, 34268, 2304, 3215, 15844, 16472, 3224, 9154, 27188, 43042, 1975, 8315, 18871, 16254, 9673, 12984, 6027, 5296, 10721, 1829, 17889, 23758, 27188, 43042, 20449, 8315, 3224, 16472, 4948, 2423, 8310, 2031, 294, 3175, 288, 294, 20666, 3794, 995, 45865, 8608, 5172, 2288, 37037, 15730, 13672, 1829, 8608, 995, 22488, 8315, 8978, 21542, 1211, 37037, 15730, 8717, 9307, 7435, 39896, 9154, 37279, 16572, 5172, 17889, 4238, 426, 24976, 4948, 23758, 31439, 2423, 26890, 3224, 7649], "avg_logprob": -0.2503858083560143, "compression_ratio": 1.6568627450980393, "no_speech_prob": 0.0, "words": [{"start": 3342.29, "end": 3343.21, "word": "البرهان", "probability": 0.81231689453125}, {"start": 3343.21, "end": 3343.41, "word": " هنا", "probability": 0.40673828125}, {"start": 3343.41, "end": 3343.99, "word": " اعتمد", "probability": 0.9072265625}, {"start": 3343.99, "end": 3344.21, "word": " على", "probability": 0.64697265625}, {"start": 3344.21, "end": 3344.83, "word": " انه", "probability": 0.8681640625}, {"start": 3344.83, "end": 3345.25, "word": " من", "probability": 0.900390625}, {"start": 3345.25, "end": 3345.71, "word": " الفرض", "probability": 0.91943359375}, {"start": 3345.71, "end": 3345.99, "word": " انا", "probability": 0.6375732421875}, {"start": 3345.99, "end": 3346.39, "word": " عندي", "probability": 0.90869140625}, {"start": 3346.39, "end": 3347.03, "word": " المثال", "probability": 0.7124837239583334}, {"start": 3347.03, "end": 3347.97, "word": " لأي", "probability": 0.8489583333333334}, {"start": 3347.97, "end": 3348.55, "word": " epsilon", "probability": 0.302490234375}, {"start": 3348.55, "end": 3349.35, "word": " هذا", "probability": 0.8701171875}, {"start": 3349.35, "end": 3349.89, "word": " الفرض", "probability": 0.96484375}, {"start": 3349.89, "end": 3352.33, "word": " معناه", "probability": 0.9129231770833334}, {"start": 3352.33, "end": 3352.59, "word": " ان", "probability": 0.88916015625}, {"start": 3352.59, "end": 3353.01, "word": " limit", "probability": 0.5302734375}, {"start": 3353.01, "end": 3353.35, "word": " ال", "probability": 0.341552734375}, {"start": 3353.35, "end": 3353.83, "word": " sequence", "probability": 0.94677734375}, {"start": 3353.83, "end": 3354.23, "word": " x", "probability": 0.7099609375}, {"start": 3354.23, "end": 3354.55, "word": " in", "probability": 0.1817626953125}, {"start": 3354.55, "end": 3355.01, "word": " minus", "probability": 0.875}, {"start": 3355.01, "end": 3355.43, "word": " y", "probability": 0.96435546875}, {"start": 3355.43, "end": 3355.91, "word": " in", "probability": 0.95458984375}, {"start": 3355.91, "end": 3357.39, "word": " بالساوي", "probability": 0.71551513671875}, {"start": 3357.39, "end": 3358.11, "word": " سفر", "probability": 0.8494466145833334}, {"start": 3358.11, "end": 3359.11, "word": " وهذا", "probability": 0.8828125}, {"start": 3359.11, "end": 3359.25, "word": " اللي", "probability": 0.890380859375}, {"start": 3359.25, "end": 3359.65, "word": " ساعدنا", "probability": 0.97509765625}, {"start": 3359.65, "end": 3359.77, "word": " في", "probability": 0.908203125}, {"start": 3359.77, "end": 3360.11, "word": " الحل", "probability": 0.915283203125}, {"start": 3360.11, "end": 3362.81, "word": " وهذا", "probability": 0.80126953125}, {"start": 3362.81, "end": 3363.25, "word": " ناتج", "probability": 0.8780924479166666}, {"start": 3363.25, "end": 3363.49, "word": " هي", "probability": 0.452880859375}, {"start": 3363.49, "end": 3363.69, "word": " من", "probability": 0.8076171875}, {"start": 3363.69, "end": 3364.29, "word": " تعريف", "probability": 0.9894205729166666}, {"start": 3364.29, "end": 3364.77, "word": " epsilon", "probability": 0.9208984375}, {"start": 3364.77, "end": 3365.25, "word": " capital", "probability": 0.771484375}, {"start": 3365.25, "end": 3365.51, "word": " N", "probability": 0.60693359375}, {"start": 3365.51, "end": 3365.71, "word": " لل", "probability": 0.64404296875}, {"start": 3365.71, "end": 3366.03, "word": " limit", "probability": 0.802734375}, {"start": 3366.03, "end": 3367.93, "word": " هذا", "probability": 0.2169189453125}, {"start": 3367.93, "end": 3368.39, "word": " هو", "probability": 0.96484375}, {"start": 3368.39, "end": 3369.19, "word": " البرهان", "probability": 0.9571533203125}], "temperature": 1.0}], "language": "ar", "language_probability": 1.0, "duration": 3370.005, "duration_after_vad": 2846.7312499999853}
\ No newline at end of file
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/XjWoXKhuE-o_raw.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/XjWoXKhuE-o_raw.srt
new file mode 100644
index 0000000000000000000000000000000000000000..bab5f28a2adfd8f4029718766f9db09b2bd9b959
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/XjWoXKhuE-o_raw.srt
@@ -0,0 +1,1772 @@
+1
+00:00:21,090 --> 00:00:26,570
+إذن في المحاضرة هذه ان شاء الله هنحل بعض التمرين
+
+2
+00:00:26,570 --> 00:00:35,250
+لل homework اللي تابع ل section تلاتة واحد و تلاتة
+
+3
+00:00:35,250 --> 00:00:44,390
+اتنين فأزملتكم تسأل عن ال .. نحن نحل السؤال 13
+
+4
+00:00:44,390 --> 00:00:46,030
+section تلاتة واحد
+
+5
+00:00:49,430 --> 00:01:15,310
+نكتب السؤال على اللوحة سيكشن
+
+6
+00:01:15,310 --> 00:01:25,990
+السؤال 13section تلاتة واحد انا
+
+7
+00:01:25,990 --> 00:01:31,150
+عندي بي is real number اكبر من سفر اصغر من واحد
+
+8
+00:01:31,150 --> 00:01:36,010
+وبينا
+
+9
+00:01:36,010 --> 00:01:40,990
+نثبت show انه ال limit
+
+10
+00:01:44,880 --> 00:01:53,020
+للـ sequence اللي الحد العام تبعها n في d to n لما
+
+11
+00:01:53,020 --> 00:02:03,560
+n تقول infinity بساوي سفر والكتاب جايلكم
+
+12
+00:02:03,560 --> 00:02:08,840
+use ال binomial theorem كما في مثال 3 1 11 الجزء
+
+13
+00:02:08,840 --> 00:02:16,010
+ديفلو حاولتوا تستخدموا نفس أسلوب البرهان تبع
+
+14
+00:02:16,010 --> 00:02:20,350
+المثال اللي استخدمنا فيه ال binomial theorem
+
+15
+00:02:20,350 --> 00:02:28,550
+فهتصلوا للنتيجة فهي البرهان نشوف
+
+16
+00:02:28,550 --> 00:02:32,170
+كيف نستخدم ال binomial theorem في الوصول إلى
+
+17
+00:02:32,170 --> 00:02:40,300
+المطلوبأنا عندي من الفرض سفر أصغر من بي أصغر من
+
+18
+00:02:40,300 --> 00:02:48,300
+واحد هذا بيؤدي ان واحد على بي أكبر من واحد
+
+19
+00:02:48,300 --> 00:02:55,500
+وبالتالي هذا بيؤدي ان واحد على بي سالب واحد أكبر
+
+20
+00:02:55,500 --> 00:03:08,230
+من سفر اذا ناخد letlet a خلّيني اعرف عدد a على انه
+
+21
+00:03:08,230 --> 00:03:13,850
+العدد الموجب واحد على بي سالب واحد طبعا هذا عدد
+
+22
+00:03:13,850 --> 00:03:20,110
+موجب حسب ما شوفنا وهذا
+
+23
+00:03:20,110 --> 00:03:28,210
+بيقدّي انه العدد لو حليت المعادلة هذه في بي فهيطلع
+
+24
+00:03:28,210 --> 00:03:38,250
+بي بساوي واحدعلى واحد زائد ال a وبالتالي
+
+25
+00:03:38,250 --> 00:03:49,150
+so by ال binomial باستخدام
+
+26
+00:03:49,150 --> 00:03:59,120
+ال binomial theorem انا عنديواحد زائد a الكل قص n
+
+27
+00:03:59,120 --> 00:04:09,300
+بيساوي واحد زائد n في a زائد نص n في n سالب واحد
+
+28
+00:04:09,300 --> 00:04:17,800
+في a تردية زائد و هكذا تمام
+
+29
+00:04:17,800 --> 00:04:24,890
+إلى آخر حد طبعا هيكون a to nهذا بالظبط زي ما عملنا
+
+30
+00:04:24,890 --> 00:04:31,950
+في مثال سادة وبالتالي
+
+31
+00:04:31,950 --> 00:04:42,090
+هذا بيقدي من هنا هذا
+
+32
+00:04:42,090 --> 00:04:51,620
+المجموعة بيطلع أكبر من أو ساوي نصفي n في n زي سالب
+
+33
+00:04:51,620 --> 00:05:00,000
+واحد في ايه ترميها يعني أنا أخدت بس الحد التالت من
+
+34
+00:05:00,000 --> 00:05:05,120
+المجموعة ده المجموعة طبعا مجموعة أعداد موجة بكلها
+
+35
+00:05:05,120 --> 00:05:10,180
+فالمجموعة ده بالتأكيد أكبر من أو ساوي الحد التالت
+
+36
+00:05:10,180 --> 00:05:14,740
+في ايه هذا صحيح مافي مشكلة تمام
+
+37
+00:05:17,950 --> 00:05:33,450
+وبالتالي اذا n في b أس n ايش بيساوي بيساوي n على
+
+38
+00:05:33,450 --> 00:05:43,480
+واحد زائد a لكل أس n صح؟هذه B ف B أس N بساوي واحد
+
+39
+00:05:43,480 --> 00:05:50,900
+على واحد زاد A to N و أضرب في N فبصير هيك طيب
+
+40
+00:05:50,900 --> 00:05:58,560
+من هنا مقلوب واحد زاد A لكل أس N هيطلع أصغر من أوي
+
+41
+00:05:58,560 --> 00:06:09,580
+ساوي مقلوب العدد هذا اذا هذا أصغر من أوي ساوي N
+
+42
+00:06:14,820 --> 00:06:20,480
+على N في
+
+43
+00:06:20,480 --> 00:06:28,380
+N سالب واحد في .. في N سالب واحد في ايه تربية على
+
+44
+00:06:28,380 --> 00:06:38,980
+اتنين وفي عندنا كمان N العكس
+
+45
+00:06:38,980 --> 00:06:39,600
+العكس
+
+46
+00:06:46,180 --> 00:06:55,020
+هي عندى n ومقلوب هدا بطلع اتنين n في n سالب واحد
+
+47
+00:06:55,020 --> 00:07:01,760
+في a تربية تمام؟ إذا هدا إيجا من هنا الان بختصر ال
+
+48
+00:07:01,760 --> 00:07:12,620
+n مع ال n فهدا بطلع اتنين على n سالب واحد في a
+
+49
+00:07:12,620 --> 00:07:14,040
+تربية تمام؟
+
+50
+00:07:16,500 --> 00:07:23,140
+الان هذا الكلام صحيح لكل n أكبر من واحد طبعا ممنوع
+
+51
+00:07:23,140 --> 00:07:27,100
+أخد n بساوي واحد لأنه في الحالة هذه بيصير في قسم
+
+52
+00:07:27,100 --> 00:07:31,780
+على سفر لأن لكل الأعداد الطبيعية n أكبر من واحد n
+
+53
+00:07:31,780 --> 00:07:36,980
+في bios n بيطلع أصغر منه يساوي اثنين على n سالب
+
+54
+00:07:36,980 --> 00:07:44,420
+واحد في a تربية الان تعالوا نثبت ان ال limit لل
+
+55
+00:07:44,420 --> 00:07:46,240
+sequence هذه بساوي سفر
+
+56
+00:07:50,790 --> 00:07:57,390
+هنستخدم تعريف epsilon capital N لـ limit لذن let
+
+57
+00:07:57,390 --> 00:08:02,090
+epsilon let
+
+58
+00:08:02,090 --> 00:08:12,830
+epsilon أكبر من سبل be given by
+
+59
+00:08:12,830 --> 00:08:19,210
+Archimedean property by Archimedeanproperty حسب
+
+60
+00:08:19,210 --> 00:08:25,750
+خاصية Archimedes يوجد نقدر نلاقي عدد طبيعي capital
+
+61
+00:08:25,750 --> 00:08:34,530
+N ينتمي إلى N يعتمد طبعا على إبسلون بحيث أنه مقلوب
+
+62
+00:08:34,530 --> 00:08:41,590
+capital N أصغر من A تربية في إبسلون على 2
+
+63
+00:08:49,130 --> 00:08:54,230
+الـ A تربي عدد موجب إبسلون على 2 عدد موجب إذا هذا
+
+64
+00:08:54,230 --> 00:09:00,830
+عدد موجب الـ Archimedean property بتقول لأي عدد
+
+65
+00:09:00,830 --> 00:09:05,450
+موجب زي هذا بقدر ألاقي عدد طبيعي capital N مقلوب
+
+66
+00:09:05,450 --> 00:09:09,250
+وأصغر من العدد الموجب وبالتالي capital N هذا زي ما
+
+67
+00:09:09,250 --> 00:09:13,510
+أنتوا شايفين مرتبط بإبسلون بالمتباينة هذه وبالتالي
+
+68
+00:09:13,510 --> 00:09:18,970
+capital N هذا depends أو يعتمد على إبسلونOkay إذا
+
+69
+00:09:18,970 --> 00:09:22,510
+هذا من الـ Archimedean Property طب ليش أنا أختارت
+
+70
+00:09:22,510 --> 00:09:29,930
+هذا العدد عشان نخل المسافة بين Xm و 0 أصغر من Y
+
+71
+00:09:29,930 --> 00:09:38,290
+فرطبناها أو ركبناها عساس نصل لإيه الهدف هذا تعالى
+
+72
+00:09:38,290 --> 00:09:46,950
+نشوف إذا hence وبالتالي hence بناء على ذلكلو أخدت
+
+73
+00:09:46,950 --> 00:09:57,950
+n أكبر من capital N هذا بيقدي ان n سالب واحد أكبر
+
+74
+00:09:57,950 --> 00:10:08,270
+من أو ساوي capital N وهذا بيقدي ان absolute n في b
+
+75
+00:10:08,270 --> 00:10:16,690
+to n سالب صفرإيش هذا بيساوي بيساوي n في بيقص n لأن
+
+76
+00:10:16,690 --> 00:10:26,710
+هذا عدد موجب ومن هنا من هنا n في بيقص n أصغر من أو
+
+77
+00:10:26,710 --> 00:10:33,690
+يساوي اتنين على n
+
+78
+00:10:33,690 --> 00:10:40,750
+سالب واحد في a تربية وهذا
+
+79
+00:10:40,750 --> 00:10:42,390
+أصغر من أو يساوي
+
+80
+00:10:50,580 --> 00:11:00,000
+هذا أصغر من أوي ساوي واحد على capital M في اتنين
+
+81
+00:11:00,000 --> 00:11:11,660
+على A تربية يعني
+
+82
+00:11:11,660 --> 00:11:19,140
+أنا من هنا منواحد على N سالب واحد مقلوب N سالب
+
+83
+00:11:19,140 --> 00:11:27,000
+واحد هيطلع أعظم أسئلة مقلوب capital N وهذا
+
+84
+00:11:27,000 --> 00:11:33,400
+عبارة عن واحد على N سالب واحد اتنين على A تربية
+
+85
+00:11:36,970 --> 00:11:42,010
+فمقلوب n سالب واحد اصغر من او ساوي مقلوب capital N
+
+86
+00:11:42,010 --> 00:11:51,630
+في اتنين على ا تربية تمام؟ شفتوا من اين اتيت؟
+
+87
+00:11:51,630 --> 00:11:58,690
+طيب انا من هنا من هنا واحد مقلوب capital N اصغر من
+
+88
+00:11:58,690 --> 00:12:09,470
+ا تربية في ابسلون على اتنين ضربتنين على اي تربية
+
+89
+00:12:09,470 --> 00:12:13,630
+اذا شوفت ليه اخدت ان هنا اي تربية في ابسلون على
+
+90
+00:12:13,630 --> 00:12:19,210
+اتنين عشان اختصر اي تربية مع اي تربية واتنين مع
+
+91
+00:12:19,210 --> 00:12:26,870
+اتنين ويبقى ابسلون اذا
+
+92
+00:12:26,870 --> 00:12:35,170
+ماذا اثبتنا اثبتت انه لأي given ابسلون عدد موجب
+
+93
+00:12:35,980 --> 00:12:42,520
+يوجد capital N تعتمد على epsilon بحيث لكل N أكبر
+
+94
+00:12:42,520 --> 00:12:48,260
+من capital N طلع عندي المسافة بين الحد العام لل
+
+95
+00:12:48,260 --> 00:12:51,900
+sequence اللي هو N في BN وال limit المنشودة اللي
+
+96
+00:12:51,900 --> 00:12:57,860
+هي سفر المسافة بينهم طلعت أصغر من epsilonإذا حسب
+
+97
+00:12:57,860 --> 00:13:03,100
+تعريف epsilon capital N لل limit هذا معناه أنه ال
+
+98
+00:13:03,100 --> 00:13:06,720
+limit بما أن هذا صحيح لأي epsilon، epsilon was
+
+99
+00:13:06,720 --> 00:13:11,540
+arbitrary إذاً هيك ممكن أثبتنا إن limit N في B to
+
+100
+00:13:11,540 --> 00:13:16,760
+N as N tends to infinity بساوي سفر وهو المطلوب
+
+101
+00:13:16,760 --> 00:13:22,640
+okay تمام؟ إذاً
+
+102
+00:13:22,640 --> 00:13:27,950
+هنا استخدمنا ال binomial theorem ساعدتنيفي الوصول
+
+103
+00:13:27,950 --> 00:13:33,590
+للمتباينة هذه و الوصول للمتباينة هذه اللي احنا
+
+104
+00:13:33,590 --> 00:13:42,730
+استخدمناها في البرهان سهلة البرهان تمام بفهم
+
+105
+00:13:42,730 --> 00:13:45,970
+الخطوة هذه اقول ان ال limit يعني اخد ال limit
+
+106
+00:13:45,970 --> 00:13:49,750
+للترفين اقول ان واحد على n نقص الواحد ماهي cost
+
+107
+00:13:49,750 --> 00:13:55,840
+zero اذا ال limit المقدراتمن أنهي المتباينة؟ هذه؟
+
+108
+00:13:55,840 --> 00:14:02,160
+بنفع اه بنفع يعني انت عندك هنا ممكن واحد يستخدم ال
+
+109
+00:14:02,160 --> 00:14:08,980
+sandwich او ال squeeze theorem فبدل ما نستخدم
+
+110
+00:14:08,980 --> 00:14:15,680
+تعريف epsilon capital N نيجي نقول انه الان انا
+
+111
+00:14:15,680 --> 00:14:25,150
+عندي هاي Nفي b to n طلعت أصغر من أو ساوي اتنين على
+
+112
+00:14:25,150 --> 00:14:31,450
+n سالب واحد في a تردية وطبعا بالتأكيد هذا أكبر من
+
+113
+00:14:31,450 --> 00:14:35,390
+أو ساوي سفر لأن ال n عدد موجب و ال b to n عدد موجب
+
+114
+00:14:35,390 --> 00:14:43,530
+وهذا صحيح لكل n أكبر من واحدالان هذا عبارة عن
+
+115
+00:14:43,530 --> 00:14:47,410
+sequence هي الحد العام تبعها لما N تقول ل infinity
+
+116
+00:14:47,410 --> 00:14:52,230
+مقلوق N سالب واحد تقول ل infinity وبالتالي مقلوبها
+
+117
+00:14:52,230 --> 00:14:55,990
+يقول ل infinity فى ثابت موجة باتنين على A تربية
+
+118
+00:14:55,990 --> 00:15:01,570
+عفوا لما N تقول ل infinity المقام بيروح ل infinity
+
+119
+00:15:01,570 --> 00:15:07,110
+وبالتالي مقلوب وبروح ل سفر تمام؟
+
+120
+00:15:16,990 --> 00:15:22,190
+إذن هذه الـ sequence تقول لـ 0 نهايتها 0 وهذه ال
+
+121
+00:15:22,190 --> 00:15:26,970
+constant sequence 0 نهايتها 0 إذن by squeeze
+
+122
+00:15:26,970 --> 00:15:30,410
+theorem limit ال sequence هذه بيساوي 0 وبلاش
+
+123
+00:15:30,410 --> 00:15:35,350
+نستخدم تعريف epsilon capital N لكن هذا السؤال في
+
+124
+00:15:35,350 --> 00:15:39,750
+section 3-1 ماكناش ماخدين ال squeeze theorem فلازم
+
+125
+00:15:39,750 --> 00:15:43,770
+اتحاليها على طريقة باستخدام ال definition لكن لو
+
+126
+00:15:43,770 --> 00:15:48,910
+في الامتحانو ممكن ماتفرجش انت متعلم ال definition
+
+127
+00:15:48,910 --> 00:15:52,630
+و متعلم ال exquisite theorem و استخدم أي طريقة
+
+128
+00:15:52,630 --> 00:15:58,330
+تعجبك okay تمام في
+
+129
+00:15:58,330 --> 00:16:01,010
+أسئلة تانية في حد عنده أي سؤال تاني في section
+
+130
+00:16:01,010 --> 00:16:07,890
+تلاتة واحد و تلاتة اتنين تفضلي في أي section تلاتة
+
+131
+00:16:07,890 --> 00:16:10,510
+واحد طيب ماشي الحالة
+
+132
+00:16:50,410 --> 00:17:09,310
+السؤال عشرة section تلاتة واحد السؤال هذا بيقول if
+
+133
+00:17:09,310 --> 00:17:20,060
+limit sequence x in بساوي xوالـ X هذا أكبر من
+
+134
+00:17:20,060 --> 00:17:24,880
+السفر then
+
+135
+00:17:24,880 --> 00:17:29,340
+then
+
+136
+00:17:29,340 --> 00:17:36,780
+there exist يوجد capital N عدد طبيعي او capital M
+
+137
+00:17:36,780 --> 00:17:48,170
+natural number عدد طبيعي such thatxn أكبر من السفر
+
+138
+00:17:48,170 --> 00:18:08,950
+لكل n أكبر من أو ساوي م لت
+
+139
+00:18:08,950 --> 00:18:13,250
+y أكبر من السفر be given
+
+140
+00:18:17,620 --> 00:18:23,600
+خد أي إبسلون أكبر من الصفر إذن
+
+141
+00:18:23,600 --> 00:18:30,900
+إبسلون على اتنين برضه بيطلع عدد موجة طيب
+
+142
+00:18:30,900 --> 00:18:38,880
+احنا فرضين ان limit xn بيساوي x إذن since xn
+
+143
+00:18:38,880 --> 00:18:44,960
+converges to xوهي إبسلون على اتنين عدد أكبر من
+
+144
+00:18:44,960 --> 00:18:54,480
+السفر إذا يوجد capital M عدد طبيعي يعتمد على
+
+145
+00:18:54,480 --> 00:18:58,840
+إبسلون عدد
+
+146
+00:18:58,840 --> 00:19:05,140
+طبيعي بحيث أنه لكل N أكبر من أو ساوي capital M
+
+147
+00:19:05,140 --> 00:19:33,800
+تطلع المسافة من XNهو ال X أصغر من Y أتنين طيب
+
+148
+00:19:33,800 --> 00:19:34,980
+أنا ال epsilon هذا
+
+149
+00:19:37,520 --> 00:19:44,200
+ممكن اخده انا عندي من الفرض x اكبر من 0 فممكن اخد
+
+150
+00:19:44,200 --> 00:19:49,640
+ال epsilon هذا بساوي x بساوي
+
+151
+00:19:49,640 --> 00:19:56,480
+x انا
+
+152
+00:19:56,480 --> 00:20:04,400
+ممكن اخد ال epsilon بساوي x او حتى x ع 2 او x ع 2
+
+153
+00:20:04,400 --> 00:20:10,380
+هذا بالتأكيدالإبسلون هذا هي عدد موجب اعتبره هو
+
+154
+00:20:10,380 --> 00:20:15,660
+given وبالتالي
+
+155
+00:20:15,660 --> 00:20:20,580
+انا اخدت الأن إبسلون X عدد موجب إذا X عتنين عدد
+
+156
+00:20:20,580 --> 00:20:26,270
+موجب واخدت إبسلون عبارة عن X عتنينفاعتبر هذا given
+
+157
+00:20:26,270 --> 00:20:31,070
+إبسلون إبسلون مُعطى مُسبَخًا فحسب التعريف بما أن X
+
+158
+00:20:31,070 --> 00:20:34,390
+in converge ل X إذا يوجد عدد طبيعي يعتمد على
+
+159
+00:20:34,390 --> 00:20:38,710
+إبسلون بحيث لكل in أكبر من أو ساوي capital N
+
+160
+00:20:38,710 --> 00:20:45,730
+المسافة هذه أصغر من إبسلون الآن عوض عن إبسلون
+
+161
+00:20:45,730 --> 00:20:54,490
+بساوي X ع 2 فهذا بيؤديالان فك ال absolute value
+
+162
+00:20:54,490 --> 00:21:03,070
+فبطلع عندي xn سالب x أصغر من x ع 2 أكبر من سالب x
+
+163
+00:21:03,070 --> 00:21:08,570
+ع 2، مظبوط؟
+
+164
+00:21:08,570 --> 00:21:15,370
+طب
+
+165
+00:21:15,370 --> 00:21:17,790
+لو أخدت هذا الجزء من المتباينة
+
+166
+00:21:20,790 --> 00:21:28,770
+فبصير عندي xn أكبر من ودي x على الناحية التالية
+
+167
+00:21:28,770 --> 00:21:38,050
+أكبر من x سالب x على 2 وبالتالي
+
+168
+00:21:38,050 --> 00:21:46,710
+إذا أنا عندي هاي xn أكبر من x على 2 وهذا أكبر من
+
+169
+00:21:46,710 --> 00:21:57,210
+السفرتمام؟ وهذا صحيح إذا طلع عندي xn أكبر من السفر
+
+170
+00:21:57,210 --> 00:22:07,170
+وهذا صحيح لكل n أكبر من أو ساوي capital M وهو
+
+171
+00:22:07,170 --> 00:22:12,630
+المطلوبتمام إذا هنا استخدمنا تعريف epsilon capital
+
+172
+00:22:12,630 --> 00:22:19,690
+M وهنا استنتجنا إن لازم xn يطلع أكبر من السفر لكل
+
+173
+00:22:19,690 --> 00:22:32,210
+M أكبر من أو يساوي capital M تمام واضح البرهان طيب
+
+174
+00:22:32,210 --> 00:22:34,110
+في أي أسئلة تانية؟
+
+175
+00:22:37,830 --> 00:22:48,330
+section تلاتة اتنين مين
+
+176
+00:22:48,330 --> 00:22:54,390
+عنده سؤال اي سؤال في اي section تلاتة اتنين تلاتة
+
+177
+00:22:54,390 --> 00:23:03,070
+اتنين سبعتاش
+
+178
+00:23:03,070 --> 00:23:05,150
+section تلاتة اتنين
+
+179
+00:23:40,180 --> 00:23:44,200
+أنا في عندي هنا sequence of positive real numbers
+
+180
+00:23:44,200 --> 00:23:55,680
+إذا xn حدود عموجة بقى لكل n such
+
+181
+00:23:55,680 --> 00:24:00,560
+that limit ل
+
+182
+00:24:00,560 --> 00:24:11,550
+xn زاد واحد على xn لما n تقول infinityبساوي عدد ال
+
+183
+00:24:11,550 --> 00:24:20,550
+أكبر من واحد و المقلوب show اثبت في الحالة هذه ان
+
+184
+00:24:20,550 --> 00:24:25,750
+ال sequence
+
+185
+00:24:25,750 --> 00:24:30,170
+xm is
+
+186
+00:24:30,170 --> 00:24:34,350
+unbounded is not bounded
+
+187
+00:24:38,480 --> 00:24:46,100
+and hence not
+
+188
+00:24:46,100 --> 00:24:53,460
+convergent لأن لو كانت convergent تطلع bounded
+
+189
+00:25:13,370 --> 00:25:17,190
+يعني من الشرط هذا ممكن تباطم الـ sequence
+
+190
+00:25:17,190 --> 00:25:21,290
+increasing متزايدة
+
+191
+00:26:05,950 --> 00:26:08,750
+أه ..
+
+192
+00:26:31,500 --> 00:26:38,240
+ممكن نعمل برهان بال .. بالتناقض افرم
+
+193
+00:26:38,240 --> 00:26:48,640
+انها bounded وممكن نصل لتناقض من تعريف ال .. هنا
+
+194
+00:26:48,640 --> 00:26:56,680
+ال sequence هذه of quotient convergent لعدد L أكبر
+
+195
+00:26:56,680 --> 00:27:00,220
+من واحد ممكن باستخدامه
+
+196
+00:27:02,850 --> 00:27:14,250
+باستخدام تعريف الـ convergence زاد او
+
+197
+00:27:14,250 --> 00:27:18,390
+ممكن من الفرض هذا لثبت انه ال sequence unbounded
+
+198
+00:27:18,390 --> 00:27:22,870
+او ممكن بالتناقض اما باستخدام تعريف epsilon
+
+199
+00:27:22,870 --> 00:27:29,600
+capital N من ال convergence هذانعمل برهان بالتناقض
+
+200
+00:27:29,600 --> 00:27:35,560
+لنصل إلى هاجة يعني تتناقض مع الفرض اللي هنا على أي
+
+201
+00:27:35,560 --> 00:27:40,540
+حال انا هسيب في حد حل السؤال هذا طيب انا هسيبكم
+
+202
+00:27:40,540 --> 00:27:45,320
+تفكروا فيه و تقرؤوا برهان شوفوا برهان انا في
+
+203
+00:27:45,320 --> 00:27:49,380
+البرهان النظرية هذه اللي كانت قلتلكم اقرؤوا
+
+204
+00:27:49,380 --> 00:27:54,650
+فحاولوا انك تتسفيدوا من البرهان تبع النظريةاللي
+
+205
+00:27:54,650 --> 00:27:57,930
+كانت بتقول إن لو كانت ال limit هذه بساوي L أصغر من
+
+206
+00:27:57,930 --> 00:28:03,370
+واحد فبتطلع ال sequence convergent للصفر فإقرأوا
+
+207
+00:28:03,370 --> 00:28:08,710
+البرهان تبع النظرية هذه وشوفوا كيف يعني النظرية
+
+208
+00:28:08,710 --> 00:28:12,750
+هذه أثبتت وشوفوا لو كان ال L أكبر من واحد كيف
+
+209
+00:28:12,750 --> 00:28:17,450
+بيطلع البرهان إيش اللي بيخل البرهان هذا يبطل صحيح
+
+210
+00:28:18,870 --> 00:28:23,230
+أه فعيدوا قراءته و حالكم تحلوه و إذا ما حلتوهوش
+
+211
+00:28:23,230 --> 00:28:27,290
+يعني المرة الجاية ممكن تحلوا مع بعض أه ماشي الحال
+
+212
+00:28:27,290 --> 00:28:30,470
+فإقرأوا
+
+213
+00:28:30,470 --> 00:28:35,150
+برهان النظرية اللي سيبنا قولنالكم البرهانها موجود
+
+214
+00:28:35,150 --> 00:28:38,030
+في الكتاب و بدي أكم تقرأوا تفهموا هل قرأتوا
+
+215
+00:28:38,030 --> 00:28:45,010
+البرهان؟حاولوا تقرأ ايه حاولوا تتعملوا ايه تشوفوا
+
+216
+00:28:45,010 --> 00:28:50,070
+وين في البرهان ال ال اكبر من واحد بتخلي البرهان
+
+217
+00:28:50,070 --> 00:28:55,050
+يبطل صح وين المشكلة وشوفوا
+
+218
+00:28:55,050 --> 00:28:58,210
+اذا كانوا تقدروا تحلو ولا لأ اذا انا هاسيبكم
+
+219
+00:28:58,210 --> 00:29:02,610
+تفكروا فيه مرة تانية و تحاولوا تحلوه اذا ماعرفتهوش
+
+220
+00:29:02,610 --> 00:29:09,110
+ممكن نحله مرة تانية او في المرة القادمة نعم مين
+
+221
+00:29:09,110 --> 00:29:13,190
+اللي بتحكي هذهماحدش لو سمحته تحكي إلا غير ترفع
+
+222
+00:29:13,190 --> 00:29:18,790
+إيدها الأول و بعدين أقزمها طيب إذا هذا السؤال
+
+223
+00:29:18,790 --> 00:29:22,510
+هنسيبكم يتفكروا فيه مرة تانية في أي أسئلة تانية
+
+224
+00:29:22,510 --> 00:29:26,710
+section تلاتة اتنين أو تلاتة واحد
+
+225
+00:29:45,050 --> 00:29:50,450
+في حد عندها سؤال في نفس
+
+226
+00:29:50,450 --> 00:29:55,770
+ال section نعم فالقادة ماعطينا sequence انه احنا
+
+227
+00:29:55,770 --> 00:29:59,390
+نشوف اذا هي تتجوز و لا تتجوز استخدمت ال ratio test
+
+228
+00:29:59,390 --> 00:30:04,310
+نعم طلعت ال limit بتساوي واحد و احنا الشرط ان تكون
+
+229
+00:30:04,310 --> 00:30:09,790
+ال limit اقل من واحد صح فالقادة هذه بتطلع تطلع ال
+
+230
+00:30:09,790 --> 00:30:12,430
+limit ل sequence لو معطنيها تساوي zero
+
+231
+00:30:15,730 --> 00:30:21,110
+لأ لازم يكون أصغر من واحد مابتساويش الواحد معناته
+
+232
+00:30:21,110 --> 00:30:26,150
+ال test بيفشل لأ هي سوى واحد إذا بالساوية واحد
+
+233
+00:30:26,150 --> 00:30:33,430
+ارجعي لهي تمرين 16 بقول إذا كانت ال limit بالساوية
+
+234
+00:30:33,430 --> 00:30:38,710
+واحد فممكن
+
+235
+00:30:38,710 --> 00:30:41,650
+تكون ال sequence convergent أو divergent يعني هذا
+
+236
+00:30:41,650 --> 00:30:46,920
+ال test ال ratio test بيفشلهي في سؤال 16 هتجيب
+
+237
+00:30:46,920 --> 00:30:52,480
+بمثالين اول شي اذا كانت ال limit هذه بالساوي واحد
+
+238
+00:30:52,480 --> 00:30:59,740
+فهتجيب بمثالين ال limit تبع ال quotient تبع كل
+
+239
+00:30:59,740 --> 00:31:03,220
+واحدة بالساوي واحد لكن واحدة convergent واحدة
+
+240
+00:31:03,220 --> 00:31:08,140
+divergent وبالتالي ال test هذا بيفشل اذا كانت ال L
+
+241
+00:31:08,140 --> 00:31:12,420
+بالساوي واحد اما لو كانت ال L اصغر من واحدفال
+
+242
+00:31:12,420 --> 00:31:16,400
+sequence xn تطلع convergent للصفر إذا كان ال L
+
+243
+00:31:16,400 --> 00:31:21,740
+أكبر من 1 فال sequence تطلع divergent okay تمام
+
+244
+00:31:21,740 --> 00:31:30,340
+هذا هو ال ratio test فهل جبت أمثلة؟ كويس ممتاز طيب
+
+245
+00:31:30,340 --> 00:31:36,220
+إيش دخل دي؟ دي معناته بدك تستخدم طريقة تانية غير
+
+246
+00:31:36,220 --> 00:31:43,260
+ال ratio test صحيح لأن حسب سؤال 16الـ test بيفشل
+
+247
+00:31:43,260 --> 00:31:48,320
+إذا كانت limit ال ratio ال ratio test بيفشل إذا
+
+248
+00:31:48,320 --> 00:31:53,020
+كانت limit لل ratio بساوي واحد وبالتالي بدك تبحث
+
+249
+00:31:53,020 --> 00:31:54,300
+عن طريقة تانية
+
+250
+00:32:12,840 --> 00:32:31,940
+طيب في أسئلة تانية في
+
+251
+00:32:31,940 --> 00:32:35,300
+section تلاتة واحد و تلاتة اتنين في عندكم أي سؤال
+
+252
+00:32:35,300 --> 00:32:42,490
+مافيش أسئلة لسه مش دارسين مش محاضرينكان واحدة بس
+
+253
+00:32:42,490 --> 00:32:51,430
+لدرسة و هم اللي بيسألوا الأسئلة والباقي مستمع طيب
+
+254
+00:32:51,430 --> 00:32:54,930
+بتحبوا نرجع لأسئلة chapter اتنين في أسئلة في
+
+255
+00:32:54,930 --> 00:33:01,130
+chapter اتنين اذا
+
+256
+00:33:01,130 --> 00:33:10,470
+في عندكم أسئلة في section اتنين
+
+257
+00:33:10,470 --> 00:33:11,010
+اربعة
+
+258
+00:33:26,250 --> 00:33:35,710
+السؤال هذا يعني في الكتاب أعطيكم hint كيف
+
+259
+00:33:35,710 --> 00:33:41,790
+يعني تحلوه موجود في نهاية الكتاب فحاولوا تقرا أيه
+
+260
+00:33:41,790 --> 00:33:46,530
+تقرا ال hint هذا و تستفيدي منه و تشوفي يعني هذا
+
+261
+00:33:46,530 --> 00:33:54,780
+أكيد هساعدك في حل السؤال شفتيه قبل هيك؟طيب طلعي
+
+262
+00:33:54,780 --> 00:33:59,360
+خلف الكتاب فيه hint او ارشادات لبعض التمرين
+
+263
+00:33:59,360 --> 00:34:06,680
+بيعطيكي يعني طريقة مقتضبة لحل او بحط رجلك على طريق
+
+264
+00:34:06,680 --> 00:34:12,840
+الحل فحاولي تقرا ايه و تستفيدي منه و اذا فهمتي
+
+265
+00:34:12,840 --> 00:34:19,640
+الارشاد هذا ممكن تحل السؤال فانتي و زمايلكتطلعوا
+
+266
+00:34:19,640 --> 00:34:23,580
+على الإرشادات هذه تبعت التمرين أو بعض الحلول
+
+267
+00:34:23,580 --> 00:34:28,240
+المختصرة و حاولوا تستفيدوا منها و تفصلوها و تكتبوا
+
+268
+00:34:28,240 --> 00:34:35,340
+الحل بطريقة واضحة و كاملة فهسيبكم
+
+269
+00:34:35,340 --> 00:34:42,440
+تقرؤوا الإرشاد و تحاولوا تستفيدوا منه أي أسئلة
+
+270
+00:34:42,440 --> 00:34:49,980
+تانية في section 2 4 2 3 2 2إن واحد الجزء اللي
+
+271
+00:34:49,980 --> 00:34:56,460
+داخل الامتحان، في عندكم أي سؤال فيه؟ منين في عندها
+
+272
+00:34:56,460 --> 00:35:00,260
+سؤال؟
+
+273
+00:35:00,260 --> 00:35:07,020
+في أسئلة كتير حلوة ومهمة ويا بدوا أنكم مش مدرسين
+
+274
+00:35:07,020 --> 00:35:08,680
+ولا حتى مستعدين للامتحان
+
+275
+00:35:16,700 --> 00:35:20,800
+في اي اسلة في chapter 2 او chapter 3 الجزء الداخل
+
+276
+00:35:20,800 --> 00:35:21,960
+في الامتحان
+
+277
+00:36:04,610 --> 00:36:11,090
+فيش أسئلة؟ طيب
+
+278
+00:36:11,090 --> 00:36:15,390
+أنا هحللكم يعني كمان سؤالين واحد من section تلاتة
+
+279
+00:36:15,390 --> 00:36:21,070
+واحد وواحد من تلاتة اتنين
+
+280
+00:36:21,070 --> 00:36:28,670
+خليني
+
+281
+00:36:28,670 --> 00:36:29,830
+أحل السؤال
+
+282
+00:36:46,350 --> 00:36:58,770
+يعني مثلا يعني
+
+283
+00:36:58,770 --> 00:37:04,090
+مثلا السؤال الخامسة
+
+284
+00:37:04,090 --> 00:37:10,530
+السؤال
+
+285
+00:37:10,530 --> 00:37:16,320
+الخامسة الفرح دي section تلاتة واحدuse definition
+
+286
+00:37:16,320 --> 00:37:25,660
+use definition of limit to
+
+287
+00:37:25,660 --> 00:37:33,880
+establish انه
+
+288
+00:37:33,880 --> 00:37:37,800
+ال limit لإن
+
+289
+00:37:37,800 --> 00:37:44,970
+تربية سالب واحد علىتنين انتر بيه زائد تلاتة ال
+
+290
+00:37:44,970 --> 00:37:52,850
+sequence اللي حد العم تبعها الكاسر هذا بيساوي نص و
+
+291
+00:37:52,850 --> 00:37:56,410
+بيثبت ان ال sequence هذي convergence و نهايتها نص
+
+292
+00:37:56,410 --> 00:38:00,390
+بيستخدم ال definition ماهو ال definition المقصود
+
+293
+00:38:00,390 --> 00:38:06,700
+في هنااللي هو تعريف epsilon capital N لل limit أو
+
+294
+00:38:06,700 --> 00:38:21,360
+للنهاية تعريف epsilon capital N طيب انا
+
+295
+00:38:21,360 --> 00:38:27,300
+في النهاية في نهاية المطاف تعريف epsilon capital N
+
+296
+00:38:31,470 --> 00:38:36,710
+عايزني أثبت أن المسافة بين xn اللي هو enter بيها
+
+297
+00:38:36,710 --> 00:38:42,510
+سالب واحد على اتنين enter بيها زائد تلاتة سالب نص
+
+298
+00:38:42,510 --> 00:38:47,270
+بدنا هذا يكون أصغر من أي given epsilon عدد موجه
+
+299
+00:38:47,270 --> 00:38:53,950
+لكل n أكبر من أو ساوي capital N حيث capital N عدد
+
+300
+00:38:53,950 --> 00:39:00,410
+طبيعي هنجيبه ويعتمد على ال epsilonفنشوف مع بعض هذا
+
+301
+00:39:00,410 --> 00:39:07,410
+إيه من الآخر طيب إذا هنا solution إذا بقول أنا
+
+302
+00:39:07,410 --> 00:39:12,490
+عايز في النهاية absolute interview سالب واحد على
+
+303
+00:39:12,490 --> 00:39:17,970
+اتنين interview زائد تلاتة سالب مصر بسأل نفسي متى
+
+304
+00:39:17,970 --> 00:39:24,570
+هذا بيكون أصغر من أي epsilon موجب هذا بكافئ
+
+305
+00:39:27,220 --> 00:39:34,160
+الـ absolute value بين واحد المقامات هي اتنين في
+
+306
+00:39:34,160 --> 00:39:40,260
+اتنين انت ربيع الزائد تلاتة و بيصير عندنا اتنين
+
+307
+00:39:40,260 --> 00:39:46,720
+انت ربيع سالب اتنين تضرب هذا في اتنين سالب اتنين
+
+308
+00:39:46,720 --> 00:39:53,680
+انت ربيع موجة بتلاتة لان هذا المقدار اللي فوق
+
+309
+00:39:53,680 --> 00:39:55,580
+بيبقى اصغر من epsilon
+
+310
+00:39:58,630 --> 00:40:02,730
+طيب أنا عندي اتنين in تربية و هاي سالب اتنين in
+
+311
+00:40:02,730 --> 00:40:07,450
+تربية بروحوا مع بعض و عندي سالب اتنين و السالب
+
+312
+00:40:07,450 --> 00:40:10,630
+تلاتة بطلع خمسة يعني دلوقتي بصير absolute سالب
+
+313
+00:40:10,630 --> 00:40:16,330
+خمسة على اتنين في
+
+314
+00:40:16,330 --> 00:40:19,230
+اتنين in تربية زائد تلاتة
+
+315
+00:40:24,890 --> 00:40:31,830
+بدي هذا يكون أصغر من ي طيب
+
+316
+00:40:31,830 --> 00:40:38,990
+هاد عبارة عن خمسة هاد
+
+317
+00:40:38,990 --> 00:40:48,570
+عبارة عن خمسة على اتنين اتنين enter بيها زي
+
+318
+00:40:48,570 --> 00:40:49,370
+التلاتة
+
+319
+00:40:52,780 --> 00:41:02,080
+متى بيكون هذا أصغر من epsilon هذا
+
+320
+00:41:02,080 --> 00:41:09,220
+بكافئ هذا
+
+321
+00:41:09,220 --> 00:41:15,900
+بكافئ ان اقول واحد متى بيكون واحد على اتنين انتر
+
+322
+00:41:15,900 --> 00:41:30,390
+بيه زائد تلاتة أصغر منإتنين على خمسة إبسلون طيب
+
+323
+00:41:30,390 --> 00:41:35,550
+إذا
+
+324
+00:41:35,550 --> 00:41:42,470
+أنا ممكن أستخدم ال Archimedean property إذا هنا
+
+325
+00:41:42,470 --> 00:41:49,690
+let إبسلون أكبر من السفر بجبل
+
+326
+00:41:51,720 --> 00:41:57,880
+نبدأ بأبسلون أكبر من السفر تعريف epsilon capital N
+
+327
+00:41:57,880 --> 00:42:02,160
+بيقول ابدا بأبسلون أكبر من السفر و جيب capital N
+
+328
+00:42:03,440 --> 00:42:07,880
+بحيث أن المسافة بين XN و X أصغر من إمسون لكل N
+
+329
+00:42:07,880 --> 00:42:15,440
+أكبر من ما يستوى capital N بحيث أن المسافة بين XN
+
+330
+00:42:15,440 --> 00:42:15,440
+و X أصغر من إمسون لكل N أكبر من ما يستوى capital N
+
+331
+00:42:15,440 --> 00:42:17,600
+بحيث أن المسافة بين XN و X أصغر من إمسون لكل N
+
+332
+00:42:17,600 --> 00:42:17,940
+أكبر من ما يستوى capital N بحيث أن المسافة بين XN
+
+333
+00:42:17,940 --> 00:42:20,660
+و X أصغر من إمسون لكل N أكبر من ما يستوى capital N
+
+334
+00:42:20,660 --> 00:42:21,360
+بحيث أن المسافة بين XN و X أصغر من إمسون لكل N
+
+335
+00:42:21,360 --> 00:42:23,640
+أكبر من ما يستوى capital N بحيث أن المسافة بين XN
+
+336
+00:42:23,640 --> 00:42:28,200
+و X أصغر
+
+337
+00:42:28,200 --> 00:42:35,040
+من إمسون لكل N أكبر من ما يستوى capital N بحit
+
+338
+00:42:35,040 --> 00:42:43,620
+choose it choose طبعا
+
+339
+00:42:43,620 --> 00:42:51,500
+by Archimedean property capital
+
+340
+00:42:51,500 --> 00:43:01,200
+N عدد طبيعي بحيث انه واحد علىإتنين في capital N
+
+341
+00:43:01,200 --> 00:43:07,820
+تربية زائد تلاتة أصغر من اتنين على خمسة epsilon
+
+342
+00:43:07,820 --> 00:43:20,180
+ممكن
+
+343
+00:43:20,180 --> 00:43:26,070
+ألاقي capital N عدد طبيعيمقنوب 2 في مربع زائد
+
+344
+00:43:26,070 --> 00:43:32,170
+تلاتة طبعا تلاتة مش epsilon واحد على اتنين enter
+
+345
+00:43:32,170 --> 00:43:41,290
+key زائد تلاتة اصغر من اتنين على خمسة epsilon الان
+
+346
+00:43:41,290 --> 00:43:46,770
+اذا لو اخدت small n اكبر من أوسع ال capital N هذا
+
+347
+00:43:46,770 --> 00:44:00,110
+بيقدي انه واحد علىتنين انت ربيع زائد تلاتة او بلاش
+
+348
+00:44:00,110 --> 00:44:09,230
+absolute اه بيقدي ان absolute طيب
+
+349
+00:44:09,230 --> 00:44:16,750
+هذا بيقدي ان الكلام هذا اصغر من او يساوي واحد على
+
+350
+00:44:16,750 --> 00:44:25,510
+اتنين capital enter بيه زائد تلاتةوبالتالي هذا
+
+351
+00:44:25,510 --> 00:44:31,390
+بيقدي ان ال absolute value لان تربية سالب واحد على
+
+352
+00:44:31,390 --> 00:44:42,670
+اتنين ان تربية سالب تلاتة سالب نص طلع هذا
+
+353
+00:45:09,580 --> 00:45:16,580
+خمسة على اتنين
+
+354
+00:45:16,580 --> 00:45:20,140
+في اتنين enter بي عزائى التلاتة
+
+355
+00:45:28,400 --> 00:45:34,680
+وهذا هيطلع أصغر منه ويسوي خمسة على اتنين في اتنين
+
+356
+00:45:34,680 --> 00:45:42,000
+capital Interbias زاد تلاتة ومن هنا هذا أصغر من
+
+357
+00:45:42,000 --> 00:45:47,320
+خمسة
+
+358
+00:45:47,320 --> 00:45:54,280
+على اتنين ضرب اتنين على خمسة في epsilon اللي هو
+
+359
+00:45:54,280 --> 00:45:55,160
+بيطلع epsilon
+
+360
+00:45:59,840 --> 00:46:03,560
+أذن هذه لأي epsilon أكبر من صفر لجيت فيه capital N
+
+361
+00:46:03,560 --> 00:46:08,200
+مرتبطة لcapital N هي في epsilon depends on epsilon
+
+362
+00:46:08,200 --> 00:46:12,280
+بتعتمد على epsilon بحيث لكل n أكبر من او سوى
+
+363
+00:46:12,280 --> 00:46:17,920
+capital N طلع absolute xn minus x أصغر من epsilon
+
+364
+00:46:19,350 --> 00:46:24,350
+طبعا إذا هذا حسب تعريف by definition of epsilon
+
+365
+00:46:24,350 --> 00:46:29,770
+capital N of limit بطلع عندي limit N تربيع سالب
+
+366
+00:46:29,770 --> 00:46:34,750
+واحد على اتنين N تربيع زائد تلاتة لما N تقول
+
+367
+00:46:34,750 --> 00:46:37,830
+infinity بساوي نص
+
+368
+00:46:44,620 --> 00:46:48,560
+بالمثل ممكن نحل باقى التمرين اللى هى الفروع A وB
+
+369
+00:46:48,560 --> 00:46:54,940
+وC باستخدام التعريف فحاولوا تتدربوا على التمرين
+
+370
+00:46:54,940 --> 00:47:02,700
+هادى و تحلوا أسئلة زيها فى حد عنده أي سؤال تانى فى
+
+371
+00:47:02,700 --> 00:47:07,260
+هذا ال section طيب
+
+372
+00:47:07,260 --> 00:47:12,220
+نحل كمان سؤال فى section تلاتة اتنين
+
+373
+00:47:27,570 --> 00:47:34,750
+في انكم أي سؤال بسكتشن تلاتة اتنين اخر
+
+374
+00:47:34,750 --> 00:47:35,250
+سؤال
+
+375
+00:47:57,080 --> 00:48:03,480
+هي سؤال واحد وعشرين section تلاتة
+
+376
+00:48:03,480 --> 00:48:13,760
+اتنين suppose
+
+377
+00:48:13,760 --> 00:48:24,980
+افترضي ان ال sequence x in converge ل x and ال
+
+378
+00:48:24,980 --> 00:48:33,200
+sequence y inand yn is such that is a sequence
+
+379
+00:48:33,200 --> 00:48:40,900
+such that for any epsilon for
+
+380
+00:48:40,900 --> 00:48:46,240
+any epsilon أكبر من السفر يوجد
+
+381
+00:48:46,240 --> 00:48:53,780
+m بحيث يوجد عدد m such that
+
+382
+00:48:56,580 --> 00:49:06,460
+absolute xn minus yn أصغر من إبسلون لكل N أكبر من
+
+383
+00:49:06,460 --> 00:49:14,260
+أو ساو كابتل N فالسؤال
+
+384
+00:49:14,260 --> 00:49:19,060
+does it
+
+385
+00:49:19,060 --> 00:49:22,820
+follow هل
+
+386
+00:49:22,820 --> 00:49:34,030
+ينتج من ذلكهل ال sequence yn تطلع
+
+387
+00:49:34,030 --> 00:49:44,210
+convergent فنشوف
+
+388
+00:49:44,210 --> 00:49:44,930
+مع بعض
+
+389
+00:49:53,440 --> 00:49:59,260
+كمان مرة اندي two sequences واحدة x in واحدة y in
+
+390
+00:49:59,260 --> 00:50:04,280
+ال sequence x in مُعطَى انها convergent to some x
+
+391
+00:50:04,280 --> 00:50:08,880
+إلى عدد ما x ال limit تبقى تاكس وال sequence y in
+
+392
+00:50:08,880 --> 00:50:14,600
+بتحقق الشرط هذا وهو
+
+393
+00:50:14,600 --> 00:50:19,600
+انه لأي epsilon أكبر من سفر في عدد طبيعي حتى هذا
+
+394
+00:50:19,600 --> 00:50:27,790
+عدد طبيعي المفروض يكونبنشر ال number بحيث انه لكل
+
+395
+00:50:27,790 --> 00:50:31,810
+n أكبر من ما يستوى capital N المسافة بين xn وyn
+
+396
+00:50:31,810 --> 00:50:35,510
+أصغر من نفسها هل هذا بيقدم ال sequence yn
+
+397
+00:50:35,510 --> 00:50:40,870
+convergent؟ هنشوف الآن أن فعلا تطلع ال sequence yn
+
+398
+00:50:40,870 --> 00:50:46,130
+convergent ونهايتها هي نفس نهاية ال sequence xn
+
+399
+00:50:46,130 --> 00:50:51,270
+لأن هنا الإجابة yes
+
+400
+00:50:53,550 --> 00:51:01,270
+and y in converge to x لكن
+
+401
+00:51:01,270 --> 00:51:07,570
+هذا بيده برهان اذا
+
+402
+00:51:07,570 --> 00:51:11,370
+to see this
+
+403
+00:51:11,370 --> 00:51:16,610
+نبدأ
+
+404
+00:51:16,610 --> 00:51:18,610
+بإبسلون أكبر من السفر
+
+405
+00:51:36,810 --> 00:51:44,450
+let by hypothesis من الفرض من
+
+406
+00:51:44,450 --> 00:51:50,820
+الفرض من ال hypothesisأنا عندي absolute xn minus
+
+407
+00:51:50,820 --> 00:51:54,860
+yn أصغر
+
+408
+00:51:54,860 --> 00:52:03,700
+من إبسلون أكبر من أو ساوي سفر وهذا صحيح لكل n أكبر
+
+409
+00:52:03,700 --> 00:52:10,440
+من أو ساوي capital M وهذا
+
+410
+00:52:10,440 --> 00:52:15,380
+الكلام صحيح لكل إبسلون أكبر من السفر
+
+411
+00:52:24,820 --> 00:52:36,980
+فمن هنا فمن
+
+412
+00:52:36,980 --> 00:52:45,680
+هنا بهدف بيقدي ان ال limit ل xn minus yn لما n
+
+413
+00:52:45,680 --> 00:52:49,420
+تقول infinity بساوي سفر
+
+414
+00:52:54,150 --> 00:52:58,570
+مش شرط هذا انا
+
+415
+00:52:58,570 --> 00:53:03,950
+عندي ال ..
+
+416
+00:53:03,950 --> 00:53:08,010
+ما معناه ان limit ال sequence هذه بساوة سفر؟ معناه
+
+417
+00:53:08,010 --> 00:53:16,620
+لأي epsilon أكبر من السفر يوجد capital Mعدد طبيعي
+
+418
+00:53:16,620 --> 00:53:21,840
+يعتمد على إبسلن بحيث أنه لكل n أكبر من أو ساوي
+
+419
+00:53:21,840 --> 00:53:28,860
+capital N هذا بيقدي أن absolute xn minus yn minus
+
+420
+00:53:28,860 --> 00:53:34,700
+الصفر أصغر من إبسلنهي معنى ان limit ال sequence
+
+421
+00:53:34,700 --> 00:53:40,740
+للفرق بساوي سفر ايش معنى هذا لأي epsilon أكبر من
+
+422
+00:53:40,740 --> 00:53:46,660
+سفر يوجد capital M يعتمد على N عدد طبيعي يعتمد على
+
+423
+00:53:46,660 --> 00:53:51,020
+ال epsilon بحيث لكل N أكبر من أو ساوي capital N
+
+424
+00:53:51,020 --> 00:53:55,540
+المسافة بين الحد العام لل sequence و limit اللي هي
+
+425
+00:53:55,540 --> 00:54:00,140
+سفر أصغر من epsilon هذا الكلام هى متحقق هنا هى
+
+426
+00:54:00,140 --> 00:54:04,850
+متحققتامام؟ إذا هذا بنحصل عليه وبالتالي limit xn
+
+427
+00:54:04,850 --> 00:54:14,070
+minus yn بساوي سفر ومنها الآن أنا عندي ال yn ممكن
+
+428
+00:54:14,070 --> 00:54:20,870
+كتبتها على صورة yn
+
+429
+00:54:20,870 --> 00:54:32,610
+سالب xn موجب xnوهذا بيساوي سالب Xn سالب Yn زاد Xn
+
+430
+00:54:32,610 --> 00:54:40,630
+تمام؟ إذا ال limit ل Yn as n tends to infinity
+
+431
+00:54:40,630 --> 00:54:49,110
+بيساوي limit الطرف اليمين ف limit Xn سالب Yn
+
+432
+00:54:49,110 --> 00:54:56,410
+مضروبة في سالب واحد بيطلع برا ال limitزائد limit
+
+433
+00:54:56,410 --> 00:55:03,770
+xn لما n تقول لإنفينيتي وهنا
+
+434
+00:55:03,770 --> 00:55:08,770
+لسه احنا مثبتين هذا عبارة عن سالب limit sequence
+
+435
+00:55:08,770 --> 00:55:16,570
+xn minus yn بالساوية سفر، سالد واحد في سفر
+
+436
+00:55:19,990 --> 00:55:26,850
+زاد limit xn اللي هي x تمام اذا limit ال sequence
+
+437
+00:55:26,850 --> 00:55:32,370
+yn تطلع بالساوي x اذا
+
+438
+00:55:32,370 --> 00:55:37,210
+هنا اثبتنا ان ال sequence yn تطلع convergent وال
+
+439
+00:55:37,210 --> 00:55:44,210
+limit تبعتها بالساوي x تمامالبرهان هنا اعتمد على
+
+440
+00:55:44,210 --> 00:55:49,890
+انه من الفرض انا عندي المثال لأي epsilon هذا الفرض
+
+441
+00:55:49,890 --> 00:55:57,390
+معناه ان limit ال sequence x in minus y in بالساوي
+
+442
+00:55:57,390 --> 00:56:04,290
+سفر وهذا اللي ساعدنا في الحل وهذا ناتج هي من تعريف
+
+443
+00:56:04,290 --> 00:56:09,190
+epsilon capital N لل limit هذا هو البرهان
+
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/YiGM8L9BEY0_postprocess.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/YiGM8L9BEY0_postprocess.srt
new file mode 100644
index 0000000000000000000000000000000000000000..b721c3641efd40dbbe2eee3192f0cadcb68f263f
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/YiGM8L9BEY0_postprocess.srt
@@ -0,0 +1,1332 @@
+1
+00:00:20,920 --> 00:00:26,360
+بسم الله الرحمن الرحيم اليوم هناخد أخر لقاء في ال
+
+2
+00:00:26,360 --> 00:00:31,460
+course وهو تكملة section خمسة أربعة في الكتاب
+
+3
+00:00:31,460 --> 00:00:38,850
+المقرر اللي بتكلم عن ال uniform continuityفي
+
+4
+00:00:38,850 --> 00:00:45,330
+المحاضرة السابقة عرفنا الاتصال المنتظم وشوفنا
+
+5
+00:00:45,330 --> 00:00:49,930
+أثبتنا نظريات
+
+6
+00:00:49,930 --> 00:00:54,170
+مهمة عن الاتصال المنتظم أو عدم الاتصال المنتظم
+
+7
+00:00:54,170 --> 00:00:58,850
+فأخدنا ال non uniform continuity criterion اللي
+
+8
+00:00:58,850 --> 00:01:04,770
+حسبها أو ممكن نستخدمها في اثبات أن دالة محددة ليست
+
+9
+00:01:04,770 --> 00:01:09,750
+uniform ل continuous على مجموعةمحددة جزئية من
+
+10
+00:01:09,750 --> 00:01:13,330
+الأعداد الحقيقية فكان في عندي non uniform
+
+11
+00:01:13,330 --> 00:01:18,270
+continuity criterion و آخر نظرية أثبتنا نظرية مهمة
+
+12
+00:01:18,270 --> 00:01:22,490
+هو هي ال uniform continuity criterion اللي بتقول
+
+13
+00:01:22,490 --> 00:01:27,650
+أنه لو كانت ال function تبعتي متصلة
+
+14
+00:01:28,780 --> 00:01:31,940
+على المجال تبعها والمجال تبعها closed bounded
+
+15
+00:01:31,940 --> 00:01:38,760
+interval فالاتصال يتحول الى اتصال منتظم طبعا احنا
+
+16
+00:01:38,760 --> 00:01:43,120
+شفنا في المحاضرة السابقة انه دايما الاتصال المنتظم
+
+17
+00:01:43,120 --> 00:01:47,200
+اقوى من الاتصال العادى لو كانت الدالة uniform ل
+
+18
+00:01:47,200 --> 00:01:50,960
+continuous فبتكون continuous لكن العكس ليس صحيح
+
+19
+00:01:52,640 --> 00:02:00,080
+فخدنا مثال على دالة function دالة واحد على X شفنا
+
+20
+00:02:00,080 --> 00:02:04,920
+أنها متصلة continuous على الفترة المفتوحة من سفر
+
+21
+00:02:04,920 --> 00:02:09,620
+إلى ملا نهاية but it was not uniformly continuous
+
+22
+00:02:09,620 --> 00:02:15,780
+على نفس الفترة وبالتالي الاتصال العادي لا يؤدي
+
+23
+00:02:15,780 --> 00:02:22,820
+للاتصال المنطماليوم هنتعرف على نوع جديد من ال
+
+24
+00:02:22,820 --> 00:02:27,040
+functions وهو لبسش functions و ال functions هدول
+
+25
+00:02:27,040 --> 00:02:32,200
+هتكونوا دائما كلهم uniformly continuous فنعرف لبسش
+
+26
+00:02:32,200 --> 00:02:38,640
+function definition a
+
+27
+00:02:38,640 --> 00:02:42,680
+function f
+
+28
+00:02:42,680 --> 00:02:44,740
+from a to r
+
+29
+00:02:47,770 --> 00:02:58,050
+إذ لبسش .. بنسميها
+
+30
+00:02:58,050 --> 00:03:03,010
+لبسش on
+
+31
+00:03:03,010 --> 00:03:10,310
+a إذا وجد if there exists k positive number such
+
+32
+00:03:10,310 --> 00:03:13,510
+that absolute f of x
+
+33
+00:03:36,970 --> 00:03:40,090
+وطبعا ممكن اثبات بكل سهولة
+
+34
+00:03:49,120 --> 00:03:55,860
+الأن هنثبت و هنشوف أنه كل لبسش function أو كل
+
+35
+00:03:55,860 --> 00:04:00,320
+function بتحقق لبسش condition اللي هو الشرط هذا
+
+36
+00:04:07,690 --> 00:04:12,250
+كل function بتحقق لبسش condition أو .. أو سمنها
+
+37
+00:04:12,250 --> 00:04:17,910
+لبسش function بتكون uniformly continuous فنشوف
+
+38
+00:04:17,910 --> 00:04:23,390
+المرحلة دالك إذا هنا every أو
+
+39
+00:04:23,390 --> 00:04:33,270
+if .. if from a to r is لبسش is
+
+40
+00:04:33,270 --> 00:04:47,300
+لبشسon a then it is uniformly continuous
+
+41
+00:04:47,300 --> 00:04:56,080
+on a proof
+
+42
+00:04:56,080 --> 00:05:00,840
+assume
+
+43
+00:05:03,530 --> 00:05:10,310
+إذا كان لبسش على
+
+44
+00:05:10,310 --> 00:05:20,250
+a ثم حسب التعريف هناك كمية positive كمية كمية كامة
+
+45
+00:05:20,250 --> 00:05:20,470
+كمية كمية كمية كمية كمية كمية كمية كمية كمية كمية
+
+46
+00:05:20,470 --> 00:05:20,690
+كمية كمية كمية كمية كمية كمية كمية كمية كمية كمية
+
+47
+00:05:20,690 --> 00:05:21,490
+كمية كمية كمية كمية كمية كمية كمية كمية كمية كمية
+
+48
+00:05:21,490 --> 00:05:21,530
+كمية كمية كمية كمية كمية كمية كمية كمية كمية كمية
+
+49
+00:05:21,530 --> 00:05:21,690
+كمية كمية كمية كمية كمية كمية كمية كمية كمية كمية
+
+50
+00:05:21,690 --> 00:05:28,390
+كمية كمية كمية
+
+51
+00:05:28,390 --> 00:05:38,840
+كمk times absolute x minus u for all x where u and
+
+52
+00:05:38,840 --> 00:05:52,180
+a طيب
+
+53
+00:05:52,180 --> 00:05:55,760
+لتسمي ال condition هذا star
+
+54
+00:05:58,650 --> 00:06:02,570
+let epsilon أكبر من السفر بيجبن let epsilon أكبر
+
+55
+00:06:02,570 --> 00:06:05,490
+من السفر بيجبن let epsilon أكبر من السفر بيجبن let
+
+56
+00:06:05,490 --> 00:06:06,290
+epsilon أكبر من السفر بيجبن let epsilon أكبر من
+
+57
+00:06:06,290 --> 00:06:06,550
+السفر بيجبن let epsilon أكبر من السفر بيجبن let
+
+58
+00:06:06,550 --> 00:06:06,570
+epsilon أكبر من السفر بيجبن let epsilon أكبر من
+
+59
+00:06:06,570 --> 00:06:07,810
+السفر بيجبن let epsilon أكبر من السفر بيجبن let
+
+60
+00:06:07,810 --> 00:06:09,990
+epsilon أكبر من السفر بيجبن
+
+61
+00:06:23,960 --> 00:06:29,040
+عدد موجة إبسلون على K بيطلع عدد موجة وبالتالي إذن
+
+62
+00:06:29,040 --> 00:06:35,920
+هنا أثبتت إن user Delta تعتمد على إبسلون فقط فلهذه
+
+63
+00:06:35,920 --> 00:06:42,280
+الإبسلون then لو كانت X و U موجودين في A و
+
+64
+00:06:42,280 --> 00:06:47,840
+Absolute X minus U أصغر من Delta فهذا هيقدّي إن
+
+65
+00:06:47,840 --> 00:07:01,000
+Absolute F of X-f of u باي ستار حسب المتباينة star
+
+66
+00:07:01,000 --> 00:07:06,600
+هذا بيطلع أصغر منه ساوي absolute x minus u
+
+67
+00:07:13,020 --> 00:07:18,320
+وانا عندي absolute x minus u أصغر من دلتا اذا هذا
+
+68
+00:07:18,320 --> 00:07:25,240
+أصغر عفوا by star في عندي هنا k ضرب absolute x
+
+69
+00:07:25,240 --> 00:07:31,780
+minus u الان انا عندي absolute x minus u أصغر من
+
+70
+00:07:31,780 --> 00:07:36,840
+دلتا لأن هذا أصغر من k في دلتا وانا عندي ماخد دلتا
+
+71
+00:07:36,840 --> 00:07:45,940
+بالساوي y على kأصبح هذا أصغر من إبسلون لأي
+
+72
+00:07:45,940 --> 00:07:52,500
+إبسلون أكبر من 0 يوجد delta تعتمد على إبسلون فقط
+
+73
+00:07:52,500 --> 00:07:59,340
+بحيث أنه لكل x و u في a المسافة بينهم أصغر من
+
+74
+00:07:59,340 --> 00:08:02,820
+delta طلع المسافة بين ال images أصغر من إبسلون
+
+75
+00:08:06,120 --> 00:08:11,800
+epsilon أكبر من السفر was arbitrary، إذن هذا صحيح
+
+76
+00:08:11,800 --> 00:08:15,580
+لكل epsilon وبالتالي by definition، إذن ال
+
+77
+00:08:15,580 --> 00:08:20,780
+function f is uniformly continuous
+
+78
+00:08:20,780 --> 00:08:30,340
+on E، وهو المطلوب إذن هناك أثبتنا إن كل لبسش
+
+79
+00:08:30,340 --> 00:08:34,120
+function is uniformly continuous
+
+80
+00:08:36,360 --> 00:08:45,020
+لكن العكس ليس صحيحا .. العكس ليس صحيحا remark ..
+
+81
+00:08:45,020 --> 00:08:55,400
+remark the
+
+82
+00:08:55,400 --> 00:09:00,740
+canvas .. the canvas of above theorem
+
+83
+00:09:05,350 --> 00:09:11,730
+is false for
+
+84
+00:09:11,730 --> 00:09:16,970
+example على سبيل المثال يعني معنى آخر لو كانت ال
+
+85
+00:09:16,970 --> 00:09:24,750
+function uniform ل continuous مش شرط تكون لبسش على
+
+86
+00:09:24,750 --> 00:09:31,370
+نفس ال function على نفس ال .. for example consider
+
+87
+00:09:35,170 --> 00:09:48,950
+Consider الـ function f of x بساوي جدر الـ x هو
+
+88
+00:09:48,950 --> 00:09:54,790
+x ينتمي ل I بساوي closed interval من صفر لاثنين
+
+89
+00:10:07,930 --> 00:10:14,030
+by exercise فى exercise أخدناه اللى هو جبنالكم
+
+90
+00:10:14,030 --> 00:10:23,090
+إياه سؤال فى الامتحان ال exercise هذا كان .. خلينا
+
+91
+00:10:23,090 --> 00:10:23,730
+نشوف
+
+92
+00:10:39,170 --> 00:10:45,750
+أو ممكن اثبات أن الدالة هذه is continuous طيب
+
+93
+00:10:45,750 --> 00:10:52,030
+اه
+
+94
+00:10:52,030 --> 00:10:55,910
+by exercise
+
+95
+00:10:55,910 --> 00:11:04,250
+في chapter اربعة اربعة واحد question تمام اه اربعة
+
+96
+00:11:04,250 --> 00:11:11,540
+واحد مظبوط صحيحby exercise تمامية section اربعة
+
+97
+00:11:11,540 --> 00:11:16,540
+واحد ال
+
+98
+00:11:16,540 --> 00:11:24,940
+function if is continuous على الفترة لأن في هداك
+
+99
+00:11:24,940 --> 00:11:32,480
+ال exercise هتتبتو انه limit جدر ال X لما X تقول ل
+
+100
+00:11:32,480 --> 00:11:42,910
+C بساوي جدر ال Cلكل C أكبر من أو ساوي السفر طبعا
+
+101
+00:11:42,910 --> 00:11:48,030
+في ال exercise ماخد C أكبر من السفر لكن لما C
+
+102
+00:11:48,030 --> 00:11:52,950
+بساوي السفر فهذا trivial وبالتالي هذا معناه أن
+
+103
+00:11:52,950 --> 00:11:59,890
+دالة F هذا
+
+104
+00:11:59,890 --> 00:12:05,330
+معناه شرط الاتصال عن C متحقق فهذا معناه أن F is
+
+105
+00:12:05,330 --> 00:12:12,720
+continuousAt C وده صحيح لكل C أكبر من أوسعها سفر
+
+106
+00:12:12,720 --> 00:12:20,520
+وبالتالي اذا F is continuous على الفترة من سفر إلى
+
+107
+00:12:20,520 --> 00:12:25,140
+ملا نهاية وبالتالي متصلة على الفترة من سفر إلى
+
+108
+00:12:25,140 --> 00:12:33,460
+اتنين اللي هي جزوة منها okay تمام طيب اذا
+
+109
+00:12:40,220 --> 00:12:48,440
+إذا by طيب since I بساو الفترة من الزفر لإتنين
+
+110
+00:12:48,440 --> 00:12:58,140
+الفترة هذه is closed and bounded و
+
+111
+00:12:58,140 --> 00:13:05,080
+if continuous عليها then by
+
+112
+00:13:05,080 --> 00:13:09,520
+uniform continuity theorem
+
+113
+00:13:11,440 --> 00:13:14,900
+نظرية الاتصال المنتظم بتقول إذا كان في عندي
+
+114
+00:13:14,900 --> 00:13:20,060
+function f متصلة على فترة مغلقة أو محدودة فالاتصال
+
+115
+00:13:20,060 --> 00:13:25,320
+هذا بيكون اتصال منتظم uniform continuity ففي عندي
+
+116
+00:13:25,320 --> 00:13:32,800
+by uniform continuity theorem تطلع f is uniformly
+
+117
+00:13:32,800 --> 00:13:39,840
+continuous
+
+118
+00:13:41,540 --> 00:13:48,120
+على الفترة I إذاً هي مثال على function uniform ل
+
+119
+00:13:48,120 --> 00:13:52,880
+continuous على المجال تبعها هنشوف الآن إن هذه ال
+
+120
+00:13:52,880 --> 00:14:07,060
+function ما هياش لبسش على نفس الفترة إذا
+
+121
+00:14:07,060 --> 00:14:07,640
+ال claim
+
+122
+00:14:13,370 --> 00:14:23,650
+if is not .. if is not لبسش على
+
+123
+00:14:23,650 --> 00:14:28,590
+الفترة I فلبرحان
+
+124
+00:14:28,590 --> 00:14:33,990
+ذلك assume
+
+125
+00:14:33,990 --> 00:14:37,950
+on
+
+126
+00:14:37,950 --> 00:14:38,670
+contrary
+
+127
+00:14:43,190 --> 00:14:48,550
+assume on contrary that
+
+128
+00:14:48,550 --> 00:15:01,650
+if is لبسش on
+
+129
+00:15:01,650 --> 00:15:04,090
+I then
+
+130
+00:15:06,570 --> 00:15:14,610
+there exists k أكبر من السفر بحيث أنه absolute f
+
+131
+00:15:14,610 --> 00:15:28,610
+of x minus f of u أصغر منها ساوي k في absolute x
+
+132
+00:15:28,610 --> 00:15:36,950
+minus u لكل xهو you تنتمي للفترة I اللى هى الفترة
+
+133
+00:15:36,950 --> 00:15:39,970
+المغلطة من سفر لاتنين
+
+134
+00:15:59,060 --> 00:16:04,840
+إذا ان هنا فرضنا ال contrary و يطلع ان بيطلع عندى
+
+135
+00:16:04,840 --> 00:16:09,160
+فيه huge العدد موجة بحيث كان أنا ادم اتحقق وهذا
+
+136
+00:16:09,160 --> 00:16:15,340
+بيقدر ان absolute f of x لو خدنا u بساوة سفر minus
+
+137
+00:16:15,340 --> 00:16:24,910
+f of 0 أصغر لو ساوة k فabsolute xوهذا صحيح لكل x
+
+138
+00:16:24,910 --> 00:16:32,150
+تنتمي للفترة I إذا أنا هنا أخدت U بساوي سفر و
+
+139
+00:16:32,150 --> 00:16:39,070
+السفر ينتمي للفترة I طيب أنا عندي F صفر بساوي سفر
+
+140
+00:16:39,070 --> 00:16:44,030
+إذا
+
+141
+00:16:44,030 --> 00:16:47,130
+بطلع عندي absolute
+
+142
+00:16:48,770 --> 00:16:58,510
+f of x أصغر من أو يساوي k في absolute ال X وهذا
+
+143
+00:16:58,510 --> 00:17:04,470
+صحيح لكل X تم تمي لفترة I هي الفترة المغلقة من
+
+144
+00:17:04,470 --> 00:17:09,690
+السفر لفترة بس
+
+145
+00:17:09,690 --> 00:17:14,810
+هذا هيدي لتناقل طيب
+
+146
+00:17:15,530 --> 00:17:28,330
+تاك لو أخدت x بساوي واحد على n تربيها فهذا
+
+147
+00:17:28,330 --> 00:17:33,990
+عبارة عن .. هذا ينتمي للفترة .. للفترة المغلفة من
+
+148
+00:17:33,990 --> 00:17:40,410
+سفر لإتنين اللي هي I لأن هذا عدد موجب لكل n ينتمي
+
+149
+00:17:40,410 --> 00:17:46,820
+ل N لكل عدد طبيعي هذا بطلعينتمي للفترة هذه
+
+150
+00:17:46,820 --> 00:17:54,300
+وبالتالي إذا المفروض يطلع absolute if لواحد على N
+
+151
+00:17:54,300 --> 00:18:02,060
+تربية أصغر من أو يساوي K في absolute واحد على N
+
+152
+00:18:02,060 --> 00:18:10,840
+تربية هذا صحيح لكل N في Nطيب if واحد على ان تربية
+
+153
+00:18:10,840 --> 00:18:16,760
+بيطلع بساوي الجدر التربية إلى واحد على ان تربية
+
+154
+00:18:16,760 --> 00:18:20,200
+اللي
+
+155
+00:18:20,200 --> 00:18:26,600
+هو عبارة عن واحد على ان ف absolute واحد على ان
+
+156
+00:18:26,600 --> 00:18:34,100
+بيطلع واحد على ان أصغر من أو ساوي كفي واحد على ان
+
+157
+00:18:34,100 --> 00:18:43,520
+تربية هذا صحيحلكل N في N اضرب
+
+158
+00:18:43,520 --> 00:18:50,880
+المتدينة هذه في N تربية فبطلع عندي N أصغر من أو
+
+159
+00:18:50,880 --> 00:18:59,320
+ساوي K for all N في N وهذا بتناقض مع ال
+
+160
+00:18:59,320 --> 00:19:06,040
+Archimedean property which contradicts
+
+161
+00:19:07,330 --> 00:19:11,130
+التي تتناقض
+
+162
+00:19:11,130 --> 00:19:19,250
+مع مين؟ التي تتناقض مع الـ Archimedean property
+
+163
+00:19:23,770 --> 00:19:28,110
+خاصية Archimedes لأن خاصية Archimedes بتقوللي لأي
+
+164
+00:19:28,110 --> 00:19:35,690
+عدد K عدد موجب أو أي عدد حقيقي K يوجد N0 عدد طبيعي
+
+165
+00:19:35,690 --> 00:19:45,660
+لحيث أن N0 أكبر من K صح؟و من هنا كل الأعداد
+
+166
+00:19:45,660 --> 00:19:53,360
+الطبيعية من ضمنها N0 أشملها أصغر من أو يساوي ال K
+
+167
+00:19:53,360 --> 00:20:00,740
+فبطلع N0 أكبر من N0 contradiction إذا السبب ال
+
+168
+00:20:00,740 --> 00:20:04,640
+contradiction هذا أنه إيه ال assumption الفرض
+
+169
+00:20:04,640 --> 00:20:12,440
+تبعنا ال assumption تبعنا أن F is لبسش on I okay
+
+170
+00:20:14,100 --> 00:20:19,120
+إذاً هذا بتثبت هذا ال contradiction بتثبت أن ال is
+
+171
+00:20:19,120 --> 00:20:29,820
+عفوًا if is not لبسش on
+
+172
+00:20:29,820 --> 00:20:37,120
+a أو i وهو المطلوب إذاً هذا مثال على function
+
+173
+00:20:37,120 --> 00:20:44,810
+uniformly continuous على set معينةلكنها ليست لبسش
+
+174
+00:20:44,810 --> 00:20:50,450
+لكن أثبتنا قبل ايه ان كل لبسش function is always
+
+175
+00:20:50,450 --> 00:20:57,750
+uniformly continuous ناخد
+
+176
+00:20:57,750 --> 00:20:58,770
+بعض الأمثلة
+
+177
+00:21:26,120 --> 00:21:35,220
+example led f of x بساوي x تربية و x ينتمي
+
+178
+00:21:35,220 --> 00:21:41,800
+للمجموعة a اللي هي الفترة المغلقة من سفر إلى بي
+
+179
+00:21:41,800 --> 00:21:50,410
+حيث بي أي عدد موجب بي أي عدد موجببنأثبت أن الـ
+
+180
+00:21:50,410 --> 00:21:59,950
+function هذه تطلع uniformly continuous show
+
+181
+00:21:59,950 --> 00:22:05,950
+that show
+
+182
+00:22:05,950 --> 00:22:11,670
+أن f is uniformly continuous
+
+183
+00:22:11,670 --> 00:22:23,720
+on a ففيه برهنيناو حالين proof one حال الأول since
+
+184
+00:22:23,720 --> 00:22:28,400
+if
+
+185
+00:22:28,400 --> 00:22:37,400
+is continuous on a being
+
+186
+00:22:37,400 --> 00:22:39,440
+a polynomial
+
+187
+00:22:44,780 --> 00:22:47,300
+لأنها polynomial و احنا قلنا كل polynomial
+
+188
+00:22:47,300 --> 00:22:51,800
+function متصل على R وبالتالي على أي مجموعة جزئية
+
+189
+00:22:51,800 --> 00:22:59,760
+من R زي المجموعة A اللي هي الفترة المغلقة من سفر
+
+190
+00:22:59,760 --> 00:23:05,620
+إلى الـ B ف
+
+191
+00:23:05,620 --> 00:23:10,920
+if is continuous على A كونها polynomial and بما
+
+192
+00:23:10,920 --> 00:23:19,480
+انه and sinceالـ set A هذه اللي هي عبارة عن الفترة
+
+193
+00:23:19,480 --> 00:23:29,320
+المغلقة من سفر لـ B is closed and bounded and
+
+194
+00:23:29,320 --> 00:23:34,500
+bounded interval
+
+195
+00:23:34,500 --> 00:23:44,980
+then by uniform continuity theorem
+
+196
+00:23:47,180 --> 00:23:53,380
+حسب نظرية الاتصال المنتظم اللى بتقول لو كان فيها
+
+197
+00:23:53,380 --> 00:23:57,140
+function مجالها closed bounded interval و ال
+
+198
+00:23:57,140 --> 00:24:03,120
+function متصل عليها فالاتصال بتحول الى اتصال منتظم
+
+199
+00:24:03,120 --> 00:24:08,040
+اذا ال function f is uniformly
+
+200
+00:24:10,270 --> 00:24:16,870
+continuous on a وهذا برهان لأنه ممكن نستخدم ال
+
+201
+00:24:16,870 --> 00:24:20,550
+uniform continuity theorem لإثبات أنه function
+
+202
+00:24:20,550 --> 00:24:24,970
+اللي زي هذه الدالة التربية uniform continuous على
+
+203
+00:24:24,970 --> 00:24:32,550
+أي فترة مغلقة زي الفترة هذه الحل
+
+204
+00:24:32,550 --> 00:24:37,830
+التاني ممكن نثبت أن الدالة هذه لمساش برضه و أستخدم
+
+205
+00:24:37,830 --> 00:24:38,710
+نظرية هذه
+
+206
+00:24:41,510 --> 00:24:48,950
+انشوف مع بعض، هنا البرهان تاني أو برهان رقم اتنين،
+
+207
+00:24:48,950 --> 00:24:58,810
+proof اتنين claim
+
+208
+00:24:58,810 --> 00:25:03,950
+انه F is لبسش
+
+209
+00:25:08,200 --> 00:25:18,740
+on a التي هي الفترة المغلفة من سفر إلى بى
+
+210
+00:25:18,740 --> 00:25:26,260
+فالإثبات هذا الكلام تعالى نشوف هي absolute f of x
+
+211
+00:25:26,260 --> 00:25:35,000
+minus f of u ايش بيساوي absolute x
+
+212
+00:25:35,820 --> 00:25:43,960
+تربية minus U تربية بساوي absolute X زائد U في
+
+213
+00:25:43,960 --> 00:25:51,860
+absolute X minus U وهذا بساوي absolute X زائد U في
+
+214
+00:25:51,860 --> 00:25:58,640
+absolute X negative U و by triangle inequality
+
+215
+00:25:58,640 --> 00:26:04,330
+absolute X زائد U أصغر من أو ساوي absolute Xزاد
+
+216
+00:26:04,330 --> 00:26:13,910
+absolute u كل هذا مضروف absolute x minus u الان
+
+217
+00:26:13,910 --> 00:26:19,970
+ال u و ال x ينتموا للمجال تبع الدولة وبالتالي
+
+218
+00:26:19,970 --> 00:26:29,990
+كلاهما عداد غير سالفة و كلاهما أصغر من أو يساوي ال
+
+219
+00:26:29,990 --> 00:26:37,320
+b صح؟إن هذا أصغر من أوي ساوي بي زائد بي في
+
+220
+00:26:37,320 --> 00:26:45,420
+absolute x minus u for all x و u ينتموا للمجموع
+
+221
+00:26:45,420 --> 00:26:52,520
+اللي هي الفترة المغلفة من سفر إلى بي طبعا هذا
+
+222
+00:26:52,520 --> 00:27:01,660
+بساوياتنين بي في absolute x minus u for all x و u
+
+223
+00:27:01,660 --> 00:27:08,820
+تنتمي الى a اذا هاي شرط لبسيش اتحقق with k بيساو
+
+224
+00:27:08,820 --> 00:27:17,340
+اتنين بيعدد موجب اذا هنا take k
+
+225
+00:27:17,340 --> 00:27:21,800
+بيساو اتنين بيعدد موجب
+
+226
+00:27:36,690 --> 00:27:38,510
+Okay طبعا
+
+227
+00:27:54,180 --> 00:28:09,940
+واضح البران في اي سؤال او استفسار في
+
+228
+00:28:09,940 --> 00:28:17,160
+عندي نظرية تتعلق بال uniform الها علاقة بال
+
+229
+00:28:17,160 --> 00:28:22,180
+uniform continuity وهي النظرية التالية
+
+230
+00:28:38,260 --> 00:28:49,440
+فيرم if if from a to r is uniformly is uniformly
+
+231
+00:28:49,440 --> 00:28:52,540
+continuous
+
+232
+00:28:52,540 --> 00:29:00,780
+on a then
+
+233
+00:29:03,000 --> 00:29:09,640
+For any Cauchy Sequence
+
+234
+00:29:09,640 --> 00:29:18,780
+xn contained in A The
+
+235
+00:29:18,780 --> 00:29:22,700
+sequence f
+
+236
+00:29:22,700 --> 00:29:31,580
+of xn اللي هي ال image لسيقونس xn is Cauchy
+
+237
+00:29:33,110 --> 00:29:40,090
+in R that
+
+238
+00:29:40,090 --> 00:29:45,950
+is that
+
+239
+00:29:45,950 --> 00:29:56,030
+is هذا يعني هذا يعني هذا يعني انه uniformly
+
+240
+00:29:56,030 --> 00:30:01,090
+uniformly continuous
+
+241
+00:30:04,960 --> 00:30:18,040
+functions preserve كوشي
+
+242
+00:30:18,040 --> 00:30:22,460
+sequences
+
+243
+00:30:27,960 --> 00:30:34,420
+يعني الدوال اللي بتكون متصل اتصال منتظم بتحافظ على
+
+244
+00:30:34,420 --> 00:30:39,960
+cushy sequences بمعنى انه لو كانت xn cushy
+
+245
+00:30:39,960 --> 00:30:46,260
+sequence في المجال تبع الدالة A فصورتها هتطلع
+
+246
+00:30:46,260 --> 00:30:52,080
+cushy sequence في المجال المقابل R والبرهان سهل
+
+247
+00:30:53,210 --> 00:30:57,390
+طبعا هذا بس صحيح لل uniform ل continuous functions
+
+248
+00:30:57,390 --> 00:31:02,350
+أما لو كانت ال function بس continuous فمش شرط
+
+249
+00:31:02,350 --> 00:31:07,510
+اتحافظ على كوشي sequences والبرهان
+
+250
+00:31:07,510 --> 00:31:18,670
+سهل بسيط prove let
+
+251
+00:31:18,670 --> 00:31:29,880
+if from A to Rب uniformly continuous on
+
+252
+00:31:29,880 --> 00:31:44,240
+a and let x in contained in a,b كوشي كوشي sequence
+
+253
+00:31:44,240 --> 00:31:50,420
+و بدنا نثبت ان ال image لل sequence x in بتطلع
+
+254
+00:31:50,420 --> 00:31:57,060
+كوشي طيبto show ان
+
+255
+00:31:57,060 --> 00:32:09,340
+ال image لسيكوينس XN is Cauchy لبرهان
+
+256
+00:32:09,340 --> 00:32:15,180
+ان ال sequence هذه ال image لسيكوينس XN is Cauchy
+
+257
+00:32:15,180 --> 00:32:24,130
+نحاول نطبق تعريف Cauchysequence او نحاول نحقق شرط
+
+258
+00:32:24,130 --> 00:32:31,290
+كوشي فكيف نحقق قولت epsilon أكبر من السفر بيه جبن
+
+259
+00:32:31,290 --> 00:32:39,210
+وبينا نرد عليها بcapital N تحققلي شرط كوشي طيب
+
+260
+00:32:39,210 --> 00:32:44,110
+since if
+
+261
+00:32:44,110 --> 00:32:45,390
+is uniformly
+
+262
+00:32:47,670 --> 00:32:55,510
+continuous on a إذا لأي إبسلون موجبة زي هذه يوجد
+
+263
+00:32:55,510 --> 00:33:02,650
+إذا
+
+264
+00:33:02,650 --> 00:33:09,410
+لأي إبسلون زي هذهمع ان if uniform continuous اذا
+
+265
+00:33:09,410 --> 00:33:13,670
+لأي epsilon حسب تعريف ال uniform continuity يوجد
+
+266
+00:33:13,670 --> 00:33:21,510
+delta تعتمد على epsilon عدد موجة بحيث انه لو كان x
+
+267
+00:33:24,390 --> 00:33:30,090
+و U موجودين في A و Absolute X minus U أصغر من
+
+268
+00:33:30,090 --> 00:33:37,570
+Delta فهذا بعدي أن Absolute F of X minus F of U
+
+269
+00:33:37,570 --> 00:33:47,250
+أصغر من Y نسمي ال implication head star الان
+
+270
+00:33:47,250 --> 00:33:53,610
+since ال sequence X in is Cauchy
+
+271
+00:33:58,410 --> 00:34:02,810
+then و delta and
+
+272
+00:34:02,810 --> 00:34:11,090
+delta أكبر من السفر طبعا هذه given is given ال
+
+273
+00:34:11,090 --> 00:34:13,850
+delta هذه قلنا يوجد delta عدد موجة بما أن هذه
+
+274
+00:34:13,850 --> 00:34:20,650
+تعتبر given delta فلل delta هذه اللي هنا عدد موجة
+
+275
+00:34:20,650 --> 00:34:27,960
+بما أن xn is Cauchyإذا there exist يوجد capital N
+
+276
+00:34:27,960 --> 00:34:37,220
+يعتمد على delta عدد طبيعي بحيث انه شرط كوشي يتحقق
+
+277
+00:34:37,220 --> 00:34:43,140
+وهو لكل N و M أكبر من أو ساوي capital N بطل عندي
+
+278
+00:34:43,140 --> 00:34:49,040
+absolute xn minus xm أصغر من delta
+
+279
+00:34:52,280 --> 00:35:01,060
+بنسمي هذه double star now star and double star
+
+280
+00:35:01,060 --> 00:35:14,240
+بيقدّوا أنه يوجد capital N يعتمد على epsilon لأن
+
+281
+00:35:14,240 --> 00:35:18,800
+ال delta بتعتمد على epsilonالـ delta بتعتمد على
+
+282
+00:35:18,800 --> 00:35:26,260
+إبسلون، ملاحظة الحال ف N هذه نفسها N of delta
+
+283
+00:35:26,260 --> 00:35:33,360
+بيساوي N of إبسلون بتتمي ل N بحيث أنه لو كان N و M
+
+284
+00:35:33,360 --> 00:35:40,480
+أكبر من أوي ساوي capital N فهذا بيقدّي أنه by
+
+285
+00:35:40,480 --> 00:35:48,270
+double starهذا بيقدم absolute xn minus xm أصغر من
+
+286
+00:35:48,270 --> 00:35:55,450
+دلتا وحسب ال star by star ال star بتقول لو كان
+
+287
+00:35:55,450 --> 00:36:01,030
+عندي x و u المسافة بينهم أصغر من دلتا فالمسافة بين
+
+288
+00:36:01,030 --> 00:36:08,790
+صورهم اللي هي xn هنا وصورة ال xm تطلع أصغر من إبسم
+
+289
+00:36:10,550 --> 00:36:16,550
+تمام؟ إذا هنا أثبتت لأي إبسلون أكبر من السفر يوجد
+
+290
+00:36:16,550 --> 00:36:20,830
+capital N يعتمد على إبسلون عدد طبيعي بحيث لكل M M
+
+291
+00:36:20,830 --> 00:36:25,510
+أكبر من أو ساوي capital N طلع المسافة بين F of X M
+
+292
+00:36:25,510 --> 00:36:31,650
+و F of X M أصغر من إبسلون إذا بما أنه since إبسلون
+
+293
+00:36:31,650 --> 00:36:39,100
+أكبر من السفر was arbitraryإذا الـ sequence f of x
+
+294
+00:36:39,100 --> 00:36:45,120
+in is Cauchy تطلع الـ sequence هذه Cauchy وهو
+
+295
+00:36:45,120 --> 00:36:54,080
+المطلوب okay تمام ممكن نستخدم النظرية هذه ممكن
+
+296
+00:36:54,080 --> 00:37:01,400
+نستخدم النظرية هذه في ال ..
+
+297
+00:37:01,400 --> 00:37:06,320
+ان نثبت ان function معينة ليستuniform and
+
+298
+00:37:06,320 --> 00:37:16,200
+continuous هاي example use
+
+299
+00:37:16,200 --> 00:37:20,100
+above theorem
+
+300
+00:37:20,100 --> 00:37:31,860
+to show ال function f of x بالسعر واحد على x is
+
+301
+00:37:31,860 --> 00:37:32,280
+not
+
+302
+00:37:35,480 --> 00:37:43,220
+uniformly continuous on a بساوي الفترة المفتوحة من
+
+303
+00:37:43,220 --> 00:37:44,680
+صفر إلى ملا نهار
+
+304
+00:37:57,800 --> 00:38:01,060
+لحظة ان النظرية دي ايش بتقول لو كانت ال function
+
+305
+00:38:01,060 --> 00:38:05,140
+uniform ل continuous فلازم تحافظ على كوشي sequence
+
+306
+00:38:05,140 --> 00:38:09,440
+طب لو محافظتش على كوشي sequence مش ممكن تكون
+
+307
+00:38:09,440 --> 00:38:17,260
+uniform ل continuous صح؟ مظبوط؟ اذا هنا proof
+
+308
+00:38:17,260 --> 00:38:25,680
+by above theorem حسب النظرية على it suffices
+
+309
+00:38:28,360 --> 00:38:36,220
+to show يكفي اثبات ان f is .. if does not .. if
+
+310
+00:38:36,220 --> 00:38:46,960
+does .. does not preserve .. preserve Cauchy
+
+311
+00:38:46,960 --> 00:38:52,860
+sequences ف
+
+312
+00:38:52,860 --> 00:38:53,500
+consider
+
+313
+00:38:56,930 --> 00:39:03,270
+consider ال sequence xn اللي هي بساوي واحد على ن
+
+314
+00:39:03,270 --> 00:39:11,470
+ال sequence هذه converge لصفر وبالتالي
+
+315
+00:39:11,470 --> 00:39:25,130
+اذا xn is Cauchy تمام but صورة ال xn
+
+316
+00:39:28,660 --> 00:39:37,460
+أيش بتطلع؟ صورة الواحد على ان تطلع ال sequence in
+
+317
+00:39:37,460 --> 00:39:43,620
+صح؟ و ال sequence هذه properly divergent to
+
+318
+00:39:43,620 --> 00:39:49,760
+infinity، اذا I'm divergent، اذا I'm not Cauchy
+
+319
+00:39:49,760 --> 00:39:54,220
+تمام؟
+
+320
+00:39:57,130 --> 00:40:01,450
+Okay؟ وبالتالي إذا هاي في عندي .. هاي في عندي ..
+
+321
+00:40:01,450 --> 00:40:08,790
+إذا if لا تحافظ على ال koshi sequences إذا if does
+
+322
+00:40:08,790 --> 00:40:13,210
+not preserve
+
+323
+00:40:13,210 --> 00:40:19,050
+.. preserve koshi
+
+324
+00:40:26,330 --> 00:40:31,610
+sequences وبالتالي حسب النظرية الأخيرة مابتكونش
+
+325
+00:40:31,610 --> 00:40:34,630
+uniformly continuous لأن لو كانت uniformly
+
+326
+00:40:34,630 --> 00:40:38,370
+continuous فالمفروض تاخد كوشي sequence زي هذه
+
+327
+00:40:38,370 --> 00:40:42,730
+تعطينا صورتها كوشي sequence وهذا مستحيل okay تمام
+
+328
+00:40:42,730 --> 00:40:47,970
+واضح في أي سؤال اي استفسار اذا هيك نكتفي بهذا
+
+329
+00:40:47,970 --> 00:40:52,540
+القدر من section خمسة اربعة وزي ما حكينا سابقاهذا
+
+330
+00:40:52,540 --> 00:40:57,260
+كان آخر section هناخده في المقرر و بالتالي هيكون
+
+331
+00:40:57,260 --> 00:41:03,400
+يعني .. يعني ان شاء الله أنهينا ال course كما هو
+
+332
+00:41:03,400 --> 00:41:10,600
+موضح على ال syllabus فشكرا لكم و شكرا لحسن إصداركم
+
+333
+00:41:10,600 --> 00:41:13,580
+و يعطيكم ألف عافية
+
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/ZfnDnf4RR5M.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/ZfnDnf4RR5M.srt
new file mode 100644
index 0000000000000000000000000000000000000000..8e8c2f15e6a11cb27c37e7234311162eac55e35b
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/ZfnDnf4RR5M.srt
@@ -0,0 +1,1931 @@
+1
+00:00:20,670 --> 00:00:26,650
+السلام عليكم اليوم إن شاء الله هنكمل section أربعة
+
+2
+00:00:26,650 --> 00:00:35,990
+اللي عرفنا فيه ال limits و ال functions أخذنا 
+
+3
+00:00:35,990 --> 00:00:40,650
+المرة الأولى التلاتة تعريف epsilon delta ل limit of
+
+4
+00:00:40,650 --> 00:00:45,990
+function وشوفنا أن هذا بكافة تعريف في neighborhood
+
+5
+00:00:45,990 --> 00:00:51,040
+definition ل limit of function على النقطة و بدنا
+
+6
+00:00:51,040 --> 00:00:57,500
+ناخد أمثلة كيف نستخدم تعريف epsilon delta في إثبات
+
+7
+00:00:57,500 --> 00:01:03,360
+أن ال limit لدالة معينة عن نقطة معينة بتساوي عدد
+
+8
+00:01:03,360 --> 00:01:08,310
+محدد، فخدنا بعض الأمثلة اليوم هنستمر، هنعطي مزيد من
+
+9
+00:01:08,310 --> 00:01:12,930
+الأمثلة و بندرس خواص ال limits لـ ال functions
+
+10
+00:01:12,930 --> 00:01:19,490
+فالمثال اللي وصلناه له رقم تلاتة عايزين نثبت إن ال
+
+11
+00:01:19,490 --> 00:01:25,730
+limit لدالة  x تربيع لما x تقول لـ c بساوي c
+
+12
+00:01:25,730 --> 00:01:29,330
+تربيع فـ solution
+
+13
+00:01:33,240 --> 00:01:40,260
+ناخد f of x بالساوي x تربيع هيفرض x ينتمي الى r
+
+14
+00:01:40,260 --> 00:01:43,880
+واحنا
+
+15
+00:01:43,880 --> 00:01:50,600
+عايزين من الآخر نثبت إن ال absolute value لـ f of x 
+
+16
+00:01:50,600 --> 00:01:58,400
+ناقص c تربيع أصغر من أي given epsilon عدد موجب
+
+17
+00:01:59,300 --> 00:02:04,780
+عندما الـ x تكون قريبة من النقطة c أو تقع في جوار
+
+18
+00:02:04,780 --> 00:02:13,600
+delta معينة للعدد c طيب هذا عبارة عن Absolute x 
+
+19
+00:02:13,600 --> 00:02:22,620
+تربيع ناقص c تربيع بتحلل إلى Absolute x ناقص c في
+
+20
+00:02:22,620 --> 00:02:24,620
+x زائد c
+
+21
+00:02:27,570 --> 00:02:33,430
+إذاً هذا عبارة عن absolute x زائد c في absolute x 
+
+22
+00:02:33,430 --> 00:02:37,830
+ناقص c الآن
+
+23
+00:02:37,830 --> 00:02:42,430
+بدي أحاول أخلي هذا أصغر من أو يساوي عدد موجب بـ
+
+24
+00:02:42,430 --> 00:02:47,290
+فبحاول
+
+25
+00:02:47,290 --> 00:02:52,830
+آخذ فيه قيمة delta، لتكن delta بالساوي واحد 
+
+26
+00:03:00,710 --> 00:03:12,430
+then أنا عندي absolute x زائد c هذا أصغر
+
+27
+00:03:12,430 --> 00:03:19,910
+من أو يساوي absolute x زائد absolute c في absolute x
+
+28
+00:03:19,910 --> 00:03:25,450
+ناقص c باستخدام ال triangle inequality، absolute x 
+
+29
+00:03:25,450 --> 00:03:30,430
+زائد c أصلاً لو ساوي absolute x زائد absolute c الآن
+
+30
+00:03:30,430 --> 00:03:39,530
+absolute x بيساوي absolute x ناقص c زائد زائد
+
+31
+00:03:39,530 --> 00:03:44,270
+c ممكن أطرح من الـ x c وأرجعها، وباستخدام الـ
+
+32
+00:03:44,270 --> 00:03:49,030
+triangular equality هذا أصغر لو يساوي absolute x 
+
+33
+00:03:49,030 --> 00:03:58,150
+زائد c زائد absolute c، فلو كان absolute x ناقص c 
+
+34
+00:03:58,150 --> 00:04:04,070
+أصغر من delta اللي هي بيساوي واحد، إذا كان خلّينا
+
+35
+00:04:04,070 --> 00:04:07,190
+ناخد delta بيساوي واحد، إذا كان absolute x ناقص c 
+
+36
+00:04:07,190 --> 00:04:13,370
+أصغر من delta اللي أنا ماخدها واحد، فهذا بيطلع أصغر
+
+37
+00:04:13,370 --> 00:04:21,490
+من واحد زائد absolute c وبالتالي 
+
+38
+00:04:21,490 --> 00:04:32,370
+absolute x تربيع ناقص c تربيع بيطلع أصغر من absolute 
+
+39
+00:04:32,370 --> 00:04:35,150
+x اللي هي واحد زائد
+
+40
+00:04:37,400 --> 00:04:43,720
+اتنين في absolute c في absolute x ناقص c
+
+41
+00:04:48,510 --> 00:04:51,770
+كمان مرة، احنا توصلنا إلى إن ال absolute value
+
+42
+00:04:51,770 --> 00:04:57,150
+للفرق هذا أصغر من أو يساوي absolute x زائد absolute
+
+43
+00:04:57,150 --> 00:05:01,290
+c في absolute x ناقص c، أخدنا delta بالساوي واحد
+
+44
+00:05:01,290 --> 00:05:05,190
+وقلنا لو كان absolute x ناقص c أصغر من delta اللي
+
+45
+00:05:05,190 --> 00:05:09,190
+هي واحد بتطلع absolute x أصغر من واحد زائد absolute
+
+46
+00:05:09,190 --> 00:05:13,830
+c وبالتالي absolute الفرق هذا هي أصغر من أو يساوي
+
+47
+00:05:14,180 --> 00:05:18,500
+absolute x هي أصغر من واحد زائد absolute c وانتي
+
+48
+00:05:18,500 --> 00:05:23,820
+absolute c فأصغر من واحد زائد اتنين فـ absolute c ضرب
+
+49
+00:05:23,820 --> 00:05:29,460
+absolute x ناقص c، الآن بدي أخلي هذا أصغر من
+
+50
+00:05:29,460 --> 00:05:39,160
+epsilon، هذا بدي أخليه أصغر من epsilon لما
+
+51
+00:05:39,160 --> 00:05:40,920
+يكون هذا أصغر من delta
+
+52
+00:05:48,400 --> 00:05:52,700
+فباخد إذا لما يكون هذا أصغر من delta فهذا بصير أصغر
+
+53
+00:05:52,700 --> 00:05:58,880
+من واحد زائد اتنين absolute c في delta لما يكون ال
+
+54
+00:05:58,880 --> 00:06:03,000
+absolute value لـ x ناقص c أصغر من delta فهذا بيطلع
+
+55
+00:06:03,000 --> 00:06:07,740
+أصغر من واحد زائد اتنين في absolute c في delta الآن
+
+56
+00:06:07,740 --> 00:06:16,040
+متى بيكون هذا أصغر من epsilon؟ لما delta إذا كانت delta 
+
+57
+00:06:16,040 --> 00:06:25,240
+هذه أصغر من أو يساوي epsilon على واحد زائد اتنين 
+
+58
+00:06:25,240 --> 00:06:29,960
+في absolute of c، إذا هاي قيمة تانية لـ delta، هاي
+
+59
+00:06:29,960 --> 00:06:35,660
+ندي delta بيساوي واحد و delta أصغر من أو يساوي
+
+60
+00:06:35,660 --> 00:06:39,760
+epsilon على واحد زائد اتنين في absolute of c، إذا
+
+61
+00:06:39,760 --> 00:06:49,590
+باجي بقول let epsilon أكبر من الصفر be given a
+
+62
+00:06:49,590 --> 00:06:57,690
+choose delta بيساوي ال minimum الأصغر بين القيمتين
+
+63
+00:06:57,690 --> 00:07:07,050
+واحد و epsilon على واحد زائد اتنين في absolute c، ال
+
+64
+00:07:07,050 --> 00:07:13,660
+delta هذه الآن عدد موجب ويعتمد على epsilon، إذا لهذه
+
+65
+00:07:13,660 --> 00:07:24,140
+الـ delta لو كان x ينتمي لـ r اللي هو مجال الدالة و
+
+66
+00:07:24,140 --> 00:07:31,780
+absolute x ناقص c أكبر من صفر أصغر من delta فهذا
+
+67
+00:07:31,780 --> 00:07:36,960
+بيؤدي طبعاً الـ delta هذه هي الأصغر من العددين هذوله
+
+68
+00:07:36,960 --> 00:07:40,920
+وبالتالي أصغر من أو يساوي واحد وأصغر من أو يساوي
+
+69
+00:07:40,920 --> 00:07:46,820
+كسر هذا، فالـ delta أكيد أصغر من أو يساوي الواحد،
+
+70
+00:07:46,820 --> 00:07:52,360
+لما الـ delta أصغر من أو يساوي الواحد، هذا بيقدر
+
+71
+00:07:52,360 --> 00:07:56,940
+أن absolute x
+
+72
+00:07:56,940 --> 00:08:04,800
+أصغر من واحد زائد absolute c وكمان هذا بيؤدي إنه
+
+73
+00:08:04,800 --> 00:08:11,000
+absolute x تربيع ناقص c تربيع أصغر من أو يساوي
+
+74
+00:08:11,000 --> 00:08:23,400
+absolute x زائد absolute c، absolute 
+
+75
+00:08:23,400 --> 00:08:28,640
+x ناقص c وبالتالي هذا أصغر من أو يساوي واحد زائد
+
+76
+00:08:28,640 --> 00:08:39,840
+اتنين absolute c وهذا أصغر من delta و
+
+77
+00:08:39,840 --> 00:08:49,520
+الآن الـ delta هذه طبعاً
+
+78
+00:08:49,520 --> 00:08:55,420
+هذا أصغر من delta والـ delta قلنا أصغر منها و
+
+79
+00:08:55,420 --> 00:08:58,600
+يساوي epsilon، هاي واحد زائد اتنين 
+
+80
+00:09:19,770 --> 00:09:26,670
+بشكل صحيح، بما أن epsilon أكبر من الصفر was
+
+81
+00:09:26,670 --> 00:09:27,550
+arbitrary
+
+82
+00:09:31,810 --> 00:09:35,410
+إذاً هيك بنكون أثباتنا، لكل epsilon أكبر من الصفر
+
+83
+00:09:35,410 --> 00:09:41,150
+يوجد delta تعتمد على epsilon عدد موجب، ب half لكل x
+
+84
+00:09:41,150 --> 00:09:46,710
+المسافة مختلفة عن الـ c والمسافة بينها وبين الـ c
+
+85
+00:09:46,710 --> 00:09:52,590
+أصغر من delta بتطلع المسافة بين f of x و c تربيع
+
+86
+00:09:52,590 --> 00:10:01,190
+أصغر من epsilon، إذاً we have By definition إن ال
+
+87
+00:10:01,190 --> 00:10:11,390
+limit لـ x تربيع لما x تقول إلى c بيساوي c تربيع وهو
+
+88
+00:10:11,390 --> 00:10:17,410
+المطلوب، okay؟ إذن هذا هو برهان إن ال limit للدالة
+
+89
+00:10:17,410 --> 00:10:23,300
+التربيعية عن c بيساوي c تربيع، استخدمنا تعريف epsilon
+
+90
+00:10:23,300 --> 00:10:28,260
+delta وشوفنا إن delta هنا لازم تكون الأصغر من
+
+91
+00:10:28,260 --> 00:10:34,520
+القيمتين اللي هو الواحد والكسر اللي هناك، هي أي
+
+92
+00:10:34,520 --> 00:10:41,020
+سؤال، خلينا ناخد كمان مثال مشابه لهذا وفيه
+
+93
+00:10:41,020 --> 00:10:44,560
+الـ delta برضه بتساوي ال minimum لكمتين
+
+94
+00:10:53,440 --> 00:11:02,620
+المثال رقم أربعة، show أنه ال limit لواحد على x
+
+95
+00:11:02,620 --> 00:11:14,020
+لما x تقول إلى zero، لأ
+
+96
+00:11:14,020 --> 00:11:20,340
+ال limit لواحد على x لما x تقول إلى أي عدد c بيساوي
+
+97
+00:11:20,340 --> 00:11:27,420
+واحد على c حيث c أكبر من 0، فهنا
+
+98
+00:11:27,420 --> 00:11:30,520
+بناخد ال function تبعتي الدالة اللي بتحدد ال
+
+99
+00:11:30,520 --> 00:11:36,800
+limit هي عبارة عن f of x بيساوي واحد على x حيث x
+
+100
+00:11:36,800 --> 00:11:41,720
+موجبة، إذا المجال تبع الدالة هذه الفترة المفتوحة من
+
+101
+00:11:41,720 --> 00:11:49,410
+صفر إلى ما لا نهاية، و c عدد موجب، طيب أنا عايز
+
+102
+00:11:49,410 --> 00:11:57,190
+أثبت إن absolute f of x ناقص واحد على c بدي هذا
+
+103
+00:11:57,190 --> 00:12:02,470
+يكون أصغر من أي given epsilon عندما x تكون قريبة
+
+104
+00:12:02,470 --> 00:12:12,230
+من ال c أو في جوار delta لل c فهذا طبعاً إيش بيساوي، هي
+
+105
+00:12:12,230 --> 00:12:17,790
+absolute واحد على x ناقص واحد على c وهذا بيساوي
+
+106
+00:12:17,790 --> 00:12:27,950
+absolute c ناقص x على x في c وهذا بيساوي واحد على 
+
+107
+00:12:27,950 --> 00:12:33,270
+x في c ضرب absolute x ناقص c
+
+108
+00:12:38,060 --> 00:12:45,780
+الآن بدي أحاول أجيب upper bound عدد 
+
+109
+00:12:45,780 --> 00:12:53,720
+موجب بـ m بحيث الـ 1 على x في c يكون أصغر من أو يساوي
+
+110
+00:12:53,720 --> 00:12:57,360
+الـ m تعالوا نشوف كيف نجيب ال upper bound هذا أو ال
+
+111
+00:12:57,360 --> 00:13:06,330
+bound، أنا عندي الـ take الأول، take أنا عندي الـ c عدد 
+
+112
+00:13:06,330 --> 00:13:12,190
+موجب، take delta بيساوي c على اتنين هذا عدد موجب
+
+113
+00:13:12,190 --> 00:13:16,050
+then
+
+114
+00:13:16,050 --> 00:13:23,510
+absolute x ناقص c أصغر من delta اللي هو بيساوي c ع
+
+115
+00:13:23,510 --> 00:13:32,030
+2 بيؤدي إن x أصغر من تلاتة c ع 2 أكبر من c ع 2
+
+116
+00:13:32,030 --> 00:13:43,430
+وهذا بيؤدي إن واحد على x في c أصغر من اتنين على c
+
+117
+00:13:43,430 --> 00:13:44,270
+تربيع
+
+118
+00:13:50,000 --> 00:13:54,920
+الـ x أكبر من c على 2، إذا مقلوب الـ x أصغر من 2 على
+
+119
+00:13:54,920 --> 00:14:01,220
+c، مقلوب الـ x وأضربها في 1 على c بيطلع أصغر من 2
+
+120
+00:14:01,220 --> 00:14:07,100
+على c تربيع وبالتالي 
+
+121
+00:14:07,100 --> 00:14:13,540
+هذا العدد هذا هو الـ m عدد
+
+122
+00:14:13,540 --> 00:14:16,320
+موجب إذاً
+
+123
+00:14:18,670 --> 00:14:28,290
+في الحالة هذه، في الحالة 
+
+124
+00:14:28,290 --> 00:14:34,910
+هذه بصير عندي هذا أصغر من اتنين على c تربيع وطبعاً
+
+125
+00:14:34,910 --> 00:14:39,430
+هذا أصغر من delta، absolute x ناقص c طبعاً بيكون
+
+126
+00:14:39,430 --> 00:14:44,690
+أصغر من delta الآن عشان يكون هذا أصغر من أو يساوي
+
+127
+00:14:44,690 --> 00:14:52,820
+epsilon فنختار choose الـ delta أصغر من أو يساوي حل 
+
+128
+00:14:52,820 --> 00:14:57,140
+المتباينة هذه في الـ delta فالـ delta ستصبح أصغر
+
+129
+00:14:57,140 --> 00:15:04,540
+من أو يساوي c تربيع على 2 epsilon فهي قيمة تانية لـ 
+
+130
+00:15:04,540 --> 00:15:09,440
+delta فبأخد الـ delta ال minimum للقيمة الأولى
+
+131
+00:15:10,560 --> 00:15:16,040
+والقيمة التانية، هذا هيخلي إنه لكل x المسافة بين أو
+
+132
+00:15:16,040 --> 00:15:20,320
+بين c أصغر من delta هتخلي المسافة بين f of x و واحد
+
+133
+00:15:20,320 --> 00:15:26,280
+على c أصغر من ال given epsilon، نكتب الكلام هذا، let 
+
+134
+00:15:26,280 --> 00:15:29,240
+epsilon be given، choose delta بالساوي ال minimum
+
+135
+00:15:29,240 --> 00:15:36,260
+نختار 
+
+136
+00:15:36,260 --> 00:15:42,030
+delta ال minimum للعدد الموجب بـ c ع 2، والعدد 
+
+137
+00:15:42,030 --> 00:15:49,000
+التاني ده هو c تربيع ع 2 في epsilon طبعاً هذا عدد 
+
+138
+00:15:49,000 --> 00:15:52,740
+أكيد عدد موجب لأن هذا موجب وهذا موجب والأصغر بينهم
+
+139
+00:15:52,740 --> 00:15:56,820
+هييطلع موجب واتنين بيعتمدوا على epsilon، إذن delta 
+
+140
+00:15:56,820 --> 00:16:00,620
+عدد موجب بيعتمد على epsilon، إذا لأي epsilon أكبر 
+
+141
+00:16:00,620 --> 00:16:04,360
+من سفر هين أثبتت يوجد delta تعتمد على epsilon عدد
+
+142
+00:16:04,360 --> 00:16:11,680
+موجب بحيث أنه لكل x ينتمي إلى المجال هنا اللي هو
+
+143
+00:16:11,680 --> 00:16:19,300
+الفترة المفتوحة من سفر إلى دالة نهاية و absolute x
+
+144
+00:16:19,300 --> 00:16:25,400
+minus c أكبر من سفر أصغر من ال delta هذا بيقود إلى أن
+
+145
+00:16:25,400 --> 00:16:33,260
+ال delta هذه أصغر من أو يساوي c ع 2 فلما ال delta
+
+146
+00:16:33,260 --> 00:16:39,280
+تكون أصغر من أو يساوي c ع 2 هذا بيقود إلى أنه واحد على
+
+147
+00:16:39,280 --> 00:16:42,460
+واحد
+
+148
+00:16:42,460 --> 00:16:52,060
+على x في c أصغر من اثنين على c تربيع وهذا بدوره
+
+149
+00:16:52,060 --> 00:17:00,940
+يقدم absolute واحد على x minus واحد على c يساوي
+
+150
+00:17:00,940 --> 00:17:06,480
+واحد على x في c في absolute x minus c أصغر من
+
+151
+00:17:06,480 --> 00:17:14,850
+اثنين على c تربيع في delta و ال delta هذه الآن أصغر
+
+152
+00:17:14,850 --> 00:17:19,310
+من أو يساوي ال delta هذه هي ال delta اللي فوق
+
+153
+00:17:19,310 --> 00:17:25,010
+أصغر من أو يساوي العدد هذا أيه والعدد الثاني لأنها
+
+154
+00:17:25,010 --> 00:17:31,130
+الأصغر بين اثنين لأن هي اثنين على c تربيع ضرب c
+
+155
+00:17:31,130 --> 00:17:36,390
+تربيع اثنين في epsilon هذا يروح مع هذا مخلوق بعض
+
+156
+00:17:36,390 --> 00:17:41,030
+يظل عندي epsilon since
+
+157
+00:17:43,190 --> 00:17:50,970
+Y أكبر من سفر was arbitrary إذا أنا لكل Y أكبر
+
+158
+00:17:50,970 --> 00:17:56,850
+من سفر جبت Delta تعتمد على Y بحيث لكل X مختلفة
+
+159
+00:17:56,850 --> 00:18:00,570
+عن ال C المسافة بينها وبين ال C أصغر من Delta كل
+
+160
+00:18:00,570 --> 00:18:05,450
+المسافة بين F of X و 1 على C أصغر من Y إذا by
+
+161
+00:18:05,450 --> 00:18:06,010
+definition
+
+162
+00:18:09,260 --> 00:18:14,820
+by definition of limit يطلع عندي ال limit لل
+
+163
+00:18:14,820 --> 00:18:20,740
+function واحد على X لما X تقول إلى C يساوي واحد
+
+164
+00:18:20,740 --> 00:18:24,240
+على C وهو المطلوب
+
+165
+00:18:26,860 --> 00:18:31,720
+واضح في أي سؤال؟ في كمان مثال آخر زي هدف الكتاب،
+
+166
+00:18:31,720 --> 00:18:37,860
+هسيبكم تقرؤوه لأن الفكرة شبيهة بالفكرة في المثال
+
+167
+00:18:37,860 --> 00:18:47,720
+الأخير وبالتالي ما فيش إشي جديد ننتقل إلى دراسة
+
+168
+00:18:52,750 --> 00:18:56,370
+ال sequential criterion هي حاجة اسمها sequential
+
+169
+00:18:56,370 --> 00:19:09,570
+criterion حاجة .. حاجة بتكافئ التعريف sequential
+
+170
+00:19:09,570 --> 00:19:12,750
+criterion
+
+171
+00:19:45,930 --> 00:19:53,970
+العبارات التالية متكافئة Limit f of x as x tends to
+
+172
+00:19:53,970 --> 00:20:03,290
+c يساوي عدد L بحيث عدد حقيقي اثنين for every
+
+173
+00:20:06,410 --> 00:20:14,330
+for every sequence xn contained in A حدودها
+
+174
+00:20:14,330 --> 00:20:25,090
+مختلفة عن ال C such that limit xn يساوي C we
+
+175
+00:20:25,090 --> 00:20:31,470
+have limit ال image لسيكوينس xn as n tends to
+
+176
+00:20:31,470 --> 00:20:34,410
+infinity يساوي العدد L
+
+177
+00:20:39,210 --> 00:20:42,690
+إن ال sequential criterion هذه بتقول إن عشان أثبت
+
+178
+00:20:42,690 --> 00:20:46,470
+إن ال limit لل function f and x يساوي c يساوي
+
+179
+00:20:46,470 --> 00:20:52,090
+العدد L  هدا بكافة إن أنا أثبت إنه لو أخدت أي
+
+180
+00:20:52,090 --> 00:20:59,010
+sequence نهايتها C فلازم يكون نهاية صورتها يساوي
+
+181
+00:20:59,010 --> 00:21:04,140
+العدد L لو قدرت أعمل هذا في الكلام فبقى هذا بيكافئ
+
+182
+00:21:04,140 --> 00:21:08,880
+أن نقول إن ال limit ل f of x يعني ال x يساوي
+
+183
+00:21:08,880 --> 00:21:15,060
+c يساوي العدد ال .. نثبت النظرية هذه تروف one
+
+184
+00:21:15,060 --> 00:21:23,020
+implies two assume one
+
+185
+00:21:23,020 --> 00:21:28,480
+i.e
+
+186
+00:21:30,950 --> 00:21:37,470
+ال limit ل F of X لما X تقول ل C يساوي العدد L
+
+187
+00:21:37,470 --> 00:21:44,070
+عايزين
+
+188
+00:21:44,070 --> 00:21:48,450
+نثبت عشان
+
+189
+00:21:48,450 --> 00:21:55,250
+نثبت اثنين عشان نثبت اثنين صحيح to
+
+190
+00:21:55,250 --> 00:21:58,190
+prove two holds
+
+191
+00:22:00,720 --> 00:22:05,500
+to prove two holds let
+
+192
+00:22:05,500 --> 00:22:17,380
+Xn be a sequence in A حدودها مختلفة عن ال C such
+
+193
+00:22:17,380 --> 00:22:26,960
+that limit Xn يساوي C we claim
+
+194
+00:22:30,360 --> 00:22:45,320
+أن ال limit ل f of x ل f of xn لما
+
+195
+00:22:45,320 --> 00:22:52,340
+n تقول ل infinity دي يساوي L لبرهان ذلك let epsilon
+
+196
+00:22:52,340 --> 00:22:55,400
+أكبر
+
+197
+00:22:55,400 --> 00:22:57,020
+من صفر be given
+
+198
+00:23:02,180 --> 00:23:08,440
+since xnxnxn
+
+199
+00:23:10,770 --> 00:23:16,490
+بما أننا فرضنا limit f of x لما x تقول ل c يساوي
+
+200
+00:23:16,490 --> 00:23:21,450
+L من تعريف epsilon دلتا لل limit إذا يوجد دلتا
+
+201
+00:23:21,450 --> 00:23:27,770
+تعتمد على epsilon عدد موجب بحيث أنه لكل x ينتمي
+
+202
+00:23:27,770 --> 00:23:33,730
+إلى a و absolute x minus c أكبر من صفر أصغر من دلتا
+
+203
+00:23:42,740 --> 00:23:52,080
+epsilon دلتا للنهايات نسمي
+
+204
+00:23:52,080 --> 00:23:53,760
+ال implication هذه star
+
+205
+00:24:01,580 --> 00:24:07,300
+and the limit xn يساوي c احنا فرضنا ان في انديو
+
+206
+00:24:07,300 --> 00:24:14,840
+سيكوينس xn ونهايتها c then
+
+207
+00:24:14,840 --> 00:24:26,910
+for the above دلتا الموجبة يوجد دلتا موجبة أخذت دلتا
+
+208
+00:24:26,910 --> 00:24:31,910
+هذه الموجبة وطبقت تعريف epsilon capital N ل limit of
+
+209
+00:24:31,910 --> 00:24:36,590
+sequence فبما أن ال sequence هذه نهايتها C إذا لأي
+
+210
+00:24:36,590 --> 00:24:42,070
+دلتا أو epsilon عدد موجب there exists capital N
+
+211
+00:24:42,070 --> 00:24:46,710
+يعتمد على ال Delta طبعا ال Delta تعتمد على
+
+212
+00:24:46,710 --> 00:24:51,370
+epsilon إذا ال N هذه يعتمد على epsilon عدد طبيعي،
+
+213
+00:24:51,370 --> 00:24:56,650
+بحيث أنه لكل N أكبر من أو يساوي capital N، تطلع
+
+214
+00:24:56,650 --> 00:25:02,130
+عندي absolute X N minus C أصغر من Delta، نسمي ال
+
+215
+00:25:02,130 --> 00:25:03,990
+implication هذه double star
+
+216
+00:25:07,580 --> 00:25:16,300
+now star and double star بيؤدوا إلى ما يلي لو كان
+
+217
+00:25:16,300 --> 00:25:23,340
+n أكبر من أو يساوي capital N هذا بيقود إلى أن absolute
+
+218
+00:25:23,340 --> 00:25:27,740
+Xn
+
+219
+00:25:27,740 --> 00:25:36,440
+minus C أصغر من Delta هذا باستخدام double star صح؟
+
+220
+00:25:39,750 --> 00:25:44,230
+لو كانت n أكبر من أو يساوي capital N فبيطلع absolute
+
+221
+00:25:44,230 --> 00:25:52,050
+xn minus c أصغر من delta و من ال star لو كان عندي
+
+222
+00:25:52,050 --> 00:25:59,130
+xn طبعا xn هذا موجود في a ال xn موجود في a مختلف
+
+223
+00:25:59,130 --> 00:25:59,810
+عن ال c
+
+224
+00:26:02,830 --> 00:26:07,750
+فلو كان absolute of xn minus c badly except xn
+
+225
+00:26:07,750 --> 00:26:13,850
+أصغر من delta فحسب ال star هذا بقدر absolute of f
+
+226
+00:26:13,850 --> 00:26:22,590
+of xn minus L أصغر من epsilon الآن بما أن هذا صحيح
+
+227
+00:26:22,590 --> 00:26:28,270
+بما أن since epsilon أكبر من صفر was arbitrary
+
+228
+00:26:30,740 --> 00:26:42,380
+إن إحنا أثبتنا هيك لكل epsilon يوجد
+
+229
+00:26:42,380 --> 00:26:50,250
+capital N يعتمد على epsilon عدد طبيعي لكل n أكبر من
+
+230
+00:26:50,250 --> 00:26:55,310
+أو يساوي capital N absolute f of xn minus L أصغر من
+
+231
+00:26:55,310 --> 00:27:00,150
+epsilon إذا by epsilon capital N definition لل limit
+
+232
+00:27:00,150 --> 00:27:06,050
+of sequence بيطلع عندي limit ل sequence
+
+233
+00:27:06,050 --> 00:27:12,910
+f of xn as n tends to infinity يساوي L وبالتالي
+
+234
+00:27:12,910 --> 00:27:21,850
+هيك بيكون إذا two holds هكذا أثبتنا أن واحد يؤدي
+
+235
+00:27:21,850 --> 00:27:26,610
+إلى اثنين اثنين
+
+236
+00:27:26,610 --> 00:27:30,270
+بيقول for every sequence فهي اللي أخدت arbitrary
+
+237
+00:27:30,270 --> 00:27:36,810
+sequence في a minus c وبشرط بحيث أن ال sequence هي
+
+238
+00:27:36,810 --> 00:27:37,710
+اللي نهايتها c
+
+239
+00:27:42,350 --> 00:27:46,210
+و أثبتنا أن ال limit لل image لل sequence يساوي L
+
+240
+00:27:46,210 --> 00:27:51,710
+هذا بالظبط اللي هو العبارة اثنين لأن هيك يكون
+
+241
+00:27:51,710 --> 00:27:58,270
+أثبتنا واحد بيقود إلى اثنين واضح مفهوم اللي هو نثبت
+
+242
+00:27:58,270 --> 00:28:02,510
+العكس نثبت أن اثنين بيقود لواحد
+
+243
+00:28:16,210 --> 00:28:22,870
+بالنسبة للعبارة اثنين بتقود إلى العبارة واحد فالاثنان
+
+244
+00:28:22,870 --> 00:28:27,270
+ذالف بالمناسبة الأخوات اللي قاعدات ورا دولة إيش
+
+245
+00:28:27,270 --> 00:28:31,510
+بتعملوا أنتو؟ معليش أوقف تصوير إيش مجاعتكم أنتو
+
+246
+00:28:31,510 --> 00:28:34,830
+أنا أول حاجة و ثاني حاجة؟ إيش بتتكلمون؟ دكتور معك
+
+247
+00:28:34,830 --> 00:28:38,070
+لأ لأ لما هم بتتكلم عامليننا أزعاج لأ باحكوا إذا
+
+248
+00:28:38,070 --> 00:28:40,550
+انتو بتتكلموا لأ باحكي على اندر ده ليش مصورة إن
+
+249
+00:28:40,550 --> 00:28:44,510
+الوضع هو وضع نفسه لأ بنتكلميش لأ باحكي عن البرادة
+
+250
+00:28:44,510 --> 00:28:49,530
+اللي ورا دولة في بنات بتتكلموا، أنتو اللي ورا
+
+251
+00:28:49,530 --> 00:28:55,390
+بتتكلموا ولا في ناس غيرك؟ في حد بتتكلم و أنا بشرح
+
+252
+00:28:55,390 --> 00:29:00,090
+تتكلم و هذا عمللي أزعاج كثير، فلو سمحتوا إذا أنتو
+
+253
+00:29:00,090 --> 00:29:04,890
+قاعدين تتكلموا ورا اطلعوا في حديقة اتكلموا فيها،
+
+254
+00:29:04,890 --> 00:29:10,670
+حتى لو باسم المحاضرة ممنوع تتكلموا، شوية أزعاج هو
+
+255
+00:29:10,670 --> 00:29:13,850
+مين اللي بتتكلم؟ إذا أنت اللي بتتكلم من قعدته
+
+256
+00:29:13,850 --> 00:29:20,350
+وراك بتتكلم ما تتكلمش لأن غير ترفع يدك، ارفع يدك و
+
+257
+00:29:20,350 --> 00:29:24,670
+تقعد لسانك، ما تتكلمي مع الجنب بدون اسم، لأن هذا
+
+258
+00:29:24,670 --> 00:29:28,150
+عندنا قاعدة في المحاضرة، ممنوع حد يتكلم مع الجنب و
+
+259
+00:29:28,150 --> 00:29:34,030
+تتحدث مع حد شخص آخر إلا إذا عندك سؤال، ترفع يدك،
+
+260
+00:29:34,030 --> 00:29:37,990
+تستنى لما أقول من عنده سؤال من عنده حاجة صار، ترفع
+
+261
+00:29:37,990 --> 00:29:41,790
+يدك و بجاوب كانك ما بتقدر أنت تعمليني قصة مع اللغة،
+
+262
+00:29:41,790 --> 00:29:48,330
+قوم أنت .. أنت .. قوم يقعد في مطعم، يبقوا عالم،
+
+263
+00:29:48,330 --> 00:29:51,190
+فلو سرحت أنك تتكلم مش مع بعض، هانديك السفسة
+
+264
+00:30:01,910 --> 00:30:04,950
+ممنوع حد يتكلم مع الجنب في المحاضرة، أنا بعمل
+
+265
+00:30:04,950 --> 00:30:08,850
+إزعاج، بدك أنت في السفسار، عندك أي إيش أنا بواجب،
+
+266
+00:30:08,850 --> 00:30:13,990
+بقول من عنده سؤال، من عنده حاجة، يتفضل يسأل
+
+267
+00:30:13,990 --> 00:30:21,850
+ساعتها، بس لا تتكلم وأنا ضايق طرابك،
+
+268
+00:30:21,850 --> 00:30:24,230
+يقولنا الكلام قدر مئة مرة في المحاضرة، ممنوع
+
+269
+00:30:24,230 --> 00:30:25,690
+الكلام الجامل
+
+270
+00:30:35,880 --> 00:30:40,860
+تفضل يا أبو حمزة إذا
+
+271
+00:30:40,860 --> 00:30:45,380
+الآن بدنا نكمل البرنامج بإثبات الاثنين بيقود لواحد
+
+272
+00:30:45,380 --> 00:30:51,740
+الإثبات الاثنين بيقود لواحد بدنا نثبت we prove ال
+
+273
+00:30:51,740 --> 00:30:59,120
+contrapositive we prove not واحد implies not two
+
+274
+00:31:01,070 --> 00:31:04,730
+هذا هو ال contrapositive للعبارة لل implication
+
+275
+00:31:04,730 --> 00:31:16,630
+هذه فإذا assume .. assume not واحد ف not واحد معناه
+
+276
+00:31:16,630 --> 00:31:27,190
+ال limit ل F of X لما X تقول ل C لا تساوي L
+
+277
+00:31:30,200 --> 00:31:32,020
+this means هذا يعني
+
+278
+00:31:35,090 --> 00:31:40,190
+الآن نرجع لتعريف ال limit أو ال function شفنا
+
+279
+00:31:40,190 --> 00:31:42,530
+المرة السادسة في تعريفين في epsilon delta
+
+280
+00:31:42,530 --> 00:31:46,270
+definition و في neighborhood definition ال 
+
+281
+00:31:46,270 --> 00:31:49,610
+neighborhood definition بيقول إذا كان عشان تكون
+
+282
+00:31:49,610 --> 00:31:53,770
+limit لـ f of x من x أو لـ c بالساوي عدد L هذا
+
+283
+00:31:53,770 --> 00:31:57,210
+بيكافئ أنه لكل epsilon neighborhood لـ L يوجد delta
+
+284
+00:31:57,210 --> 00:32:01,130
+neighborhood للـ C بحيث لكل x في ال delta
+
+285
+00:32:01,130 --> 00:32:04,630
+neighborhood صورته لازم تطلع في الـ epsilon
+
+286
+00:32:04,630 --> 00:32:08,290
+neighborhood الآن أن في الكلام هذا ما معنى أن ال
+
+287
+00:32:08,290 --> 00:32:13,570
+limit of x at c بيستويش لعدد L معناه بدل لكل epsilon
+
+288
+00:32:13,570 --> 00:32:17,930
+neighborhood لـ L there exists there exists epsilon
+
+289
+00:32:17,930 --> 00:32:25,330
+zero neighborhood of L بسميه
+
+290
+00:32:25,330 --> 00:32:32,110
+V epsilon zero neighborhood لـ L بحيث أنه لكل 
+
+291
+00:32:33,900 --> 00:32:43,060
+Delta neighborhood V Delta أو C يوجد X يعتمد على
+
+292
+00:32:43,060 --> 00:32:50,540
+Delta ينتمي إلى A ومختلف عن الـ C وموجود في الـ
+
+293
+00:32:50,540 --> 00:32:55,560
+Delta neighborhood بحيث
+
+294
+00:32:55,560 --> 00:33:01,100
+أن صورة الـ X Delta
+
+295
+00:33:05,360 --> 00:33:16,380
+لا تنتمي للإبسلون zero neighborhood لـ LL طيب
+
+296
+00:33:16,380 --> 00:33:26,140
+لو أخذنا take لكل N في N take delta بساوي واحد على
+
+297
+00:33:26,140 --> 00:33:31,600
+N then
+
+298
+00:33:31,600 --> 00:33:32,540
+there exists
+
+299
+00:33:37,520 --> 00:33:47,100
+دلتا تعتمد على n دلتا تعتمد على n دلتا تعتمد على n
+
+300
+00:33:47,100 --> 00:33:50,360
+دلتا 
+
+301
+00:33:50,360 --> 00:33:56,520
+تعتمد
+
+302
+00:33:56,520 --> 00:33:57,620
+على
+
+303
+00:34:00,880 --> 00:34:10,260
+و بحيث أن F لـ Xm لا ينتمي لإبسلون Zero
+
+304
+00:34:10,260 --> 00:34:18,300
+neighborhood لـ L طب ما هذا الأخير معناه أو بيقدّي
+
+305
+00:34:26,330 --> 00:34:34,390
+this implies هذا بيقدّي نكون أثبتنا أن لكل n يوجد
+
+306
+00:34:34,390 --> 00:34:42,490
+xn في a إذا يوجد sequence xn موجودة في ال set A
+
+307
+00:34:42,490 --> 00:34:47,170
+حدودها مختلفة عن ال C كل ال xn مختلفة عن ال C
+
+308
+00:34:47,170 --> 00:35:01,980
+وموجودة في v1 على n of c بحيث أن f ل xn لا تنتمي ل
+
+309
+00:35:01,980 --> 00:35:11,900
+v epsilon zero ل n لكل n هذا معناه أن يوجد
+
+310
+00:35:11,900 --> 00:35:20,580
+sequence xn contained in a minus c بحيث أن لاحظوا
+
+311
+00:35:20,580 --> 00:35:26,240
+الـ sequence Xn تنتمي لـ V 1 على N of C اللي هو
+
+312
+00:35:26,240 --> 00:35:30,960
+عبارة عن الفترة C سالب واحد على N C موجب واحد على
+
+313
+00:35:30,960 --> 00:35:37,360
+N لكل N هذا معناه أن absolute Xn minus C أصغر من
+
+314
+00:35:37,360 --> 00:35:42,420
+واحد على N أصغر
+
+315
+00:35:42,420 --> 00:35:47,020
+من واحد على N لكل N في N and
+
+316
+00:35:50,460 --> 00:35:55,500
+F of Xn لا تنتمي للـY0 neighborhood الـY0
+
+317
+00:35:55,500 --> 00:35:59,720
+neighborhood هذا عبارة عن الفترة المفتوحة L minus
+
+318
+00:35:59,720 --> 00:36:08,000
+Y0 L زائد Y0 فF of Xn لا تنتمي للفترة المفتوحة هذه
+
+319
+00:36:08,000 --> 00:36:15,720
+معناه absolute المسافة بين F of Xn وL أكبر من أو
+
+320
+00:36:15,720 --> 00:36:18,460
+ساوي Y0 لكل N
+
+321
+00:36:21,430 --> 00:36:26,750
+هذا الكلام معناه أن
+
+322
+00:36:26,750 --> 00:36:32,190
+يوجد sequence x in موجودة في a حدودها مختلفة عن ال
+
+323
+00:36:32,190 --> 00:36:41,030
+c وهذا الكلام معناه such that limit x in بساوي c
+
+324
+00:36:43,330 --> 00:36:51,410
+حسب نظرية 2.4.2
+
+325
+00:36:51,410 --> 00:36:54,970
+4.2 4.2 4.2 4.2
+
+326
+00:36:54,970 --> 00:36:55,590
+4.2 4.2 4.2 4.2 4
+
+327
+00:36:55,590 --> 00:36:55,730
+4.2 4.2 4.2 4.2 4
+
+328
+00:36:55,730 --> 00:36:56,990
+2.4.2 4.2 4.2 4.2 4.2
+
+329
+00:36:56,990 --> 00:36:59,130
+4.2 4.2 4.2 4.2 4
+
+330
+00:36:59,130 --> 00:37:08,450
+2.4.2 4.2 4.2 4
+
+331
+00:37:12,640 --> 00:37:16,920
+الـ limit لـ
+
+332
+00:37:16,920 --> 00:37:20,760
+sequence f of xn لما n تقول الـ infinity مش ممكن
+
+333
+00:37:20,760 --> 00:37:26,540
+تساوي العدد L لأن لو ال limit لـ f of xn بيساوي
+
+334
+00:37:26,540 --> 00:37:30,220
+العدد L، المفروض ال absolute value للفرق ده تكون
+
+335
+00:37:30,220 --> 00:37:36,660
+أصغر من أي epsilon zero لكل N من capital N و أنت
+
+336
+00:37:36,660 --> 00:37:40,930
+طالع، لكن هذا الكلام مش صحيح Okay إن هذا بالظبط
+
+337
+00:37:40,930 --> 00:37:48,210
+العبارة الأخيرة which which
+
+338
+00:37:48,210 --> 00:37:55,690
+is نفي العبارة 2 هذه
+
+339
+00:37:55,690 --> 00:37:58,450
+العبارة الأخيرة هي نفي العبارة 2 هذه العبارة
+
+340
+00:37:58,450 --> 00:38:06,320
+2 ال statement 2 بيقول لكل sequence بحيث أن
+
+341
+00:38:06,320 --> 00:38:09,100
+ال limit بتاعتها C، ال limit لل image بتاعتها
+
+342
+00:38:09,100 --> 00:38:13,660
+بالساولة L هنا اتوصلنا أن there exist بدل for all
+
+343
+00:38:13,660 --> 00:38:18,660
+there exists sequence نهايتها C لكن نهاية صورتها
+
+344
+00:38:18,660 --> 00:38:25,020
+لا تساول L إذا هيك بنكون أثبتنا أنه لا إذا we
+
+345
+00:38:25,020 --> 00:38:29,800
+proved not
+
+346
+00:38:31,130 --> 00:38:39,390
+not 1 implies not 2 therefore 2 implies 1
+
+347
+00:38:39,390 --> 00:38:46,610
+وهذا يكمل البرهان واضح؟ في أي سؤال؟ في أي استفسار؟
+
+348
+00:38:46,610 --> 00:38:53,590
+يبدو أننا كملنا برهان النظرية في أي استفسار؟
+
+349
+00:38:55,700 --> 00:39:03,780
+الآن من النظرية هذه ينتج مباشرة نظرية مهمة لتقل
+
+350
+00:39:03,780 --> 00:39:13,660
+عنها أهمية ويلها اسم divergence
+
+351
+00:39:13,660 --> 00:39:16,900
+criteria
+
+352
+00:39:25,650 --> 00:39:36,650
+لت if the function from A to R and see the cluster
+
+353
+00:39:36,650 --> 00:39:39,750
+point
+
+354
+00:39:39,750 --> 00:39:45,850
+of A then 1
+
+355
+00:39:47,360 --> 00:39:54,460
+الـ limit لـ f of x لما x تقول لـ c لا تساوي ال f
+
+356
+00:39:54,460 --> 00:40:01,440
+and only if there exists a sequence xm contained in
+
+357
+00:40:01,440 --> 00:40:10,180
+a حدودها مختلفة عن ال c such that limit xm بتساوي
+
+358
+00:40:10,180 --> 00:40:20,490
+c but limit f of x in لا تساوي n الكرتيريا
+
+359
+00:40:20,490 --> 00:40:25,750
+التانية اللي هي عشان
+
+360
+00:40:25,750 --> 00:40:31,930
+نقول limit f of x لما x تقولها c does not exist in
+
+361
+00:40:31,930 --> 00:40:43,690
+R هذا بكافئ أن هناك sequence Xn محتوى A حدودها
+
+362
+00:40:43,690 --> 00:40:50,870
+مختلفة عن C بحيث أن نهايتها بساوي
+
+363
+00:40:50,870 --> 00:41:00,310
+C ولكن نهاية صورتها لا
+
+364
+00:41:00,310 --> 00:41:02,670
+توجد في R
+
+365
+00:41:16,230 --> 00:41:21,250
+كمان النظرية هذه برهانها ينتج مباشرة من النظرية
+
+366
+00:41:21,250 --> 00:41:27,990
+اللي فوق مثلا هي عندي لإثبات ال band الأول عشان
+
+367
+00:41:27,990 --> 00:41:31,130
+أثبت limit f of x  مستويش L and C
+
+368
+00:41:34,380 --> 00:41:38,880
+يعني كإني بقول نفي العبارة 1 هذا هو نفي العبارة
+
+369
+00:41:38,880 --> 00:41:42,560
+1 طب احنا لسه بثبتين ان 1 بكافي 2
+
+370
+00:41:42,560 --> 00:41:46,560
+وبالتالي نفي العبارة 1 بكافي نفي الـ 2 فنفي
+
+371
+00:41:46,560 --> 00:41:51,100
+الـ 2 هذا هو يوجد a sequence تتقارب لـ C لكن صورة
+
+372
+00:41:51,100 --> 00:41:56,720
+تلاتة تتقارب لـ L إذا برهان الجزء الأول نتيجة مباشرة
+
+373
+00:41:56,720 --> 00:42:02,130
+على مضارية ال form والجزء الثاني زيه بدل هنا عشان
+
+374
+00:42:02,130 --> 00:42:06,070
+أقول أن ال limit هذه does not exist يعني لو أخدت
+
+375
+00:42:06,070 --> 00:42:12,650
+أي عدد L فال limit هنا لا تساوي L معناه أنه في
+
+376
+00:42:12,650 --> 00:42:18,050
+sequence والكلام هذا ال limit هذه ما تسويش أي L أي
+
+377
+00:42:18,050 --> 00:42:23,890
+عدد حقيقي إذا النظرية هذه نتيجة مباشرة على النظرية
+
+378
+00:42:23,890 --> 00:42:27,880
+sequential criterion النظرية التي سبقتها الآن هذه
+
+379
+00:42:27,880 --> 00:42:31,560
+النظرية هنستخدمها في إثبات إن ال limit لدالة
+
+380
+00:42:31,560 --> 00:42:36,000
+معينة، عن نقطة معينة غير موجودة، فهي بعض الأمثلة
+
+381
+00:42:36,000 --> 00:42:39,140
+كيف
+
+382
+00:42:39,140 --> 00:42:42,500
+نستخدم ال divergence كتير، كيف نثبت ال divergence
+
+383
+00:42:42,500 --> 00:42:48,020
+أو عدم وجود limit لدالة معينة عن نقطة معينة، فمثلا
+
+384
+00:42:48,020 --> 00:43:02,210
+ناخد أول مثال show that limit لـ 1 على x لما x تقول
+
+385
+00:43:02,210 --> 00:43:09,470
+إلى صفر does not exist in R فلبرهان
+
+386
+00:43:09,470 --> 00:43:16,870
+ذلك let
+
+387
+00:43:16,870 --> 00:43:24,050
+f of x بساوي 1 على x وده أخد الـ x موجبة يعني
+
+388
+00:43:24,050 --> 00:43:27,130
+نعتبر أن ال domain للدالة هذه اللي هو الفترة A
+
+389
+00:43:27,130 --> 00:43:31,270
+بساوي الفترة مفتوحة من صفر لما لا نهاية و نثبت
+
+390
+00:43:31,270 --> 00:43:34,750
+أن الدالة هذه ما ليهاش limit عند الصفر أو عند الصفر
+
+391
+00:43:34,750 --> 00:43:40,650
+من اليمين فلإثبات أن ال limit للدالة هذه عند الصفر
+
+392
+00:43:40,650 --> 00:43:44,470
+ماهياش موجودة حسب ال divergence criteria يعني بدي
+
+393
+00:43:44,470 --> 00:43:48,210
+أثبت أن يوجد .. بدي أجيب sequence نهايتها صفر لكن
+
+394
+00:43:48,210 --> 00:43:52,490
+نهاية صورتها مش موجودة فال sequence إذا هنا
+
+395
+00:43:52,490 --> 00:43:59,560
+consider الـ sequence التي تفي بهذا الغرض اللي هي
+
+396
+00:43:59,560 --> 00:44:06,400
+xn بساوي واحد على n لكل n في n فواضح أنه limit
+
+397
+00:44:06,400 --> 00:44:16,720
+xn بساوي limit واحد على n بتساوي صفر وواضح أنه
+
+398
+00:44:16,720 --> 00:44:22,800
+xn contained in a اللي هي الفترة هذه معدى الصفر
+
+399
+00:44:22,800 --> 00:44:31,310
+صح؟ وعندي ال limit لل image لل sequence xn بساوي ال
+
+400
+00:44:31,310 --> 00:44:38,250
+limit لـ 1 على xn لما n تقوى ل infinity بساوي ال
+
+401
+00:44:38,250 --> 00:44:43,410
+limit لـ n لما n تقوى ل infinity بساوي infinity
+
+402
+00:44:43,410 --> 00:44:49,730
+وهذه طبعا ال infinity does not exist in R ليست عدد
+
+403
+00:44:49,730 --> 00:44:55,980
+حقيقي النهاية نجحت في إيجاد sequence موجودة في A
+
+404
+00:44:55,980 --> 00:45:00,840
+وحدودها مختلفة عن الصفر ونهايتها صفر لكن نهاية
+
+405
+00:45:00,840 --> 00:45:06,960
+صورتها مش موجودة في R وبالتالي therefore by
+
+406
+00:45:06,960 --> 00:45:14,020
+divergence criterion limit
+
+407
+00:45:14,020 --> 00:45:23,240
+لـ F of X أو 1 على X لما x س تقول إلى 0 does not
+
+408
+00:45:23,240 --> 00:45:28,860
+exist in R وفي حقيقة الأمر أثبتنا أن limit 1 على x
+
+409
+00:45:28,860 --> 00:45:34,180
+لما x س تقول إلى 0 من اليمين غير موجودة لأن أخذنا
+
+410
+00:45:34,180 --> 00:45:42,620
+المجال كل الأعداد الموجودة بالمثل ممكن إثبات أن
+
+411
+00:45:42,620 --> 00:45:50,390
+limit لـ 1 على x لما X تقول إلى صفر من اليسار does
+
+412
+00:45:50,390 --> 00:45:55,540
+not exist أن أنا أخد المرة هذه ال X هنا في الدالة
+
+413
+00:45:55,540 --> 00:46:00,560
+هذه ال domain تبعها الفترة من سالب ما له نهاية إلى
+
+414
+00:46:00,560 --> 00:46:05,720
+صفر وأقول أن ال X هنا أصغر من صفر ونفس البرهان
+
+415
+00:46:05,720 --> 00:46:09,820
+هيطلع عندي ال limit لما X تقول صفر من اليسار does
+
+416
+00:46:09,820 --> 00:46:13,500
+not exist وبالتالي ال limit عند ال X من الجهتين
+
+417
+00:46:13,500 --> 00:46:19,820
+does not exist تمام okay هذا مثال مثال ثاني واضح
+
+418
+00:46:19,820 --> 00:46:21,440
+فيه أي سفصار فيه أي سؤال
+
+419
+00:46:25,390 --> 00:46:35,810
+ناخد مثال ثاني show
+
+420
+00:46:35,810 --> 00:46:42,590
+that limit للـ signum function signum x لما x تقول 
+
+421
+00:46:42,590 --> 00:46:48,930
+إلى صفر does not exist where حيث و ال signum
+
+422
+00:46:48,930 --> 00:46:52,450
+function where
+
+423
+00:46:57,580 --> 00:47:02,460
+where signum x هي عبارة عن function في x بنعرفها
+
+424
+00:47:02,460 --> 00:47:07,000
+على أنها واحد إذا كان x أكبر من صفر صفر إذا كان x
+
+425
+00:47:07,000 --> 00:47:12,300
+بساوى صفر سالب واحد إذا كان x أصغر من صفر وهي
+
+426
+00:47:12,300 --> 00:47:13,360
+الرسمة تبعتها
+
+427
+00:47:24,680 --> 00:47:28,920
+فالدالة لما x أكبر من صفر بيستوي ثابت واحد عند
+
+428
+00:47:28,920 --> 00:47:34,400
+الصفر بيستوي صفر و لما x أصغر من صفر بيستوي سالب
+
+429
+00:47:34,400 --> 00:47:38,040
+واحد طيب
+
+430
+00:47:38,040 --> 00:47:48,360
+note that لاحظوا أن الدالة هذه sigma of x بتساوي
+
+431
+00:47:48,360 --> 00:47:52,120
+x على absolute x fx
+
+432
+00:47:53,900 --> 00:47:59,820
+لا تساوي صفر إذا كان x بساوي صفر فدالة sigma بها
+
+433
+00:47:59,820 --> 00:48:07,900
+نفس x على absolute x نفس .. نفس الحاجة طيب الآن
+
+434
+00:48:07,900 --> 00:48:13,400
+إثبات أن ال limit لدالها عند صفر مش موجودة طبعا
+
+435
+00:48:13,400 --> 00:48:17,440
+في تفاضل ألف في برهان في تفاضل ألف بيقول أن هي
+
+436
+00:48:17,440 --> 00:48:21,380
+الدالة لما X تؤول إلى صفر من اليمين ال limit لها واحد
+
+437
+00:48:21,380 --> 00:48:25,500
+لما X تؤول إلى صفر من اليسار نهايتها سالب واحد ال limit
+
+438
+00:48:25,500 --> 00:48:28,040
+من اليمين ليستوي ال limit من اليسار إذا ال limit
+
+439
+00:48:28,040 --> 00:48:33,690
+لدالها عند صفر does not exist برهان accurate صحيح
+
+440
+00:48:33,690 --> 00:48:37,030
+مئة بالمئة ما في مشكلة لكن لو بدنا نعطي برهان
+
+441
+00:48:37,030 --> 00:48:41,810
+باستخدام ال divergence criterion فالبرهان هيكون
+
+442
+00:48:41,810 --> 00:48:46,270
+كالتالي consider
+
+443
+00:48:46,270 --> 00:48:51,410
+بدنا نجيب sequence xn
+
+444
+00:48:54,550 --> 00:48:58,490
+التي نهايتها مختلفة عن الصفر نهايتها صفر لكن نهايت
+
+445
+00:48:58,490 --> 00:49:03,950
+صورتها بساوي صفر ف consider ال sequence اللي هي Xn
+
+446
+00:49:03,950 --> 00:49:09,110
+الحد العام تبعها Xn بساوي سالب واحد أس ان على N
+
+447
+00:49:09,110 --> 00:49:19,190
+لكل N في N ال sequence هذه تنتمي إلى A اللي هو R
+
+448
+00:49:19,190 --> 00:49:21,890
+بعد الصفر
+
+449
+00:49:26,050 --> 00:49:29,530
+موجودة في المجال تبع الدالة المجال تبع الدالة دي
+
+450
+00:49:29,530 --> 00:49:37,570
+كل الأعداد اللي حصلت فيها معدد R صح؟ وعندي و ال
+
+451
+00:49:37,570 --> 00:49:44,610
+limit و ال limit ل XM as M tends to infinity بساوى
+
+452
+00:49:44,610 --> 00:49:50,150
+و ال limitلسالب واحد أس ان على ان لما ان تقول
+
+453
+00:49:50,150 --> 00:49:55,110
+infinity ال limit لل sequence دي ايش بيساوي بيساوي
+
+454
+00:49:55,110 --> 00:50:03,470
+صفر by squeeze theorem او
+
+455
+00:50:03,470 --> 00:50:08,050
+by sandwich theorem but
+
+456
+00:50:08,050 --> 00:50:15,650
+لكن تعالوا نشوف ال limitلـ f of xn as n tends to
+
+457
+00:50:15,650 --> 00:50:19,810
+infinity شو بيساوي؟ بيساوي الـ limit as n tends to
+
+458
+00:50:19,810 --> 00:50:25,750
+infinity احنا عندي الـ xn هنا بيستويش صفر وبالتالي
+
+459
+00:50:25,750 --> 00:50:30,250
+الـ f of x تبعتي اللي هي الـ signum function فهذا
+
+460
+00:50:30,250 --> 00:50:34,050
+بيساوي limit signum xn
+
+461
+00:50:36,620 --> 00:50:41,540
+مظبوط و ال x in قلنا هنا بساويش 0 وبالتالي هذا
+
+462
+00:50:41,540 --> 00:50:47,000
+عبارة عن limit as n tends to infinity ال signal ل
+
+463
+00:50:47,000 --> 00:50:55,420
+x in بساوي x in على absolute x in فهذا
+
+464
+00:50:55,420 --> 00:51:02,210
+بساوي ال limit as n tends to infinity لـ xn عبارة
+
+465
+00:51:02,210 --> 00:51:09,190
+عن سالب واحد أس n على n على absolute xn absolute
+
+466
+00:51:09,190 --> 00:51:16,530
+xn بساوي واحد على n أصحبت؟ الآن نجسم ونبسط ال limit
+
+467
+00:51:16,530 --> 00:51:23,750
+as n tends to infinity بطلع سالب واحد أس n وال
+
+468
+00:51:23,750 --> 00:51:27,210
+sequence هذه ال limit تبعتها أثبتنا قبل هيك
+
+469
+00:51:28,730 --> 00:51:33,410
+بطريقتين على الأقل أن ال limit هذه does not exist
+
+470
+00:51:33,410 --> 00:51:44,830
+does not exist وبالتالي إذا either by the
+
+471
+00:51:44,830 --> 00:51:47,630
+divergence criterion
+
+472
+00:51:50,230 --> 00:51:54,070
+هي أثبتت أن الـ use and sequence موجودة في المجال
+
+473
+00:51:54,070 --> 00:51:58,970
+تبع الدالة معدى الصفر نهايتها صفر لكن نهاية صورتها
+
+474
+00:51:58,970 --> 00:52:03,270
+does not exist إذا by ال band الثاني من ال
+
+475
+00:52:03,270 --> 00:52:11,590
+divergence criterion ال limit لل
+
+476
+00:52:11,590 --> 00:52:17,490
+signum function لما X تقول الصفر does not exist
+
+477
+00:52:17,490 --> 00:52:18,570
+غير موجودة
+
+478
+00:52:20,890 --> 00:52:26,890
+Okay تمام واضح واضح البرهان في أي استفسار في أي
+
+479
+00:52:26,890 --> 00:52:34,470
+سؤال Okay
+
+480
+00:52:34,470 --> 00:52:39,470
+نوقف هنا وإن شاء الله بنكمل المرة الجاية في بعض
+
+481
+00:52:39,470 --> 00:52:45,290
+مثالين الموجودة في الكتاب حاولوا تقرؤهم أو مثال
+
+482
+00:52:46,220 --> 00:52:50,740
+الشباب بالمثال هذا حاولوا تقرؤوا والمرة الجاية
+
+483
+00:52:50,740 --> 00:52:52,580
+هنبدأ section جديد 
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/_28CmIWMuzY.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/_28CmIWMuzY.srt
new file mode 100644
index 0000000000000000000000000000000000000000..eff803b8fce75ff778dd59227c57dfd2d0d939f9
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/_28CmIWMuzY.srt
@@ -0,0 +1,1615 @@
+1
+00:00:21,330 --> 00:00:27,290
+اليوم طبعا هنكمل الشرح
+
+2
+00:00:27,290 --> 00:00:30,650
+أو 
+
+3
+00:00:30,650 --> 00:00:35,610
+بعض الملاحظات على النظرية اللي أخذناها في المحاضرة 
+
+4
+00:00:35,610 --> 00:00:42,910
+السابقة النظرية هذه بتتحدث عن nested interval
+
+5
+00:00:42,910 --> 00:00:48,620
+property أو خاصية الفترات المتداخلة وشفنا في 
+
+6
+00:00:48,620 --> 00:00:54,720
+النظرية هذه أن لو في عندي sequence of nested
+
+7
+00:00:54,720 --> 00:00:58,660
+intervals الفترات هذه كلهم nested يعني كل واحدة 
+
+8
+00:00:58,660 --> 00:01:05,820
+تحتوي اللي بعدها مباشرة، زائد أن الفترات هذه كلهم 
+
+9
+00:01:05,820 --> 00:01:14,580
+closed كلهم closed و bounded ففي 
+
+10
+00:01:14,580 --> 00:01:20,210
+الحالة هذه التقاطع تبع الـ sequence of intervals لا 
+
+11
+00:01:20,210 --> 00:01:24,310
+يساوي  ∅ يعني في على الأقل عنصر واحد بالتقاطع 
+
+12
+00:01:24,310 --> 00:01:30,510
+شفنا برضه لو في الشرط الإضافي هذا اتحقق وهو 
+
+13
+00:01:30,510 --> 00:01:35,570
+لاحظوا أن هذه عبارة عن ℕ  فهذه sequence من 
+
+14
+00:01:35,570 --> 00:01:42,690
+العداد الطبيعية الغير سالبة، و بالمناسبة الـ infimum واضح 
+
+15
+00:01:42,690 --> 00:01:48,940
+أنه lower bound للمجموعة هذه، صح؟ لكن مش شرط أن 
+
+16
+00:01:48,940 --> 00:01:54,780
+الـ infimum يكون هو الـ infimum للمجموعة هذه، فإذا كان الـ
+
+17
+00:01:54,780 --> 00:01:57,960
+infimum للمجموعة هذه اللي هو أكبر lower bound هو 
+
+18
+00:01:57,960 --> 00:02:06,440
+الـ infimum فالتقاطع فيها عنصر واحد، okay تمام وشفنا 
+
+19
+00:02:06,440 --> 00:02:11,800
+مرتين على البرهان المرة اللي فاتت، و أعتقد أن 
+
+20
+00:02:11,800 --> 00:02:16,860
+البرهان مكتوب بالتفصيل واضح ومرينا عليه جزء جزء 
+
+21
+00:02:16,860 --> 00:02:22,000
+فأرجو أن تكونوا قرأتوها كمان مرة وفهمتوها، في حد 
+
+22
+00:02:22,000 --> 00:02:27,860
+عنده استفسار على البرهان أو النظرية هذه؟ طيب الآن 
+
+23
+00:02:27,860 --> 00:02:35,820
+النظرية هذه، نرجع للنظرية كمان مرة الآن 
+
+24
+00:02:35,820 --> 00:02:41,480
+في ملاحظة بتقول أن لو أنا في النظرية هذه الفترات 
+
+25
+00:02:41,480 --> 00:02:49,780
+هذه، الفرض أن الفترات in مغلقة closed، لو حذفت، شيلت 
+
+26
+00:02:49,780 --> 00:03:01,600
+الفرض هذا فالنظرية هذه بتبطل تكون صحيحة، فالنظرية 
+
+27
+00:03:01,600 --> 00:03:05,000
+هذه بتبطل تكون صحيحة، وحنشوف counter example يوضح 
+
+28
+00:03:05,000 --> 00:03:07,460
+عدم صحتها، كذلك 
+
+29
+00:03:09,100 --> 00:03:13,220
+طب افرض أن هذا الشرط متحقق في الفترات، لكن اللي مش 
+
+30
+00:03:13,220 --> 00:03:17,680
+متحقق اللي هو الـ boundedness، يعني الفترات هذه ليست 
+
+31
+00:03:17,680 --> 00:03:21,420
+محدودة، ليست bounded، برضه في الحالة هذه النظرية 
+
+32
+00:03:21,420 --> 00:03:26,620
+تفشل، و في counter example يوضح فشلها، okay إذا 
+
+33
+00:03:26,620 --> 00:03:30,640
+حنشوف two counter examples، خليني نشوفهم مع بعض 
+
+34
+00:03:36,610 --> 00:03:39,790
+إذا هذه الـ remark اللي أنا اتحدث عنها قلت أن it 
+
+35
+00:03:39,790 --> 00:03:44,090
+should be noted يجب ملاحظة أن generally بصورة عامة 
+
+36
+00:03:44,090 --> 00:03:48,030
+a nested sequence of intervals need not have a
+
+37
+00:03:48,030 --> 00:03:51,290
+common point يعني لو فيه nested sequence من 
+
+38
+00:03:51,290 --> 00:03:57,010
+الفترات المتداخلة مش شرط تقاطعهم يكون في  ∅ يعني 
+
+39
+00:03:57,010 --> 00:04:02,650
+أي نقطة أو نقطة مشاركة، يعني مش شرط أن التقاطع لها 
+
+40
+00:04:02,650 --> 00:04:11,000
+يساوي  ∅ فالآن  هذه هي اللي حكينا عنها أول 
+
+41
+00:04:11,000 --> 00:04:18,500
+مثال، هذه 
+
+42
+00:04:18,500 --> 00:04:23,080
+في المثال الأول الفرض the hypothesis الفرض أن الـ 
+
+43
+00:04:23,080 --> 00:04:28,940
+intervals I<sub>n</sub> في نظرية 22 be closed cannot be
+
+44
+00:04:28,940 --> 00:04:34,800
+dropped يعني لا يمكن حذفه، لا يمكن الاستغناء عنه 
+
+45
+00:04:34,800 --> 00:04:41,180
+وتبقى النظرية نظرية صحيحة، for example على سبيل 
+
+46
+00:04:41,180 --> 00:04:49,120
+المثال لو أخذت الفترات I<sub>n</sub>، الفترة I<sub>n</sub> هي الفترة 
+
+47
+00:04:49,120 --> 00:04:55,580
+المفتوحة من 0 لـ 1/n حيث n عدد طبيعي، فواضح أن 
+
+48
+00:04:55,580 --> 00:05:00,460
+الفترات هذه nested صح؟ لأن الفترة الأولى هتكون 
+
+49
+00:05:00,460 --> 00:05:04,820
+الفترة المفتوحة من 0 لـ 1، الفترة الثانية الفترة المفتوحة 
+
+50
+00:05:04,820 --> 00:05:12,180
+من 0 لـ 1/2، وهذه محتوى في I<sub>1</sub>، و I<sub>3</sub> الفترة 
+
+51
+00:05:12,180 --> 00:05:16,540
+المفتوحة من 0 لـ 1/3 محتوى داخل I<sub>2</sub>، و هكذا لذلك 
+
+52
+00:05:16,540 --> 00:05:21,720
+واضح أن الـ sequence of open intervals I<sub>n</sub> is nested
+
+53
+00:05:21,720 --> 00:05:27,560
+sequence كذلك عناصر الـ sequence هذه bounded، هذه 
+
+54
+00:05:27,560 --> 00:05:33,710
+فترات محصورة لكن الفترات هذه not closed مش closed
+
+55
+00:05:33,710 --> 00:05:38,630
+يعني عبارة عن مجموعات مفتوحة ليست مغلقة، إذن شرط 
+
+56
+00:05:38,630 --> 00:05:45,910
+الإغلاق هنا انحذف، وبالتالي نتيجة النظرية مش شرط 
+
+57
+00:05:45,910 --> 00:05:50,750
+تكون صحيحة، إذن التقاطع هنا لنفس الـ sequence هذه 
+
+58
+00:05:50,750 --> 00:05:54,410
+بيطلع بيساوي ∅ مافيش common point، مافيش نقطة 
+
+59
+00:05:54,410 --> 00:05:59,950
+مشتركة في هذه الفترات، طبعا هذا مش واضح 
+
+60
+00:06:04,230 --> 00:06:08,470
+هذا تقاطع الفترات المفتوحة هذه بيساوي ∅ هذا مش 
+
+61
+00:06:08,470 --> 00:06:14,310
+واضح يحتاج إلى برهان، هي البرهان بين قوسين مربعين
+
+62
+00:06:14,310 --> 00:06:21,470
+تعالوا نبرهن أن تقاطع الفترات هذه بيساوي ∅ to see 
+
+63
+00:06:21,470 --> 00:06:27,670
+this to see this معناه لبرهان ذلك مش لرأي ذلك، هذا 
+
+64
+00:06:27,670 --> 00:06:34,040
+تعبير مجازي استخدمه لبرهان الشيء، العبارة اللي احنا 
+
+65
+00:06:34,040 --> 00:06:38,400
+عايزينها، ف to see this suppose in the contrary
+
+66
+00:06:38,400 --> 00:06:43,320
+بنفترض على النقيض أن التقاطع هذا بيساويش ∅ يعني في 
+
+67
+00:06:43,320 --> 00:06:48,100
+على الأقل عنصر x في التقاطع بنصل لتناقض، طيب الـ x 
+
+68
+00:06:48,100 --> 00:06:53,360
+موجود في التقاطع معناته x موجود في I<sub>n</sub> لكل n، إذن x
+
+69
+00:06:53,360 --> 00:06:58,310
+موجود في كل واحدة من الفترات I<sub>n</sub>، طيب x موجود في 
+
+70
+00:06:58,310 --> 00:07:03,510
+الفترة I<sub>n</sub> معناته x أكبر من 0 أصغر من 1/n 
+
+71
+00:07:03,510 --> 00:07:09,970
+أصغر من 1/n، أصغر من 1/n تمام 
+
+72
+00:07:09,970 --> 00:07:13,970
+وبالتالي 
+
+73
+00:07:13,970 --> 00:07:20,430
+حسب الـ Archimedean property، هذا عبارة عن أحد صور 
+
+74
+00:07:20,430 --> 00:07:25,750
+الـ Archimedean property بتقول بما أن x هذا عدد 
+
+75
+00:07:25,750 --> 00:07:33,530
+موجب، الـ x هذا عدد موجب، إذا يوجد عدد طبيعي n<sub>0</sub> 
+
+76
+00:07:33,530 --> 00:07:39,150
+مقلوبه وأصغر من العدد الموجب وهذا بتديني تناقض،
+
+77
+00:07:39,150 --> 00:07:47,370
+هذا بتديني تناقض، هذا بتناقض مع كون الـ x أصغر من 1 
+
+78
+00:07:47,370 --> 00:07:53,170
+على n لكل n، يعني الـ x هذه أصغر من 1/n<sub>0</sub> وهي في 
+
+79
+00:07:53,170 --> 00:07:57,210
+نفس الوقت أكبر من 1/n<sub>0</sub>، لأن هذا بتديني تناقض
+
+80
+00:07:57,210 --> 00:08:04,250
+لأن التناقض هذا سبب الـ assumption تبعنا أن يوجد x 
+
+81
+00:08:04,250 --> 00:08:09,210
+في التقاطع، لأن الصح أن التقاطع هذا مافيش فيه ولا 
+
+82
+00:08:09,210 --> 00:08:16,140
+عنصر يعني is the empty set، إن هذا مثال بيورجي أو 
+
+83
+00:08:16,140 --> 00:08:21,900
+بيوضح أنه لو حذفنا شرط أن الفترات في نظرية 22 
+
+84
+00:08:21,900 --> 00:08:26,980
+closed فبتطلع النظرية، النظرية تفشل، بتبطل النظرية 
+
+85
+00:08:26,980 --> 00:08:32,720
+و هذا مثال بيوضح فشلها، الآن المثال الثاني نفس 
+
+86
+00:08:32,720 --> 00:08:38,480
+الحاجة، الفرض أن الفترات في نظرية 22 be bounded
+
+87
+00:08:40,090 --> 00:08:43,690
+بتكون محدودة cannot be dropped لا يمكن إسقاطه
+
+88
+00:08:43,690 --> 00:08:48,250
+لا يمكن إهماله، فعشان 
+
+89
+00:08:48,250 --> 00:08:52,750
+نوضح هذا الكلام بـ counter example، ف for example
+
+90
+00:08:52,750 --> 00:08:56,750
+على سبيل المثال هناخد الفترات المغلقة I<sub>n</sub>، فترة 
+
+91
+00:08:56,750 --> 00:09:03,190
+مغلقة من n إلى ما لا نهاية حيث n عدد طبيعي، هذه 
+
+92
+00:09:03,190 --> 00:09:10,150
+الفترات كل هذه فترة مغلقة، كل فترة على الصورة هذه 
+
+93
+00:09:10,150 --> 00:09:17,010
+مغلقة إذا شرط الإغلاق متحقق، بعدين الفترات هذه nested 
+
+94
+00:09:17,010 --> 00:09:20,430
+لحظة، أول فترة هي الفترة المغلقة من واحد لما لا 
+
+95
+00:09:20,430 --> 00:09:24,450
+نهاية، الثانية فترة مغلقة من اثنين لما لا نهاية
+
+96
+00:09:24,450 --> 00:09:30,110
+وهذه محتوى في I<sub>1</sub>، الفترة الثالثة الفترة المغلقة 
+
+97
+00:09:30,110 --> 00:09:33,410
+من ثلاثة لما لا نهاية وهذه محتوى في I<sub>2</sub> وهكذا 
+
+98
+00:09:33,410 --> 00:09:38,730
+فالفترات هذه nested and closed مغلقة لكن ماهي 
+
+99
+00:09:38,730 --> 00:09:42,190
+bounded مش محصورة، it's not bounded .. هذه كمجموعة 
+
+100
+00:09:42,190 --> 00:09:48,870
+is not bounded above، كمجموعة ليس لها supremum، is 
+
+101
+00:09:48,870 --> 00:09:52,390
+not bounded above، إذن شرط الـ boundedness اختل 
+
+102
+00:09:52,390 --> 00:09:57,970
+وبالتالي نتيجة النظرية هتختلف، إذا الفترات هذه 
+
+103
+00:09:57,970 --> 00:10:03,410
+closed but unbounded وإذا هنجد أن تقاطع الفترات 
+
+104
+00:10:03,410 --> 00:10:08,930
+هذه مافيش فيه ولا نقطة، تقاطع هذا بيساوي ∅ كمان 
+
+105
+00:10:08,930 --> 00:10:15,350
+مرة، المساواة هذه بدها مش واضحة ليست واضحة، فبدنا 
+
+106
+00:10:15,350 --> 00:10:20,730
+نثبت صحة المساواة هذه كمان مرة، نعمل برهان بالتناقض
+
+107
+00:10:20,730 --> 00:10:24,370
+نعمل برهان بالتناقض 
+
+108
+00:10:29,770 --> 00:10:34,830
+فافرض أن التقاطع هذا لا يساوي ∅، وبالتالي يوجد 
+
+109
+00:10:34,830 --> 00:10:40,670
+x في التقاطع، إذا x موجود في الفترة I<sub>n</sub> لكل n، هذا 
+
+110
+00:10:40,670 --> 00:10:46,950
+من تعريف التقاطع، x موجودة في I<sub>n</sub> معناته x أكبر من 
+
+111
+00:10:46,950 --> 00:10:53,870
+أو يساوي n وهذا صحيح لكل n، هذا بتناقض مع الـ 
+
+112
+00:10:53,870 --> 00:10:58,510
+Archimedean property نظرية الأساسية نظرية 15 
+
+113
+00:10:58,510 --> 00:11:05,450
+في الشبطرة، دي اللي بتقول لأي عدد حقيقي x ينتمي إلى 
+
+114
+00:11:05,450 --> 00:11:16,530
+ℝ بتؤدي أن يوجد n<sub>0</sub> ينتمي إلى ℕ بحيث أن x أصغر من 
+
+115
+00:11:16,530 --> 00:11:17,330
+n<sub>0</sub> 
+
+116
+00:11:21,190 --> 00:11:27,210
+هذه هي الـ Archimedean property الأساسية، طيب أنا 
+
+117
+00:11:27,210 --> 00:11:32,850
+عندي الآن من الـ Archimedean property عندي يوجد عدد 
+
+118
+00:11:32,850 --> 00:11:42,060
+طبيعي n<sub>0</sub> لـ  n<sub>0</sub> أكبر من x، وعندي هنا أن x أكبر من أو 
+
+119
+00:11:42,060 --> 00:11:47,340
+يساوي n لكل n في ℕ وبالتالي x أكبر من أو يساوي n 
+
+120
+00:11:47,340 --> 00:11:51,820
+<sub>0</sub> لأن n<sub>0</sub> ينتمي إلى ℕ، فإذا عندي هنا x أكبر 
+
+121
+00:11:51,820 --> 00:11:56,180
+من أو يساوي n<sub>0</sub> و x أصغر من n<sub>0</sub>، هذا بيديني 
+
+122
+00:11:56,180 --> 00:12:02,840
+تناقض، إذا في عندي contradiction، إذا هذا العنصر غير 
+
+123
+00:12:02,840 --> 00:12:07,870
+موجود such an x does not exist يعني التقاطع هذا 
+
+124
+00:12:07,870 --> 00:12:13,550
+بيساوى ∅ كما هو مطلوب، تمام؟ واضح البرهان؟ إذن هذه 
+
+125
+00:12:13,550 --> 00:12:17,690
+مثال ثاني بوضح أن شرط الـ boundedness لا يمكن 
+
+126
+00:12:17,690 --> 00:12:25,730
+إسقاطه وتبقى nested intervals theorem صحيحة، okay؟ 
+
+127
+00:12:25,730 --> 00:12:31,710
+في نظرية ثانية يمكن هذه مرت معاكم في المبادئ لكن 
+
+128
+00:12:31,710 --> 00:12:37,170
+اليوم هنعطيلها برهان يعتمد على الـ nested intervals 
+
+129
+00:12:37,170 --> 00:12:40,610
+theorem أو nested intervals property برهان جديد 
+
+130
+00:12:40,610 --> 00:12:48,730
+غير اللي أخذته في مبادئ الرياضيات، فالنظرية 
+
+131
+00:12:48,730 --> 00:12:54,590
+هذه 24 بتتحدث عن الـ uncountability of the real
+
+132
+00:12:54,590 --> 00:12:59,560
+numbers فبكل بساطة النظرية هذه بتقول أن مجموعة 
+
+133
+00:12:59,560 --> 00:13:04,340
+الاعداد الحقيقية is uncountable the set ℝ of all 
+
+134
+00:13:04,340 --> 00:13:09,460
+real numbers is uncountable طيب 
+
+135
+00:13:09,460 --> 00:13:15,460
+ماذا يعني أن الـ set تكون countable؟ في حد فيكم 
+
+136
+00:13:15,460 --> 00:13:21,380
+بيُعرف؟ الـ set A أو S، definition
+
+137
+00:13:24,240 --> 00:13:31,920
+definition تعريف S is countable if
+
+138
+00:13:31,920 --> 00:13:46,700
+and only if  كتّب في المبادئ either أما S is finite or
+
+139
+00:13:46,700 --> 00:13:50,040
+أو 
+
+140
+00:13:50,040 --> 00:13:58,450
+S is denumerable أو في بيجيكشن one to one 
+
+141
+00:13:58,450 --> 00:14:03,850
+correspondence بينها وبين الأعداد الطبيعية يعني
+
+142
+00:14:03,850 --> 00:14:16,970
+هذا معناه it is denumerable قابلة للترقيم طيب
+
+143
+00:14:16,970 --> 00:14:23,330
+إذا كانت ال set ليست 
+
+144
+00:14:23,330 --> 00:14:29,090
+finite وليست in one to one correspondence with
+
+145
+00:14:29,090 --> 00:14:33,550
+the natural numbers أو ليست denumerable فبنسميها
+
+146
+00:14:33,550 --> 00:14:38,410
+uncountable غير قابلة للعد countable قابلة للعد
+
+147
+00:14:38,410 --> 00:14:44,750
+uncountable غير قابلة للعد طيب
+
+148
+00:14:44,750 --> 00:14:52,150
+ال
+
+149
+00:14:52,150 --> 00:14:52,390
+..
+
+150
+00:14:55,180 --> 00:15:03,200
+معروف في مبادئ رياضيات درسنا أن ال interval هذه و
+
+151
+00:15:03,200 --> 00:15:08,220
+ال interval هذه كلاهما uncountable الفترة
+
+152
+00:15:08,220 --> 00:15:11,120
+المفتوحة من صفر لواحد infinite set أول حاجة
+
+153
+00:15:11,120 --> 00:15:15,800
+infinite set و
+
+154
+00:15:15,800 --> 00:15:18,900
+طبعًا ممكن تثبت أنها uncountable
+
+155
+00:15:21,370 --> 00:15:26,370
+و طبعًا هذه الفترة المغلقة تحتوي هذه الفترة
+
+156
+00:15:26,370 --> 00:15:29,110
+المفتوحة فإذا كانت هذه uncountable هذه تكون
+
+157
+00:15:29,110 --> 00:15:35,530
+uncountable وهذا البرهان موجود في نظرية في الكتاب
+
+158
+00:15:35,530 --> 00:15:42,010
+المقرر textbook الكتاب المقرر
+
+159
+00:15:42,010 --> 00:15:47,110
+طبعًا
+
+160
+00:15:47,110 --> 00:15:50,410
+طيب
+
+161
+00:15:57,430 --> 00:16:05,570
+الـ احنا عايزين نثبت عشان نثبت أن ال set هذه ال R
+
+162
+00:16:05,570 --> 00:16:14,770
+لاحظوا أن ال R is
+
+163
+00:16:14,770 --> 00:16:18,490
+in one to one correspondence مع الفترة المفتوحة أو
+
+164
+00:16:18,490 --> 00:16:22,630
+المغلقة حتى في
+
+165
+00:16:22,630 --> 00:16:28,250
+byjection بينها وبين الفترة المفتوحة المغلقة 01
+
+166
+00:16:28,250 --> 00:16:36,890
+وبرضه المفتوحة الآن لو أثبتنا أن الفترة هذه
+
+167
+00:16:36,890 --> 00:16:44,150
+uncountable فهذه
+
+168
+00:16:44,150 --> 00:16:50,530
+الـ R in one to one correspondence معها فال R هذه
+
+169
+00:16:50,530 --> 00:16:54,400
+تطلع uncountable هذه نظرية موجودة في مبادئ
+
+170
+00:16:54,400 --> 00:16:58,080
+الرياضيات إذا كانت هناك مجموعتين
+
+171
+00:16:58,080 --> 00:17:02,860
+equivalent to each other فهناك بيجيكشن بينهم إذا
+
+172
+00:17:02,860 --> 00:17:06,540
+كانت واحدة countable فال أخرى تظهر countable إذا
+
+173
+00:17:06,540 --> 00:17:10,380
+كانت واحدة countable فال أخرى تظهر countable إذا
+
+174
+00:17:10,380 --> 00:17:14,140
+كانت هذه finite تظهر هذه finite إذا كانت هذه
+
+175
+00:17:14,140 --> 00:17:19,440
+infinite تظهر هذه infinite كل هذه نظريات موجودة في
+
+176
+00:17:19,440 --> 00:17:24,350
+مبادئ الرياضيات إذا لو أثبتنا أن الفترة هادي
+
+177
+00:17:24,350 --> 00:17:31,010
+uncountable فبتطلع R uncountable طيب
+
+178
+00:17:31,010 --> 00:17:42,050
+لإثبات ذلك هنا عايزين نثبت أن الفترة هادي نثبت أن 
+
+179
+00:17:42,050 --> 00:17:47,030
+الفترة هادي uncountable لبرهان ذلك نعمل برهان
+
+180
+00:17:47,030 --> 00:17:47,770
+بالتناقض
+
+181
+00:17:57,100 --> 00:18:01,160
+بنثبت أن الفترة المغلقة هذه uncountable نفرض
+
+182
+00:18:01,160 --> 00:18:04,940
+المقيد
+
+183
+00:18:04,940 --> 00:18:08,780
+أن الفترة هذه countable لاحظوا أن الفترة هذه
+
+184
+00:18:08,780 --> 00:18:14,500
+infinite والآن countable إذا بتطلع equipotent أو
+
+185
+00:18:14,500 --> 00:18:17,640
+in one to one correspondence with natural numbers
+
+186
+00:18:22,850 --> 00:18:26,550
+الآن في الحالة هذه I in one to one correspondence
+
+187
+00:18:26,550 --> 00:18:31,570
+with real numbers أو بنسميها innumerable صح؟
+
+188
+00:18:33,280 --> 00:18:36,560
+الآن ال set I denumerable يعني ممكن ترقيمها
+
+189
+00:18:36,560 --> 00:18:41,840
+ بالأعداد الطبيعية إذا ممكن نسمي عناصرها xn حيث n عدد
+
+190
+00:18:41,840 --> 00:18:46,340
+طبيعي اللي هي x1, x2, x3  أي set denumerable
+
+191
+00:18:46,340 --> 00:18:49,900
+أو in one to one correspondence with natural
+
+192
+00:18:49,900 --> 00:18:55,140
+numbers ممكن ترقيم عناصرها ك list by the natural
+
+193
+00:18:55,140 --> 00:18:59,200
+numbers إذا I هي كل عناصرها رقمناهم بالأعداد
+
+194
+00:18:59,200 --> 00:18:59,800
+الطبيعية
+
+195
+00:19:05,090 --> 00:19:08,350
+لحظة احنا بدنا نصل الى تناقض احنا بدنا القرآن
+
+196
+00:19:08,350 --> 00:19:15,650
+وفرضنا ال contrary هو Assume ال contrary ان I is
+
+197
+00:19:15,650 --> 00:19:19,870
+countable بدنا نصل الى تناقض طيب هي الفترة I هي
+
+198
+00:19:19,870 --> 00:19:26,890
+الفترة I هي الفترة I هذه
+
+199
+00:19:26,890 --> 00:19:31,550
+I  وفي اندكس اي هاد الفترة اي هاد يعني عناصرها هي
+
+200
+00:19:31,550 --> 00:19:37,390
+مرقمة اكس واحد اكس اتنين الى ما لا نهاية افرض ان اكس
+
+201
+00:19:37,390 --> 00:19:46,510
+واحد موجود هنا اول عنصر في الفترة موجود هنا فممكن
+
+202
+00:19:46,510 --> 00:19:54,530
+اختار فترة مغلقة اسميها اي واحد ممكن اختار فترة
+
+203
+00:19:54,530 --> 00:20:03,260
+مغلقة أسميها I1 بحيث أن ال X1 هذه لا تنتمي للفترة
+
+204
+00:20:03,260 --> 00:20:07,520
+I1 وممكن
+
+205
+00:20:07,520 --> 00:20:13,100
+اختار فترة مغلقة ثانية طب افرضي أن X2 موجودة هنا
+
+206
+00:20:13,100 --> 00:20:19,680
+العنصر الثاني في الفترة I ايه رقم X2 موجود هنا أو
+
+207
+00:20:19,680 --> 00:20:27,400
+هنا أو هنا فبقدر اختار فترة مغلقة ثانية نسميها I2
+
+208
+00:20:27,400 --> 00:20:36,120
+اللي هي الفترة هذه بحيث أن X2 لا تنتمي ل I2 و
+
+209
+00:20:36,120 --> 00:20:42,400
+الفترة I2 هي فترة جزئية من I1 طيب افرض أن X3
+
+210
+00:20:42,400 --> 00:20:50,120
+موجودة هنا أو هنا أو هنا أو أي مكان ثاني فبقدر
+
+211
+00:20:50,120 --> 00:20:58,310
+اختار فترة مغلقة تسميها I3 اللي هي الفترة هذه بحيث
+
+212
+00:20:58,310 --> 00:21:05,450
+أن X3 لا تنتمي للفترة I3 والفترة I3 جزئية من
+
+213
+00:21:05,450 --> 00:21:12,490
+الفترة I2 ممكن نستمر على هذا النمط هنحصل على
+
+214
+00:21:12,490 --> 00:21:21,110
+sequence of intervals اللي هي I1 تحتوي I2 تحتوي I3
+
+215
+00:21:22,570 --> 00:21:27,550
+و هكذا ممكن نستمر إلى ما لا نهاية و كل الفترات هذول
+
+216
+00:21:27,550 --> 00:21:32,570
+محتوى .. كل واحدة منهم محتوى داخل ال I و كل واحدة
+
+217
+00:21:32,570 --> 00:21:38,710
+من الفترات هذه صممناها بحيث أن XN لا ينتمي إلى IN
+
+218
+00:21:38,710 --> 00:21:47,190
+لكل N بيساوي واحد اثنين إلى ما لا نهاية صح؟ إذا لو
+
+219
+00:21:47,190 --> 00:21:53,130
+استمرنا في العملية هذه هنحصل على sequence of nested
+
+220
+00:21:53,130 --> 00:21:57,170
+intervals و ال intervals هدول كلهم closed و
+
+221
+00:21:57,170 --> 00:22:01,570
+bounded كلهم closed و bounded و كل الفترات هذه
+
+222
+00:22:01,570 --> 00:22:07,190
+محتوية داخل الفترة I و X in العنصر X in من ال
+
+223
+00:22:07,190 --> 00:22:14,470
+sequence هذه لا ينتمي ل I in لكل N الآن ممكن نطبق
+
+224
+00:22:14,470 --> 00:22:18,090
+nested interval property theorem اللي هي theorem
+
+225
+00:22:20,050 --> 00:22:23,550
+بتقول ده في عندي sequence of nested intervals و
+
+226
+00:22:23,550 --> 00:22:29,030
+كلهم closed و bounded فالتقاطع تبعهم لا يساوي في I
+
+227
+00:22:29,030 --> 00:22:34,650
+إذا التقاطع تبعهم موجود في نقطة واحدة دعنا نسميها
+
+228
+00:22:34,650 --> 00:22:43,030
+ساى و الفترات هذه كلها كل الفترات هذه موجودة داخل
+
+229
+00:22:43,030 --> 00:22:47,390
+الفترة I داخل الفترة I
+
+230
+00:22:53,490 --> 00:23:04,810
+ماشي هنا اه
+
+231
+00:23:04,810 --> 00:23:07,630
+ايش صار؟ هي فوق صار
+
+232
+00:23:12,680 --> 00:23:17,360
+إذا حسب نفس ال interval theorem يوجد نقطة psi في
+
+233
+00:23:17,360 --> 00:23:22,080
+تقاطع الفترات الفترات كلهم موجودين داخل I إذا
+
+234
+00:23:22,080 --> 00:23:29,540
+تقاطعهم موجود داخل I وبالتالي النقطة psi هذه
+
+235
+00:23:29,540 --> 00:23:37,060
+موجودة في كل الفترات I in لأن موجودة في تقاطعهم
+
+236
+00:23:37,060 --> 00:23:43,850
+كلهم صح إذا النقطة psi موجودة في I in لكل N و الفترة
+
+237
+00:23:43,850 --> 00:23:52,690
+I N هذه لا تحتوي من هنا الفترة I N لا تحتوي X N
+
+238
+00:23:52,690 --> 00:23:58,690
+والآن تحتوي psi إذا psi لا تساوي X N الكلام هذا
+
+239
+00:23:58,690 --> 00:24:04,430
+صحيح لكل N بظبط هذا عنصر في الفترة هذه و X N ليس
+
+240
+00:24:04,430 --> 00:24:07,970
+عنصر فيها إذا مش ممكن يكونوا متساويين صح؟
+
+241
+00:24:10,780 --> 00:24:19,120
+ال Psi قلنا هي تنتمي إلى I ال Psi موجودة في I و
+
+242
+00:24:19,120 --> 00:24:27,620
+الفترة I رقمنا عناصرها قبل شوية انا
+
+243
+00:24:27,620 --> 00:24:36,300
+في اندكس يوجد عدد حقيقي Psi ينتمي ل I و في نفس
+
+244
+00:24:36,300 --> 00:24:42,430
+الوجهة الفترة I هي كل عناصرها مُرقّمة بالعداد
+
+245
+00:24:42,430 --> 00:24:48,030
+الطبيعي عناصرها X1 و X2 إلى ما لا نهائي والآن في
+
+246
+00:24:48,030 --> 00:24:59,090
+عندي Psi مختلف عن كل عناصر الفترة I فهذا
+
+247
+00:24:59,090 --> 00:25:04,510
+بيدّي أن ال sequence أو ال set هذه is not a
+
+248
+00:25:04,510 --> 00:25:10,330
+complete enumeration of I ليست ترقيم كامل للفترة I
+
+249
+00:25:10,330 --> 00:25:15,750
+وهذا تناقض يعني
+
+250
+00:25:15,750 --> 00:25:20,530
+احنا قلنا الفترة I هذه تطلع countable و infinite
+
+251
+00:25:20,530 --> 00:25:26,830
+إذا ممكن نرقم إذا denumerable يعني ممكن نرقم عن
+
+252
+00:25:26,830 --> 00:25:31,770
+اصرها كلها بالأعداد الطبيعية وبالتالي كل عنصرها X
+
+253
+00:25:33,730 --> 00:25:43,170
+تمام؟ الآن في البرهان هذا وجدنا أن في صي عنصر جديد
+
+254
+00:25:43,170 --> 00:25:49,310
+في I مختلف عن كل عناصرها معناته هذه ال list ليست
+
+255
+00:25:49,310 --> 00:25:54,650
+ترقيم كامل ل I في عناصر أخرى زي صي مش مرقمة و هذا
+
+256
+00:25:54,650 --> 00:25:59,950
+تناقض لأن إحنا عندنا ال set I هذه countable و
+
+257
+00:25:59,950 --> 00:26:04,210
+infinite و denomable إذن هي list تبع كل عناصرها
+
+258
+00:26:04,210 --> 00:26:11,230
+فكيف طلع فيه عنصر جديد مختلف عن عناصرها فهذا تناقض
+
+259
+00:26:11,230 --> 00:26:18,140
+إذن هذا التناقض بيثبت أن فرضنا أن الفترة I كانت
+
+260
+00:26:18,140 --> 00:26:22,140
+countable كان فرض خاطئ وبالتالي الفترة I تطلع
+
+261
+00:26:22,140 --> 00:26:27,520
+uncountable إذا الآن الفترة I uncountable وأنا
+
+262
+00:26:27,520 --> 00:26:36,140
+عندي R equivalent لفترة مغلقة 01 ممكن نوجد
+
+263
+00:26:36,140 --> 00:26:40,860
+bijection بينهم إذا ال R تطلع uncountable كما هو
+
+264
+00:26:40,860 --> 00:26:45,080
+مطلوب إذا 
+
+265
+00:26:45,080 --> 00:26:50,750
+هذا هو برهان النظرية اللي أفادت هي طبعًا برهان بيعتمد
+
+266
+00:26:50,750 --> 00:26:55,430
+على شوفنا ال nested interval theorem وبالتالي هذا
+
+267
+00:26:55,430 --> 00:26:58,510
+برهان مختلف عن البرهان اللي بنعطيه في مبادئ
+
+268
+00:26:58,510 --> 00:27:05,270
+الرياضيات في برهان ثاني برضه لنظرية هذه يعطى في
+
+269
+00:27:05,270 --> 00:27:10,710
+مبادئ الرياضيات اللي هو باستخدام counter diagonal
+
+270
+00:27:10,710 --> 00:27:14,690
+argument مشهور
+
+271
+00:27:14,690 --> 00:27:20,990
+يعني البرهان يرجع إلى العالم الرياضي Cantor يسمى
+
+272
+00:27:20,990 --> 00:27:24,750
+Cantor دي أقنع ال argument بيثبت أن الفترة المفتوحة
+
+273
+00:27:24,750 --> 00:27:29,330
+من صفر لواحد is uncountable وبالتالي R is
+
+274
+00:27:29,330 --> 00:27:33,310
+uncountable لأن R في byjection بينها وبين ال open
+
+275
+00:27:33,310 --> 00:27:37,670
+interval من صفر لواحد الآن في نتيجة على نظرية
+
+276
+00:27:37,670 --> 00:27:42,490
+الأخيرة هذه ال set هذه ال R minus Q اللي هي ال 
+
+277
+00:27:42,490 --> 00:27:46,590
+set of all irrationals أيضًا is uncountable
+
+278
+00:27:46,590 --> 00:27:50,690
+والبرهان هنا نفس البرهان اللي بنعطيه في المبادئ
+
+279
+00:27:50,690 --> 00:27:55,470
+برهان by contradiction assume and contrary أن ال
+
+280
+00:27:55,470 --> 00:28:02,110
+set R minus Q is countable وإحنا عندنا نظرية في 
+
+281
+00:28:02,110 --> 00:28:07,640
+المبادئ أخذنا بتقول إن لو في عندي مجموعتين A وB وكل 
+
+282
+00:28:07,640 --> 00:28:14,140
+واحدة منهم countable فاتحادهم بيطلع countable الآن
+
+283
+00:28:14,140 --> 00:28:17,640
+أنا في عندي Q countable معروف أن Q is countable
+
+284
+00:28:17,640 --> 00:28:24,160
+والآن احنا فرضنا أن R-Q is countable إذا اتحاد
+
+285
+00:28:24,160 --> 00:28:28,420
+المجموعتين هدول اللي هو R بيطلع countable وهذا
+
+286
+00:28:28,420 --> 00:28:31,420
+بتناقض مع النتيجة اللي لسه مثبتينها في النظرية
+
+287
+00:28:31,420 --> 00:28:36,240
+السابقة Okay إذا في عندي contradiction إذا الفرض
+
+288
+00:28:36,240 --> 00:28:39,780
+أنه الست هذه countable كان خاطئ إذا الصح أنه الست
+
+289
+00:28:39,780 --> 00:28:45,280
+هذه اللي هي ال irrational number is uncountable
+
+290
+00:28:45,280 --> 00:28:57,120
+okay تمام إذا الـ مع
+
+291
+00:28:57,120 --> 00:29:01,620
+انتهاء النتيجة هذه هيك بنكون غطينا section 2
+
+292
+00:29:01,620 --> 00:29:08,660
+خمسة وهاي التمرين المطلوب تحلوها مش عايز أبدأ
+
+293
+00:29:08,660 --> 00:29:14,020
+section جديد عايز أن احنا نستغل الوقت المتبقي من
+
+294
+00:29:14,020 --> 00:29:19,160
+المحاضرة في حل أسئلة discussion يعني مناقشة فأي
+
+295
+00:29:19,160 --> 00:29:22,360
+واحدة فيكم عندها مناقشة احنا أنا عارف أن انتوا
+
+296
+00:29:22,360 --> 00:29:28,100
+هتحضروا حالكم ليوم السبت المناقشة لكن برضه أكيد
+
+297
+00:29:28,100 --> 00:29:32,040
+يعني في بينكم ناس محضرين فلو عندكم أسئلة في
+
+298
+00:29:32,040 --> 00:29:36,680
+section 2 3 أو 2 4 أو section
+
+299
+00:29:36,680 --> 00:29:41,160
+2 2 أو 2 1 فممكن نحاول نحلها في
+
+300
+00:29:41,160 --> 00:29:47,080
+الوقت المتبقي من المحاضرة ماشي الحال فإذا مين عندها
+
+301
+00:29:47,080 --> 00:29:53,540
+أي سؤال في الـ .. المحاضرات
+
+302
+00:29:53,540 --> 00:30:03,220
+السابقة أو تمارين المحاضرات السابقة من
+
+303
+00:30:03,220 --> 00:30:08,540
+لديها سؤال؟ في عندنا أسئلة كتيرة في المحاضرات
+
+304
+00:30:08,540 --> 00:30:15,470
+السابقة homework كتير مين لديها سؤال؟ مين عندها
+
+305
+00:30:15,470 --> 00:30:23,170
+سؤال؟ مين بتحلم السؤال؟ ولا
+
+306
+00:30:23,170 --> 00:30:29,690
+واحدة عندها سؤال؟ تفضل طيب
+
+307
+00:30:30,890 --> 00:30:35,570
+طبعا هذه إشارة غير يعني غير إيجابية أو إشارة سلبية
+
+308
+00:30:35,570 --> 00:30:42,530
+يعني لحد الآن أنتوا مش المادة ما بتدرسهاش دراسة
+
+309
+00:30:42,530 --> 00:30:49,530
+حقيقية وهذا معناه أن أنتوا مش ماخدينها بجد يعني
+
+310
+00:30:49,530 --> 00:30:59,690
+كما أجب وهذا دليل عليكم تحلوش مسألة ف أنا
+
+311
+00:30:59,690 --> 00:31:04,600
+هسأل عنكم خليني أحل لكم كام سؤال هاي section 2
+
+312
+00:31:04,600 --> 00:31:29,640
+3 هنا هاي
+
+313
+00:31:29,640 --> 00:31:31,000
+مثلا سؤال 4
+
+314
+00:31:35,030 --> 00:31:43,690
+هي السؤال 4 section 2 3 أنا
+
+315
+00:31:43,690 --> 00:31:51,850
+عندي set S 4 بيساوي كل الأعداد 1- -1-
+
+316
+00:31:51,850 --> 00:32:03,910
+1  /  N  حيث N عدد طبيعي والمطلوب
+
+317
+00:32:03,910 --> 00:32:04,490
+find
+
+318
+00:32:07,290 --> 00:32:17,550
+Find الـ Supremum أو الـ infimum ل S4  و أيضا الـ 
+
+319
+00:32:17,550 --> 00:32:29,950
+Supremum ل S4 طيب
+
+320
+00:32:29,950 --> 00:32:34,370
+احنا أخذنا في مثال في الـ section هذا
+
+321
+00:32:37,360 --> 00:32:39,940
+خلنا ننام هنا ولا لسه؟
+
+322
+00:33:14,820 --> 00:33:21,960
+Solution أخذنا احنا مثال بيقول أنه الـ .. لو كان في
+
+323
+00:33:21,960 --> 00:33:25,320
+.. في الـ section اللي بعد وممكن الحل باستخدام
+
+324
+00:33:25,320 --> 00:33:32,500
+المثال رقم A يعني by example
+
+325
+00:33:42,050 --> 00:33:52,450
+2 4 1 الجزء A أنا عندي الـ supremum ل A
+
+326
+00:33:52,450 --> 00:33:58,990
+زاد S A عدد حقيقي S مجموعة جزئية من R أثبتنا أن
+
+327
+00:33:58,990 --> 00:34:09,970
+هذا بيساوي A زائد supremum الـ S فلو
+
+328
+00:34:09,970 --> 00:34:24,340
+بدي أحل الجزء B ف let S بيساوي مجموعة ..
+
+329
+00:34:24,340 --> 00:34:29,740
+let
+
+330
+00:34:29,740 --> 00:34:36,560
+S بيساوي مجموعة الأعداد -1 
+
+331
+00:34:36,560 --> 00:34:42,140
+أس N على N حيث N عدد طبيعي
+
+332
+00:34:48,030 --> 00:34:54,310
+ممكن نحول السلب لموجة هاد عبارة عن -1 ..
+
+333
+00:34:54,310 --> 00:35:05,610
+-1 و نص و -1 تلت و -1 ربع و كده
+
+334
+00:35:17,860 --> 00:35:29,700
+فممكن اثبات أن الـ super mom تبع السيدتها دي
+
+335
+00:35:29,700 --> 00:35:33,820
+أستاذ
+
+336
+00:35:33,820 --> 00:35:40,840
+نعم هنحن لو سالب اللي برا دخلناه يصير -1
+
+337
+00:35:40,840 --> 00:35:45,980
+plus 1 plus 1 على أنا ممكن اه ممكن ناخد -
+
+338
+00:35:45,980 --> 00:35:50,560
+هذا فهذا أكبر أنصر هو الواحد صح هيك بتنحل الصح
+
+339
+00:35:50,560 --> 00:35:56,620
+برضه هذا ممكن فبيصير عندي هنا 1 - اول أنصر
+
+340
+00:35:56,620 --> 00:36:03,960
+1- نص فالصبر ممكن يكون 1 بعدين 1- تلت
+
+341
+00:36:03,960 --> 00:36:12,760
+ربع و هكذا فالـ supremum إذاً الـ supremum ل S بيساوي
+
+342
+00:36:12,760 --> 00:36:17,480
+هاي اللي .. لاحظ أن الأكبر عدد في الست هذه هو
+
+343
+00:36:17,480 --> 00:36:23,840
+الواحد 1 أكبر من أو يساوي كل الأعداد هذه وهو
+
+344
+00:36:23,840 --> 00:36:27,020
+أصغر upper bound إذاً الواحد upper bound للست هذه
+
+345
+00:36:27,020 --> 00:36:32,400
+هي أكبر من أو يساوي كل عناصرها وهو أصغر upper bound
+
+346
+00:36:32,400 --> 00:36:41,930
+إذاً هذا بيساوي 1 لـ س .. إيش بس يعني؟
+
+347
+00:36:41,930 --> 00:36:48,850
+ما
+
+348
+00:36:48,850 --> 00:36:54,370
+هو أصغر؟ طلع 2 2 أستاذ الـ super 2 مش
+
+349
+00:36:54,370 --> 00:37:00,770
+هي على حسب القاعدة نحن نحط أي واحد بيصير
+
+350
+00:37:00,770 --> 00:37:04,520
+2؟ لا لا احنا بنحكي عن الست هذه اللي هنا مش
+
+351
+00:37:04,520 --> 00:37:11,020
+اللي هناك هذه S وهذه S4 فبيختلفوا عن بعض الست هذه
+
+352
+00:37:11,020 --> 00:37:15,760
+هذا هي أنصرها فما
+
+353
+00:37:15,760 --> 00:37:21,700
+هو بيناجب lower bound أو أكبر lower bound أكبر 
+
+354
+00:37:21,700 --> 00:37:30,120
+lower bound طب نلاحظ -1/2 أصغر من -1/4 أصغر 
+
+355
+00:37:30,120 --> 00:37:47,710
+من بعد هيك -1/6 اه فاعتقد
+
+356
+00:37:47,710 --> 00:37:52,010
+أن هذا هيطلع -1/2 هذا أكبر lower bound
+
+357
+00:37:58,870 --> 00:38:04,470
+طيب لو طبقنا النظرية هذه أنا أخدت S بيساوي الكلام
+
+358
+00:38:04,470 --> 00:38:12,830
+هذا و A بيساوي 1 إذا
+
+359
+00:38:12,830 --> 00:38:24,470
+الـ supremum ل S 4 بيساوي A زائد الـ supremum ل S
+
+360
+00:38:24,470 --> 00:38:34,260
+صح؟ والـ a بيساوي 1 والـ suprem لـ s بيساوي 1
+
+361
+00:38:34,260 --> 00:38:43,460
+فبيطلع الـ suprem لـ S 4 بيساوي 2 تمام؟ الآن
+
+362
+00:38:43,460 --> 00:38:53,660
+بنجيب الـ infimum لـ S 4 بنفس الطريقة ممكن
+
+363
+00:38:53,660 --> 00:38:54,340
+إثبات
+
+364
+00:39:00,490 --> 00:39:10,070
+إذا هنا similar
+
+365
+00:39:10,070 --> 00:39:17,590
+example
+
+366
+00:39:17,590 --> 00:39:24,090
+similar
+
+367
+00:39:24,090 --> 00:39:32,030
+example 2 4 1 أي ممكن من خلاله نثبت أن
+
+368
+00:39:32,030 --> 00:39:38,330
+الـ infimum أن 
+
+369
+00:39:38,330 --> 00:39:44,490
+الـ infimum لـ set A زائد S بيساوي A زائد الـ infimum لـ S
+
+370
+00:39:44,490 --> 00:39:49,310
+وبالتالي
+
+371
+00:39:49,310 --> 00:39:53,390
+أن
+
+372
+00:39:53,390 --> 00:40:00,480
+أنا لو بدي أجرب على جزء A ف الـ infimum لـ S 4
+
+373
+00:40:00,480 --> 00:40:13,780
+بيساوي الـ infimum لـ A زائد S اللي هو الـ infimum لـ 
+
+374
+00:40:13,780 --> 00:40:22,630
+1 زائد S وهذا بيساوي 1 زائد infimum لـ S و
+
+375
+00:40:22,630 --> 00:40:28,770
+هذا بيساوي 1 زائد infimum الـ S -1/2 فبيطلع 1/2
+
+376
+00:40:28,770 --> 00:40:36,210
+okay أن الـ infimum لـ set S 4 بيطلع - بيطلع 1/2
+
+377
+00:40:36,210 --> 00:40:41,910
+هذا حل حل ثاني أن أنا يعني أحاول
+
+378
+00:40:47,360 --> 00:40:54,460
+اه يعني أكتب عناصر المجموعة هذه أفرطها وأحاول
+
+379
+00:40:54,460 --> 00:40:59,600
+أشوف وين أصغر عنصر ووين أكبر عنصر ووين هيكون في
+
+380
+00:40:59,600 --> 00:41:04,640
+عندي upper bounds و lower bounds ونحاول نثبت أنه
+
+381
+00:41:04,640 --> 00:41:12,060
+الـ .. يعني بالطريقة هذه يعني ممكن نحن نحل الأسئلة 
+
+382
+00:41:12,060 --> 00:41:20,320
+بطريقة ثانية فهذا حلو يعني
+
+383
+00:41:20,320 --> 00:41:25,900
+هذا الـ set ممكن نكتب عناصرها يعني أول عنصر لما N
+
+384
+00:41:25,900 --> 00:41:33,680
+بيساوي 1 1- -1-
+
+385
+00:41:33,680 --> 00:41:44,350
+-1 يعني 0 الانصر اللي بعده 1-1/2
+
+386
+00:41:44,350 --> 00:41:56,450
+بيطلع 1/2 اللي بعده بيطلع 1- -1/3 يعني
+
+387
+00:41:56,450 --> 00:42:03,270
+1/3 يعني قد ايه 4/3 اللي بعده 1
+
+388
+00:42:03,270 --> 00:42:07,210
+موجب 1/4 بيطلع قد ايه
+
+389
+00:42:09,700 --> 00:42:17,520
+5/4 و هكذا فهنلاحظ
+
+390
+00:42:17,520 --> 00:42:24,700
+أن الـ 2 2 upper bound لأن هو هيكون أكبر
+
+391
+00:42:24,700 --> 00:42:31,100
+عنصر و ننتبه للست لو في أي upper bound ثاني لو في
+
+392
+00:42:31,100 --> 00:42:33,320
+أي upper bound
+
+393
+00:42:37,830 --> 00:42:45,630
+of S4 فهذا بيقودى أن 2 أصغر من أو يساوي الـ V
+
+394
+00:42:45,630 --> 00:42:50,890
+لأنه 2 عنصر في الست S4 صح؟ إذا 2 upper
+
+395
+00:42:50,890 --> 00:42:54,810
+bound واضح أن 2 أكبر من أو يساوي كل عناصر S4
+
+396
+00:42:54,810 --> 00:43:04,320
+صح؟ ولو أخدت أي upper bound لـ S4 فبما أن V هو upper
+
+397
+00:43:04,320 --> 00:43:09,200
+bound لـ S4 و 2 عنصر في S4 إذن 2 أصغر من أو 
+
+398
+00:43:09,200 --> 00:43:14,640
+يساوي V إذن هنا أثبتنا أن 2 upper bound لـ S4
+
+399
+00:43:14,640 --> 00:43:19,500
+و 2 أصغر من أو يساوي أي upper bound لـ S4 إذن
+
+400
+00:43:19,500 --> 00:43:23,440
+2 هو الـ supremum بالمثل ممكن نثبت أن النص هو
+
+401
+00:43:23,440 --> 00:43:28,140
+الـ infimum إذن هذا برهان ثاني أنا اتعمدت أعطيكم البرهان
+
+402
+00:43:28,140 --> 00:43:32,260
+هذا عشان القوانين هذه نشوف كيف نطبقها برضه هذا
+
+403
+00:43:32,260 --> 00:43:38,520
+برهان صعب ناجح الحل؟ okay؟
+
+404
+00:43:38,520 --> 00:43:42,080
+في أي سؤال أو استفسار؟ إذا احنا هنكتفي بهذا القدر
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/_28CmIWMuzY_raw.json b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/_28CmIWMuzY_raw.json
new file mode 100644
index 0000000000000000000000000000000000000000..5ddb8442d2ba8ee33958f302fce9740ac9a98797
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/_28CmIWMuzY_raw.json
@@ -0,0 +1 @@
+{"segments": [{"id": 1, "seek": 4617, "start": 21.33, "end": 46.17, "text": "اليوم طبعا هنكمل الشرح أو بعض الملاحظات على النظرية اللي أخدناها في المحاضرة السابقة النظرية هذه بتتحدث عن nested interval property أو خاصية الفترات المتداخلة", "tokens": [6027, 1829, 20498, 23032, 3555, 3615, 995, 8032, 1863, 24793, 1211, 25124, 2288, 5016, 34051, 45030, 11242, 9673, 15040, 5016, 19913, 9307, 15844, 28239, 19913, 2288, 10632, 13672, 1829, 5551, 9778, 3215, 8315, 11296, 8978, 9673, 5016, 46958, 25720, 21136, 16758, 28671, 28239, 19913, 2288, 10632, 29538, 39894, 2655, 24401, 12984, 18871, 15646, 292, 15035, 4707, 34051, 16490, 33546, 10632, 27188, 2655, 2288, 9307, 9673, 2655, 28259, 9778, 37977], "avg_logprob": -0.08766741092715945, "compression_ratio": 1.542857142857143, "no_speech_prob": 0.0, "words": [{"start": 21.33, "end": 22.45, "word": "اليوم", "probability": 0.97119140625}, {"start": 22.45, "end": 22.93, "word": " طبعا", "probability": 0.9915771484375}, {"start": 22.93, "end": 23.77, "word": " هنكمل", "probability": 0.973876953125}, {"start": 23.77, "end": 27.29, "word": " الشرح", "probability": 0.7639973958333334}, {"start": 27.29, "end": 30.65, "word": " أو", "probability": 0.716796875}, {"start": 30.65, "end": 30.91, "word": " بعض", "probability": 0.953369140625}, {"start": 30.91, "end": 31.85, "word": " الملاحظات", "probability": 0.98818359375}, {"start": 31.85, "end": 32.49, "word": " على", "probability": 0.861328125}, {"start": 32.49, "end": 33.15, "word": " النظرية", "probability": 0.892333984375}, {"start": 33.15, "end": 34.41, "word": " اللي", "probability": 0.723388671875}, {"start": 34.41, "end": 34.99, "word": " أخدناها", "probability": 0.86259765625}, {"start": 34.99, "end": 35.09, "word": " في", "probability": 0.95751953125}, {"start": 35.09, "end": 35.61, "word": " المحاضرة", "probability": 0.9625244140625}, {"start": 35.61, "end": 36.25, "word": " السابقة", "probability": 0.98828125}, {"start": 36.25, "end": 38.43, "word": " النظرية", "probability": 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" في", "probability": 0.98046875}, {"start": 120.9, "end": 121.38, "word": " أنصر", "probability": 0.861328125}, {"start": 121.38, "end": 121.9, "word": " واحد", "probability": 0.953369140625}, {"start": 121.9, "end": 124.56, "word": " okay", "probability": 0.400146484375}, {"start": 124.56, "end": 125.4, "word": " تمام", "probability": 0.908447265625}, {"start": 125.4, "end": 126.44, "word": " وشوفنا", "probability": 0.82568359375}, {"start": 126.44, "end": 126.9, "word": " مرين", "probability": 0.88427734375}, {"start": 126.9, "end": 127.14, "word": " على", "probability": 0.92333984375}, {"start": 127.14, "end": 127.78, "word": " البرهان", "probability": 0.9022216796875}, {"start": 127.78, "end": 129.56, "word": " المرة", "probability": 0.91357421875}, {"start": 129.56, "end": 129.74, "word": " اللي", "probability": 0.84228515625}, {"start": 129.74, "end": 130.24, "word": " فاتت", "probability": 0.9871419270833334}, {"start": 130.24, "end": 130.52, "word": " و", "probability": 0.7666015625}, {"start": 130.52, "end": 131.6, "word": " أعتقد", "probability": 0.7762858072916666}, {"start": 131.6, "end": 131.8, "word": " أن", "probability": 0.65771484375}, {"start": 131.8, "end": 132.36, "word": " البرهان", "probability": 0.9254150390625}, {"start": 132.36, "end": 132.76, "word": " مكتوب", "probability": 0.9898681640625}, {"start": 132.76, "end": 133.44, "word": " بالتفصيل", "probability": 0.9771484375}], "temperature": 1.0}, {"id": 5, "seek": 16210, "start": 134.12, "end": 162.1, "text": "واضح ومرنا عليه جزء جزء فأرجعكم تكونوا قرأتهوا كمان مرة وفهمتهوا في حد عنده استفسار على المرهانة النظرية هذه طيب الآن النظرية هذه نرجع للنظرية كمان مرة الآن في ملاحظة بتقول انه لو انا في النظرية هذه الفترات هذه", "tokens": [2407, 46958, 5016, 4032, 29973, 8315, 47356, 10874, 11622, 38207, 10874, 11622, 38207, 6156, 10721, 47341, 3615, 24793, 6055, 30544, 14407, 12174, 2288, 10721, 47395, 14407, 9122, 2304, 7649, 3714, 25720, 4032, 5172, 16095, 47395, 14407, 8978, 11331, 3215, 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"probability": 0.71728515625}, {"start": 138.1, "end": 138.58, "word": " تكونوا", "probability": 0.9269205729166666}, {"start": 138.58, "end": 139.2, "word": " قرأتهوا", "probability": 0.88017578125}, {"start": 139.2, "end": 139.48, "word": " كمان", "probability": 0.8963216145833334}, {"start": 139.48, "end": 139.76, "word": " مرة", "probability": 0.968017578125}, {"start": 139.76, "end": 140.74, "word": " وفهمتهوا", "probability": 0.8712890625}, {"start": 140.74, "end": 141.7, "word": " في", "probability": 0.489501953125}, {"start": 141.7, "end": 142.0, "word": " حد", "probability": 0.98486328125}, {"start": 142.0, "end": 142.26, "word": " عنده", "probability": 0.933349609375}, {"start": 142.26, "end": 142.96, "word": " استفسار", "probability": 0.7740071614583334}, {"start": 142.96, "end": 143.18, "word": " على", "probability": 0.69921875}, {"start": 143.18, "end": 144.9, "word": " المرهانة", "probability": 0.797607421875}, {"start": 144.9, "end": 145.28, "word": " النظرية", "probability": 0.8377685546875}, {"start": 145.28, "end": 145.64, "word": " هذه", "probability": 0.89306640625}, {"start": 145.64, "end": 147.4, "word": " طيب", "probability": 0.95751953125}, {"start": 147.4, "end": 147.86, "word": " الآن", "probability": 0.658935546875}, {"start": 147.86, "end": 148.48, "word": " النظرية", "probability": 0.9752197265625}, {"start": 148.48, "end": 148.9, "word": " هذه", "probability": 0.92138671875}, {"start": 148.9, "end": 149.56, "word": " نرجع", "probability": 0.90966796875}, {"start": 149.56, "end": 150.0, "word": " للنظرية", "probability": 0.87021484375}, {"start": 150.0, "end": 150.34, "word": " كمان", "probability": 0.9775390625}, {"start": 150.34, "end": 150.72, "word": " مرة", "probability": 0.9912109375}, {"start": 150.72, "end": 155.82, "word": " الآن", "probability": 0.6685791015625}, {"start": 155.82, "end": 156.2, "word": " في", "probability": 0.9130859375}, {"start": 156.2, "end": 157.42, "word": " ملاحظة", "probability": 0.9763671875}, {"start": 157.42, "end": 158.58, "word": " بتقول", "probability": 0.970947265625}, {"start": 158.58, "end": 159.06, "word": " انه", "probability": 0.6796875}, {"start": 159.06, "end": 159.28, "word": " لو", "probability": 0.92236328125}, {"start": 159.28, "end": 159.62, "word": " انا", "probability": 0.77001953125}, {"start": 159.62, "end": 159.8, "word": " في", "probability": 0.87744140625}, {"start": 159.8, "end": 160.34, "word": " النظرية", "probability": 0.9862060546875}, {"start": 160.34, "end": 160.7, "word": " هذه", "probability": 0.94677734375}, {"start": 160.7, "end": 161.48, "word": " الفترات", "probability": 0.958740234375}, {"start": 161.48, "end": 162.1, "word": " هذه", "probability": 0.9345703125}], "temperature": 1.0}, {"id": 6, "seek": 18746, "start": 163.94, "end": 187.46, "text": "الفرض ان الفترات in مغلقة closed لو حذفت شيلت الفرض هذا فالنظرية هذه بتبطل تكون صحيحة فالنظرية هذه بتبطل تكون صحيحة وحنشوف counter example يوضح عدم صحتها كذلك", "tokens": [6027, 5172, 43042, 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"طب افرضه ان هذا شرط متحقق في الفترات لكن اللي مش متحقق اللي هو ال boundedness يعني الفترات هذه ليست محدودة ليست bounded برضه في الحالة هذه المظهرية تفشل و في counter example يوضح فشلها okay اذا حنشوف two counter examples خليني نشوفهم مع بعض", "tokens": [9566, 3555, 1975, 5172, 43042, 3224, 16472, 23758, 13412, 2288, 9566, 44650, 5016, 4587, 4587, 8978, 27188, 2655, 2288, 9307, 44381, 13672, 1829, 37893, 44650, 5016, 4587, 4587, 13672, 1829, 31439, 2423, 37498, 1287, 37495, 22653, 27188, 2655, 2288, 9307, 29538, 32239, 14851, 3714, 24401, 23328, 3660, 32239, 14851, 37498, 4724, 43042, 3224, 8978, 21542, 6027, 3660, 29538, 9673, 19913, 3224, 2288, 10632, 6055, 5172, 8592, 1211, 4032, 8978, 5682, 1365, 7251, 2407, 11242, 5016, 6156, 8592, 1211, 11296, 1392, 1975, 15730, 11331, 1863, 8592, 38688, 732, 5682, 5110, 16490, 1211, 9957, 1829, 8717, 8592, 38688, 16095, 20449, 45030, 11242], "avg_logprob": -0.16383044790513446, "compression_ratio": 1.76036866359447, "no_speech_prob": 0.0, 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في", "probability": 0.91357421875}, {"start": 200.48, "end": 200.8, "word": " الحالة", "probability": 0.9710286458333334}, {"start": 200.8, "end": 201.0, "word": " هذه", "probability": 0.7158203125}, {"start": 201.0, "end": 201.42, "word": " المظهرية", "probability": 0.6744140625}, {"start": 201.42, "end": 201.96, "word": " تفشل", "probability": 0.9453125}, {"start": 201.96, "end": 202.94, "word": " و", "probability": 0.95703125}, {"start": 202.94, "end": 203.1, "word": " في", "probability": 0.475830078125}, {"start": 203.1, "end": 203.48, "word": " counter", "probability": 0.693359375}, {"start": 203.48, "end": 204.08, "word": " example", "probability": 0.955078125}, {"start": 204.08, "end": 204.74, "word": " يوضح", "probability": 0.96923828125}, {"start": 204.74, "end": 205.66, "word": " فشلها", "probability": 0.96630859375}, {"start": 205.66, "end": 206.42, "word": " okay", "probability": 0.716796875}, {"start": 206.42, "end": 206.62, "word": " اذا", "probability": 0.55419921875}, {"start": 206.62, "end": 207.02, "word": " حنشوف", "probability": 0.7491455078125}, {"start": 207.02, "end": 207.22, "word": " two", "probability": 0.91650390625}, {"start": 207.22, "end": 207.74, "word": " counter", "probability": 0.8515625}, {"start": 207.74, "end": 208.86, "word": " examples", "probability": 0.828125}, {"start": 208.86, "end": 209.64, "word": " خليني", "probability": 0.851806640625}, {"start": 209.64, "end": 210.12, "word": " نشوفهم", "probability": 0.9012451171875}, {"start": 210.12, "end": 210.28, "word": " مع", "probability": 0.98876953125}, {"start": 210.28, "end": 210.64, "word": " بعض", "probability": 0.99169921875}], "temperature": 1.0}, {"id": 8, "seek": 24343, "start": 216.61, "end": 243.43, "text": "إذا هدى ال remark اللى انا اتحدث عنها قلت انه it should be noted يجب ملاحظة ان journal بصورة عامة instant sequence of intervals need not have a common point يعني لو فيه ending sequence من الفترات المتداخلة مش شرط تقاطعهم يكون في في يعني اى نقطة او نقطة مشاركة يعني مش شرط ان التقاطع لها يساوي فيه", "tokens": [28814, 15730, 8032, 3215, 7578, 2423, 7942, 13672, 7578, 1975, 8315, 1975, 2655, 24401, 12984, 18871, 11296, 12174, 1211, 2655, 16472, 3224, 309, 820, 312, 12964, 7251, 7435, 3555, 3714, 15040, 5016, 19913, 3660, 16472, 6708, 4724, 9381, 13063, 3660, 6225, 10943, 3660, 9836, 8310, 295, 26651, 643, 406, 362, 257, 2689, 935, 37495, 22653, 45164, 8978, 3224, 8121, 8310, 9154, 27188, 2655, 2288, 9307, 9673, 2655, 3215, 47283, 37977, 37893, 13412, 2288, 9566, 6055, 4587, 41193, 3615, 16095, 7251, 30544, 8978, 8978, 37495, 22653, 1975, 7578, 8717, 47432, 3660, 1975, 2407, 8717, 47432, 3660, 37893, 9640, 4117, 3660, 37495, 22653, 37893, 13412, 2288, 9566, 16472, 16712, 4587, 41193, 3615, 5296, 11296, 7251, 3794, 995, 45865, 8978, 3224], "avg_logprob": -0.20365020958315425, "compression_ratio": 1.6383763837638377, "no_speech_prob": 0.0, "words": [{"start": 216.61, "end": 216.95, "word": "إذا", "probability": 0.557373046875}, {"start": 216.95, 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"probability": 0.930908203125}, {"start": 241.29, "end": 241.51, "word": " مش", "probability": 0.9873046875}, {"start": 241.51, "end": 241.79, "word": " شرط", "probability": 0.99560546875}, {"start": 241.79, "end": 241.89, "word": " ان", "probability": 0.703125}, {"start": 241.89, "end": 242.45, "word": " التقاطع", "probability": 0.93212890625}, {"start": 242.45, "end": 242.65, "word": " لها", "probability": 0.6429443359375}, {"start": 242.65, "end": 243.03, "word": " يساوي", "probability": 0.9388427734375}, {"start": 243.03, "end": 243.43, "word": " فيه", "probability": 0.76220703125}], "temperature": 1.0}, {"id": 9, "seek": 27166, "start": 244.58, "end": 271.66, "text": "فالأنثى لها دى هدا هى اللى حكينا عنها اول مثال هاى فى المثال الاول الفرض the hypothesis الفرض ان ال intervals I in فى نظرية 22 be closed cannot be dropped يعني لا يمكن حذفه", "tokens": [5172, 6027, 33456, 12984, 7578, 5296, 11296, 11778, 7578, 8032, 28259, 8032, 7578, 13672, 7578, 11331, 4117, 1829, 8315, 18871, 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"word": " عنها", "probability": 0.9931640625}, {"start": 248.9, "end": 251.0, "word": " اول", "probability": 0.767578125}, {"start": 251.0, "end": 252.92, "word": " مثال", "probability": 0.991455078125}, {"start": 252.92, "end": 258.5, "word": " هاى", "probability": 0.5193684895833334}, {"start": 258.5, "end": 258.6, "word": " فى", "probability": 0.77099609375}, {"start": 258.6, "end": 259.04, "word": " المثال", "probability": 0.96337890625}, {"start": 259.04, "end": 259.54, "word": " الاول", "probability": 0.91064453125}, {"start": 259.54, "end": 260.8, "word": " الفرض", "probability": 0.895751953125}, {"start": 260.8, "end": 261.44, "word": " the", "probability": 0.245361328125}, {"start": 261.44, "end": 262.0, "word": " hypothesis", "probability": 0.82275390625}, {"start": 262.0, "end": 262.6, "word": " الفرض", "probability": 0.9189453125}, {"start": 262.6, "end": 262.92, "word": " ان", "probability": 0.89404296875}, {"start": 262.92, "end": 263.08, "word": " ال", "probability": 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270.1, "word": " لا", "probability": 0.85595703125}, {"start": 270.1, "end": 270.58, "word": " يمكن", "probability": 0.96875}, {"start": 270.58, "end": 271.66, "word": " حذفه", "probability": 0.8817138671875}], "temperature": 1.0}, {"id": 10, "seek": 30210, "start": 273.26, "end": 302.1, "text": "لا يمكن الاستجناء عنه وتبقى النظرية نظرية صحيحة for example على سبيل المثال لو أخدت الفترات I N الفترة I N هي الفترة المفتوحة من 0 ل 1 على N حيث N عدد طبيعي فواضح ان الفترات هدي nested صح؟ لأن الفترة الأولى هتكون الفترة مفتوحة من 0 ل 1", "tokens": [15040, 7251, 43020, 42963, 14851, 7435, 8315, 38207, 18871, 3224, 34683, 3555, 4587, 7578, 28239, 19913, 2288, 10632, 8717, 19913, 2288, 10632, 20328, 5016, 1829, 5016, 3660, 337, 1365, 15844, 8608, 21292, 1211, 9673, 12984, 6027, 45164, 5551, 9778, 3215, 2655, 27188, 2655, 2288, 9307, 286, 426, 27188, 2655, 25720, 286, 426, 39896, 27188, 2655, 25720, 9673, 5172, 2655, 2407, 5016, 3660, 9154, 1958, 5296, 502, 15844, 426, 11331, 1829, 12984, 426, 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{"start": 292.4, "end": 292.8, "word": " عدد", "probability": 0.9684244791666666}, {"start": 292.8, "end": 293.4, "word": " طبيعي", "probability": 0.97216796875}, {"start": 293.4, "end": 295.42, "word": " فواضح", "probability": 0.955322265625}, {"start": 295.42, "end": 295.58, "word": " ان", "probability": 0.495849609375}, {"start": 295.58, "end": 296.18, "word": " الفترات", "probability": 0.990966796875}, {"start": 296.18, "end": 296.52, "word": " هدي", "probability": 0.48193359375}, {"start": 296.52, "end": 297.2, "word": " nested", "probability": 0.653076171875}, {"start": 297.2, "end": 298.54, "word": " صح؟", "probability": 0.6971028645833334}, {"start": 298.54, "end": 299.1, "word": " لأن", "probability": 0.84716796875}, {"start": 299.1, "end": 299.6, "word": " الفترة", "probability": 0.98291015625}, {"start": 299.6, "end": 299.96, "word": " الأولى", "probability": 0.9773763020833334}, {"start": 299.96, "end": 300.46, "word": " هتكون", "probability": 0.82763671875}, {"start": 300.46, "end": 300.86, "word": " الفترة", "probability": 0.9423828125}, {"start": 300.86, "end": 301.28, "word": " مفتوحة", "probability": 0.9508463541666666}, {"start": 301.28, "end": 301.4, "word": " من", "probability": 0.99267578125}, {"start": 301.4, "end": 301.64, "word": " 0", "probability": 0.958984375}, {"start": 301.64, "end": 301.84, "word": " ل", "probability": 0.9248046875}, {"start": 301.84, "end": 302.1, "word": " 1", "probability": 0.927734375}], "temperature": 1.0}, {"id": 11, "seek": 32910, "start": 303.04, "end": 329.1, "text": "الفترة التانية الفترة مفتوحة من سفر لنص وهذه محتوى في I واحد و I تلاتة الفترة مفتوحة من سفر لتلت محتوى داخل I اتنين و هكذا لذلك واضح ان ال sequence of open intervals IN is nested sequence كذلك عناصر ال sequence هذه bounded هذه فترات محصورة", "tokens": [6027, 5172, 2655, 25720, 16712, 7649, 10632, 27188, 2655, 25720, 3714, 5172, 2655, 2407, 5016, 3660, 9154, 8608, 5172, 2288, 5296, 1863, 9381, 37037, 24192, 3714, 33753, 2407, 7578, 8978, 286, 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{"start": 325.4, "end": 325.76, "word": " sequence", "probability": 0.9794921875}, {"start": 325.76, "end": 326.38, "word": " هذه", "probability": 0.60791015625}, {"start": 326.38, "end": 326.96, "word": " bounded", "probability": 0.92822265625}, {"start": 326.96, "end": 327.56, "word": " هذه", "probability": 0.69921875}, {"start": 327.56, "end": 328.02, "word": " فترات", "probability": 0.9794921875}, {"start": 328.02, "end": 329.1, "word": " محصورة", "probability": 0.972265625}], "temperature": 1.0}, {"id": 12, "seek": 35995, "start": 330.33, "end": 359.95, "text": "لكن الفترات هذه not closed مش closed يعني عبارة عن مجموعات مفتوحة ليست مغلقة، إذن شرط الإغلاق هنا انحذف وبالتالي نتيجة النظرية مش شرط تكون صحيحة، إذن التقاطع هنا لنفس ال sequence هذه بيطلع بساوي fine مافيش common point مافيش نقطة مشتركة في هذه الفترات طبعا هذا مش واضح", "tokens": [1211, 19452, 27188, 2655, 2288, 9307, 29538, 406, 5395, 37893, 5395, 37495, 22653, 6225, 3555, 9640, 3660, 18871, 3714, 7435, 2304, 45367, 9307, 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576.01, "end": 594.97, "text": "and closed مغلقة لكن ماهياش bounded مش محصورة it's not bounded .. هذه كمجموعة is not bounded above، كمجموعة ليس لها supreme، is not bounded above، اذا شرط ال boundedness اختل وبالتالي نتيجة النظرية هتختل", "tokens": [474, 5395, 3714, 17082, 1211, 28671, 44381, 19446, 3224, 1829, 33599, 37498, 37893, 3714, 5016, 9381, 13063, 3660, 309, 311, 406, 37498, 4386, 29538, 9122, 2304, 7435, 2304, 2407, 27884, 307, 406, 37498, 3673, 12399, 9122, 2304, 7435, 2304, 2407, 27884, 32239, 3794, 5296, 11296, 27756, 12399, 307, 406, 37498, 3673, 12399, 1975, 15730, 13412, 2288, 9566, 2423, 37498, 1287, 1975, 46456, 1211, 46599, 6027, 2655, 6027, 1829, 8717, 31371, 7435, 3660, 28239, 19913, 2288, 10632, 8032, 2655, 46456, 1211], "avg_logprob": -0.21334876248865953, "compression_ratio": 1.6, "no_speech_prob": 0.0, "words": [{"start": 576.01, "end": 576.47, "word": "and", "probability": 0.6904296875}, {"start": 576.47, "end": 577.03, "word": " closed", "probability": 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" N", "probability": 0.9736328125}, {"start": 672.41, "end": 673.63, "word": " بحيث", "probability": 0.95654296875}, {"start": 673.63, "end": 673.93, "word": " ان", "probability": 0.8671875}, {"start": 673.93, "end": 675.63, "word": " X", "probability": 0.85791015625}, {"start": 675.63, "end": 676.29, "word": " أصغر", "probability": 0.9532470703125}, {"start": 676.29, "end": 676.53, "word": " من", "probability": 0.97998046875}, {"start": 676.53, "end": 677.33, "word": " N0", "probability": 0.968505859375}], "temperature": 1.0}, {"id": 25, "seek": 69781, "start": 681.19, "end": 697.81, "text": "هذه هي الـ Archimedean property الأساسية طيب أنا عندي الأن من ال Archimedean property عندي يوجد عدد طبيعي N0 لصد أكبر من X وعندي هنا", "tokens": [3224, 24192, 39896, 2423, 39184, 10984, 332, 4858, 282, 4707, 16247, 3794, 32277, 10632, 23032, 1829, 3555, 41850, 18871, 16254, 16247, 1863, 9154, 2423, 10984, 332, 4858, 282, 4707, 18871, 16254, 7251, 29245, 3215, 6225, 3215, 3215, 23032, 21292, 3615, 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"end": 718.48, "word": " عندي", "probability": 0.81298828125}, {"start": 718.48, "end": 719.1, "word": " contradiction", "probability": 0.6220703125}, {"start": 719.1, "end": 720.7, "word": " إذا", "probability": 0.803955078125}, {"start": 720.7, "end": 721.6, "word": " هذا", "probability": 0.7578125}, {"start": 721.6, "end": 722.42, "word": " العنصر", "probability": 0.9306640625}, {"start": 722.42, "end": 722.84, "word": " غير", "probability": 0.99365234375}, {"start": 722.84, "end": 723.4, "word": " موجود", "probability": 0.98779296875}], "temperature": 1.0}, {"id": 27, "seek": 74723, "start": 724.51, "end": 747.23, "text": "such an x does not exist يعني التقاطة هذا بساوي في كما هو مطلوب، تمام؟ واضح البرهن؟ إذن هذه مثال تاني بوضح أن شرط ال boundedness لا يمكن اسقاطه وتبقى nested intervals theorem صحيحة، okay؟ في نظرية تانية", "tokens": [39974, 364, 2031, 775, 406, 2514, 37495, 22653, 16712, 4587, 41193, 3660, 23758, 4724, 3794, 995, 45865, 8978, 9122, 15042, 31439, 3714, 9566, 1211, 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{"start": 726.79, "end": 727.05, "word": " يعني", "probability": 0.70166015625}, {"start": 727.05, "end": 727.63, "word": " التقاطة", "probability": 0.8746337890625}, {"start": 727.63, "end": 727.87, "word": " هذا", "probability": 0.499755859375}, {"start": 727.87, "end": 728.33, "word": " بساوي", "probability": 0.811279296875}, {"start": 728.33, "end": 728.59, "word": " في", "probability": 0.68505859375}, {"start": 728.59, "end": 728.95, "word": " كما", "probability": 0.8828125}, {"start": 728.95, "end": 729.13, "word": " هو", "probability": 0.97412109375}, {"start": 729.13, "end": 730.07, "word": " مطلوب،", "probability": 0.7853515625}, {"start": 730.07, "end": 731.15, "word": " تمام؟", "probability": 0.8483072916666666}, {"start": 731.15, "end": 731.39, "word": " واضح", "probability": 0.9583333333333334}, {"start": 731.39, "end": 733.03, "word": " البرهن؟", "probability": 0.805859375}, {"start": 733.03, "end": 733.41, "word": " إذن", "probability": 0.694091796875}, {"start": 733.41, "end": 733.55, "word": " هذه", "probability": 0.294677734375}, {"start": 733.55, "end": 733.93, "word": " مثال", "probability": 0.972412109375}, {"start": 733.93, "end": 734.31, "word": " تاني", "probability": 0.9444986979166666}, {"start": 734.31, "end": 734.77, "word": " بوضح", "probability": 0.912353515625}, {"start": 734.77, "end": 734.93, "word": " أن", "probability": 0.46630859375}, {"start": 734.93, "end": 735.47, "word": " شرط", "probability": 0.97412109375}, {"start": 735.47, "end": 736.31, "word": " ال", "probability": 0.75244140625}, {"start": 736.31, "end": 737.13, "word": " boundedness", "probability": 0.798828125}, {"start": 737.13, "end": 737.33, "word": " لا", "probability": 0.6572265625}, {"start": 737.33, "end": 737.69, "word": " يمكن", "probability": 0.9638671875}, {"start": 737.69, "end": 738.35, "word": " اسقاطه", "probability": 0.7177734375}, {"start": 738.35, "end": 739.45, "word": " وتبقى", "probability": 0.870361328125}, {"start": 739.45, "end": 740.15, "word": " nested", "probability": 0.85546875}, {"start": 740.15, "end": 740.83, "word": " intervals", "probability": 0.791015625}, {"start": 740.83, "end": 741.39, "word": " theorem", "probability": 0.8212890625}, {"start": 741.39, "end": 742.85, "word": " صحيحة،", "probability": 0.8179931640625}, {"start": 742.85, "end": 745.73, "word": " okay؟", "probability": 0.77734375}, {"start": 745.73, "end": 745.95, "word": " في", "probability": 0.70361328125}, {"start": 745.95, "end": 746.59, "word": " نظرية", "probability": 0.97607421875}, {"start": 746.59, "end": 747.23, "word": " تانية", "probability": 0.99658203125}], "temperature": 1.0}, {"id": 28, "seek": 77515, "start": 748.93, "end": 775.15, "text": "يمكن هذه مرت معاكم في المبادئ لكن اليوم هنعطيلها برهان يعتمد على ال nested intervals theorem او nested intervals property برهان جديد غير اللي أخدته في مبادئ الرياضيات فالنظرية هذه 24 تتحدث عن ال uncountability of the real numbers", "tokens": [32640, 19452, 29538, 3714, 43500, 20449, 995, 24793, 8978, 9673, 3555, 18513, 19986, 44381, 45595, 20498, 8032, 1863, 3615, 9566, 26895, 11296, 4724, 2288, 3224, 7649, 7251, 34268, 2304, 3215, 15844, 2423, 15646, 292, 26651, 20904, 1975, 2407, 15646, 292, 26651, 4707, 4724, 2288, 3224, 7649, 10874, 16254, 3215, 32771, 13546, 13672, 1829, 5551, 9778, 3215, 47395, 8978, 3714, 3555, 18513, 19986, 2423, 16572, 46958, 1829, 9307, 6156, 6027, 1863, 19913, 2288, 10632, 29538, 4022, 6055, 2655, 24401, 12984, 18871, 2423, 6219, 792, 2310, 295, 264, 957, 3547], "avg_logprob": -0.20488062333524898, "compression_ratio": 1.5244444444444445, "no_speech_prob": 0.0, "words": [{"start": 748.93, "end": 749.45, "word": "يمكن", "probability": 0.67138671875}, {"start": 749.45, "end": 749.69, "word": " هذه", "probability": 0.2509765625}, {"start": 749.69, "end": 750.03, "word": " مرت", "probability": 0.629150390625}, {"start": 750.03, "end": 750.45, "word": " معاكم", "probability": 0.974609375}, {"start": 750.45, "end": 750.61, "word": " في", "probability": 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of all real numbers is uncountable طيب ما معنى أن ال set تكون countable؟ في حد فيكم بتعرف؟ ال set A أو S definition", "tokens": [5172, 3555, 28820, 4724, 3794, 41193, 3660, 28239, 19913, 2288, 10632, 29538, 39894, 39648, 14739, 3714, 7435, 2304, 2407, 27884, 18863, 3215, 18513, 21542, 38436, 4587, 10632, 307, 6219, 792, 712, 264, 992, 497, 295, 439, 957, 3547, 307, 6219, 792, 712, 23032, 1829, 3555, 19446, 20449, 1863, 7578, 14739, 2423, 992, 6055, 30544, 1207, 712, 22807, 8978, 11331, 3215, 8978, 24793, 39894, 3615, 28480, 22807, 2423, 992, 316, 34051, 318, 7123], "avg_logprob": -0.21511129647085112, "compression_ratio": 1.4278350515463918, "no_speech_prob": 0.0, "words": [{"start": 776.4, "end": 777.02, "word": "فبكل", "probability": 0.6334228515625}, {"start": 777.02, "end": 777.58, "word": " بساطة", "probability": 0.9864501953125}, {"start": 777.58, "end": 778.16, "word": " النظرية", "probability": 0.925537109375}, {"start": 778.16, "end": 778.36, "word": " هذه", "probability": 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{"id": 31, "seek": 86369, "start": 835.63, "end": 863.69, "text": "أو في بيجيكشن one to one correspondence بينها وبين الأعداد الطبيعية يعني هذا معناه is denumerable قابلة للترقيم طيب إذا كانت ال set ماهياش finite", "tokens": [10721, 2407, 8978, 4724, 1829, 7435, 1829, 4117, 8592, 1863, 472, 281, 472, 38135, 49374, 11296, 46599, 9957, 16247, 22488, 18513, 41950, 21292, 3615, 10632, 37495, 22653, 23758, 20449, 8315, 3224, 307, 1441, 15583, 712, 12174, 16758, 37977, 24976, 2655, 2288, 4587, 32640, 23032, 1829, 3555, 11933, 15730, 25961, 2655, 2423, 992, 19446, 3224, 1829, 33599, 19362], "avg_logprob": -0.16540948147403783, "compression_ratio": 1.3703703703703705, "no_speech_prob": 0.0, "words": [{"start": 835.63, "end": 836.09, "word": "أو", "probability": 0.748046875}, {"start": 836.09, "end": 836.39, "word": " في", "probability": 0.90869140625}, {"start": 836.39, "end": 837.43, "word": " بيجيكشن", "probability": 0.7362234933035714}, {"start": 837.43, "end": 838.15, "word": " one", 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"probability": 0.6717122395833334}, {"start": 847.31, "end": 849.69, "word": " قابلة", "probability": 0.8917643229166666}, {"start": 849.69, "end": 851.23, "word": " للترقيم", "probability": 0.96845703125}, {"start": 851.23, "end": 856.97, "word": " طيب", "probability": 0.9710286458333334}, {"start": 856.97, "end": 857.25, "word": " إذا", "probability": 0.908935546875}, {"start": 857.25, "end": 857.89, "word": " كانت", "probability": 0.992919921875}, {"start": 857.89, "end": 859.41, "word": " ال", "probability": 0.9208984375}, {"start": 859.41, "end": 859.81, "word": " set", "probability": 0.88427734375}, {"start": 859.81, "end": 863.33, "word": " ماهياش", "probability": 0.927734375}, {"start": 863.33, "end": 863.69, "word": " finite", "probability": 0.98046875}], "temperature": 1.0}, {"id": 32, "seek": 89239, "start": 865.07, "end": 892.39, "text": "وماهياش in one to one correspondence with the natural numbers او ماهياش denumerable فبنسميها uncountable غير قابلة للعد countable قابلة للعد uncountable غير قابلة للعد طيب ال ..", "tokens": [2407, 15042, 3224, 1829, 33599, 294, 472, 281, 472, 38135, 365, 264, 3303, 3547, 1975, 2407, 19446, 3224, 1829, 33599, 1441, 15583, 712, 6156, 3555, 1863, 38251, 1829, 11296, 6219, 792, 712, 32771, 13546, 12174, 16758, 37977, 24976, 22488, 1207, 712, 12174, 16758, 37977, 24976, 22488, 6219, 792, 712, 32771, 13546, 12174, 16758, 37977, 24976, 22488, 23032, 1829, 3555, 2423, 4386], "avg_logprob": -0.16519657906986052, "compression_ratio": 1.5973154362416107, "no_speech_prob": 0.0, "words": [{"start": 865.0699999999999, "end": 866.03, "word": "وماهياش", "probability": 0.78505859375}, {"start": 866.03, "end": 866.57, "word": " in", "probability": 0.62109375}, {"start": 866.57, "end": 866.85, "word": " one", "probability": 0.91064453125}, {"start": 866.85, "end": 867.15, "word": " to", "probability": 0.85400390625}, {"start": 867.15, "end": 867.37, "word": " one", "probability": 0.95068359375}, {"start": 867.37, "end": 868.29, "word": " 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ان ال interval هذي و ال interval هذي كلا هما uncountable الفترة المفتوحة من سفر لواحد infinite set اول حاجة infinite set و طبعا ممكن تثبت انها uncountable", "tokens": [2304, 3615, 32887, 5172, 8978, 3714, 3555, 18513, 19986, 12602, 1829, 46958, 1829, 9307, 11778, 2288, 3794, 8315, 16472, 2423, 15035, 8032, 8848, 1829, 4032, 2423, 15035, 8032, 8848, 1829, 9122, 15040, 8032, 15042, 6219, 792, 712, 27188, 2655, 25720, 9673, 5172, 2655, 2407, 5016, 3660, 9154, 8608, 5172, 2288, 5296, 14407, 24401, 13785, 992, 1975, 12610, 11331, 26108, 3660, 13785, 992, 4032, 23032, 3555, 3615, 995, 3714, 43020, 6055, 12984, 3555, 2655, 16472, 11296, 6219, 792, 712], "avg_logprob": -0.19649921301044995, "compression_ratio": 1.5747126436781609, "no_speech_prob": 0.0, "words": [{"start": 895.18, "end": 896.18, "word": "معروف", "probability": 0.9410400390625}, {"start": 896.18, "end": 897.18, "word": " في", "probability": 0.71435546875}, {"start": 897.18, "end": 897.82, "word": " مبادئ", "probability": 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957.43, "end": 984.87, "text": "الـ احنا عايزين نثبت عشان نثبت ان ال set هذه ال R لاحظوا ان ال R is in one to one correspondence مع الفترة المفتوحة او المغلقة حتى في byjection بينه وبين الفترة", "tokens": [6027, 39184, 1975, 5016, 8315, 6225, 47302, 11622, 9957, 8717, 12984, 3555, 2655, 6225, 8592, 7649, 8717, 12984, 3555, 2655, 16472, 2423, 992, 29538, 2423, 497, 20193, 5016, 19913, 14407, 16472, 2423, 497, 307, 294, 472, 281, 472, 38135, 20449, 27188, 2655, 25720, 9673, 5172, 2655, 2407, 5016, 3660, 1975, 2407, 9673, 17082, 1211, 28671, 11331, 49975, 8978, 538, 1020, 313, 49374, 3224, 46599, 9957, 27188, 2655, 25720], "avg_logprob": -0.2290534420289855, "compression_ratio": 1.5121951219512195, "no_speech_prob": 0.0, "words": [{"start": 957.4300000000001, "end": 958.3100000000001, "word": "الـ", "probability": 0.26239013671875}, {"start": 958.3100000000001, "end": 959.19, "word": " احنا", "probability": 0.7806803385416666}, {"start": 959.19, "end": 959.71, "word": " عايزين", 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"probability": 0.84259033203125}, {"start": 1032.58, "end": 1032.84, "word": " هذه", "probability": 0.87060546875}, {"start": 1032.84, "end": 1033.28, "word": " finite", "probability": 0.9873046875}, {"start": 1033.28, "end": 1033.8, "word": " إذا", "probability": 0.78662109375}, {"start": 1033.8, "end": 1034.14, "word": " كانت", "probability": 0.917236328125}, {"start": 1034.14, "end": 1034.14, "word": " هذه", "probability": 0.97998046875}, {"start": 1034.14, "end": 1034.48, "word": " infinite", "probability": 0.69921875}, {"start": 1034.48, "end": 1034.92, "word": " تظهر", "probability": 0.974853515625}, {"start": 1034.92, "end": 1035.14, "word": " هذه", "probability": 0.94580078125}, {"start": 1035.14, "end": 1035.54, "word": " infinite", "probability": 0.97216796875}, {"start": 1035.54, "end": 1037.8, "word": " كل", "probability": 0.6376953125}, {"start": 1037.8, "end": 1038.02, "word": " هذه", "probability": 0.9765625}, {"start": 1038.02, "end": 1038.58, "word": " نظريات", 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36145, 27188, 2655, 25720, 8032, 18513, 7578, 6219, 792, 712, 5296, 26890, 3224, 7649, 29910, 23275, 8717, 25957, 1211, 4724, 2288, 3224, 7649, 20666, 2655, 8315, 4587, 11242], "avg_logprob": -0.2523619265750397, "compression_ratio": 1.8775510204081634, "no_speech_prob": 0.0, "words": [{"start": 1042.47, "end": 1042.79, "word": "إذا", "probability": 0.640625}, {"start": 1042.79, "end": 1043.01, "word": " لو", "probability": 0.51806640625}, {"start": 1043.01, "end": 1043.41, "word": " أثبتنا", "probability": 0.96845703125}, {"start": 1043.41, "end": 1043.57, "word": " إن", "probability": 0.446533203125}, {"start": 1043.57, "end": 1044.07, "word": " الفترة", "probability": 0.96142578125}, {"start": 1044.07, "end": 1044.35, "word": " هادى", "probability": 0.4852294921875}, {"start": 1044.35, "end": 1045.11, "word": " uncountable", "probability": 0.7174886067708334}, {"start": 1045.11, "end": 1045.65, "word": " فبطلع", "probability": 0.8146484375}, {"start": 1045.65, "end": 1045.93, "word": " R", "probability": 0.372802734375}, {"start": 1045.93, "end": 1046.73, "word": " uncountable", "probability": 0.7298177083333334}, {"start": 1046.73, "end": 1051.01, "word": " طيب", "probability": 0.79736328125}, {"start": 1051.01, "end": 1051.55, "word": " لإثبات", "probability": 0.841357421875}, {"start": 1051.55, "end": 1052.21, "word": " ذلك", "probability": 0.969482421875}, {"start": 1052.21, "end": 1054.21, "word": " هنا", "probability": 0.09002685546875}, {"start": 1054.21, "end": 1054.79, "word": " عايزين", "probability": 0.9334716796875}, {"start": 1054.79, "end": 1055.75, "word": " نثبت", "probability": 0.9866943359375}, {"start": 1055.75, "end": 1058.45, "word": " إن", "probability": 0.50390625}, {"start": 1058.45, "end": 1059.21, "word": " الفترة", "probability": 0.9822591145833334}, {"start": 1059.21, "end": 1060.73, "word": " هادى", "probability": 0.8961588541666666}, {"start": 1060.73, "end": 1061.91, "word": " نثبت", "probability": 0.77587890625}, {"start": 1061.91, "end": 1062.05, "word": " إن", "probability": 0.49755859375}, {"start": 1062.05, "end": 1062.53, "word": " الفترة", "probability": 0.9781901041666666}, {"start": 1062.53, "end": 1062.97, "word": " هادى", "probability": 0.9654947916666666}, {"start": 1062.97, "end": 1063.83, "word": " uncountable", "probability": 0.8444010416666666}, {"start": 1063.83, "end": 1064.39, "word": " لبرهان", "probability": 0.79754638671875}, {"start": 1064.39, "end": 1064.83, "word": " ذلك", "probability": 0.992919921875}, {"start": 1064.83, "end": 1066.63, "word": " نعمل", "probability": 0.8401692708333334}, {"start": 1066.63, "end": 1067.03, "word": " برهان", "probability": 0.975830078125}, {"start": 1067.03, "end": 1067.77, "word": " بالتناقض", "probability": 0.96455078125}], "temperature": 1.0}, {"id": 39, "seek": 109764, "start": 1077.1, "end": 1097.64, "text": "بنثبت ان الفترة المغلقة هذي uncountable نفرض المقيد ان الفترة هذي countable لاحظوا ان الفترة هذي infinite والان countable اذا بتطلع equipotent او in one to one correspondence with natural numbers", "tokens": [3555, 1863, 12984, 3555, 2655, 16472, 27188, 2655, 25720, 9673, 17082, 1211, 28671, 8032, 8848, 1829, 6219, 792, 712, 8717, 5172, 43042, 9673, 4587, 25708, 16472, 27188, 2655, 25720, 8032, 8848, 1829, 1207, 712, 20193, 5016, 19913, 14407, 16472, 27188, 2655, 25720, 8032, 8848, 1829, 13785, 16070, 7649, 1207, 712, 1975, 15730, 39894, 9566, 1211, 3615, 5037, 310, 317, 1975, 2407, 294, 472, 281, 472, 38135, 365, 3303, 3547], "avg_logprob": -0.315625005534717, "compression_ratio": 1.5941176470588236, "no_speech_prob": 0.0, "words": [{"start": 1077.1, "end": 1077.76, "word": "بنثبت", "probability": 0.652099609375}, {"start": 1077.76, "end": 1077.86, "word": " ان", "probability": 0.3935546875}, {"start": 1077.86, "end": 1078.26, "word": " الفترة", "probability": 0.9762369791666666}, {"start": 1078.26, "end": 1078.72, "word": " المغلقة", "probability": 0.89501953125}, {"start": 1078.72, "end": 1079.02, "word": " هذي", "probability": 0.48583984375}, {"start": 1079.02, "end": 1079.84, "word": " uncountable", "probability": 0.7830403645833334}, {"start": 1079.84, "end": 1081.16, "word": " نفرض", "probability": 0.4908040364583333}, {"start": 1081.16, "end": 1084.94, "word": " المقيد", "probability": 0.713134765625}, {"start": 1084.94, "end": 1085.46, "word": " ان", "probability": 0.4931640625}, {"start": 1085.46, "end": 1085.94, "word": " الفترة", "probability": 0.9840494791666666}, {"start": 1085.94, "end": 1086.28, "word": " هذي", "probability": 0.81689453125}, {"start": 1086.28, "end": 1086.94, "word": " countable", "probability": 0.967041015625}, {"start": 1086.94, "end": 1087.92, "word": " لاحظوا", "probability": 0.793548583984375}, {"start": 1087.92, "end": 1088.02, "word": " ان", "probability": 0.79931640625}, {"start": 1088.02, "end": 1088.44, "word": " الفترة", "probability": 0.98193359375}, {"start": 1088.44, "end": 1088.78, "word": " هذي", "probability": 0.90283203125}, {"start": 1088.78, "end": 1089.32, "word": " infinite", "probability": 0.74560546875}, {"start": 1089.32, "end": 1090.88, "word": " والان", "probability": 0.6458740234375}, {"start": 1090.88, "end": 1091.72, "word": " countable", "probability": 0.955078125}, {"start": 1091.72, "end": 1092.7, "word": " اذا", "probability": 0.5946044921875}, {"start": 1092.7, "end": 1093.26, "word": " بتطلع", "probability": 0.899169921875}, {"start": 1093.26, "end": 1094.26, "word": " equipotent", "probability": 0.7721354166666666}, {"start": 1094.26, "end": 1094.5, "word": " او", "probability": 0.907958984375}, {"start": 1094.5, "end": 1094.68, "word": " in", "probability": 0.7099609375}, {"start": 1094.68, "end": 1094.88, "word": " one", "probability": 0.9287109375}, {"start": 1094.88, "end": 1094.98, "word": " to", "probability": 0.7041015625}, {"start": 1094.98, "end": 1095.1, "word": " one", "probability": 0.94384765625}, {"start": 1095.1, "end": 1095.76, "word": " correspondence", "probability": 0.94921875}, {"start": 1095.76, "end": 1096.34, "word": " with", "probability": 0.92041015625}, {"start": 1096.34, "end": 1096.9, "word": " natural", "probability": 0.51904296875}, {"start": 1096.9, "end": 1097.64, "word": " numbers", "probability": 0.84619140625}], "temperature": 1.0}, {"id": 40, "seek": 111157, "start": 1102.85, "end": 1111.57, "text": "الأن في الحالة هذه I in one to one correspondence within actual numbers أو بنسميها innumerable صح؟", "tokens": [6027, 33456, 8978, 21542, 6027, 3660, 29538, 286, 294, 472, 281, 472, 38135, 1951, 3539, 3547, 34051, 44945, 38251, 1829, 11296, 7714, 15583, 712, 20328, 5016, 22807], "avg_logprob": -0.3024553528853825, "compression_ratio": 1.050420168067227, "no_speech_prob": 0.0, "words": [{"start": 1102.85, "end": 1103.21, "word": "الأن", "probability": 0.4774169921875}, {"start": 1103.21, "end": 1103.33, "word": " في", "probability": 0.84423828125}, {"start": 1103.33, "end": 1103.73, "word": " الحالة", "probability": 0.98681640625}, {"start": 1103.73, "end": 1104.03, "word": " هذه", "probability": 0.8701171875}, {"start": 1104.03, "end": 1104.49, "word": " I", "probability": 0.666015625}, {"start": 1104.49, "end": 1105.31, "word": " in", "probability": 0.77392578125}, {"start": 1105.31, "end": 1105.59, "word": " one", "probability": 0.95263671875}, {"start": 1105.59, "end": 1105.71, "word": " to", "probability": 0.833984375}, {"start": 1105.71, "end": 1105.89, "word": " one", "probability": 0.94873046875}, {"start": 1105.89, "end": 1106.55, "word": " correspondence", "probability": 0.93701171875}, {"start": 1106.55, "end": 1107.11, "word": " within", "probability": 0.34326171875}, {"start": 1107.11, "end": 1107.55, "word": " actual", "probability": 0.76318359375}, {"start": 1107.55, "end": 1108.17, "word": " numbers", "probability": 0.85205078125}, {"start": 1108.17, "end": 1109.03, "word": " أو", "probability": 0.59130859375}, {"start": 1109.03, "end": 1109.77, "word": " بنسميها", "probability": 0.8753662109375}, {"start": 1109.77, "end": 1110.43, "word": " innumerable", "probability": 0.5974934895833334}, {"start": 1110.43, "end": 1111.57, "word": " صح؟", "probability": 0.9039713541666666}], "temperature": 1.0}, {"id": 41, "seek": 113980, "start": 1113.28, "end": 1139.8, "text": "الان ال set I denominable يعني ممكن ترقيمها بالأعداد الطبيعية إذا ممكن نسمي عناصرها xn-n عدد طبيعي اللي هي x1, x2, x3 الاخرى أي set denominable أو in one to one correspondence with natural numbers ممكن ترقيم عناصرها ك list by the natural numbers إذا I هي كل عناصرها رقمناهم بالأعداد الطبيعية", "tokens": [6027, 7649, 2423, 992, 286, 16244, 712, 37495, 22653, 3714, 43020, 6055, 2288, 4587, 32640, 11296, 20666, 10721, 22488, 18513, 41950, 21292, 3615, 10632, 11933, 15730, 3714, 43020, 8717, 38251, 1829, 18871, 33546, 2288, 11296, 2031, 77, 12, 77, 6225, 3215, 3215, 23032, 21292, 3615, 1829, 13672, 1829, 39896, 2031, 16, 11, 2031, 17, 11, 2031, 18, 2423, 47283, 2288, 7578, 36632, 992, 16244, 712, 34051, 294, 472, 281, 472, 38135, 365, 3303, 3547, 3714, 43020, 6055, 2288, 4587, 32640, 18871, 33546, 2288, 11296, 9122, 1329, 538, 264, 3303, 3547, 11933, 15730, 286, 39896, 28242, 18871, 33546, 2288, 11296, 12602, 4587, 2304, 8315, 16095, 20666, 10721, 22488, 18513, 41950, 21292, 3615, 10632], "avg_logprob": -0.22856747365630833, "compression_ratio": 1.7890295358649788, "no_speech_prob": 0.0, "words": [{"start": 1113.28, "end": 1113.74, "word": "الان", "probability": 0.8251953125}, {"start": 1113.74, "end": 1113.94, "word": " ال", "probability": 0.69482421875}, {"start": 1113.94, "end": 1114.18, "word": " set", "probability": 0.720703125}, {"start": 1114.18, "end": 1114.42, "word": " I", "probability": 0.4423828125}, {"start": 1114.42, "end": 1115.06, "word": " denominable", "probability": 0.66552734375}, {"start": 1115.06, "end": 1115.38, "word": " يعني", "probability": 0.845458984375}, {"start": 1115.38, "end": 1115.7, "word": " ممكن", "probability": 0.982177734375}, {"start": 1115.7, "end": 1116.56, "word": " ترقيمها", "probability": 0.96064453125}, {"start": 1116.56, "end": 1117.32, "word": " بالأعداد", "probability": 0.80841064453125}, {"start": 1117.32, "end": 1118.0, "word": " الطبيعية", "probability": 0.955078125}, {"start": 1118.0, "end": 1118.34, "word": " إذا", "probability": 0.50225830078125}, {"start": 1118.34, "end": 1118.76, "word": " ممكن", "probability": 0.913818359375}, {"start": 1118.76, "end": 1119.34, "word": " نسمي", "probability": 0.87109375}, {"start": 1119.34, "end": 1119.98, "word": " عناصرها", "probability": 0.8487548828125}, {"start": 1119.98, "end": 1120.9, "word": " xn", "probability": 0.4947509765625}, {"start": 1120.9, "end": 1121.58, "word": "-n", "probability": 0.4942626953125}, {"start": 1121.58, "end": 1121.84, "word": " عدد", "probability": 0.7550455729166666}, {"start": 1121.84, "end": 1122.22, "word": " طبيعي", "probability": 0.9515380859375}, {"start": 1122.22, "end": 1122.38, "word": " اللي", "probability": 0.60467529296875}, {"start": 1122.38, "end": 1122.5, "word": " هي", "probability": 0.7880859375}, {"start": 1122.5, "end": 1122.94, "word": " x1,", "probability": 0.805908203125}, {"start": 1123.1, "end": 1123.52, "word": " x2,", "probability": 0.888916015625}, {"start": 1123.64, "end": 1124.08, "word": " x3", "probability": 0.9501953125}, {"start": 1124.08, "end": 1125.12, "word": " الاخرى", "probability": 0.596466064453125}, {"start": 1125.12, "end": 1125.3, "word": " أي", "probability": 0.4072265625}, {"start": 1125.3, "end": 1125.64, "word": " set", "probability": 0.97119140625}, {"start": 1125.64, "end": 1126.34, "word": " denominable", "probability": 0.821044921875}, {"start": 1126.34, "end": 1127.1, "word": " أو", "probability": 0.740234375}, {"start": 1127.1, "end": 1127.8, "word": " in", "probability": 0.66552734375}, {"start": 1127.8, "end": 1128.04, "word": " one", "probability": 0.9091796875}, {"start": 1128.04, "end": 1128.2, "word": " to", "probability": 0.7451171875}, {"start": 1128.2, "end": 1128.4, "word": " one", "probability": 0.9501953125}, {"start": 1128.4, "end": 1129.14, "word": " correspondence", "probability": 0.95458984375}, {"start": 1129.14, "end": 1129.5, "word": " with", "probability": 0.919921875}, {"start": 1129.5, "end": 1129.9, "word": " natural", "probability": 0.425537109375}, {"start": 1129.9, "end": 1130.54, "word": " numbers", "probability": 0.8564453125}, {"start": 1130.54, "end": 1131.32, "word": " ممكن", "probability": 0.981689453125}, {"start": 1131.32, "end": 1131.92, "word": " ترقيم", "probability": 0.97216796875}, {"start": 1131.92, "end": 1132.5, "word": " عناصرها", "probability": 0.9798583984375}, {"start": 1132.5, "end": 1132.76, "word": " ك", "probability": 0.97900390625}, {"start": 1132.76, "end": 1133.16, "word": " list", "probability": 0.88330078125}, {"start": 1133.16, "end": 1134.54, "word": " by", "probability": 0.9072265625}, {"start": 1134.54, "end": 1134.76, "word": " the", "probability": 0.85595703125}, {"start": 1134.76, "end": 1135.14, "word": " natural", "probability": 0.841796875}, {"start": 1135.14, "end": 1135.68, "word": " numbers", "probability": 0.79443359375}, {"start": 1135.68, "end": 1136.0, "word": " إذا", "probability": 0.5262451171875}, {"start": 1136.0, "end": 1136.72, "word": " I", "probability": 0.8681640625}, {"start": 1136.72, "end": 1137.1, "word": " هي", "probability": 0.408203125}, {"start": 1137.1, "end": 1137.38, "word": " كل", "probability": 0.89453125}, {"start": 1137.38, "end": 1137.96, "word": " عناصرها", "probability": 0.9888916015625}, {"start": 1137.96, "end": 1138.66, "word": " رقمناهم", "probability": 0.901953125}, {"start": 1138.66, "end": 1139.2, "word": " بالأعداد", "probability": 0.941650390625}, {"start": 1139.2, "end": 1139.8, "word": " الطبيعية", "probability": 0.9822998046875}], "temperature": 1.0}, {"id": 42, "seek": 116721, "start": 1145.09, "end": 1167.21, "text": "لحظة احنا بدنا نصل الى تناقض احنا بدنا القرآن وفرضنا ال contrary هيو Assume ال contrary ان I is countable بدنا نصل الى تناقض طيب هي الفترة I هي الفترة I هي الفترة I هذه I", "tokens": [1211, 5016, 19913, 3660, 1975, 5016, 8315, 47525, 8315, 8717, 36520, 2423, 7578, 6055, 8315, 4587, 11242, 1975, 5016, 8315, 47525, 8315, 25062, 2288, 148, 48506, 4032, 5172, 43042, 8315, 2423, 19506, 39896, 2407, 6281, 2540, 2423, 19506, 16472, 286, 307, 1207, 712, 47525, 8315, 8717, 36520, 2423, 7578, 6055, 8315, 4587, 11242, 23032, 1829, 3555, 39896, 27188, 2655, 25720, 286, 39896, 27188, 2655, 25720, 286, 39896, 27188, 2655, 25720, 286, 29538, 286], "avg_logprob": -0.26478039493431915, "compression_ratio": 1.8541666666666667, "no_speech_prob": 0.0, "words": [{"start": 1145.09, "end": 1145.61, "word": "لحظة", "probability": 0.80047607421875}, {"start": 1145.61, "end": 1145.85, "word": " احنا", "probability": 0.6822509765625}, {"start": 1145.85, "end": 1146.09, "word": " بدنا", "probability": 0.8466796875}, {"start": 1146.09, "end": 1146.41, "word": " نصل", "probability": 0.95458984375}, {"start": 1146.41, "end": 1146.65, "word": " الى", "probability": 0.77734375}, {"start": 1146.65, "end": 1147.21, "word": " تناقض", "probability": 0.958984375}, {"start": 1147.21, "end": 1147.47, "word": " احنا", "probability": 0.8396809895833334}, {"start": 1147.47, "end": 1147.77, "word": " بدنا", "probability": 0.87109375}, {"start": 1147.77, "end": 1148.35, "word": " القرآن", "probability": 0.6682586669921875}, {"start": 1148.35, "end": 1149.97, "word": " وفرضنا", "probability": 0.8033447265625}, {"start": 1149.97, "end": 1150.11, "word": " ال", "probability": 0.91552734375}, {"start": 1150.11, "end": 1150.65, "word": " contrary", "probability": 0.5712890625}, {"start": 1150.65, "end": 1151.23, "word": " هيو", "probability": 0.4146728515625}, {"start": 1151.23, "end": 1152.91, "word": " Assume", "probability": 0.60528564453125}, {"start": 1152.91, "end": 1153.31, "word": " ال", "probability": 0.90625}, {"start": 1153.31, "end": 1154.01, "word": " contrary", "probability": 0.8408203125}, {"start": 1154.01, "end": 1154.97, "word": " ان", "probability": 0.75244140625}, {"start": 1154.97, "end": 1155.35, "word": " I", "probability": 0.424560546875}, {"start": 1155.35, "end": 1155.65, "word": " is", "probability": 0.923828125}, {"start": 1155.65, "end": 1156.25, "word": " countable", "probability": 0.95751953125}, {"start": 1156.25, "end": 1156.55, "word": " بدنا", "probability": 0.849365234375}, {"start": 1156.55, "end": 1156.83, "word": " نصل", "probability": 0.985595703125}, {"start": 1156.83, "end": 1157.07, "word": " الى", "probability": 0.96630859375}, {"start": 1157.07, "end": 1157.69, "word": " تناقض", "probability": 0.991455078125}, {"start": 1157.69, "end": 1158.61, "word": " طيب", "probability": 0.7513020833333334}, {"start": 1158.61, "end": 1158.75, "word": " هي", "probability": 0.402099609375}, {"start": 1158.75, "end": 1159.27, "word": " الفترة", "probability": 0.9811197916666666}, {"start": 1159.27, "end": 1159.57, "word": " I", "probability": 0.99072265625}, {"start": 1159.57, "end": 1159.87, "word": " هي", "probability": 0.381103515625}, {"start": 1159.87, "end": 1160.41, "word": " الفترة", "probability": 0.98095703125}, {"start": 1160.41, "end": 1160.77, "word": " I", "probability": 0.986328125}, {"start": 1160.77, "end": 1162.41, "word": " هي", "probability": 0.55078125}, {"start": 1162.41, "end": 1162.93, "word": " الفترة", "probability": 0.9900716145833334}, {"start": 1162.93, "end": 1163.85, "word": " I", "probability": 0.9970703125}, {"start": 1163.85, "end": 1166.89, "word": " هذه", "probability": 0.371337890625}, {"start": 1166.89, "end": 1167.21, "word": " I", "probability": 0.99072265625}], "temperature": 1.0}, {"id": 43, "seek": 119510, "start": 1168.25, "end": 1195.11, "text": "و في اندس اي هاد الفترة اي هاد يعني عناصرها هي مرقمة اكس واحد اكس اتنين الى ملا نهاية افرض ان اكس واحد موجود هان اول عنصر في الفترة موجود هان فممكن اختار فترة مغلقة اسميها اي واحد ممكن اختار فترة مغلقة", "tokens": [2407, 8978, 16472, 3215, 3794, 1975, 1829, 8032, 18513, 27188, 2655, 25720, 1975, 1829, 8032, 18513, 37495, 22653, 18871, 33546, 2288, 11296, 39896, 3714, 2288, 4587, 46007, 1975, 4117, 3794, 36764, 24401, 1975, 4117, 3794, 1975, 2655, 1863, 9957, 2423, 7578, 3714, 15040, 8717, 11296, 10632, 1975, 5172, 43042, 16472, 1975, 4117, 3794, 36764, 24401, 3714, 29245, 23328, 8032, 7649, 1975, 12610, 18871, 9381, 2288, 8978, 27188, 2655, 25720, 3714, 29245, 23328, 8032, 7649, 6156, 2304, 43020, 1975, 46456, 9640, 6156, 2655, 25720, 3714, 17082, 1211, 28671, 24525, 2304, 1829, 11296, 1975, 1829, 36764, 24401, 3714, 43020, 1975, 46456, 9640, 6156, 2655, 25720, 3714, 17082, 1211, 28671], "avg_logprob": -0.2122395840232019, "compression_ratio": 2.228395061728395, "no_speech_prob": 0.0, "words": [{"start": 1168.25, "end": 1168.47, "word": "و", "probability": 0.5712890625}, {"start": 1168.47, "end": 1168.59, "word": " في", "probability": 0.30126953125}, {"start": 1168.59, "end": 1168.99, "word": " اندس", "probability": 0.6846516927083334}, 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"probability": 0.97607421875}, {"start": 1180.89, "end": 1181.03, "word": " في", "probability": 0.60888671875}, {"start": 1181.03, "end": 1181.55, "word": " الفترة", "probability": 0.99072265625}, {"start": 1181.55, "end": 1182.59, "word": " موجود", "probability": 0.9825846354166666}, {"start": 1182.59, "end": 1182.89, "word": " هان", "probability": 0.907958984375}, {"start": 1182.89, "end": 1186.51, "word": " فممكن", "probability": 0.9181315104166666}, {"start": 1186.51, "end": 1187.11, "word": " اختار", "probability": 0.9791666666666666}, {"start": 1187.11, "end": 1187.55, "word": " فترة", "probability": 0.9915364583333334}, {"start": 1187.55, "end": 1188.15, "word": " مغلقة", "probability": 0.9659423828125}, {"start": 1188.15, "end": 1190.35, "word": " اسميها", "probability": 0.9002685546875}, {"start": 1190.35, "end": 1190.69, "word": " اي", "probability": 0.761474609375}, {"start": 1190.69, "end": 1191.19, "word": " واحد", "probability": 0.990478515625}, {"start": 1191.19, "end": 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"probability": 0.975341796875}, {"start": 1212.0, "end": 1212.84, "word": " موجودة", "probability": 0.9541015625}, {"start": 1212.84, "end": 1213.1, "word": " هنا", "probability": 0.697265625}, {"start": 1213.1, "end": 1215.94, "word": " العنصر", "probability": 0.9488525390625}, {"start": 1215.94, "end": 1216.36, "word": " التاني", "probability": 0.9064127604166666}, {"start": 1216.36, "end": 1216.44, "word": " في", "probability": 0.8642578125}, {"start": 1216.44, "end": 1216.88, "word": " الفترة", "probability": 0.9822591145833334}, {"start": 1216.88, "end": 1217.18, "word": " I", "probability": 0.9912109375}, {"start": 1217.18, "end": 1217.46, "word": " ايه", "probability": 0.7827962239583334}, {"start": 1217.46, "end": 1217.74, "word": " رقم", "probability": 0.96533203125}, {"start": 1217.74, "end": 1218.18, "word": " X2", "probability": 0.96533203125}, {"start": 1218.18, "end": 1218.68, "word": " موجود", "probability": 0.92626953125}, {"start": 1218.68, "end": 1218.94, "word": " هنا", "probability": 0.8935546875}, {"start": 1218.94, "end": 1219.68, "word": " او", "probability": 0.802978515625}, {"start": 1219.68, "end": 1219.88, "word": " هنا", "probability": 0.8505859375}, {"start": 1219.88, "end": 1220.26, "word": " او", "probability": 0.917236328125}, {"start": 1220.26, "end": 1220.56, "word": " هنا", "probability": 0.970703125}, {"start": 1220.56, "end": 1222.56, "word": " فبقدر", "probability": 0.9603271484375}, {"start": 1222.56, "end": 1223.22, "word": " اختار", "probability": 0.9695638020833334}, {"start": 1223.22, "end": 1223.7, "word": " فترة", "probability": 0.9866536458333334}, {"start": 1223.7, "end": 1224.2, "word": " مغلقة", "probability": 0.9833984375}, {"start": 1224.2, "end": 1224.74, "word": " تانية", "probability": 0.9943033854166666}], "temperature": 1.0}, {"id": 45, "seek": 125424, "start": 1225.76, "end": 1254.24, "text": "نسميها I2 اللي هي الفترة هذه بحيث ان X2 لا تنتمي ل I2 و الفترة I2 هي فترة جزئية من I1 طيب افرض ان X3 موجودة هنا او 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{"start": 1531.51, "end": 1531.77, "word": " X", "probability": 0.634765625}], "temperature": 1.0}, {"id": 56, "seek": 155525, "start": 1533.73, "end": 1555.25, "text": "تمام؟ الآن في البرهان هذا وجدنا ان في صي أنصر جديد في I مختلف عن كل عناصرها معناته هذه ال list ليست ترقيم كامل ل I في عناصر أخرى زي صي مش مرقمة و هذا تناقض", "tokens": [2655, 15042, 2304, 22807, 6024, 48506, 8978, 2423, 26890, 3224, 7649, 23758, 49610, 3215, 8315, 16472, 8978, 20328, 1829, 14739, 9381, 2288, 10874, 16254, 3215, 8978, 286, 3714, 46456, 46538, 18871, 28242, 18871, 33546, 2288, 11296, 20449, 8315, 47395, 29538, 2423, 1329, 32239, 14851, 6055, 2288, 4587, 32640, 9122, 10943, 1211, 5296, 286, 8978, 18871, 33546, 2288, 5551, 34740, 7578, 30767, 1829, 20328, 1829, 37893, 3714, 2288, 4587, 46007, 4032, 23758, 6055, 8315, 4587, 11242], "avg_logprob": -0.19274259868421054, "compression_ratio": 1.6204819277108433, "no_speech_prob": 0.0, "words": [{"start": 1533.73, "end": 1534.69, "word": "تمام؟", "probability": 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0.72509765625}, {"start": 1543.65, "end": 1544.67, "word": " مختلف", "probability": 0.98779296875}, {"start": 1544.67, "end": 1544.83, "word": " عن", "probability": 0.96240234375}, {"start": 1544.83, "end": 1545.07, "word": " كل", "probability": 0.990234375}, {"start": 1545.07, "end": 1545.71, "word": " عناصرها", "probability": 0.975830078125}, {"start": 1545.71, "end": 1547.55, "word": " معناته", "probability": 0.7947591145833334}, {"start": 1547.55, "end": 1547.93, "word": " هذه", "probability": 0.4912109375}, {"start": 1547.93, "end": 1548.29, "word": " ال", "probability": 0.93359375}, {"start": 1548.29, "end": 1548.67, "word": " list", "probability": 0.95751953125}, {"start": 1548.67, "end": 1549.31, "word": " ليست", "probability": 0.943115234375}, {"start": 1549.31, "end": 1549.87, "word": " ترقيم", "probability": 0.97900390625}, {"start": 1549.87, "end": 1550.35, "word": " كامل", "probability": 0.994140625}, {"start": 1550.35, "end": 1550.45, "word": " ل", "probability": 0.947265625}, {"start": 1550.45, "end": 1550.65, "word": " I", "probability": 0.919921875}, {"start": 1550.65, "end": 1550.85, "word": " في", "probability": 0.42431640625}, {"start": 1550.85, "end": 1551.33, "word": " عناصر", "probability": 0.9933268229166666}, {"start": 1551.33, "end": 1551.87, "word": " أخرى", "probability": 0.8728841145833334}, {"start": 1551.87, "end": 1552.53, "word": " زي", "probability": 0.916015625}, {"start": 1552.53, "end": 1552.79, "word": " صي", "probability": 0.937255859375}, {"start": 1552.79, "end": 1552.99, "word": " مش", "probability": 0.98876953125}, {"start": 1552.99, "end": 1553.61, "word": " مرقمة", "probability": 0.8980712890625}, {"start": 1553.61, "end": 1554.47, "word": " و", "probability": 0.458984375}, {"start": 1554.47, "end": 1554.65, "word": " هذا", "probability": 0.89453125}, {"start": 1554.65, "end": 1555.25, "word": " تناقض", "probability": 0.97021484375}], "temperature": 1.0}, {"id": 57, "seek": 157525, "start": 1556.63, "end": 1575.25, "text": "لأن إحنا عندنا ال set I هذي countable و infinite و denomable إذن هي list تبع كل عناصرها فكيف طلع فيه أنصر جديد مختلف عن عناصرها فهذا تناقض إذن هذا التناقض بثبت أن فرضنا أن الفترة I", "tokens": [1211, 33456, 11933, 5016, 8315, 43242, 8315, 2423, 992, 286, 8032, 8848, 1829, 1207, 712, 4032, 13785, 4032, 1441, 298, 712, 11933, 8848, 1863, 39896, 1329, 6055, 3555, 3615, 28242, 18871, 33546, 2288, 11296, 6156, 4117, 33911, 23032, 1211, 3615, 8978, 3224, 14739, 9381, 2288, 10874, 16254, 3215, 3714, 46456, 46538, 18871, 18871, 33546, 2288, 11296, 6156, 3224, 15730, 6055, 8315, 4587, 11242, 11933, 8848, 1863, 23758, 16712, 8315, 4587, 11242, 4724, 12984, 3555, 2655, 14739, 6156, 43042, 8315, 14739, 27188, 2655, 25720, 286], "avg_logprob": -0.2531249915852266, "compression_ratio": 1.5210526315789474, "no_speech_prob": 0.0, "words": [{"start": 1556.63, "end": 1557.03, "word": "لأن", "probability": 0.573974609375}, {"start": 1557.03, "end": 1557.33, "word": " إحنا", 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"word": " هي", "probability": 0.52392578125}, {"start": 1562.29, "end": 1562.77, "word": " list", "probability": 0.7666015625}, {"start": 1562.77, "end": 1563.23, "word": " تبع", "probability": 0.9225260416666666}, {"start": 1563.23, "end": 1563.55, "word": " كل", "probability": 0.9521484375}, {"start": 1563.55, "end": 1564.21, "word": " عناصرها", "probability": 0.923095703125}, {"start": 1564.21, "end": 1565.35, "word": " فكيف", "probability": 0.9215494791666666}, {"start": 1565.35, "end": 1565.65, "word": " طلع", "probability": 0.9607747395833334}, {"start": 1565.65, "end": 1565.83, "word": " فيه", "probability": 0.776123046875}, {"start": 1565.83, "end": 1566.19, "word": " أنصر", "probability": 0.9339192708333334}, {"start": 1566.19, "end": 1566.53, "word": " جديد", "probability": 0.9703776041666666}, {"start": 1566.53, "end": 1567.03, "word": " مختلف", "probability": 0.9801432291666666}, {"start": 1567.03, "end": 1567.39, "word": " عن", "probability": 0.9970703125}, {"start": 1567.39, "end": 1568.81, "word": " عناصرها", "probability": 0.9249267578125}, {"start": 1568.81, "end": 1570.65, "word": " فهذا", "probability": 0.8391927083333334}, {"start": 1570.65, "end": 1571.23, "word": " تناقض", "probability": 0.86279296875}, {"start": 1571.23, "end": 1571.93, "word": " إذن", "probability": 0.7724609375}, {"start": 1571.93, "end": 1572.09, "word": " هذا", "probability": 0.904296875}, {"start": 1572.09, "end": 1572.59, "word": " التناقض", "probability": 0.987548828125}, {"start": 1572.59, "end": 1573.11, "word": " بثبت", "probability": 0.94921875}, {"start": 1573.11, "end": 1573.31, "word": " أن", "probability": 0.56201171875}, {"start": 1573.31, "end": 1573.83, "word": " فرضنا", "probability": 0.96533203125}, {"start": 1573.83, "end": 1574.07, "word": " أن", "probability": 0.6337890625}, {"start": 1574.07, "end": 1574.95, "word": " الفترة", "probability": 0.96484375}, {"start": 1574.95, "end": 1575.25, "word": " I", "probability": 0.95947265625}], "temperature": 1.0}, {"id": 58, "seek": 160622, "start": 1577.54, "end": 1606.22, "text": "كانت countable كان فرض خاطر وبالتالي الفترة I تطلع uncountable اذا الان الفترة I uncountable وانا عندي R equivalent لفترة مغلقة 01 ممكن نوجد bijection بينهم اذا ال R تطلع uncountable كما هو مطلوب اذا هذا هو برهان", "tokens": [41361, 2655, 1207, 712, 25961, 6156, 43042, 16490, 41193, 2288, 46599, 6027, 2655, 6027, 1829, 27188, 2655, 25720, 286, 6055, 9566, 1211, 3615, 6219, 792, 712, 1975, 15730, 2423, 7649, 27188, 2655, 25720, 286, 6219, 792, 712, 4032, 7649, 995, 18871, 16254, 497, 10344, 5296, 5172, 2655, 25720, 3714, 17082, 1211, 28671, 23185, 3714, 43020, 8717, 29245, 3215, 3228, 1020, 313, 49374, 16095, 1975, 15730, 2423, 497, 6055, 9566, 1211, 3615, 6219, 792, 712, 9122, 15042, 31439, 3714, 9566, 1211, 37746, 1975, 15730, 23758, 31439, 4724, 2288, 3224, 7649], "avg_logprob": -0.23854166136847602, "compression_ratio": 1.702127659574468, "no_speech_prob": 0.0, "words": [{"start": 1577.54, "end": 1578.14, "word": "كانت", "probability": 0.58349609375}, {"start": 1578.14, "end": 1578.76, "word": " countable", "probability": 0.944580078125}, {"start": 1578.76, "end": 1579.08, "word": " كان", "probability": 0.86865234375}, {"start": 1579.08, "end": 1579.44, "word": " فرض", "probability": 0.926513671875}, {"start": 1579.44, "end": 1579.94, "word": " خاطر", "probability": 0.9602864583333334}, {"start": 1579.94, "end": 1580.82, "word": " وبالتالي", "probability": 0.93505859375}, {"start": 1580.82, "end": 1581.36, "word": " الفترة", "probability": 0.95849609375}, {"start": 1581.36, "end": 1581.58, "word": " I", "probability": 0.6787109375}, {"start": 1581.58, "end": 1582.14, "word": " تطلع", "probability": 0.8447265625}, {"start": 1582.14, "end": 1583.18, "word": " uncountable", "probability": 0.8585611979166666}, {"start": 1583.18, "end": 1584.34, "word": " اذا", "probability": 0.546142578125}, {"start": 1584.34, "end": 1584.8, "word": " الان", "probability": 0.7103271484375}, {"start": 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{"start": 1605.38, "end": 1605.64, "word": " هو", "probability": 0.97705078125}, {"start": 1605.64, "end": 1606.22, "word": " برهان", "probability": 0.891845703125}], "temperature": 1.0}, {"id": 59, "seek": 163535, "start": 1608.05, "end": 1635.35, "text": "النظرية اللي فادت هي طبعا برهان بيعتمد على شوفنا ال nested interval theorem وبالتالي هذا برهان مختلف عن البرهان اللي بنعطيه في مبادئ الرياضيات في برهان تاني برضه لنظرية هذه يعطى في مبادئ الرياضيات اللي هو باستخدام counter diagonal argument مشهور يعني البرهان", "tokens": [6027, 1863, 19913, 2288, 10632, 13672, 1829, 6156, 18513, 2655, 39896, 23032, 3555, 3615, 995, 4724, 2288, 3224, 7649, 4724, 1829, 34268, 2304, 3215, 15844, 13412, 38688, 8315, 2423, 15646, 292, 15035, 20904, 46599, 6027, 2655, 6027, 1829, 23758, 4724, 2288, 3224, 7649, 3714, 46456, 46538, 18871, 2423, 26890, 3224, 7649, 13672, 1829, 44945, 3615, 9566, 1829, 3224, 8978, 3714, 3555, 18513, 19986, 2423, 16572, 46958, 1829, 9307, 8978, 4724, 2288, 3224, 7649, 6055, 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1610.75, "end": 1610.99, "word": " على", "probability": 0.8408203125}, {"start": 1610.99, "end": 1611.61, "word": " شوفنا", "probability": 0.7111002604166666}, {"start": 1611.61, "end": 1612.25, "word": " ال", "probability": 0.57421875}, {"start": 1612.25, "end": 1612.71, "word": " nested", "probability": 0.621826171875}, {"start": 1612.71, "end": 1613.35, "word": " interval", "probability": 0.951171875}, {"start": 1613.35, "end": 1613.93, "word": " theorem", "probability": 0.80029296875}, {"start": 1613.93, "end": 1615.23, "word": " وبالتالي", "probability": 0.92783203125}, {"start": 1615.23, "end": 1615.43, "word": " هذا", "probability": 0.8564453125}, {"start": 1615.43, "end": 1615.89, "word": " برهان", "probability": 0.984375}, {"start": 1615.89, "end": 1616.45, "word": " مختلف", "probability": 0.994140625}, {"start": 1616.45, "end": 1616.71, "word": " عن", "probability": 0.97509765625}, {"start": 1616.71, "end": 1617.21, "word": " البرهان", "probability": 0.911865234375}, {"start": 1617.21, "end": 1617.37, "word": " اللي", "probability": 0.95703125}, {"start": 1617.37, "end": 1617.95, "word": " بنعطيه", "probability": 0.9080078125}, {"start": 1617.95, "end": 1618.11, "word": " في", "probability": 0.9404296875}, {"start": 1618.11, "end": 1618.51, "word": " مبادئ", "probability": 0.9285888671875}, {"start": 1618.51, "end": 1619.23, "word": " الرياضيات", "probability": 0.90078125}, {"start": 1619.23, "end": 1621.73, "word": " في", "probability": 0.6611328125}, {"start": 1621.73, "end": 1622.11, "word": " برهان", "probability": 0.9884033203125}, {"start": 1622.11, "end": 1622.45, "word": " تاني", "probability": 0.9908854166666666}, {"start": 1622.45, "end": 1622.69, "word": " برضه", "probability": 0.9573567708333334}, {"start": 1622.69, "end": 1623.27, "word": " لنظرية", "probability": 0.9033203125}, {"start": 1623.27, "end": 1623.67, "word": " هذه", "probability": 0.77392578125}, {"start": 1623.67, "end": 1625.11, "word": " يعطى", "probability": 0.70654296875}, {"start": 1625.11, "end": 1625.27, "word": " في", "probability": 0.92041015625}, {"start": 1625.27, "end": 1625.69, "word": " مبادئ", "probability": 0.99365234375}, {"start": 1625.69, "end": 1626.85, "word": " الرياضيات", "probability": 0.97734375}, {"start": 1626.85, "end": 1628.59, "word": " اللي", "probability": 0.946533203125}, {"start": 1628.59, "end": 1628.71, "word": " هو", "probability": 0.95947265625}, {"start": 1628.71, "end": 1629.53, "word": " باستخدام", "probability": 0.9842122395833334}, {"start": 1629.53, "end": 1630.13, "word": " counter", "probability": 0.86328125}, {"start": 1630.13, "end": 1630.71, "word": " diagonal", "probability": 0.84130859375}, {"start": 1630.71, "end": 1631.49, "word": " argument", "probability": 0.85693359375}, {"start": 1631.49, "end": 1634.69, "word": " مشهور", "probability": 0.9871419270833334}, {"start": 1634.69, "end": 1634.85, "word": " يعني", "probability": 0.779052734375}, {"start": 1634.85, "end": 1635.35, "word": " البرهان", "probability": 0.9036865234375}], "temperature": 1.0}, {"id": 60, "seek": 165857, "start": 1636.77, "end": 1658.57, "text": "يرجع إلى العالم الرياضي Cantor يسمى Cantor دي اقنع ال argument بثبت ان الفترة المفتوحة من سفر لواحد is uncountable وبالتالي R is uncountable لأن R في byjection بينها وبين ال open interval من سفر لواحد الآن في نتيجة على نظرية الأخيرة هذه", "tokens": [13546, 7435, 3615, 30731, 18863, 45340, 2423, 16572, 46958, 1829, 26697, 284, 7251, 38251, 7578, 26697, 284, 11778, 1829, 1975, 4587, 1863, 3615, 2423, 6770, 4724, 12984, 3555, 2655, 16472, 27188, 2655, 25720, 9673, 5172, 2655, 2407, 5016, 3660, 9154, 8608, 5172, 2288, 5296, 14407, 24401, 307, 6219, 792, 712, 46599, 6027, 2655, 6027, 1829, 497, 307, 6219, 792, 712, 5296, 33456, 497, 8978, 538, 1020, 313, 49374, 11296, 46599, 9957, 2423, 1269, 15035, 9154, 8608, 5172, 2288, 5296, 14407, 24401, 6024, 48506, 8978, 8717, 31371, 7435, 3660, 15844, 8717, 19913, 2288, 10632, 16247, 9778, 48923, 29538], "avg_logprob": -0.2892219357344569, "compression_ratio": 1.5833333333333333, "no_speech_prob": 0.0, "words": [{"start": 1636.77, "end": 1637.47, "word": "يرجع", "probability": 0.77490234375}, {"start": 1637.47, "end": 1637.61, "word": " إلى", "probability": 0.65234375}, {"start": 1637.61, "end": 1637.97, "word": " العالم", "probability": 0.9794921875}, {"start": 1637.97, "end": 1638.53, "word": " الرياضي", "probability": 0.943115234375}, {"start": 1638.53, "end": 1639.11, "word": " Cantor", "probability": 0.5838623046875}, {"start": 1639.11, "end": 1640.99, "word": " يسمى", "probability": 0.545166015625}, {"start": 1640.99, "end": 1641.61, "word": " Cantor", "probability": 0.85595703125}, {"start": 1641.61, "end": 1641.81, "word": " دي", "probability": 0.49072265625}, {"start": 1641.81, "end": 1642.15, "word": " اقنع", "probability": 0.50860595703125}, {"start": 1642.15, "end": 1642.23, "word": " ال", "probability": 0.79443359375}, {"start": 1642.23, "end": 1642.97, "word": " argument", "probability": 0.65869140625}, {"start": 1642.97, "end": 1643.75, "word": " بثبت", "probability": 0.814697265625}, {"start": 1643.75, "end": 1643.87, "word": " ان", "probability": 0.6708984375}, {"start": 1643.87, "end": 1644.27, "word": " الفترة", "probability": 0.9886067708333334}, {"start": 1644.27, "end": 1644.75, "word": " المفتوحة", "probability": 0.970703125}, {"start": 1644.75, "end": 1644.91, "word": " من", "probability": 0.97412109375}, {"start": 1644.91, "end": 1645.21, "word": " سفر", "probability": 0.7003173828125}, {"start": 1645.21, "end": 1645.79, "word": " لواحد", "probability": 0.734375}, {"start": 1645.79, "end": 1646.85, "word": " is", "probability": 0.732421875}, {"start": 1646.85, "end": 1647.79, "word": " uncountable", "probability": 0.9099934895833334}, {"start": 1647.79, "end": 1648.43, "word": " وبالتالي", "probability": 0.90712890625}, {"start": 1648.43, "end": 1648.83, "word": " R", "probability": 0.78271484375}, {"start": 1648.83, "end": 1649.33, "word": " is", "probability": 0.88671875}, {"start": 1649.33, "end": 1650.07, "word": " uncountable", "probability": 0.9361979166666666}, {"start": 1650.07, "end": 1650.35, "word": " لأن", "probability": 0.6873779296875}, {"start": 1650.35, "end": 1650.67, "word": " R", "probability": 0.359130859375}, {"start": 1650.67, "end": 1650.85, "word": " في", "probability": 0.70068359375}, {"start": 1650.85, "end": 1651.45, "word": " byjection", "probability": 0.455322265625}, {"start": 1651.45, "end": 1651.99, "word": " بينها", "probability": 0.77001953125}, {"start": 1651.99, "end": 1652.87, "word": " وبين", "probability": 0.7320556640625}, {"start": 1652.87, "end": 1652.99, "word": " ال", "probability": 0.9521484375}, {"start": 1652.99, "end": 1653.31, "word": " open", "probability": 0.8662109375}, {"start": 1653.31, "end": 1653.87, "word": " interval", "probability": 0.9814453125}, {"start": 1653.87, "end": 1654.13, "word": " من", "probability": 0.982421875}, {"start": 1654.13, "end": 1654.47, "word": " سفر", "probability": 0.9480794270833334}, {"start": 1654.47, "end": 1655.15, "word": " لواحد", "probability": 0.9020182291666666}, {"start": 1655.15, "end": 1656.37, "word": " الآن", "probability": 0.56903076171875}, {"start": 1656.37, "end": 1656.55, "word": " في", "probability": 0.71142578125}, {"start": 1656.55, "end": 1657.07, "word": " نتيجة", "probability": 0.9923095703125}, {"start": 1657.07, "end": 1657.23, "word": " على", "probability": 0.78125}, {"start": 1657.23, "end": 1657.67, "word": " نظرية", "probability": 0.872802734375}, {"start": 1657.67, "end": 1658.19, "word": " الأخيرة", "probability": 0.91748046875}, {"start": 1658.19, "end": 1658.57, "word": " هذه", "probability": 0.7216796875}], "temperature": 1.0}, {"id": 61, "seek": 168599, "start": 1659.35, "end": 1685.99, "text": "الـ set هذه الـ R minus Q اللي هي الـ set of all irrationals أيضًا is uncountable والبرهان هنا نفس البرهان اللي بنعطيه في المبادئ برهان by contradiction assume and contrary إن ال set R minus Q is countable وإحنا عندنا نظرية في المبادئ أخدنا بتقول إن لو في عندي مجموعتين A وB", "tokens": [6027, 39184, 992, 29538, 2423, 39184, 497, 3175, 1249, 13672, 1829, 39896, 2423, 39184, 992, 295, 439, 3418, 2405, 1124, 36632, 11242, 14111, 995, 307, 6219, 792, 712, 16070, 26890, 3224, 7649, 34105, 8717, 36178, 2423, 26890, 3224, 7649, 13672, 1829, 44945, 3615, 9566, 1829, 3224, 8978, 9673, 3555, 18513, 19986, 4724, 2288, 3224, 7649, 538, 34937, 6552, 293, 19506, 36145, 2423, 992, 497, 3175, 1249, 307, 1207, 712, 4032, 28814, 5016, 8315, 43242, 8315, 8717, 19913, 2288, 10632, 8978, 9673, 3555, 18513, 19986, 5551, 9778, 3215, 8315, 39894, 39648, 36145, 45164, 8978, 18871, 16254, 3714, 7435, 2304, 45367, 2655, 9957, 316, 4032, 33], "avg_logprob": -0.23586310205005465, "compression_ratio": 1.6835443037974684, "no_speech_prob": 0.0, "words": [{"start": 1659.35, "end": 1659.89, "word": "الـ", "probability": 0.5858154296875}, {"start": 1659.89, "end": 1660.11, "word": " set", "probability": 0.61865234375}, {"start": 1660.11, "end": 1660.49, "word": " هذه", "probability": 0.385498046875}, {"start": 1660.49, "end": 1660.85, "word": " الـ", "probability": 0.55322265625}, {"start": 1660.85, "end": 1661.15, "word": " R", "probability": 0.397216796875}, {"start": 1661.15, "end": 1661.57, "word": " minus", "probability": 0.30810546875}, {"start": 1661.57, "end": 1661.89, "word": " Q", "probability": 0.82568359375}, {"start": 1661.89, "end": 1662.15, "word": " اللي", "probability": 0.5694580078125}, {"start": 1662.15, "end": 1662.33, "word": " هي", "probability": 0.9013671875}, {"start": 1662.33, "end": 1662.49, "word": " الـ", "probability": 0.384765625}, {"start": 1662.49, "end": 1662.79, "word": " set", "probability": 0.80859375}, {"start": 1662.79, "end": 1663.01, "word": " of", "probability": 0.92919921875}, {"start": 1663.01, "end": 1663.33, "word": " all", "probability": 0.92236328125}, {"start": 1663.33, "end": 1664.31, "word": " irrationals", 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2231.02, "word": " هذه", "probability": 0.64404296875}, {"start": 2231.02, "end": 2231.22, "word": " هدا", "probability": 0.35906982421875}, {"start": 2231.22, "end": 2231.4, "word": " هي", "probability": 0.80126953125}, {"start": 2231.4, "end": 2232.48, "word": " أنصرها", "probability": 0.604949951171875}, {"start": 2232.48, "end": 2235.76, "word": " فما", "probability": 0.794189453125}, {"start": 2235.76, "end": 2236.14, "word": " هو", "probability": 0.99072265625}, {"start": 2236.14, "end": 2238.6, "word": " بيناجيب", "probability": 0.666015625}, {"start": 2238.6, "end": 2238.9, "word": " lower", "probability": 0.8623046875}, {"start": 2238.9, "end": 2239.38, "word": " bound", "probability": 0.8994140625}, {"start": 2239.38, "end": 2239.92, "word": " او", "probability": 0.755126953125}, {"start": 2239.92, "end": 2240.34, "word": " اكبر", "probability": 0.90234375}, {"start": 2240.34, "end": 2240.66, "word": " lower", "probability": 0.8994140625}, {"start": 2240.66, "end": 2241.2, "word": " bound", "probability": 0.90673828125}, {"start": 2241.2, "end": 2241.7, "word": " اكبر", "probability": 0.8839518229166666}, {"start": 2241.7, "end": 2241.98, "word": " lower", "probability": 0.88134765625}, {"start": 2241.98, "end": 2242.4, "word": " bound", "probability": 0.892578125}, {"start": 2242.4, "end": 2245.02, "word": " طب", "probability": 0.71923828125}, {"start": 2245.02, "end": 2245.46, "word": " نلاحظ", "probability": 0.83367919921875}, {"start": 2245.46, "end": 2245.92, "word": " سالب", "probability": 0.7376302083333334}, {"start": 2245.92, "end": 2246.36, "word": " نص", "probability": 0.986328125}, {"start": 2246.36, "end": 2248.46, "word": " اصغر", "probability": 0.8887939453125}, {"start": 2248.46, "end": 2248.64, "word": " من", "probability": 0.99755859375}, {"start": 2248.64, "end": 2249.16, "word": " سالب", "probability": 0.96923828125}, {"start": 2249.16, "end": 2249.52, "word": " ربع", "probability": 0.9052734375}, {"start": 2249.52, "end": 2250.12, "word": " اصغر", "probability": 0.9757080078125}, {"start": 2250.12, "end": 2250.36, "word": " من", "probability": 0.9970703125}], "temperature": 1.0}, {"id": 82, "seek": 227201, "start": 2251.63, "end": 2272.01, "text": "بعد هيك سالب سادس اه فاعتقد ان هذا هيطلع سالب نص هذا اكبر lower bound", "tokens": [3555, 22488, 39896, 4117, 8608, 6027, 3555, 8608, 18513, 3794, 1975, 3224, 6156, 995, 34268, 28543, 16472, 23758, 39896, 9566, 1211, 3615, 8608, 6027, 3555, 8717, 9381, 23758, 1975, 4117, 26890, 3126, 5472], "avg_logprob": -0.20875460260054646, "compression_ratio": 1.2127659574468086, "no_speech_prob": 0.0, "words": [{"start": 2251.63, "end": 2252.17, "word": "بعد", "probability": 0.6588134765625}, {"start": 2252.17, "end": 2252.53, "word": " هيك", "probability": 0.84033203125}, {"start": 2252.53, "end": 2253.07, "word": " سالب", "probability": 0.7666829427083334}, {"start": 2253.07, "end": 2254.69, "word": " سادس", "probability": 0.7607421875}, {"start": 2254.69, "end": 2257.65, 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هذه أنا أخدت S بساوي الكلام هذا و A بساوي واحد اذا ال supremum ل S أربعة بساوي A زائد ال supremum ل S صح؟", "tokens": [9566, 1829, 3555, 45164, 23032, 3555, 4587, 8315, 9673, 11242, 9640, 10632, 29538, 41850, 5551, 9778, 3215, 2655, 318, 4724, 3794, 995, 45865, 2423, 28820, 10943, 23758, 4032, 316, 4724, 3794, 995, 45865, 36764, 24401, 1975, 15730, 2423, 23710, 449, 5296, 318, 5551, 25513, 27884, 4724, 3794, 995, 45865, 316, 30767, 16373, 3215, 2423, 23710, 449, 5296, 318, 20328, 5016, 22807], "avg_logprob": -0.21055947220133198, "compression_ratio": 1.474820143884892, "no_speech_prob": 0.0, "words": [{"start": 2278.87, "end": 2279.27, "word": "طيب", "probability": 0.9557291666666666}, {"start": 2279.27, "end": 2279.47, "word": " لو", "probability": 0.97998046875}, {"start": 2279.47, "end": 2280.13, "word": " طبقنا", "probability": 0.97998046875}, {"start": 2280.13, "end": 2281.29, "word": " المضارية", "probability": 0.684295654296875}, {"start": 2281.29, "end": 2281.73, "word": " 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"probability": 0.8907877604166666}], "temperature": 1.0}, {"id": 84, "seek": 233434, "start": 2307.32, "end": 2334.34, "text": "و ال a بساوي واحد و ال suprem ل s بساوي واحد فبطلع ال suprem ل s أربعة بساوي اتنين تمام؟ الان بنجيب ال infimum ل s أربعة بنفس الطريقة ممكن اثبات", "tokens": [2407, 2423, 257, 4724, 3794, 995, 45865, 36764, 24401, 4032, 2423, 23710, 5296, 262, 4724, 3794, 995, 45865, 36764, 24401, 6156, 3555, 9566, 1211, 3615, 2423, 23710, 5296, 262, 5551, 25513, 27884, 4724, 3794, 995, 45865, 1975, 2655, 1863, 9957, 46811, 10943, 22807, 2423, 7649, 4724, 1863, 7435, 1829, 3555, 2423, 1536, 332, 449, 5296, 262, 5551, 25513, 27884, 44945, 36178, 41950, 16572, 28671, 3714, 43020, 1975, 12984, 3555, 9307], "avg_logprob": -0.19289171821634535, "compression_ratio": 1.683453237410072, "no_speech_prob": 0.0, "words": [{"start": 2307.32, "end": 2307.6, "word": "و", "probability": 0.85595703125}, {"start": 2307.6, "end": 2307.76, "word": " ال", "probability": 0.61328125}, {"start": 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"seek": 255332, "start": 2529.7, "end": 2553.32, "text": "خمس اربع و هكذا فهنلاحظ ان الاتنين اتنين upper bound لان هو هيكون اكبر عنصر و ينتبه للست لو في اي upper bound تاني لو في any upper bound", "tokens": [9778, 2304, 3794, 1975, 25513, 3615, 4032, 8032, 4117, 15730, 6156, 3224, 1863, 15040, 5016, 19913, 16472, 2423, 9307, 1863, 9957, 1975, 2655, 1863, 9957, 6597, 5472, 5296, 7649, 31439, 39896, 30544, 1975, 4117, 26890, 18871, 9381, 2288, 4032, 7251, 29399, 3555, 3224, 24976, 14851, 45164, 8978, 1975, 1829, 6597, 5472, 6055, 7649, 1829, 45164, 8978, 604, 6597, 5472], "avg_logprob": -0.17213542113701502, "compression_ratio": 1.5629629629629629, "no_speech_prob": 0.0, "words": [{"start": 2529.7, "end": 2530.2, "word": "خمس", "probability": 0.9405924479166666}, {"start": 2530.2, "end": 2530.74, "word": " اربع", "probability": 0.6211751302083334}, {"start": 2530.74, "end": 2532.32, "word": " و", "probability": 0.7744140625}, {"start": 2532.32, "end": 2532.94, "word": " هكذا", 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و", "probability": 0.94873046875}, {"start": 2547.32, "end": 2547.96, "word": " ينتبه", "probability": 0.80908203125}, {"start": 2547.96, "end": 2548.5, "word": " للست", "probability": 0.784912109375}, {"start": 2548.5, "end": 2548.8, "word": " لو", "probability": 0.94287109375}, {"start": 2548.8, "end": 2549.02, "word": " في", "probability": 0.9208984375}, {"start": 2549.02, "end": 2549.3, "word": " اي", "probability": 0.87109375}, {"start": 2549.3, "end": 2549.66, "word": " upper", "probability": 0.94140625}, {"start": 2549.66, "end": 2550.06, "word": " bound", "probability": 0.92431640625}, {"start": 2550.06, "end": 2550.52, "word": " تاني", "probability": 0.9318033854166666}, {"start": 2550.52, "end": 2550.8, "word": " لو", "probability": 0.84033203125}, {"start": 2550.8, "end": 2551.1, "word": " في", "probability": 0.71728515625}, {"start": 2551.1, "end": 2552.46, "word": " any", "probability": 0.7861328125}, {"start": 2552.46, "end": 2552.86, "word": " upper", "probability": 0.91455078125}, {"start": 2552.86, "end": 2553.32, "word": " bound", "probability": 0.923828125}], "temperature": 1.0}, {"id": 93, "seek": 257637, "start": 2557.83, "end": 2576.37, "text": "of S4 فهذا بيقدي انه اتنين اصغر من او ساوي ال V لانه اتنين عنصر في الست S4 صح؟ اذا اتنين upper bound واضح انه اتنين اكبر من او ساوي كل عناصر S4 صح؟", "tokens": [2670, 318, 19, 6156, 3224, 15730, 4724, 1829, 4587, 16254, 16472, 3224, 1975, 2655, 1863, 9957, 1975, 9381, 17082, 2288, 9154, 1975, 2407, 8608, 995, 45865, 2423, 691, 5296, 7649, 3224, 1975, 2655, 1863, 9957, 18871, 9381, 2288, 8978, 2423, 14851, 318, 19, 20328, 5016, 22807, 1975, 15730, 1975, 2655, 1863, 9957, 6597, 5472, 4032, 46958, 5016, 16472, 3224, 1975, 2655, 1863, 9957, 1975, 4117, 26890, 9154, 1975, 2407, 8608, 995, 45865, 28242, 18871, 33546, 2288, 318, 19, 20328, 5016, 22807], "avg_logprob": -0.18826219330473645, "compression_ratio": 1.704225352112676, "no_speech_prob": 2.980232238769531e-07, "words": [{"start": 2557.83, "end": 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من أو يساوي V إذن هنا أثبتنا أن اتنين upper bound ل S4 واتنين أصغر من أو يساوي أي upper bound ل S4 إذن اتنين هو ال supreme بالمثل ممكن نثبت أن النص هو الانفع إذن هذا برهان تاني", "tokens": [12610, 2407, 5551, 9778, 3215, 2655, 36632, 6597, 5472, 5296, 318, 19, 6156, 3555, 15042, 14739, 691, 31439, 6597, 5472, 5296, 318, 19, 4032, 9307, 1863, 9957, 18871, 9381, 2288, 8978, 318, 19, 11933, 8848, 1863, 1975, 2655, 1863, 9957, 5551, 9381, 17082, 2288, 9154, 34051, 7251, 3794, 995, 45865, 691, 11933, 8848, 1863, 34105, 5551, 12984, 3555, 2655, 8315, 14739, 1975, 2655, 1863, 9957, 6597, 5472, 5296, 318, 19, 4032, 9307, 1863, 9957, 5551, 9381, 17082, 2288, 9154, 34051, 7251, 3794, 995, 45865, 36632, 6597, 5472, 5296, 318, 19, 11933, 8848, 1863, 1975, 2655, 1863, 9957, 31439, 2423, 27756, 20666, 2304, 12984, 1211, 3714, 43020, 8717, 12984, 3555, 2655, 14739, 28239, 9381, 31439, 2423, 7649, 5172, 3615, 11933, 8848, 1863, 23758, 4724, 2288, 3224, 7649, 6055, 7649, 1829], "avg_logprob": 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هو", "probability": 0.9228515625}, {"start": 2600.06, "end": 2600.26, "word": " ال", "probability": 0.54345703125}, {"start": 2600.26, "end": 2600.5, "word": " supreme", "probability": 0.1339111328125}, {"start": 2600.5, "end": 2601.76, "word": " بالمثل", "probability": 0.9239501953125}, {"start": 2601.76, "end": 2602.04, "word": " ممكن", "probability": 0.941162109375}, {"start": 2602.04, "end": 2602.6, "word": " نثبت", "probability": 0.9927978515625}, {"start": 2602.6, "end": 2602.8, "word": " أن", "probability": 0.59716796875}, {"start": 2602.8, "end": 2603.22, "word": " النص", "probability": 0.861328125}, {"start": 2603.22, "end": 2603.44, "word": " هو", "probability": 0.986328125}, {"start": 2603.44, "end": 2603.96, "word": " الانفع", "probability": 0.62750244140625}, {"start": 2603.96, "end": 2604.44, "word": " إذن", "probability": 0.8167317708333334}, {"start": 2604.44, "end": 2604.6, "word": " هذا", "probability": 0.826171875}, {"start": 2604.6, "end": 2605.08, "word": " برهان", "probability": 0.888671875}, {"start": 2605.08, "end": 2605.5, "word": " تاني", "probability": 0.9220377604166666}], "temperature": 1.0}, {"id": 95, "seek": 262208, "start": 2606.6, "end": 2622.08, "text": "انا اتعمد تعطيكم البرهان هذا عشان القوانين هذه نشوف كيف نطبقها برضه هذا برهان صعيب ناجح الحل؟ okay؟ في اي سؤال او استفسار؟ اذا احنا هنكتفي بهذا القدر", "tokens": [7649, 995, 1975, 2655, 25957, 3215, 37279, 9566, 1829, 24793, 2423, 26890, 3224, 7649, 23758, 6225, 8592, 7649, 25062, 2407, 7649, 9957, 29538, 8717, 8592, 38688, 9122, 33911, 8717, 9566, 3555, 4587, 11296, 4724, 43042, 3224, 23758, 4724, 2288, 3224, 7649, 20328, 3615, 1829, 3555, 8717, 26108, 5016, 21542, 1211, 22807, 1392, 22807, 8978, 1975, 1829, 8608, 33604, 6027, 1975, 2407, 44713, 36178, 9640, 22807, 1975, 15730, 1975, 5016, 8315, 8032, 1863, 4117, 2655, 41185, 39627, 15730, 25062, 3215, 2288], "avg_logprob": -0.2438271634372664, "compression_ratio": 1.6280487804878048, "no_speech_prob": 5.960464477539063e-08, "words": [{"start": 2606.6, "end": 2606.92, "word": "انا", "probability": 0.55712890625}, {"start": 2606.92, "end": 2607.3, "word": " اتعمد", "probability": 0.77685546875}, {"start": 2607.3, "end": 2607.72, "word": " تعطيكم", "probability": 0.727783203125}, {"start": 2607.72, "end": 2608.14, "word": " البرهان", "probability": 0.882568359375}, {"start": 2608.14, "end": 2608.34, "word": " هذا", "probability": 0.30322265625}, {"start": 2608.34, "end": 2608.64, "word": " عشان", "probability": 0.9226888020833334}, {"start": 2608.64, "end": 2609.18, "word": " القوانين", "probability": 0.992919921875}, {"start": 2609.18, "end": 2609.54, "word": " هذه", "probability": 0.72705078125}, {"start": 2609.54, "end": 2610.5, "word": " نشوف", "probability": 0.8816731770833334}, {"start": 2610.5, "end": 2610.78, "word": " كيف", "probability": 0.972412109375}, {"start": 2610.78, "end": 2611.5, "word": " نطبقها", "probability": 0.94228515625}, {"start": 2611.5, "end": 2612.06, "word": " برضه", "probability": 0.8912760416666666}, {"start": 2612.06, "end": 2612.26, "word": " هذا", "probability": 0.364013671875}, {"start": 2612.26, "end": 2612.6, "word": " برهان", "probability": 0.9385986328125}, {"start": 2612.6, "end": 2613.2, "word": " صعيب", "probability": 0.8218994140625}, {"start": 2613.2, "end": 2614.88, "word": " ناجح", "probability": 0.5619303385416666}, {"start": 2614.88, "end": 2615.48, "word": " الحل؟", "probability": 0.7473958333333334}, {"start": 2615.48, "end": 2618.52, "word": " okay؟", "probability": 0.521728515625}, {"start": 2618.52, "end": 2618.7, "word": " في", "probability": 0.56591796875}, {"start": 2618.7, "end": 2619.02, "word": " اي", "probability": 0.7802734375}, {"start": 2619.02, "end": 2619.38, "word": " سؤال", "probability": 0.984375}, {"start": 2619.38, "end": 2619.6, "word": " او", "probability": 0.864990234375}, {"start": 2619.6, "end": 2620.16, "word": " استفسار؟", "probability": 0.664642333984375}, {"start": 2620.16, "end": 2620.32, "word": " اذا", "probability": 0.7900390625}, {"start": 2620.32, "end": 2620.58, "word": " احنا", "probability": 0.93896484375}, {"start": 2620.58, "end": 2621.2, "word": " هنكتفي", "probability": 0.91787109375}, {"start": 2621.2, "end": 2621.54, "word": " بهذا", "probability": 0.97119140625}, {"start": 2621.54, "end": 2622.08, "word": " القدر", "probability": 0.9949544270833334}], "temperature": 1.0}], "language": "ar", "language_probability": 1.0, "duration": 2622.694, "duration_after_vad": 2400.42062499999}
\ No newline at end of file
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/_28CmIWMuzY_raw.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/_28CmIWMuzY_raw.srt
new file mode 100644
index 0000000000000000000000000000000000000000..ff5626a8ce8ad1ea2e64ecd2bb9391d781b1a90d
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/_28CmIWMuzY_raw.srt
@@ -0,0 +1,1616 @@
+1
+00:00:21,330 --> 00:00:27,290
+اليوم طبعا هنكمل الشرح
+
+2
+00:00:27,290 --> 00:00:30,650
+أو
+
+3
+00:00:30,650 --> 00:00:35,610
+بعض الملاحظات على النظرية اللي أخدناها في المحاضرة
+
+4
+00:00:35,610 --> 00:00:42,910
+السابقة النظرية هذه بتتحدث عن nested interval
+
+5
+00:00:42,910 --> 00:00:48,620
+property أو خاصية الفترات المتداخلةوشوفنا في
+
+6
+00:00:48,620 --> 00:00:54,720
+النظرية هذه ان لو في عندي sequence of nested
+
+7
+00:00:54,720 --> 00:00:58,660
+intervals الفترات هذه كلهم nested يعني كل واحدة
+
+8
+00:00:58,660 --> 00:01:05,820
+تحتوي اللي بعدها مباشرة زائد ان الفترات هذه كلهم
+
+9
+00:01:05,820 --> 00:01:14,580
+closed كلهم closed و bounded ففي
+
+10
+00:01:14,580 --> 00:01:20,210
+الحالة هذه التقاطةتبع ال sequence of intervals لا
+
+11
+00:01:20,210 --> 00:01:24,310
+يساوي في يعني في على الأقل عنصر واحد بالتقاطة
+
+12
+00:01:24,310 --> 00:01:30,510
+شوفنا برضه لو في الشرط الإضافي هذا اتحقق وهو
+
+13
+00:01:30,510 --> 00:01:35,570
+لاحظوا أن هذه عبارة عن ساو بي ان فهذه sequence من
+
+14
+00:01:35,570 --> 00:01:42,690
+العداد السالمة الغير سالمة و بالمناسبة السفر واضح
+
+15
+00:01:42,690 --> 00:01:48,940
+أنه lower bound للمجموعة هذه صح؟لكن مش شرط أن
+
+16
+00:01:48,940 --> 00:01:54,780
+السفر يكون هو ال infimum للمجموعة هذه فإذا كان ال
+
+17
+00:01:54,780 --> 00:01:57,960
+infimum للمجموعة هذه اللي هو أكبر lower bound هو
+
+18
+00:01:57,960 --> 00:02:06,440
+السفر فالتقاط واحدة في أنصر واحد okay تمام وشوفنا
+
+19
+00:02:06,440 --> 00:02:11,800
+مرين على البرهان المرة اللي فاتت و أعتقد أن
+
+20
+00:02:11,800 --> 00:02:16,860
+البرهان مكتوب بالتفصيلواضح ومرنا عليه جزء جزء
+
+21
+00:02:16,860 --> 00:02:22,000
+فأرجعكم تكونوا قرأتهوا كمان مرة وفهمتهوا في حد
+
+22
+00:02:22,000 --> 00:02:27,860
+عنده استفسار على المرهانة النظرية هذه طيب الآن
+
+23
+00:02:27,860 --> 00:02:35,820
+النظرية هذه نرجع للنظرية كمان مرة الآن
+
+24
+00:02:35,820 --> 00:02:41,480
+في ملاحظة بتقول انه لو انا في النظرية هذه الفترات
+
+25
+00:02:41,480 --> 00:02:49,780
+هذهالفرض ان الفترات in مغلقة closed لو حذفت شيلت
+
+26
+00:02:49,780 --> 00:03:01,600
+الفرض هذا فالنظرية هذه بتبطل تكون صحيحة فالنظرية
+
+27
+00:03:01,600 --> 00:03:05,000
+هذه بتبطل تكون صحيحة وحنشوف counter example يوضح
+
+28
+00:03:05,000 --> 00:03:07,460
+عدم صحتها كذلك
+
+29
+00:03:09,100 --> 00:03:13,220
+طب افرضه ان هذا شرط متحقق في الفترات لكن اللي مش
+
+30
+00:03:13,220 --> 00:03:17,680
+متحقق اللي هو ال boundedness يعني الفترات هذه ليست
+
+31
+00:03:17,680 --> 00:03:21,420
+محدودة ليست bounded برضه في الحالة هذه المظهرية
+
+32
+00:03:21,420 --> 00:03:26,620
+تفشل و في counter example يوضح فشلها okay اذا
+
+33
+00:03:26,620 --> 00:03:30,640
+حنشوف two counter examples خليني نشوفهم مع بعض
+
+34
+00:03:36,610 --> 00:03:39,790
+إذا هدى ال remark اللى انا اتحدث عنها قلت انه it
+
+35
+00:03:39,790 --> 00:03:44,090
+should be noted يجب ملاحظة ان journal بصورة عامة
+
+36
+00:03:44,090 --> 00:03:48,030
+instant sequence of intervals need not have a
+
+37
+00:03:48,030 --> 00:03:51,290
+common point يعني لو فيه ending sequence من
+
+38
+00:03:51,290 --> 00:03:57,010
+الفترات المتداخلة مش شرط تقاطعهم يكون في في يعني
+
+39
+00:03:57,010 --> 00:04:02,650
+اى نقطة او نقطة مشاركة يعني مش شرط ان التقاطع لها
+
+40
+00:04:02,650 --> 00:04:11,000
+يساوي فيهفالأنثى لها دى هدا هى اللى حكينا عنها اول
+
+41
+00:04:11,000 --> 00:04:18,500
+مثال هاى
+
+42
+00:04:18,500 --> 00:04:23,080
+فى المثال الاول الفرض the hypothesis الفرض ان ال
+
+43
+00:04:23,080 --> 00:04:28,940
+intervals I in فى نظرية 22 be closed cannot be
+
+44
+00:04:28,940 --> 00:04:34,800
+dropped يعني لا يمكن حذفهلا يمكن الاستجناء عنه
+
+45
+00:04:34,800 --> 00:04:41,180
+وتبقى النظرية نظرية صحيحة for example على سبيل
+
+46
+00:04:41,180 --> 00:04:49,120
+المثال لو أخدت الفترات I N الفترة I N هي الفترة
+
+47
+00:04:49,120 --> 00:04:55,580
+المفتوحة من 0 ل 1 على N حيث N عدد طبيعي فواضح ان
+
+48
+00:04:55,580 --> 00:05:00,460
+الفترات هدي nested صح؟ لأن الفترة الأولى هتكون
+
+49
+00:05:00,460 --> 00:05:04,820
+الفترة مفتوحة من 0 ل 1الفترة التانية الفترة مفتوحة
+
+50
+00:05:04,820 --> 00:05:12,180
+من سفر لنص وهذه محتوى في I واحد و I تلاتة الفترة
+
+51
+00:05:12,180 --> 00:05:16,540
+مفتوحة من سفر لتلت محتوى داخل I اتنين و هكذا لذلك
+
+52
+00:05:16,540 --> 00:05:21,720
+واضح ان ال sequence of open intervals IN is nested
+
+53
+00:05:21,720 --> 00:05:27,560
+sequence كذلك عناصر ال sequence هذه bounded هذه
+
+54
+00:05:27,560 --> 00:05:33,710
+فترات محصورةلكن الفترات هذه not closed مش closed
+
+55
+00:05:33,710 --> 00:05:38,630
+يعني عبارة عن مجموعات مفتوحة ليست مغلقة، إذن شرط
+
+56
+00:05:38,630 --> 00:05:45,910
+الإغلاق هنا انحذف وبالتالي نتيجة النظرية مش شرط
+
+57
+00:05:45,910 --> 00:05:50,750
+تكون صحيحة، إذن التقاطع هنا لنفس ال sequence هذه
+
+58
+00:05:50,750 --> 00:05:54,410
+بيطلع بساوي fine مافيش common point مافيش نقطة
+
+59
+00:05:54,410 --> 00:05:59,950
+مشتركة في هذه الفترات طبعا هذا مش واضح
+
+60
+00:06:04,230 --> 00:06:08,470
+هذا تقاطع الفترات المفتوحة هذه بساوي في هذا مش
+
+61
+00:06:08,470 --> 00:06:14,310
+واضح يحتاج إلى برهان هي البرهان بين جثين مربعين
+
+62
+00:06:14,310 --> 00:06:21,470
+تعالوا نبره إن تقاطع الفترات هذه بساوي في to see
+
+63
+00:06:21,470 --> 00:06:27,670
+this to see this معناه لبرهان ذلك مش لرأي ذلك هذا
+
+64
+00:06:27,670 --> 00:06:34,040
+تعبير مجازم استخدمه لبرهانالشيء العبارة اللي احنا
+
+65
+00:06:34,040 --> 00:06:38,400
+عايزينها ف to see this suppose in the contrary
+
+66
+00:06:38,400 --> 00:06:43,320
+بنفترض على النقيد انه التقاطع هذا بسويش في يعني في
+
+67
+00:06:43,320 --> 00:06:48,100
+على الأقل عنصر X في التقاطع بنصل لتناقض طيب ال X
+
+68
+00:06:48,100 --> 00:06:53,360
+موجود في التقاطع معناته X موجود في I N لكل N إذن X
+
+69
+00:06:53,360 --> 00:06:58,310
+موجود في كل واحدة من الفترات I Nطيب X موجود في
+
+70
+00:06:58,310 --> 00:07:03,510
+الفترة I N معناته X أكبر من سفر أصغر من واحد على N
+
+71
+00:07:03,510 --> 00:07:09,970
+أصغر من واحد على N أصغر من واحد على N تمام
+
+72
+00:07:09,970 --> 00:07:13,970
+وبالتالي
+
+73
+00:07:13,970 --> 00:07:20,430
+حسب ال Archimedean property هذا عبارة عن أحد صور
+
+74
+00:07:20,430 --> 00:07:25,750
+ال Archimedean property بتقول ليبما ان X هد عدد
+
+75
+00:07:25,750 --> 00:07:33,530
+موجب، الـ X هد عدد موجب، إذا يوجد عدد طبيعي N0
+
+76
+00:07:33,530 --> 00:07:39,150
+مقلوب و أصغر من العدد الموجب وهذا بتديني تناقض،
+
+77
+00:07:39,150 --> 00:07:47,370
+هذا بتديني تناقض، هذا بتناقض مع كون ال X أصغر من 1
+
+78
+00:07:47,370 --> 00:07:53,170
+على N لكل Nيعني ال X هذه أصغر من 1 على N0 وهي في
+
+79
+00:07:53,170 --> 00:07:57,210
+نفس الواجهة أكبر من 1 على N0 لأن هذا بتديني تناقض
+
+80
+00:07:57,210 --> 00:08:04,250
+لأن التناقض هذا سبب ال assumption تبعنا أن يوجد X
+
+81
+00:08:04,250 --> 00:08:09,210
+في التقاطة لأن الصح أن التقاطع هذا مافيش فيه ولا
+
+82
+00:08:09,210 --> 00:08:16,140
+أنصر يعني is the empty setإن هذا مثال بورجي أو
+
+83
+00:08:16,140 --> 00:08:21,900
+بيوضح إنه لو حذفنا شرط إن الفترات في نظرية 22
+
+84
+00:08:21,900 --> 00:08:26,980
+closed فبتطلع الشفرة، النظرية تفشل، بتبطل النظرية
+
+85
+00:08:26,980 --> 00:08:32,720
+و هذا مثال بيوضح فشلها، الآن المثال التاني نفس
+
+86
+00:08:32,720 --> 00:08:38,480
+الحاجة، الفرض إن الفترات في نظرية 22 be bounded
+
+87
+00:08:40,090 --> 00:08:43,690
+بتكون محدودة cannot be dropped لايمكن إسخاطه
+
+88
+00:08:43,690 --> 00:08:48,250
+لايمكن إهماله فعشان
+
+89
+00:08:48,250 --> 00:08:52,750
+نوضح هذا الكلام ب counter example ف for example
+
+90
+00:08:52,750 --> 00:08:56,750
+على سبيل المثال هناخد الفترات المغلقة I N فترة
+
+91
+00:08:56,750 --> 00:09:03,190
+مغلقة من N إلى ملا نهاية حيث N عدد طبيعي هذه
+
+92
+00:09:03,190 --> 00:09:10,150
+الفتراتكل هذه فترة مغلقة كل فترة على الصورة هذه
+
+93
+00:09:10,150 --> 00:09:17,010
+مغلقة إذا شرط الإغلاق متحقق بعدين الفترات هذه نستد
+
+94
+00:09:17,010 --> 00:09:20,430
+لحظة أول فترة هي الفترة المغلقة من واحد لما لا
+
+95
+00:09:20,430 --> 00:09:24,450
+نهاية التانية فترة مغلقة من اتنين لما لا نهاية
+
+96
+00:09:24,450 --> 00:09:30,110
+وهذه محتوى في I واحد الفترة التالتة الفترة المغلقة
+
+97
+00:09:30,110 --> 00:09:33,410
+من تلاتة لما لا نهاية وهذه محتوى في I اتنين وهكذا
+
+98
+00:09:33,410 --> 00:09:38,730
+فالفترات هذه نستدand closed مغلقة لكن ماهياش
+
+99
+00:09:38,730 --> 00:09:42,190
+bounded مش محصورة it's not bounded .. هذه كمجموعة
+
+100
+00:09:42,190 --> 00:09:48,870
+is not bounded above، كمجموعة ليس لها supreme، is
+
+101
+00:09:48,870 --> 00:09:52,390
+not bounded above، اذا شرط ال boundedness اختل
+
+102
+00:09:52,390 --> 00:09:57,970
+وبالتالي نتيجة النظرية هتختلفإذا الفترات هذه
+
+103
+00:09:57,970 --> 00:10:03,410
+closed but unbounded وإذا هنجد إنه تقاطع الفترات
+
+104
+00:10:03,410 --> 00:10:08,930
+هذه مافيش فيه ولا نقطة تقاطع هذا بساوي five كمان
+
+105
+00:10:08,930 --> 00:10:15,350
+مرة المساواة هذه بدها مش واضحة ليست واضحة فبدنا
+
+106
+00:10:15,350 --> 00:10:20,730
+نثبت صحة المساواة هذه كمان مرة نعمل برهان بالتناقض
+
+107
+00:10:20,730 --> 00:10:24,370
+نعمل برهان بالتناقض
+
+108
+00:10:29,770 --> 00:10:34,830
+فافرضي أن التقاطع هذا لا يساوي في I وبالتالي يوجد
+
+109
+00:10:34,830 --> 00:10:40,670
+X في التقاطع إذا X موجود في الفترة I N لكل N هذا
+
+110
+00:10:40,670 --> 00:10:46,950
+من تعريف التقاطع X موجودة في I N معناته X أكبر من
+
+111
+00:10:46,950 --> 00:10:53,870
+أو يساوي N وهذا صحيح لكل Nهذا بتناقض مع الـ
+
+112
+00:10:53,870 --> 00:10:58,510
+Archimedean property نظرية الأساسية نظرية خمستاشر
+
+113
+00:10:58,510 --> 00:11:05,450
+في الشبطرة ده اللي بتقول لأي عدد حقيقي X ينتمي إلى
+
+114
+00:11:05,450 --> 00:11:16,530
+R بتأدي ان يوجد N0 ينتمي إلى N بحيث ان X أصغر من
+
+115
+00:11:16,530 --> 00:11:17,330
+N0
+
+116
+00:11:21,190 --> 00:11:27,210
+هذه هي الـ Archimedean property الأساسية طيب أنا
+
+117
+00:11:27,210 --> 00:11:32,850
+عندي الأن من ال Archimedean property عندي يوجد عدد
+
+118
+00:11:32,850 --> 00:11:42,060
+طبيعي N0 لصد أكبر من X وعندي هناإن X أكبر من أو
+
+119
+00:11:42,060 --> 00:11:47,340
+ساوي N لكل N في N وبالتالي X أكبر من أو ساوي N
+
+120
+00:11:47,340 --> 00:11:51,820
+Zero لأن N Zero ينتمي إلى N فإذا عندي هنا X أكبر
+
+121
+00:11:51,820 --> 00:11:56,180
+من أو ساوي N Zero و X أصغر من N Zero هذا بيديني
+
+122
+00:11:56,180 --> 00:12:02,840
+تناطق إذا في عندي contradiction إذا هذا العنصر غير
+
+123
+00:12:02,840 --> 00:12:07,870
+موجودsuch an x does not exist يعني التقاطة هذا
+
+124
+00:12:07,870 --> 00:12:13,550
+بساوي في كما هو مطلوب، تمام؟ واضح البرهن؟ إذن هذه
+
+125
+00:12:13,550 --> 00:12:17,690
+مثال تاني بوضح أن شرط ال boundedness لا يمكن
+
+126
+00:12:17,690 --> 00:12:25,730
+اسقاطه وتبقى nested intervals theorem صحيحة، okay؟
+
+127
+00:12:25,730 --> 00:12:31,710
+في نظرية تانيةيمكن هذه مرت معاكم في المبادئ لكن
+
+128
+00:12:31,710 --> 00:12:37,170
+اليوم هنعطيلها برهان يعتمد على ال nested intervals
+
+129
+00:12:37,170 --> 00:12:40,610
+theorem او nested intervals property برهان جديد
+
+130
+00:12:40,610 --> 00:12:48,730
+غير اللي أخدته في مبادئ الرياضيات فالنظرية
+
+131
+00:12:48,730 --> 00:12:54,590
+هذه 24 تتحدث عن ال uncountability of the real
+
+132
+00:12:54,590 --> 00:12:59,560
+numbersفبكل بساطة النظرية هذه بتقول أن مجموعة
+
+133
+00:12:59,560 --> 00:13:04,340
+العداد الحقيقية is uncountable the set R of all
+
+134
+00:13:04,340 --> 00:13:09,460
+real numbers is uncountable طيب
+
+135
+00:13:09,460 --> 00:13:15,460
+ما معنى أن ال set تكون countable؟ في حد فيكم
+
+136
+00:13:15,460 --> 00:13:21,380
+بتعرف؟ ال set A أو S definition
+
+137
+00:13:24,240 --> 00:13:31,920
+definition تعريف S is countable if
+
+138
+00:13:31,920 --> 00:13:46,700
+and only if كتوف المبادئ either اما S is finite or
+
+139
+00:13:46,700 --> 00:13:50,040
+او
+
+140
+00:13:50,040 --> 00:13:58,450
+S is denomableأو في بيجيكشن one to one
+
+141
+00:13:58,450 --> 00:14:03,850
+correspondence بينها وبين الأعداد الطبيعية يعني
+
+142
+00:14:03,850 --> 00:14:16,970
+هذا معناه is denumerable قابلة للترقيم طيب
+
+143
+00:14:16,970 --> 00:14:23,330
+إذا كانت ال set ماهياش
+
+144
+00:14:23,330 --> 00:14:29,090
+finiteوماهياش in one to one correspondence with
+
+145
+00:14:29,090 --> 00:14:33,550
+the natural numbers او ماهياش denumerable فبنسميها
+
+146
+00:14:33,550 --> 00:14:38,410
+uncountable غير قابلة للعد countable قابلة للعد
+
+147
+00:14:38,410 --> 00:14:44,750
+uncountable غير قابلة للعد طيب
+
+148
+00:14:44,750 --> 00:14:52,150
+ال
+
+149
+00:14:52,150 --> 00:14:52,390
+..
+
+150
+00:14:55,180 --> 00:15:03,200
+معروف في مبادئ رياضيات درسنا ان ال interval هذي و
+
+151
+00:15:03,200 --> 00:15:08,220
+ال interval هذي كلا هما uncountable الفترة
+
+152
+00:15:08,220 --> 00:15:11,120
+المفتوحة من سفر لواحد infinite set اول حاجة
+
+153
+00:15:11,120 --> 00:15:15,800
+infinite set و
+
+154
+00:15:15,800 --> 00:15:18,900
+طبعا ممكن تثبت انها uncountable
+
+155
+00:15:21,370 --> 00:15:26,370
+و طبعا هذه الفترة المغلقة تحتوي ال six هذه الفترة
+
+156
+00:15:26,370 --> 00:15:29,110
+المفتوحة فإذا كانت هذه uncountable هذه بتكون
+
+157
+00:15:29,110 --> 00:15:35,530
+uncountable وهذا البرهان موجود في نظرية في الكتاب
+
+158
+00:15:35,530 --> 00:15:42,010
+المقرر textbook الكتاب المقرر
+
+159
+00:15:42,010 --> 00:15:47,110
+طبعا
+
+160
+00:15:47,110 --> 00:15:50,410
+طيب
+
+161
+00:15:57,430 --> 00:16:05,570
+الـ احنا عايزين نثبت عشان نثبت ان ال set هذه ال R
+
+162
+00:16:05,570 --> 00:16:14,770
+لاحظوا ان ال R is
+
+163
+00:16:14,770 --> 00:16:18,490
+in one to one correspondence مع الفترة المفتوحة او
+
+164
+00:16:18,490 --> 00:16:22,630
+المغلقة حتى في
+
+165
+00:16:22,630 --> 00:16:28,250
+byjection بينه وبين الفترةالمفتوحة المغلقة 01
+
+166
+00:16:28,250 --> 00:16:36,890
+وبرضه المفتوحة الان لو أثبتنا ان الفترة هذه
+
+167
+00:16:36,890 --> 00:16:44,150
+uncountable فهذه
+
+168
+00:16:44,150 --> 00:16:50,530
+ال 6 in one to one correspondence معها فال 6 هذه R
+
+169
+00:16:50,530 --> 00:16:54,400
+تطلع uncountableهذه نظرية موجودة في مبادئ
+
+170
+00:16:54,400 --> 00:16:58,080
+الرياضيات إذا كانت هناك مجموعتين واثنتين
+
+171
+00:16:58,080 --> 00:17:02,860
+equivalent to each other فهناك بيجيكشن بينهم إذا
+
+172
+00:17:02,860 --> 00:17:06,540
+كانت واحدة countable فالتانية تظهر countable إذا
+
+173
+00:17:06,540 --> 00:17:10,380
+كانت واحدة countable فالتانية تظهر countable إذا
+
+174
+00:17:10,380 --> 00:17:14,140
+كانت هذه finite تظهر هذه finite إذا كانت هذه
+
+175
+00:17:14,140 --> 00:17:19,440
+infinite تظهر هذه infinite كل هذه نظريات موجودة في
+
+176
+00:17:19,440 --> 00:17:24,350
+مبادئ الرياضياتإذا لو أثبتنا إن الفترة هادى
+
+177
+00:17:24,350 --> 00:17:31,010
+uncountable فبطلع R uncountable طيب
+
+178
+00:17:31,010 --> 00:17:42,050
+لإثبات ذلك هنا عايزين نثبت إن الفترة هادى نثبت إن
+
+179
+00:17:42,050 --> 00:17:47,030
+الفترة هادى uncountable لبرهان ذلك نعمل برهان
+
+180
+00:17:47,030 --> 00:17:47,770
+بالتناقض
+
+181
+00:17:57,100 --> 00:18:01,160
+بنثبت ان الفترة المغلقة هذي uncountable نفرض
+
+182
+00:18:01,160 --> 00:18:04,940
+المقيد
+
+183
+00:18:04,940 --> 00:18:08,780
+ان الفترة هذي countable لاحظوا ان الفترة هذي
+
+184
+00:18:08,780 --> 00:18:14,500
+infinite والان countable اذا بتطلع equipotent او
+
+185
+00:18:14,500 --> 00:18:17,640
+in one to one correspondence with natural numbers
+
+186
+00:18:22,850 --> 00:18:26,550
+الأن في الحالة هذه I in one to one correspondence
+
+187
+00:18:26,550 --> 00:18:31,570
+within actual numbers أو بنسميها innumerable صح؟
+
+188
+00:18:33,280 --> 00:18:36,560
+الان ال set I denominable يعني ممكن ترقيمها
+
+189
+00:18:36,560 --> 00:18:41,840
+بالأعداد الطبيعية إذا ممكن نسمي عناصرها xn-n عدد
+
+190
+00:18:41,840 --> 00:18:46,340
+طبيعي اللي هي x1, x2, x3 الاخرى أي set denominable
+
+191
+00:18:46,340 --> 00:18:49,900
+أو in one to one correspondence with natural
+
+192
+00:18:49,900 --> 00:18:55,140
+numbers ممكن ترقيم عناصرها ك list by the natural
+
+193
+00:18:55,140 --> 00:18:59,200
+numbers إذا I هي كل عناصرها رقمناهم بالأعداد
+
+194
+00:18:59,200 --> 00:18:59,800
+الطبيعية
+
+195
+00:19:05,090 --> 00:19:08,350
+لحظة احنا بدنا نصل الى تناقض احنا بدنا القرآن
+
+196
+00:19:08,350 --> 00:19:15,650
+وفرضنا ال contrary هيو Assume ال contrary ان I is
+
+197
+00:19:15,650 --> 00:19:19,870
+countable بدنا نصل الى تناقض طيب هي الفترة I هي
+
+198
+00:19:19,870 --> 00:19:26,890
+الفترة I هي الفترة I هذه
+
+199
+00:19:26,890 --> 00:19:31,550
+Iو في اندس اي هاد الفترة اي هاد يعني عناصرها هي
+
+200
+00:19:31,550 --> 00:19:37,390
+مرقمة اكس واحد اكس اتنين الى ملا نهاية افرض ان اكس
+
+201
+00:19:37,390 --> 00:19:46,510
+واحد موجود هان اول عنصر في الفترة موجود هان فممكن
+
+202
+00:19:46,510 --> 00:19:54,530
+اختار فترة مغلقة اسميها اي واحد ممكن اختار فترة
+
+203
+00:19:54,530 --> 00:20:03,260
+مغلقةأسميها I1 بحيث ان ال X1 هذه لا تنتمي للفترة
+
+204
+00:20:03,260 --> 00:20:07,520
+I1 وممكن
+
+205
+00:20:07,520 --> 00:20:13,100
+اختار فترة مغلقة تانية طب افرضي ان X2 موجودة هنا
+
+206
+00:20:13,100 --> 00:20:19,680
+العنصر التاني في الفترة I ايه رقم X2 موجود هنا او
+
+207
+00:20:19,680 --> 00:20:27,400
+هنا او هنا فبقدر اختار فترة مغلقة تانيةنسميها I2
+
+208
+00:20:27,400 --> 00:20:36,120
+اللي هي الفترة هذه بحيث ان X2 لا تنتمي ل I2 و
+
+209
+00:20:36,120 --> 00:20:42,400
+الفترة I2 هي فترة جزئية من I1 طيب افرض ان X3
+
+210
+00:20:42,400 --> 00:20:50,120
+موجودة هنا او هنا او هنا او اي مكان تاني فبقدر
+
+211
+00:20:50,120 --> 00:20:58,310
+اختار فترة مغلقة تسميها I3اللي هي الفترة هذه بحيث
+
+212
+00:20:58,310 --> 00:21:05,450
+ان X3 لا تنتمي للفترة I3 والفترة I3 جزئية من
+
+213
+00:21:05,450 --> 00:21:12,490
+الفترة I2 ممكن نستمر على هذا النمط هنحصل على
+
+214
+00:21:12,490 --> 00:21:21,110
+sequence of intervals اللي هي I1 تحتوي I2 تحتوي I3
+
+215
+00:21:22,570 --> 00:21:27,550
+و هكذا ممكن نستمر إلى ملا نهاية و كل الفترات هذول
+
+216
+00:21:27,550 --> 00:21:32,570
+محتوى .. كل واحدة منهم محتوى داخل الـ I و كل واحدة
+
+217
+00:21:32,570 --> 00:21:38,710
+من الفترات هذه صممناها بحيث ان XN لا ينتمي إلى IN
+
+218
+00:21:38,710 --> 00:21:47,190
+لكل N بساوي واحد اتنين إلى ملا نهاية صح؟ إذا لو
+
+219
+00:21:47,190 --> 00:21:53,130
+استمرنا في العملية هذههنحصل على sequence of nested
+
+220
+00:21:53,130 --> 00:21:57,170
+intervals و ال intervals هدولة كلهم closed و
+
+221
+00:21:57,170 --> 00:22:01,570
+bounded كلهم closed و bounded و كل الفترات هذه
+
+222
+00:22:01,570 --> 00:22:07,190
+محتوية داخل الفترة I و X in العنصر X in من ال
+
+223
+00:22:07,190 --> 00:22:14,470
+sequence هذه لا ينتمي ل I in لكل N الان ممكن نطبق
+
+224
+00:22:14,470 --> 00:22:18,090
+nested interval property theorem اللي هي theorem
+
+225
+00:22:20,050 --> 00:22:23,550
+بتقول ده في عندي sequence of nested intervals و
+
+226
+00:22:23,550 --> 00:22:29,030
+كلهم closed و bounded فالتقاطع تبعهم لا يساوي في I
+
+227
+00:22:29,030 --> 00:22:34,650
+إذا التقاطع تبعهم موجود في نقطة واحدة دعنا نسميها
+
+228
+00:22:34,650 --> 00:22:43,030
+ساي و الفترات هذه كلها كل الفترات هذه موجودة داخل
+
+229
+00:22:43,030 --> 00:22:47,390
+الفترة I داخل الفترة I
+
+230
+00:22:53,490 --> 00:23:04,810
+ماشي هنا اه
+
+231
+00:23:04,810 --> 00:23:07,630
+ايش صار؟ هي فوق صار
+
+232
+00:23:12,680 --> 00:23:17,360
+إذا حسب نفس ال interval theorem يوجد نقطة psi في
+
+233
+00:23:17,360 --> 00:23:22,080
+تقاطة الفترات الفترات كلهم موجودين داخل I إذا
+
+234
+00:23:22,080 --> 00:23:29,540
+تقاطعهم موجود داخل I وبالتالي النقطة psi هذه
+
+235
+00:23:29,540 --> 00:23:37,060
+موجودة في كل الفترات I in لأن موجودة في تقاطعهم
+
+236
+00:23:37,060 --> 00:23:43,850
+كلهم صح إذا النقطة psi موجودة في I in لكل Nوالفترة
+
+237
+00:23:43,850 --> 00:23:52,690
+I N هذه لا تحتوي من هنا الفترة I N لا تحتوي X N
+
+238
+00:23:52,690 --> 00:23:58,690
+والان تحتوي ساي إذا ساي لا تساوي X N الكلام هذا
+
+239
+00:23:58,690 --> 00:24:04,430
+صحيح لكل N بظبط هذا عنصر في الفترة هذه و X N ليس
+
+240
+00:24:04,430 --> 00:24:07,970
+عنصر فيها إذا مش ممكن يكونوا متساويين صح؟
+
+241
+00:24:10,780 --> 00:24:19,120
+الـ Psi قلنا هي تنتمي إلى I الـ Psi موجودة في I و
+
+242
+00:24:19,120 --> 00:24:27,620
+الفترة I رقمنا عناصرها قبل شوية انا
+
+243
+00:24:27,620 --> 00:24:36,300
+في اندي يوجد عدد حقيقي Psi ينتمي ل I و في نفس
+
+244
+00:24:36,300 --> 00:24:42,430
+الوجهة الفترة I هي كل عناصرهامُرقّمة بالعداد
+
+245
+00:24:42,430 --> 00:24:48,030
+الطبيعي عناصرها X1 و X2 إلى ما لنهائي و الأن في
+
+246
+00:24:48,030 --> 00:24:59,090
+عندي Psi مختلف عن كل عناصر الفترة I فهذا
+
+247
+00:24:59,090 --> 00:25:04,510
+بيدّي أن ال sequence أو ال set هذهis not a
+
+248
+00:25:04,510 --> 00:25:10,330
+complete enumeration of I ليست ترقيم كامل للفترة I
+
+249
+00:25:10,330 --> 00:25:15,750
+وهذا تناقض يعني
+
+250
+00:25:15,750 --> 00:25:20,530
+احنا قلنا الفترة I هذه تطلع countable و infinite
+
+251
+00:25:20,530 --> 00:25:26,830
+اذا ممكن نرقم اذا denumerable يعني ممكن نرقم عن
+
+252
+00:25:26,830 --> 00:25:31,770
+صرها كلها بالاعداد الطبيعي وبالتالي كل عن صرها X
+
+253
+00:25:33,730 --> 00:25:43,170
+تمام؟ الآن في البرهان هذا وجدنا ان في صي أنصر جديد
+
+254
+00:25:43,170 --> 00:25:49,310
+في I مختلف عن كل عناصرها معناته هذه ال list ليست
+
+255
+00:25:49,310 --> 00:25:54,650
+ترقيم كامل ل I في عناصر أخرى زي صي مش مرقمة و هذا
+
+256
+00:25:54,650 --> 00:25:59,950
+تناقضلأن إحنا عندنا ال set I هذي countable و
+
+257
+00:25:59,950 --> 00:26:04,210
+infinite و denomable إذن هي list تبع كل عناصرها
+
+258
+00:26:04,210 --> 00:26:11,230
+فكيف طلع فيه أنصر جديد مختلف عن عناصرها فهذا تناقض
+
+259
+00:26:11,230 --> 00:26:18,140
+إذن هذا التناقض بثبت أن فرضنا أن الفترة Iكانت
+
+260
+00:26:18,140 --> 00:26:22,140
+countable كان فرض خاطر وبالتالي الفترة I تطلع
+
+261
+00:26:22,140 --> 00:26:27,520
+uncountable اذا الان الفترة I uncountable وانا
+
+262
+00:26:27,520 --> 00:26:36,140
+عندي R equivalent لفترة مغلقة 01 ممكن نوجد
+
+263
+00:26:36,140 --> 00:26:40,860
+bijection بينهم اذا ال R تطلع uncountable كما هو
+
+264
+00:26:40,860 --> 00:26:45,080
+مطلوب اذا
+
+265
+00:26:45,080 --> 00:26:50,750
+هذا هو برهانالنظرية اللي فادت هي طبعا برهان بيعتمد
+
+266
+00:26:50,750 --> 00:26:55,430
+على شوفنا ال nested interval theorem وبالتالي هذا
+
+267
+00:26:55,430 --> 00:26:58,510
+برهان مختلف عن البرهان اللي بنعطيه في مبادئ
+
+268
+00:26:58,510 --> 00:27:05,270
+الرياضيات في برهان تاني برضه لنظرية هذه يعطى في
+
+269
+00:27:05,270 --> 00:27:10,710
+مبادئ الرياضيات اللي هو باستخدام counter diagonal
+
+270
+00:27:10,710 --> 00:27:14,690
+argument مشهور
+
+271
+00:27:14,690 --> 00:27:20,990
+يعني البرهانيرجع إلى العالم الرياضي Cantor يسمى
+
+272
+00:27:20,990 --> 00:27:24,750
+Cantor دي اقنع ال argument بثبت ان الفترة المفتوحة
+
+273
+00:27:24,750 --> 00:27:29,330
+من سفر لواحد is uncountable وبالتالي R is
+
+274
+00:27:29,330 --> 00:27:33,310
+uncountable لأن R في byjection بينها وبين ال open
+
+275
+00:27:33,310 --> 00:27:37,670
+interval من سفر لواحد الآن في نتيجة على نظرية
+
+276
+00:27:37,670 --> 00:27:42,490
+الأخيرة هذهالـ set هذه الـ R minus Q اللي هي الـ
+
+277
+00:27:42,490 --> 00:27:46,590
+set of all irrationals أيضًا is uncountable
+
+278
+00:27:46,590 --> 00:27:50,690
+والبرهان هنا نفس البرهان اللي بنعطيه في المبادئ
+
+279
+00:27:50,690 --> 00:27:55,470
+برهان by contradiction assume and contrary إن ال
+
+280
+00:27:55,470 --> 00:28:02,110
+set R minus Q is countable وإحنا عندنا نظرية في
+
+281
+00:28:02,110 --> 00:28:07,640
+المبادئ أخدنا بتقول إن لو في عندي مجموعتين A وBوكل
+
+282
+00:28:07,640 --> 00:28:14,140
+واحدة منهم countable فاتحادهم بيطلع countable الان
+
+283
+00:28:14,140 --> 00:28:17,640
+انا في عند Q countable معروف ان Q is countable
+
+284
+00:28:17,640 --> 00:28:24,160
+والان احنا فرضين ان R-Q is countable اذا اتحاد
+
+285
+00:28:24,160 --> 00:28:28,420
+المجموعتين هدول اللي هو R بيطلع countable وهذا
+
+286
+00:28:28,420 --> 00:28:31,420
+بتناقض مع النتيجة اللي لسه مثبتينها في النظرية
+
+287
+00:28:31,420 --> 00:28:36,240
+السابقةOkay إذا في عندي contradiction إذا الفرض
+
+288
+00:28:36,240 --> 00:28:39,780
+إنه الست هذي countable كان خاطئ إذا الصح إنه الست
+
+289
+00:28:39,780 --> 00:28:45,280
+هذي اللي هي ال irrational number is is uncountable
+
+290
+00:28:45,280 --> 00:28:57,120
+okay تمام إذا ال مع
+
+291
+00:28:57,120 --> 00:29:01,620
+انتهاء النتيجة هذه هيك بنكون غطينا section اتنين
+
+292
+00:29:01,620 --> 00:29:08,660
+خمسةو هاي التمرين المطلوب تهلوها مش عايز ابدأ
+
+293
+00:29:08,660 --> 00:29:14,020
+section جديد عايز ان احنا نستغل الوقت المتبقى من
+
+294
+00:29:14,020 --> 00:29:19,160
+المحاضرة في حل اسئلة discussion يعنيمناقشة فأي
+
+295
+00:29:19,160 --> 00:29:22,360
+واحدة فيكم عندها مناقشة احنا انا عارف ان انتوا
+
+296
+00:29:22,360 --> 00:29:28,100
+هتحضروا حالكم ليوم السبت المناقشة لكن برضه اكيد
+
+297
+00:29:28,100 --> 00:29:32,040
+يعني في بينكم ناس محضرين فلو عندكم أسئلة في
+
+298
+00:29:32,040 --> 00:29:36,680
+section اتنين تلاتة او اتنين اربعة او section
+
+299
+00:29:36,680 --> 00:29:41,160
+اتنين اتنين او اتنين واحد فممكن نحاول نحلها في
+
+300
+00:29:41,160 --> 00:29:47,080
+الوقت المتبقى من المحاضرةماشي الحال فإذا مين عندها
+
+301
+00:29:47,080 --> 00:29:53,540
+أي سؤال في ال .. المحاضرات
+
+302
+00:29:53,540 --> 00:30:03,220
+السابقة أو تمارين المحاضرات السابقة من
+
+303
+00:30:03,220 --> 00:30:08,540
+لديها سؤال؟ في عندنا أسلة كتيرة في المحاضرات
+
+304
+00:30:08,540 --> 00:30:15,470
+السابقة homework كتيرمين لديها سؤال؟ مين عندها
+
+305
+00:30:15,470 --> 00:30:23,170
+سؤال؟ مين بتحلم السؤال؟ ولا
+
+306
+00:30:23,170 --> 00:30:29,690
+واحدة عندها سؤال؟ تفضل طيب
+
+307
+00:30:30,890 --> 00:30:35,570
+طبعا هذه إشارة غير يعني غير إيجابية أو إشارة سلبية
+
+308
+00:30:35,570 --> 00:30:42,530
+يعني لحد تلان أنتوا مش المادة مابتدرسهاش دراسة
+
+309
+00:30:42,530 --> 00:30:49,530
+حقيقية و هذا معناه أن أنتوا مش ماخدينها بجد يعني
+
+310
+00:30:49,530 --> 00:30:59,690
+كما أجب و هذا دليل عليكم تحلوش مسألة فانا
+
+311
+00:30:59,690 --> 00:31:04,600
+هسأل عنكمخليني أحللكم كام سؤال هاي section اتنين
+
+312
+00:31:04,600 --> 00:31:29,640
+تلاتة هنا هاي
+
+313
+00:31:29,640 --> 00:31:31,000
+مثلا سؤال أربعة
+
+314
+00:31:35,030 --> 00:31:43,690
+هي السؤال أربعة سكشن اتنين تلاتة انا
+
+315
+00:31:43,690 --> 00:31:51,850
+عندي set S أربعة بيساوي كل الأعداد واحد سالب سالب
+
+316
+00:31:51,850 --> 00:32:03,910
+واحد رصد N على N حيث N عدد طبيعي والمطلوب
+
+317
+00:32:03,910 --> 00:32:04,490
+find
+
+318
+00:32:07,290 --> 00:32:17,550
+Find الـ Supremum أو الانفمم ل S4 و ايضا ال
+
+319
+00:32:17,550 --> 00:32:29,950
+Supremum ل S4 طيب
+
+320
+00:32:29,950 --> 00:32:34,370
+احنا أخدنا في مثال في ال section هذا
+
+321
+00:32:37,360 --> 00:32:39,940
+خلنا نام هنا ولا لسه؟
+
+322
+00:33:14,820 --> 00:33:21,960
+Solution اخدنا احنا مثال بيقول انه ال .. لو كان في
+
+323
+00:33:21,960 --> 00:33:25,320
+.. في ال section اللي بعد و ممكن الحل باستخدام
+
+324
+00:33:25,320 --> 00:33:32,500
+المثال رقم A يعني by example
+
+325
+00:33:42,050 --> 00:33:52,450
+تنين اربع واحد الجزء A انا عندي ال supremum ل A
+
+326
+00:33:52,450 --> 00:33:58,990
+زاد S A عدد حقيقي S مجموعة جزئية من R اثبتنا ان
+
+327
+00:33:58,990 --> 00:34:09,970
+هذا بساوي A زاد supremum ال S فلو
+
+328
+00:34:09,970 --> 00:34:24,340
+بدى احل الجزء Bف let S بساوي مجموعة ..
+
+329
+00:34:24,340 --> 00:34:29,740
+let
+
+330
+00:34:29,740 --> 00:34:36,560
+S بساوي مجموعة الأعداد سالب
+
+331
+00:34:36,560 --> 00:34:42,140
+واحد أس N على N حيث N عدب طبيعي
+
+332
+00:34:48,030 --> 00:34:54,310
+ممكن نحول السلب لموجة هاد عبارة عن سالب واحد ..
+
+333
+00:34:54,310 --> 00:35:05,610
+سالب واحد و نص و سالب تلت و ربع و كده
+
+334
+00:35:17,860 --> 00:35:29,700
+فممكن اثبات انه ال super mom تبع السيدتها دى
+
+335
+00:35:29,700 --> 00:35:33,820
+أستاذ
+
+336
+00:35:33,820 --> 00:35:40,840
+نعم هنحن لو سالب اللي برا دخلناه يصير سالب واحد
+
+337
+00:35:40,840 --> 00:35:45,980
+plus one plus واحدعلى أنا ممكن اه ممكن ناخد سالب
+
+338
+00:35:45,980 --> 00:35:50,560
+هذا فهذا أكبر أنصر هو الواحد صح هيك بتنحل الصح
+
+339
+00:35:50,560 --> 00:35:56,620
+برضه هذا ممكن فبصير عندى هنا واحد سالب اول أنصر
+
+340
+00:35:56,620 --> 00:36:03,960
+واحد سالب نص فالصبر ممكن يكون واحد بعدين تلت سالب
+
+341
+00:36:03,960 --> 00:36:12,760
+ربع و هكذافال supremum إذاً ال supremum ل S بساوي
+
+342
+00:36:12,760 --> 00:36:17,480
+هاي اللي .. لاحظ ان الأكبر عدد في الست هذه هو
+
+343
+00:36:17,480 --> 00:36:23,840
+الواحد واحد أكبر من أو ساوي كل الأعداد هذه وهو
+
+344
+00:36:23,840 --> 00:36:27,020
+أصغر upper bound إذاً الواحد upper bound للست هذه
+
+345
+00:36:27,020 --> 00:36:32,400
+هي أكبر من أو ساوي كل عناصرها وهو أصغر upper bound
+
+346
+00:36:32,400 --> 00:36:41,930
+إذاً هذا بساوي واحدلأ س .. إيش بس يعني؟
+
+347
+00:36:41,930 --> 00:36:48,850
+ما
+
+348
+00:36:48,850 --> 00:36:54,370
+هو أصغر؟ طلع اتنين اتنين أستاذ ال super اتنين مش
+
+349
+00:36:54,370 --> 00:37:00,770
+هي على حسب القاعدة نحن نحط اي واحد بيصير
+
+350
+00:37:00,770 --> 00:37:04,520
+اتنين؟ لا لااحنا بنحكي عن ال 6 هذه اللي هانا مش
+
+351
+00:37:04,520 --> 00:37:11,020
+اللي هناك هذه S و هذه S4 فبيختلفوا عن بعض ال 6 هذه
+
+352
+00:37:11,020 --> 00:37:15,760
+هدا هي أنصرها فما
+
+353
+00:37:15,760 --> 00:37:21,700
+هو بيناجيب lower bound او اكبر lower bound اكبر
+
+354
+00:37:21,700 --> 00:37:30,120
+lower bound طب نلاحظ سالب نص اصغر من سالب ربع اصغر
+
+355
+00:37:30,120 --> 00:37:47,710
+منبعد هيك سالب سادس اه فاعتقد
+
+356
+00:37:47,710 --> 00:37:52,010
+ان هذا هيطلع سالب نص هذا اكبر lower bound
+
+357
+00:37:58,870 --> 00:38:04,470
+طيب لو طبقنا المضارية هذه أنا أخدت S بساوي الكلام
+
+358
+00:38:04,470 --> 00:38:12,830
+هذا و A بساوي واحد اذا
+
+359
+00:38:12,830 --> 00:38:24,470
+ال supremum ل S أربعة بساوي A زائد ال supremum ل S
+
+360
+00:38:24,470 --> 00:38:34,260
+صح؟و ال a بساوي واحد و ال suprem ل s بساوي واحد
+
+361
+00:38:34,260 --> 00:38:43,460
+فبطلع ال suprem ل s أربعة بساوي اتنين تمام؟ الان
+
+362
+00:38:43,460 --> 00:38:53,660
+بنجيب ال infimum ل s أربعة بنفس الطريقة ممكن
+
+363
+00:38:53,660 --> 00:38:54,340
+اثبات
+
+364
+00:39:00,490 --> 00:39:10,070
+إذا هنا similar
+
+365
+00:39:10,070 --> 00:39:17,590
+example
+
+366
+00:39:17,590 --> 00:39:24,090
+similar
+
+367
+00:39:24,090 --> 00:39:32,030
+example اتنين اربعة واحد ايهممكن من خلاله نثبت ان
+
+368
+00:39:32,030 --> 00:39:38,330
+الانفمام ان
+
+369
+00:39:38,330 --> 00:39:44,490
+الانفمام لست a زياد s بيساوي a زياد الانفمام ل s
+
+370
+00:39:44,490 --> 00:39:49,310
+وبالتالي
+
+371
+00:39:49,310 --> 00:39:53,390
+ان
+
+372
+00:39:53,390 --> 00:40:00,480
+انا لو بدي اجرب على جزء aف ال infimum ل S أربعة
+
+373
+00:40:00,480 --> 00:40:13,780
+بيساوي ال infimum ل A زائد S اللي هو ال infimum ل
+
+374
+00:40:13,780 --> 00:40:22,630
+واحد زائد S و هذا بيساوي واحد زائد infimum ل Sو
+
+375
+00:40:22,630 --> 00:40:28,770
+هذا بيساوي واحد زائد in from ال S سالب نص فبطلع نص
+
+376
+00:40:28,770 --> 00:40:36,210
+okay ان ال in from لست S أربعة بيطلع سالب بيطلع نص
+
+377
+00:40:36,210 --> 00:40:41,910
+هذا حل حل تاني ان انا يعني احاول
+
+378
+00:40:47,360 --> 00:40:54,460
+أه يعني أكتب عناصر المجموعة هذه أفرفتها و أحاول
+
+379
+00:40:54,460 --> 00:40:59,600
+أشوف وين أصغر عنصر و وين أكبر عنصر و وين هيكون في
+
+380
+00:40:59,600 --> 00:41:04,640
+عندي upper bounds و lower bounds و نحاول نثبت أنه
+
+381
+00:41:04,640 --> 00:41:12,060
+ال .. يعني بالطريقة هذه يعني ممكن نحن نحل الاسئلة
+
+382
+00:41:12,060 --> 00:41:20,320
+بطريقة تانيةفهذا حلو يعني
+
+383
+00:41:20,320 --> 00:41:25,900
+هذا ال set ممكن نكتب عناصرها يعني أول عنصر لما n
+
+384
+00:41:25,900 --> 00:41:33,680
+بساوي واحد واحد سالب سالب
+
+385
+00:41:33,680 --> 00:41:44,350
+سالب واحد يعني اتنين الانصر اللي بعدهواحد سالب نص
+
+386
+00:41:44,350 --> 00:41:56,450
+بيطلع نص اللي بعده بيطلع واحد سالب سالب تلت يعني
+
+387
+00:41:56,450 --> 00:42:03,270
+واحد تلت يعني جديش اربعة على تلاتة اللي بعده واحد
+
+388
+00:42:03,270 --> 00:42:07,210
+موجب ربع بيطلع جديش
+
+389
+00:42:09,700 --> 00:42:17,520
+خمس اربع و هكذا فهنلاحظ
+
+390
+00:42:17,520 --> 00:42:24,700
+ان الاتنين اتنين upper bound لان هو هيكون اكبر
+
+391
+00:42:24,700 --> 00:42:31,100
+عنصر و ينتبه للست لو في اي upper bound تاني لو في
+
+392
+00:42:31,100 --> 00:42:33,320
+any upper bound
+
+393
+00:42:37,830 --> 00:42:45,630
+of S4 فهذا بيقدي انه اتنين اصغر من او ساوي ال V
+
+394
+00:42:45,630 --> 00:42:50,890
+لانه اتنين عنصر في الست S4 صح؟ اذا اتنين upper
+
+395
+00:42:50,890 --> 00:42:54,810
+bound واضح انه اتنين اكبر من او ساوي كل عناصر S4
+
+396
+00:42:54,810 --> 00:43:04,320
+صح؟ولو أخدت أي upper bound ل S4 فبما أن V هو upper
+
+397
+00:43:04,320 --> 00:43:09,200
+bound ل S4 واتنين عنصر في S4 إذن اتنين أصغر من أو
+
+398
+00:43:09,200 --> 00:43:14,640
+يساوي V إذن هنا أثبتنا أن اتنين upper bound ل S4
+
+399
+00:43:14,640 --> 00:43:19,500
+واتنين أصغر من أو يساوي أي upper bound ل S4 إذن
+
+400
+00:43:19,500 --> 00:43:23,440
+اتنين هو ال supreme بالمثل ممكن نثبت أن النص هو
+
+401
+00:43:23,440 --> 00:43:28,140
+الانفع إذن هذا برهان تانيانا اتعمد تعطيكم البرهان
+
+402
+00:43:28,140 --> 00:43:32,260
+هذا عشان القوانين هذه نشوف كيف نطبقها برضه هذا
+
+403
+00:43:32,260 --> 00:43:38,520
+برهان صعيب ناجح الحل؟ okay؟
+
+404
+00:43:38,520 --> 00:43:42,080
+في اي سؤال او استفسار؟ اذا احنا هنكتفي بهذا القدر
+
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/_mc9oZHzNxs_raw.json b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/_mc9oZHzNxs_raw.json
new file mode 100644
index 0000000000000000000000000000000000000000..c288aa15bb741c1c69eef66279bb4abb3882baee
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/_mc9oZHzNxs_raw.json
@@ -0,0 +1 @@
+{"segments": [{"id": 1, "seek": 4489, "start": 20.67, "end": 44.89, "text": "بسم الله الرحمن الرحيم و السلام عليكم هنكمل ان شاء الله اليوم ال .. المثال رقم اتنين اللي بدناه في المحاضرة السابقة و ماكملناهوش فنرجعوا مع بعض بسرعة و نحاول نكمل البرهان", "tokens": [3555, 38251, 21984, 34892, 5016, 27842, 34892, 5016, 32640, 4032, 21136, 37440, 25894, 24793, 8032, 1863, 24793, 1211, 16472, 13412, 16606, 21984, 45595, 20498, 2423, 4386, 9673, 12984, 6027, 12602, 4587, 2304, 1975, 2655, 1863, 9957, 13672, 1829, 47525, 8315, 3224, 8978, 9673, 5016, 46958, 25720, 21136, 16758, 28671, 4032, 19446, 24793, 1211, 8315, 3224, 2407, 8592, 6156, 1863, 47341, 3615, 14407, 20449, 45030, 11242, 4724, 3794, 2288, 27884, 4032, 8717, 5016, 995, 12610, 8717, 24793, 1211, 2423, 26890, 3224, 7649], "avg_logprob": -0.08808117141810859, "compression_ratio": 1.8606060606060606, "no_speech_prob": 1.7881393432617188e-07, "words": [{"start": 20.67, "end": 21.03, "word": "بسم", "probability": 0.847900390625}, 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"no_speech_prob": 0.0, "words": [{"start": 608.04, "end": 608.34, "word": "إن", "probability": 0.657470703125}, {"start": 608.34, "end": 608.78, "word": " X", "probability": 0.431884765625}, {"start": 608.78, "end": 609.14, "word": " in", "probability": 0.11676025390625}, {"start": 609.14, "end": 609.94, "word": " convergence", "probability": 0.3984375}, {"start": 609.94, "end": 613.9, "word": " it", "probability": 0.5556640625}, {"start": 613.9, "end": 614.84, "word": " suffices", "probability": 0.9169921875}, {"start": 614.84, "end": 618.1, "word": " يعني", "probability": 0.797119140625}, {"start": 618.1, "end": 618.66, "word": " يكفي", "probability": 0.9576822916666666}, {"start": 618.66, "end": 619.38, "word": " إثبات", "probability": 0.90625}, {"start": 619.38, "end": 621.04, "word": " to", "probability": 0.5986328125}, {"start": 621.04, "end": 621.52, "word": " show", "probability": 0.96533203125}, {"start": 621.52, "end": 623.42, "word": " it", "probability": 0.9384765625}, {"start": 623.42, "end": 624.16, "word": " is", "probability": 0.94189453125}, {"start": 624.16, "end": 627.04, "word": " Cauchy", "probability": 0.8011067708333334}, {"start": 627.04, "end": 628.04, "word": " إذا", "probability": 0.722412109375}, {"start": 628.04, "end": 628.54, "word": " يكفي", "probability": 0.8006184895833334}, {"start": 628.54, "end": 629.1, "word": " إثبات", "probability": 0.9893798828125}, {"start": 629.1, "end": 630.3, "word": " إنها", "probability": 0.902099609375}, {"start": 630.3, "end": 630.9, "word": " Cauchy", "probability": 0.9127604166666666}], "temperature": 1.0}, {"id": 22, "seek": 65479, "start": 633.27, "end": 654.79, "text": "طيب هاي عندي .. الآن هستفيد من المتباين هذه الأخيرة لإثبات إنها كوشي طيب أنا عندي .. أنا عندي four .. قولنا M أكبر من N", "tokens": [9566, 1829, 3555, 8032, 47302, 18871, 16254, 4386, 6024, 48506, 8032, 14851, 5172, 25708, 9154, 9673, 2655, 3555, 995, 9957, 29538, 16247, 9778, 48923, 5296, 28814, 12984, 3555, 9307, 36145, 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652.59, "word": " قولنا", "probability": 0.7472330729166666}, {"start": 652.59, "end": 653.01, "word": " M", "probability": 0.533203125}, {"start": 653.01, "end": 654.17, "word": " أكبر", "probability": 0.9611002604166666}, {"start": 654.17, "end": 654.43, "word": " من", "probability": 0.99365234375}, {"start": 654.43, "end": 654.79, "word": " N", "probability": 0.9208984375}], "temperature": 1.0}, {"id": 23, "seek": 68547, "start": 657.61, "end": 685.47, "text": "أثبتنا أن أبسليوت xn نيجاتيف xm less than one over two to n minus two نسمي الانيقواليتي هذه star الان let epsilon أكبر من السفر be given", "tokens": [10721, 12984, 3555, 2655, 8315, 14739, 5551, 3555, 3794, 1211, 1829, 35473, 2031, 77, 8717, 1829, 7435, 9307, 33911, 2031, 76, 1570, 813, 472, 670, 732, 281, 297, 3175, 732, 8717, 38251, 1829, 2423, 7649, 1829, 4587, 2407, 6027, 1829, 31371, 29538, 3543, 2423, 7649, 718, 17889, 5551, 4117, 26890, 9154, 21136, 5172, 2288, 312, 2212], "avg_logprob": -0.44106360485679225, 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to", "probability": 0.81787109375}, {"start": 665.33, "end": 665.65, "word": " n", "probability": 0.94091796875}, {"start": 665.65, "end": 666.19, "word": " minus", "probability": 0.95947265625}, {"start": 666.19, "end": 666.57, "word": " two", "probability": 0.95703125}, {"start": 666.57, "end": 668.13, "word": " نسمي", "probability": 0.7096354166666666}, {"start": 668.13, "end": 670.37, "word": " الانيقواليتي", "probability": 0.614715576171875}, {"start": 670.37, "end": 670.89, "word": " هذه", "probability": 0.4521484375}, {"start": 670.89, "end": 671.53, "word": " star", "probability": 0.642578125}, {"start": 671.53, "end": 676.97, "word": " الان", "probability": 0.56549072265625}, {"start": 676.97, "end": 681.01, "word": " let", "probability": 0.7099609375}, {"start": 681.01, "end": 681.71, "word": " epsilon", "probability": 0.572265625}, {"start": 681.71, "end": 684.01, "word": " أكبر", "probability": 0.947265625}, {"start": 684.01, "end": 684.21, "word": " من", "probability": 0.99609375}, {"start": 684.21, "end": 684.83, "word": " السفر", "probability": 0.7685546875}, {"start": 684.83, "end": 685.03, "word": " be", "probability": 0.86181640625}, {"start": 685.03, "end": 685.47, "word": " given", "probability": 0.8720703125}], "temperature": 1.0}, {"id": 24, "seek": 71646, "start": 687.6, "end": 716.46, "text": "أنا بدأ أثبت إن الـ sequence تبعتي كوشي فعشان أثبت إنها كوشي ببدأ بإمسون أكبر من سفر برد عليها بcapital N بحيث إنه لكل M و N أكبر من أو يساوي capital N لازم المسافة بين XN وXM أصغر من إبسون ف let إبسون أكبر من سفر be given by Archimedean property", "tokens": [10721, 8315, 47525, 10721, 5551, 12984, 3555, 2655, 36145, 2423, 39184, 8310, 6055, 3555, 3615, 31371, 9122, 2407, 8592, 1829, 6156, 3615, 8592, 7649, 5551, 12984, 3555, 2655, 36145, 11296, 9122, 2407, 8592, 1829, 4724, 44510, 10721, 4724, 28814, 2304, 3794, 11536, 5551, 4117, 26890, 9154, 8608, 5172, 2288, 4724, 2288, 3215, 25894, 11296, 4724, 9485, 1686, 426, 4724, 5016, 1829, 12984, 36145, 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701.94, "end": 702.4, "word": " capital", "probability": 0.39697265625}, {"start": 702.4, "end": 702.76, "word": " N", "probability": 0.82958984375}, {"start": 702.76, "end": 703.76, "word": " لازم", "probability": 0.9900716145833334}, {"start": 703.76, "end": 704.28, "word": " المسافة", "probability": 0.931884765625}, {"start": 704.28, "end": 704.52, "word": " بين", "probability": 0.96630859375}, {"start": 704.52, "end": 705.18, "word": " XN", "probability": 0.5411376953125}, {"start": 705.18, "end": 706.04, "word": " وXM", "probability": 0.8515625}, {"start": 706.04, "end": 706.66, "word": " أصغر", "probability": 0.9920654296875}, {"start": 706.66, "end": 706.96, "word": " من", "probability": 0.99267578125}, {"start": 706.96, "end": 707.46, "word": " إبسون", "probability": 0.8150634765625}, {"start": 707.46, "end": 708.22, "word": " ف", "probability": 0.87353515625}, {"start": 708.22, "end": 708.54, "word": " let", "probability": 0.374755859375}, {"start": 708.54, "end": 709.02, "word": " إبسون", "probability": 0.78228759765625}, {"start": 709.02, "end": 709.4, "word": " أكبر", "probability": 0.9816080729166666}, {"start": 709.4, "end": 709.6, "word": " من", "probability": 0.99560546875}, {"start": 709.6, "end": 710.0, "word": " سفر", "probability": 0.9488932291666666}, {"start": 710.0, "end": 710.2, "word": " be", "probability": 0.8349609375}, {"start": 710.2, "end": 710.62, "word": " given", "probability": 0.884765625}, {"start": 710.62, "end": 712.94, "word": " by", "probability": 0.9404296875}, {"start": 712.94, "end": 715.52, "word": " Archimedean", "probability": 0.84490966796875}, {"start": 715.52, "end": 716.46, "word": " property", "probability": 0.83447265625}], "temperature": 1.0}, {"id": 25, "seek": 73744, "start": 718.16, "end": 737.44, "text": "من خاصية Archimedes choose ممكن نختار capital N عدد طبيعي بحيث أنه واحد على N أصغر من epsilon على أربعة", "tokens": [27842, 16490, 33546, 10632, 10984, 332, 48490, 2826, 3714, 43020, 8717, 46456, 9640, 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"word": " اربعة", "probability": 0.9637044270833334}, {"start": 1230.22, "end": 1232.14, "word": " بطلع", "probability": 0.80126953125}, {"start": 1232.14, "end": 1232.86, "word": " اتنين", "probability": 0.9588623046875}, {"start": 1232.86, "end": 1233.12, "word": " على", "probability": 0.8056640625}, {"start": 1233.12, "end": 1234.12, "word": " تلاتة", "probability": 0.9266357421875}, {"start": 1234.12, "end": 1235.1, "word": " و", "probability": 0.3681640625}, {"start": 1235.1, "end": 1235.26, "word": " ال", "probability": 0.705078125}, {"start": 1235.26, "end": 1235.56, "word": " bust", "probability": 0.378662109375}, {"start": 1235.56, "end": 1235.82, "word": " هذا", "probability": 0.783203125}, {"start": 1235.82, "end": 1236.16, "word": " هو", "probability": 0.99169921875}, {"start": 1236.16, "end": 1236.44, "word": " في", "probability": 0.68603515625}, {"start": 1236.44, "end": 1236.64, "word": " ال", "probability": 0.849609375}, {"start": 1236.64, "end": 1236.98, "word": " 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"word": " limit", "probability": 0.97607421875}, {"start": 1272.38, "end": 1273.5, "word": " x", "probability": 0.73583984375}, {"start": 1273.5, "end": 1274.38, "word": " to", "probability": 0.73828125}, {"start": 1274.38, "end": 1274.78, "word": " n", "probability": 0.72900390625}, {"start": 1274.78, "end": 1275.3, "word": " plus", "probability": 0.6845703125}, {"start": 1275.3, "end": 1275.76, "word": " one", "probability": 0.85107421875}, {"start": 1275.76, "end": 1276.24, "word": " as", "probability": 0.91845703125}, {"start": 1276.24, "end": 1276.7, "word": " n", "probability": 0.88916015625}, {"start": 1276.7, "end": 1277.56, "word": " tends", "probability": 0.81494140625}, {"start": 1277.56, "end": 1277.7, "word": " to", "probability": 0.962890625}, {"start": 1277.7, "end": 1278.16, "word": " infinity", "probability": 0.9130859375}], "temperature": 1.0}, {"id": 45, "seek": 130844, "start": 1279.12, "end": 1308.44, "text": "بساوي واحد زاد اتنين على التلاتة في ال limit الجوس واحد سالب limit واحد على اربعة أس in as in tends to infinity وهذا بساوي واحد زاد اتنين على تلاتة في واحد سالب limit واحد على اربعة in بساوي سفر فبطلع بساوي واحد", "tokens": [3555, 3794, 995, 45865, 36764, 24401, 30767, 18513, 1975, 2655, 1863, 9957, 15844, 16712, 1211, 9307, 3660, 8978, 2423, 4948, 25724, 41779, 36764, 24401, 8608, 6027, 3555, 4948, 36764, 24401, 15844, 1975, 25513, 27884, 5551, 3794, 294, 382, 294, 12258, 281, 13202, 37037, 15730, 4724, 3794, 995, 45865, 36764, 24401, 30767, 18513, 1975, 2655, 1863, 9957, 15844, 6055, 1211, 9307, 3660, 8978, 36764, 24401, 8608, 6027, 3555, 4948, 36764, 24401, 15844, 1975, 25513, 27884, 294, 4724, 3794, 995, 45865, 8608, 5172, 2288, 6156, 3555, 9566, 1211, 3615, 4724, 3794, 995, 45865, 36764, 24401], "avg_logprob": -0.18068484502269866, "compression_ratio": 2.178343949044586, "no_speech_prob": 0.0, "words": [{"start": 1279.12, "end": 1279.82, "word": "بساوي", "probability": 0.66143798828125}, {"start": 1279.82, "end": 1280.36, "word": " 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0.80126953125}, {"start": 1303.52, "end": 1304.78, "word": " بساوي", "probability": 0.9761962890625}, {"start": 1304.78, "end": 1305.36, "word": " سفر", "probability": 0.9083658854166666}, {"start": 1305.36, "end": 1307.06, "word": " فبطلع", "probability": 0.8875}, {"start": 1307.06, "end": 1307.78, "word": " بساوي", "probability": 0.9730224609375}, {"start": 1307.78, "end": 1308.44, "word": " واحد", "probability": 0.98828125}], "temperature": 1.0}, {"id": 46, "seek": 133393, "start": 1310.17, "end": 1333.93, "text": "زاد اتنين على تلاتة بساوي خمسة على تلاتة طيب هذه عبارة عن sub sequence من ال sequence xn هذه الحدود الفردية ل sequence xn طيب و انا عندي ال sequence تبعتي convergence هاي أثبتنا ان xn convergent و ال limit تبعتها بساوي العدد x", "tokens": [11622, 18513, 1975, 2655, 1863, 9957, 15844, 6055, 1211, 9307, 3660, 4724, 3794, 995, 45865, 16490, 2304, 3794, 3660, 15844, 6055, 1211, 9307, 3660, 23032, 1829, 3555, 29538, 6225, 3555, 9640, 3660, 18871, 1422, 8310, 9154, 2423, 8310, 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هذه", "probability": 0.69775390625}, {"start": 1321.07, "end": 1321.51, "word": " الحدود", "probability": 0.9881184895833334}, {"start": 1321.51, "end": 1322.15, "word": " الفردية", "probability": 0.97900390625}, {"start": 1322.15, "end": 1322.29, "word": " ل", "probability": 0.77197265625}, {"start": 1322.29, "end": 1322.67, "word": " sequence", "probability": 0.44091796875}, {"start": 1322.67, "end": 1323.27, "word": " xn", "probability": 0.980224609375}, {"start": 1323.27, "end": 1324.47, "word": " طيب", "probability": 0.9558919270833334}, {"start": 1324.47, "end": 1324.59, "word": " و", "probability": 0.896484375}, {"start": 1324.59, "end": 1324.71, "word": " انا", "probability": 0.646728515625}, {"start": 1324.71, "end": 1325.11, "word": " عندي", "probability": 0.7330322265625}, {"start": 1325.11, "end": 1325.93, "word": " ال", "probability": 0.9482421875}, {"start": 1325.93, "end": 1326.37, "word": " sequence", "probability": 0.98974609375}, {"start": 1326.37, "end": 1327.01, 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1333.57, "word": " العدد", "probability": 0.9734700520833334}, {"start": 1333.57, "end": 1333.93, "word": " x", "probability": 0.91357421875}], "temperature": 1.0}, {"id": 47, "seek": 136028, "start": 1340.16, "end": 1360.28, "text": "إذا أنا في عندي هنا since x2n plus one is a subsequence of a sequence xn and xn converges to x", "tokens": [28814, 15730, 41850, 8978, 18871, 16254, 34105, 1670, 2031, 17, 77, 1804, 472, 307, 257, 13924, 655, 295, 257, 8310, 2031, 77, 293, 2031, 77, 9652, 2880, 281, 2031], "avg_logprob": -0.3177083363135656, "compression_ratio": 1.101010101010101, "no_speech_prob": 0.0, "words": [{"start": 1340.16, "end": 1340.46, "word": "إذا", "probability": 0.33929443359375}, {"start": 1340.46, "end": 1340.66, "word": " أنا", "probability": 0.81201171875}, {"start": 1340.66, "end": 1340.88, "word": " في", "probability": 0.76904296875}, {"start": 1340.88, "end": 1341.2, "word": " عندي", "probability": 0.57000732421875}, {"start": 1341.2, "end": 1341.5, "word": " هنا", "probability": 0.97607421875}, {"start": 1341.5, "end": 1342.18, "word": " since", "probability": 0.58203125}, {"start": 1342.18, "end": 1346.0, "word": " x2n", "probability": 0.5538736979166666}, {"start": 1346.0, "end": 1346.54, "word": " plus", "probability": 0.14306640625}, {"start": 1346.54, "end": 1347.08, "word": " one", "probability": 0.8466796875}, {"start": 1347.08, "end": 1348.64, "word": " is", "probability": 0.90185546875}, {"start": 1348.64, "end": 1349.3, "word": " a", "probability": 0.97607421875}, {"start": 1349.3, "end": 1352.24, "word": " subsequence", "probability": 0.979736328125}, {"start": 1352.24, "end": 1353.84, "word": " of", "probability": 0.435546875}, {"start": 1353.84, "end": 1354.22, "word": " a", "probability": 0.60595703125}, {"start": 1354.22, "end": 1354.82, "word": " sequence", "probability": 0.97265625}, {"start": 1354.82, "end": 1355.72, "word": " xn", "probability": 0.89306640625}, {"start": 1355.72, "end": 1357.68, "word": " and", "probability": 0.8671875}, {"start": 1357.68, "end": 1358.6, "word": " xn", "probability": 0.98486328125}, {"start": 1358.6, "end": 1359.58, "word": " converges", "probability": 0.9716796875}, {"start": 1359.58, "end": 1359.84, "word": " to", "probability": 0.98583984375}, {"start": 1359.84, "end": 1360.28, "word": " x", "probability": 0.98046875}], "temperature": 1.0}, {"id": 48, "seek": 138764, "start": 1362.2, "end": 1387.64, "text": "then by previous theorem حسب نظرية السابقة إذا كانت ال sequence convergent ل x فأي subsequence منها بتكون convergent لنفس ال x إذا limit x اتنين n plus one as n tends to infinity بساوي x وبالتالي إذا x بساوي limit", "tokens": [19096, 538, 3894, 20904, 11331, 35457, 8717, 19913, 2288, 10632, 21136, 16758, 28671, 11933, 15730, 25961, 2655, 2423, 8310, 9652, 6930, 5296, 2031, 6156, 10721, 1829, 13924, 655, 9154, 11296, 39894, 30544, 9652, 6930, 5296, 1863, 36178, 2423, 2031, 11933, 15730, 4948, 2031, 1975, 2655, 1863, 9957, 297, 1804, 472, 382, 297, 12258, 281, 13202, 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"end": 1416.03, "word": " كما", "probability": 0.915283203125}, {"start": 1416.03, "end": 1416.37, "word": " هو", "probability": 0.98974609375}, {"start": 1416.37, "end": 1417.01, "word": " مطلوب", "probability": 0.948974609375}], "temperature": 1.0}, {"id": 50, "seek": 143659, "start": 1417.55, "end": 1436.59, "text": "Okay إذا هيك بنكون إحنا برهننا إن ال sequence في المثال هذا اللي معرفة بطريقة استقرائية is convergent ونهايتها خمسة تلاتة Okay تمام؟ المفهوم واضح؟ في أي استفسار؟ في أي سؤال؟", "tokens": [8297, 11933, 15730, 39896, 4117, 44945, 30544, 11933, 5016, 8315, 4724, 2288, 3224, 1863, 8315, 36145, 2423, 8310, 8978, 9673, 12984, 6027, 23758, 13672, 1829, 20449, 28480, 3660, 4724, 9566, 16572, 28671, 44713, 4587, 2288, 16373, 10632, 307, 9652, 6930, 4032, 1863, 11296, 36081, 11296, 16490, 2304, 3794, 3660, 6055, 1211, 9307, 3660, 1033, 46811, 10943, 22807, 9673, 5172, 3224, 20498, 4032, 46958, 5016, 22807, 8978, 36632, 44713, 36178, 9640, 22807, 8978, 36632, 8608, 33604, 6027, 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"word": " تمام؟", "probability": 0.7688802083333334}, {"start": 1432.39, "end": 1432.87, "word": " المفهوم", "probability": 0.864501953125}, {"start": 1432.87, "end": 1433.85, "word": " واضح؟", "probability": 0.942138671875}, {"start": 1433.85, "end": 1433.85, "word": " في", "probability": 0.76171875}, {"start": 1433.85, "end": 1433.95, "word": " أي", "probability": 0.64501953125}, {"start": 1433.95, "end": 1435.31, "word": " استفسار؟", "probability": 0.945068359375}, {"start": 1435.31, "end": 1435.61, "word": " في", "probability": 0.962890625}, {"start": 1435.61, "end": 1435.77, "word": " أي", "probability": 0.9365234375}, {"start": 1435.77, "end": 1436.59, "word": " سؤال؟", "probability": 0.9847412109375}], "temperature": 1.0}, {"id": 51, "seek": 146354, "start": 1437.98, "end": 1463.54, "text": "طيب ناخد كمان مثال يمكن يبقى المثال طويل شوية لكن احنا زى ما شوفته احنا جزقناه الى تلاتة claims او three claims وكل claim كان برهانه by induction مش صعب شفنا برهان واحد منهم المرة اللى فاتت التانين برضه اسأل كمان كل claim بيخطو خطوة الى الامام بيجربنى اكتر من البرهان", "tokens": [9566, 1829, 3555, 8717, 47283, 3215, 9122, 2304, 7649, 50113, 6027, 7251, 43020, 7251, 3555, 4587, 7578, 9673, 12984, 6027, 23032, 2407, 26895, 13412, 2407, 10632, 44381, 1975, 5016, 8315, 30767, 7578, 19446, 13412, 38688, 47395, 1975, 5016, 8315, 10874, 11622, 4587, 8315, 3224, 2423, 7578, 6055, 1211, 9307, 3660, 9441, 1975, 2407, 1045, 9441, 4032, 28820, 3932, 25961, 4724, 2288, 3224, 7649, 3224, 538, 33371, 37893, 20328, 3615, 3555, 13412, 5172, 8315, 4724, 2288, 3224, 7649, 36764, 24401, 9154, 16095, 9673, 25720, 13672, 7578, 6156, 9307, 2655, 16712, 7649, 9957, 4724, 43042, 3224, 24525, 10721, 1211, 9122, 2304, 7649, 28242, 3932, 4724, 1829, 9778, 9566, 2407, 16490, 9566, 2407, 3660, 2423, 7578, 2423, 10943, 10943, 4724, 1829, 7435, 25513, 1863, 7578, 1975, 4117, 2655, 2288, 9154, 2423, 26890, 3224, 7649], "avg_logprob": -0.11854876831851223, "compression_ratio": 1.8661087866108788, 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claim", "probability": 0.986328125}, {"start": 1449.0, "end": 1449.3, "word": " كان", "probability": 0.99267578125}, {"start": 1449.3, "end": 1449.74, "word": " برهانه", "probability": 0.92578125}, {"start": 1449.74, "end": 1449.96, "word": " by", "probability": 0.95166015625}, {"start": 1449.96, "end": 1450.44, "word": " induction", "probability": 0.9814453125}, {"start": 1450.44, "end": 1450.72, "word": " مش", "probability": 0.96044921875}, {"start": 1450.72, "end": 1451.06, "word": " صعب", "probability": 0.9957682291666666}, {"start": 1451.06, "end": 1451.5, "word": " شفنا", "probability": 0.8318684895833334}, {"start": 1451.5, "end": 1452.34, "word": " برهان", "probability": 0.9937744140625}, {"start": 1452.34, "end": 1452.76, "word": " واحد", "probability": 0.99462890625}, {"start": 1452.76, "end": 1453.12, "word": " منهم", "probability": 0.995849609375}, {"start": 1453.12, "end": 1453.88, "word": " المرة", "probability": 0.831787109375}, {"start": 1453.88, "end": 1454.06, "word": " اللى", "probability": 0.953369140625}, {"start": 1454.06, "end": 1454.44, "word": " فاتت", "probability": 0.9908854166666666}, {"start": 1454.44, "end": 1454.82, "word": " التانين", "probability": 0.8497721354166666}, {"start": 1454.82, "end": 1455.16, "word": " برضه", "probability": 0.98876953125}, {"start": 1455.16, "end": 1456.2, "word": " اسأل", "probability": 0.7325846354166666}, {"start": 1456.2, "end": 1458.02, "word": " كمان", "probability": 0.92724609375}, {"start": 1458.02, "end": 1458.48, "word": " كل", "probability": 0.96875}, {"start": 1458.48, "end": 1458.92, "word": " claim", "probability": 0.99072265625}, {"start": 1458.92, "end": 1460.2, "word": " بيخطو", "probability": 0.839453125}, {"start": 1460.2, "end": 1460.64, "word": " خطوة", "probability": 0.939208984375}, {"start": 1460.64, "end": 1460.84, "word": " الى", "probability": 0.886962890625}, {"start": 1460.84, "end": 1461.34, "word": " الامام", "probability": 0.8185221354166666}, {"start": 1461.34, "end": 1462.48, "word": " بيجربنى", "probability": 0.833251953125}, {"start": 1462.48, "end": 1462.86, "word": " اكتر", "probability": 0.9393310546875}, {"start": 1462.86, "end": 1463.06, "word": " من", "probability": 0.994140625}, {"start": 1463.06, "end": 1463.54, "word": " البرهان", "probability": 0.89892578125}], "temperature": 1.0}, {"id": 52, "seek": 149409, "start": 1467.87, "end": 1494.09, "text": "ناخد مثال آخر، تالت إذا example three consider", "tokens": [1863, 47283, 3215, 50113, 6027, 19753, 34740, 12399, 6055, 6027, 2655, 11933, 15730, 1365, 1045, 1949], "avg_logprob": -0.26056985294117646, "compression_ratio": 0.8904109589041096, "no_speech_prob": 0.0, "words": [{"start": 1467.87, "end": 1468.37, "word": "ناخد", "probability": 0.9228515625}, {"start": 1468.37, "end": 1468.85, "word": " مثال", "probability": 0.976806640625}, {"start": 1468.85, "end": 1469.77, "word": " آخر،", "probability": 0.6570638020833334}, {"start": 1469.77, "end": 1470.47, "word": " تالت", "probability": 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"الحد", "probability": 0.3833821614583333}, {"start": 1498.23, "end": 1498.23, "word": " العام", "probability": 0.900146484375}, {"start": 1498.23, "end": 1498.23, "word": " بحيث", "probability": 0.482147216796875}, {"start": 1498.23, "end": 1498.25, "word": " ال", "probability": 0.324951171875}, {"start": 1498.25, "end": 1498.91, "word": " sequence", "probability": 0.09796142578125}, {"start": 1498.91, "end": 1502.55, "word": " xn", "probability": 0.3673095703125}, {"start": 1502.55, "end": 1503.31, "word": " where", "probability": 0.1898193359375}, {"start": 1503.31, "end": 1504.07, "word": " حيث", "probability": 0.67529296875}, {"start": 1504.07, "end": 1505.73, "word": " ال", "probability": 0.61083984375}, {"start": 1505.73, "end": 1506.21, "word": " term", "probability": 0.196533203125}, {"start": 1506.21, "end": 1506.69, "word": " of", "probability": 0.69921875}, {"start": 1506.69, "end": 1506.83, "word": " the", "probability": 0.5029296875}, {"start": 1506.83, "end": 1507.41, "word": " sequence", "probability": 0.93212890625}, {"start": 1507.41, "end": 1508.69, "word": " الحد", "probability": 0.310211181640625}, {"start": 1508.69, "end": 1509.21, "word": " العام", "probability": 0.9443359375}, {"start": 1509.21, "end": 1509.51, "word": " by", "probability": 0.5576171875}, {"start": 1509.51, "end": 1510.19, "word": " definition", "probability": 0.9169921875}, {"start": 1510.19, "end": 1511.49, "word": " بساوي", "probability": 0.705322265625}, {"start": 1511.49, "end": 1511.79, "word": " one", "probability": 0.60888671875}, {"start": 1511.79, "end": 1512.19, "word": " over", "probability": 0.91943359375}, {"start": 1512.19, "end": 1512.55, "word": " one", "probability": 0.9482421875}, {"start": 1512.55, "end": 1512.91, "word": " plus", "probability": 0.90625}, {"start": 1512.91, "end": 1513.21, "word": " one", "probability": 0.94091796875}, {"start": 1513.21, "end": 1513.53, "word": " over", "probability": 0.93017578125}, {"start": 1513.53, "end": 1514.07, "word": " two", "probability": 0.95751953125}], "temperature": 1.0}, {"id": 54, "seek": 153653, "start": 1515.27, "end": 1536.53, "text": "plus one over three و هكذا and so on until we get one over N حيث N ينتمي إلى N لكل N في N بنعرف XN على أنه المجموع أو مجموعة أول N", "tokens": [18954, 472, 670, 1045, 4032, 8032, 4117, 15730, 293, 370, 322, 1826, 321, 483, 472, 670, 426, 11331, 1829, 12984, 426, 7251, 29399, 2304, 1829, 30731, 426, 5296, 28820, 426, 8978, 426, 44945, 3615, 28480, 1783, 45, 15844, 14739, 3224, 9673, 7435, 2304, 45367, 34051, 3714, 7435, 2304, 2407, 27884, 5551, 12610, 426], "avg_logprob": -0.26490161209194746, "compression_ratio": 1.2765957446808511, "no_speech_prob": 0.0, "words": [{"start": 1515.27, "end": 1515.67, "word": "plus", "probability": 0.1334228515625}, {"start": 1515.67, "end": 1515.99, "word": " one", "probability": 0.68408203125}, {"start": 1515.99, "end": 1516.29, "word": " over", "probability": 0.91015625}, {"start": 1516.29, "end": 1517.07, "word": 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1563.23, "end": 1564.67, "word": " partial", "probability": 0.9677734375}, {"start": 1564.67, "end": 1566.95, "word": " ال", "probability": 0.3271484375}, {"start": 1566.95, "end": 1567.59, "word": " nth", "probability": 0.955078125}, {"start": 1567.59, "end": 1568.33, "word": " partial", "probability": 0.96240234375}, {"start": 1568.33, "end": 1570.65, "word": " sum", "probability": 0.9599609375}, {"start": 1570.65, "end": 1571.39, "word": " of", "probability": 0.98193359375}, {"start": 1571.39, "end": 1571.77, "word": " ال", "probability": 0.77783203125}, {"start": 1571.77, "end": 1572.41, "word": " harmonic", "probability": 0.95458984375}, {"start": 1572.41, "end": 1574.41, "word": " ال", "probability": 0.33056640625}, {"start": 1574.41, "end": 1575.07, "word": " harmonic", "probability": 0.9755859375}, {"start": 1575.07, "end": 1576.11, "word": " series", "probability": 0.93701171875}], "temperature": 1.0}, {"id": 56, "seek": 158788, "start": 1580.92, "end": 1587.88, "text": "اللي هي summation from k equals one to infinity ل one over k", "tokens": [6027, 20292, 39896, 28811, 490, 350, 6915, 472, 281, 13202, 5296, 472, 670, 350], "avg_logprob": -0.2776041785875956, "compression_ratio": 0.9710144927536232, "no_speech_prob": 0.0, "words": [{"start": 1580.92, "end": 1581.38, "word": "اللي", "probability": 0.52783203125}, {"start": 1581.38, "end": 1581.82, "word": " هي", "probability": 0.95849609375}, {"start": 1581.82, "end": 1583.28, "word": " summation", "probability": 0.564453125}, {"start": 1583.28, "end": 1584.66, "word": " from", "probability": 0.64697265625}, {"start": 1584.66, "end": 1585.02, "word": " k", "probability": 0.65185546875}, {"start": 1585.02, "end": 1585.52, "word": " equals", "probability": 0.3466796875}, {"start": 1585.52, "end": 1585.82, "word": " one", "probability": 0.86328125}, {"start": 1585.82, "end": 1585.98, "word": " to", "probability": 0.91943359375}, {"start": 1585.98, "end": 1586.54, "word": " infinity", "probability": 0.89013671875}, {"start": 1586.54, "end": 1586.8, "word": " ل", "probability": 0.65869140625}, {"start": 1586.8, "end": 1587.08, "word": " one", "probability": 0.68310546875}, {"start": 1587.08, "end": 1587.46, "word": " over", "probability": 0.94482421875}, {"start": 1587.46, "end": 1587.88, "word": " k", "probability": 0.8994140625}], "temperature": 1.0}, {"id": 57, "seek": 161407, "start": 1591.95, "end": 1614.07, "text": "المطلوب show أن سيكوينس XN divergence ليست convergent، is not convergent بنثبت أن سيكوينس of partial sums متتالية المجاميع الجزئية لل harmonic series بتشكل divergence sequence", "tokens": [45340, 9566, 1211, 37746, 855, 14739, 8608, 1829, 4117, 2407, 9957, 3794, 1783, 45, 47387, 32239, 14851, 9652, 6930, 12399, 307, 406, 9652, 6930, 44945, 12984, 3555, 2655, 14739, 8608, 1829, 4117, 2407, 9957, 3794, 295, 14641, 34499, 44650, 2655, 6027, 10632, 9673, 7435, 10943, 40228, 25724, 11622, 19986, 10632, 24976, 32270, 2638, 39894, 8592, 28820, 47387, 8310], 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"word": " convergent", "probability": 0.949462890625}, {"start": 1601.97, "end": 1603.63, "word": " بنثبت", "probability": 0.7620849609375}, {"start": 1603.63, "end": 1603.79, "word": " أن", "probability": 0.80859375}, {"start": 1603.79, "end": 1605.03, "word": " سيكوينس", "probability": 0.8990885416666666}, {"start": 1605.03, "end": 1606.07, "word": " of", "probability": 0.228515625}, {"start": 1606.07, "end": 1606.63, "word": " partial", "probability": 0.9111328125}, {"start": 1606.63, "end": 1607.13, "word": " sums", "probability": 0.95849609375}, {"start": 1607.13, "end": 1608.11, "word": " متتالية", "probability": 0.9625244140625}, {"start": 1608.11, "end": 1608.71, "word": " المجاميع", "probability": 0.8138427734375}, {"start": 1608.71, "end": 1609.39, "word": " الجزئية", "probability": 0.8978271484375}, {"start": 1609.39, "end": 1610.61, "word": " لل", "probability": 0.884765625}, {"start": 1610.61, "end": 1611.09, "word": " harmonic", "probability": 0.60693359375}, {"start": 1611.09, "end": 1611.65, "word": " series", "probability": 0.89990234375}, {"start": 1611.65, "end": 1612.47, "word": " بتشكل", "probability": 0.9568684895833334}, {"start": 1612.47, "end": 1613.17, "word": " divergence", "probability": 0.560546875}, {"start": 1613.17, "end": 1614.07, "word": " sequence", "probability": 0.499267578125}], "temperature": 1.0}, {"id": 58, "seek": 164762, "start": 1620.34, "end": 1647.62, "text": "by cushy criterion اذا حسب cushy criterion by cushy criterion it suffices to show يكفي اثبات", "tokens": [2322, 18422, 88, 46691, 1975, 15730, 11331, 35457, 18422, 88, 46691, 538, 18422, 88, 46691, 309, 3889, 1473, 281, 855, 7251, 4117, 41185, 1975, 12984, 3555, 9307], "avg_logprob": -0.23507253878882953, "compression_ratio": 1.3048780487804879, "no_speech_prob": 0.0, "words": [{"start": 1620.34, "end": 1620.82, "word": "by", "probability": 0.63232421875}, {"start": 1620.82, "end": 1621.62, "word": " cushy", "probability": 0.6370849609375}, {"start": 1621.62, "end": 1624.38, "word": " criterion", "probability": 0.96630859375}, {"start": 1624.38, "end": 1626.94, "word": " اذا", "probability": 0.522613525390625}, {"start": 1626.94, "end": 1627.88, "word": " حسب", "probability": 0.981689453125}, {"start": 1627.88, "end": 1629.68, "word": " cushy", "probability": 0.7127685546875}, {"start": 1629.68, "end": 1630.42, "word": " criterion", "probability": 0.984375}, {"start": 1630.42, "end": 1639.02, "word": " by", "probability": 0.72607421875}, {"start": 1639.02, "end": 1639.52, "word": " cushy", "probability": 0.96337890625}, {"start": 1639.52, "end": 1640.16, "word": " criterion", "probability": 0.9609375}, {"start": 1640.16, "end": 1641.48, "word": " it", "probability": 0.90185546875}, {"start": 1641.48, "end": 1642.48, "word": " suffices", "probability": 0.905029296875}, {"start": 1642.48, "end": 1645.74, "word": " to", "probability": 0.9501953125}, {"start": 1645.74, "end": 1646.24, "word": " show", "probability": 0.96337890625}, {"start": 1646.24, "end": 1646.92, "word": " يكفي", "probability": 0.9049479166666666}, {"start": 1646.92, "end": 1647.62, "word": " اثبات", "probability": 0.896484375}], "temperature": 1.0}, {"id": 59, "seek": 167570, "start": 1648.94, "end": 1675.7, "text": "يكفي اثبات ان ال sequence عشان نثبت ان ال sequence is divergent it suffices to show ان ال sequence xn is not كوشي لأن كوشي criterion بتقول ان ال sequence is convergent if and only if it is كوشي", "tokens": [1829, 4117, 41185, 1975, 12984, 3555, 9307, 16472, 2423, 8310, 6225, 8592, 7649, 8717, 12984, 3555, 2655, 16472, 2423, 8310, 307, 18558, 6930, 309, 3889, 1473, 281, 855, 16472, 2423, 8310, 2031, 77, 307, 406, 9122, 2407, 8592, 1829, 5296, 33456, 9122, 2407, 8592, 1829, 46691, 39894, 39648, 16472, 2423, 8310, 307, 9652, 6930, 498, 293, 787, 498, 309, 307, 9122, 2407, 8592, 1829], "avg_logprob": -0.23629807692307692, "compression_ratio": 1.6849315068493151, "no_speech_prob": 0.0, "words": [{"start": 1648.94, "end": 1649.54, "word": 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"word": " it", "probability": 0.28955078125}, {"start": 1654.94, "end": 1655.42, "word": " suffices", "probability": 0.80078125}, {"start": 1655.42, "end": 1655.68, "word": " to", "probability": 0.974609375}, {"start": 1655.68, "end": 1656.06, "word": " show", "probability": 0.955078125}, {"start": 1656.06, "end": 1656.44, "word": " ان", "probability": 0.85302734375}, {"start": 1656.44, "end": 1656.7, "word": " ال", "probability": 0.8779296875}, {"start": 1656.7, "end": 1657.2, "word": " sequence", "probability": 0.9833984375}, {"start": 1657.2, "end": 1658.0, "word": " xn", "probability": 0.4547119140625}, {"start": 1658.0, "end": 1659.28, "word": " is", "probability": 0.87646484375}, {"start": 1659.28, "end": 1659.88, "word": " not", "probability": 0.95751953125}, {"start": 1659.88, "end": 1662.6, "word": " كوشي", "probability": 0.79290771484375}, {"start": 1662.6, "end": 1667.92, "word": " لأن", "probability": 0.623291015625}, {"start": 1667.92, "end": 1668.64, "word": " كوشي", "probability": 0.8741455078125}, {"start": 1668.64, "end": 1669.12, "word": " criterion", "probability": 0.58935546875}, {"start": 1669.12, "end": 1669.68, "word": " بتقول", "probability": 0.8603515625}, {"start": 1669.68, "end": 1670.76, "word": " ان", "probability": 0.85205078125}, {"start": 1670.76, "end": 1671.12, "word": " ال", "probability": 0.92333984375}, {"start": 1671.12, "end": 1672.1, "word": " sequence", "probability": 0.97900390625}, {"start": 1672.1, "end": 1672.34, "word": " is", "probability": 0.387939453125}, {"start": 1672.34, "end": 1672.98, "word": " convergent", "probability": 0.9658203125}, {"start": 1672.98, "end": 1673.3, "word": " if", "probability": 0.939453125}, {"start": 1673.3, "end": 1673.52, "word": " and", "probability": 0.888671875}, {"start": 1673.52, "end": 1673.76, "word": " only", "probability": 0.92333984375}, {"start": 1673.76, "end": 1674.14, "word": " if", "probability": 0.97412109375}, {"start": 1674.14, "end": 1675.0, "word": " it", "probability": 0.94921875}, {"start": 1675.0, "end": 1675.22, "word": " is", "probability": 0.92529296875}, {"start": 1675.22, "end": 1675.7, "word": " كوشي", "probability": 0.8909912109375}], "temperature": 1.0}, {"id": 60, "seek": 170290, "start": 1677.18, "end": 1702.9, "text": "وبالتالي هذا بكافي ان احنا نقول ان ال sequence is not convergent if and only if it is not Cauchy عشان نثبت ان ال sequence is divergent ممكن نثبت انها is not Cauchy طيب ال .. الاثبات انها not Cauchy هنستخدم indeed", "tokens": [37746, 6027, 2655, 6027, 1829, 23758, 4724, 4117, 31845, 1829, 16472, 1975, 5016, 8315, 8717, 39648, 16472, 2423, 8310, 307, 406, 9652, 6930, 498, 293, 787, 498, 309, 307, 406, 7544, 625, 88, 6225, 8592, 7649, 8717, 12984, 3555, 2655, 16472, 2423, 8310, 307, 18558, 6930, 3714, 43020, 8717, 12984, 3555, 2655, 16472, 11296, 307, 406, 7544, 625, 88, 23032, 1829, 3555, 2423, 4386, 2423, 5718, 104, 3555, 9307, 16472, 11296, 406, 7544, 625, 88, 8032, 1863, 14851, 9778, 40448, 6451], "avg_logprob": -0.14786585384025808, "compression_ratio": 1.660919540229885, "no_speech_prob": 0.0, "words": [{"start": 1677.1799999999998, "end": 1677.82, "word": "وبالتالي", "probability": 0.93583984375}, {"start": 1677.82, "end": 1678.08, "word": " هذا", "probability": 0.7041015625}, {"start": 1678.08, "end": 1678.58, "word": " بكافي", "probability": 0.7913818359375}, {"start": 1678.58, "end": 1678.7, "word": " ان", "probability": 0.352294921875}, {"start": 1678.7, "end": 1678.92, "word": " احنا", "probability": 0.7976888020833334}, {"start": 1678.92, "end": 1679.24, "word": " نقول", "probability": 0.973876953125}, {"start": 1679.24, "end": 1679.44, "word": " ان", "probability": 0.8759765625}, {"start": 1679.44, "end": 1679.58, "word": " ال", "probability": 0.81787109375}, {"start": 1679.58, "end": 1679.86, "word": " sequence", "probability": 0.60693359375}, {"start": 1679.86, "end": 1680.32, "word": " is", "probability": 0.857421875}, {"start": 1680.32, "end": 1680.56, "word": " not", "probability": 0.95947265625}, {"start": 1680.56, "end": 1681.38, "word": " convergent", "probability": 0.96826171875}, {"start": 1681.38, "end": 1682.18, "word": " if", "probability": 0.91162109375}, {"start": 1682.18, "end": 1682.4, "word": " and", "probability": 0.94873046875}, {"start": 1682.4, "end": 1682.62, "word": " only", "probability": 0.927734375}, {"start": 1682.62, "end": 1682.88, "word": " if", "probability": 0.984375}, {"start": 1682.88, "end": 1683.04, "word": " it", "probability": 0.96826171875}, {"start": 1683.04, "end": 1683.2, "word": " is", "probability": 0.90087890625}, {"start": 1683.2, "end": 1683.46, "word": " not", "probability": 0.958984375}, {"start": 1683.46, "end": 1683.92, "word": " Cauchy", "probability": 0.7906087239583334}, {"start": 1683.92, "end": 1685.62, "word": " عشان", "probability": 0.8701171875}, {"start": 1685.62, "end": 1685.9, "word": " نثبت", "probability": 0.98291015625}, {"start": 1685.9, "end": 1686.02, "word": " ان", 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"probability": 0.7255859375}, {"start": 1692.78, "end": 1693.64, "word": " ..", "probability": 0.7255859375}, {"start": 1693.64, "end": 1696.1, "word": " الاثبات", "probability": 0.8050537109375}, {"start": 1696.1, "end": 1696.52, "word": " انها", "probability": 0.865478515625}, {"start": 1696.52, "end": 1696.82, "word": " not", "probability": 0.93115234375}, {"start": 1696.82, "end": 1697.48, "word": " Cauchy", "probability": 0.939453125}, {"start": 1697.48, "end": 1699.7, "word": " هنستخدم", "probability": 0.958984375}, {"start": 1699.7, "end": 1702.9, "word": " indeed", "probability": 0.72265625}], "temperature": 1.0}, {"id": 61, "seek": 173550, "start": 1709.18, "end": 1735.5, "text": "في حقيقة الأمر لو أخدنا لو كان M أكبر من N فهذا بيقدي ان XM minus XN ايش بيساوي؟ أنا هي عندي ال ..", "tokens": [41185, 11331, 38436, 28671, 16247, 29973, 45164, 5551, 9778, 3215, 8315, 45164, 25961, 376, 5551, 4117, 26890, 9154, 426, 6156, 3224, 15730, 4724, 1829, 4587, 16254, 16472, 1783, 44, 3175, 1783, 45, 1975, 1829, 8592, 4724, 1829, 3794, 995, 45865, 22807, 41850, 39896, 18871, 16254, 2423, 4386], "avg_logprob": -0.30631510230402154, "compression_ratio": 1.2698412698412698, "no_speech_prob": 0.0, "words": [{"start": 1709.18, "end": 1709.44, "word": "في", "probability": 0.7626953125}, {"start": 1709.44, "end": 1710.02, "word": " حقيقة", "probability": 0.9912109375}, {"start": 1710.02, "end": 1710.52, "word": " الأمر", "probability": 0.864013671875}, {"start": 1710.52, "end": 1711.96, "word": " لو", "probability": 0.6875}, {"start": 1711.96, "end": 1712.62, "word": " أخدنا", "probability": 0.9046630859375}, {"start": 1712.62, "end": 1714.9, "word": " لو", "probability": 0.56494140625}, {"start": 1714.9, "end": 1715.48, "word": " كان", "probability": 0.9814453125}, {"start": 1715.48, "end": 1716.02, "word": " M", "probability": 0.6064453125}, {"start": 1716.02, "end": 1716.7, "word": " أكبر", "probability": 0.9352213541666666}, {"start": 1716.7, "end": 1717.04, "word": " من", "probability": 0.98193359375}, {"start": 1717.04, "end": 1717.42, "word": " N", "probability": 0.955078125}, {"start": 1717.42, "end": 1721.12, "word": " فهذا", "probability": 0.9597981770833334}, {"start": 1721.12, "end": 1721.86, "word": " بيقدي", "probability": 0.7012939453125}, {"start": 1721.86, "end": 1723.38, "word": " ان", "probability": 0.50341796875}, {"start": 1723.38, "end": 1724.88, "word": " XM", "probability": 0.5374755859375}, {"start": 1724.88, "end": 1728.32, "word": " minus", "probability": 0.25537109375}, {"start": 1728.32, "end": 1729.34, "word": " XN", "probability": 0.77978515625}, {"start": 1729.34, "end": 1729.8, "word": " ايش", "probability": 0.666259765625}, {"start": 1729.8, "end": 1733.66, "word": " بيساوي؟", "probability": 0.7440592447916666}, {"start": 1733.66, "end": 1733.94, "word": " أنا", "probability": 0.45068359375}, {"start": 1733.94, "end": 1734.16, "word": " هي", "probability": 0.5517578125}, {"start": 1734.16, "end": 1734.6, "word": " عندي", "probability": 0.776123046875}, {"start": 1734.6, "end": 1735.2, "word": " ال", "probability": 0.8193359375}, {"start": 1735.2, "end": 1735.5, "word": " ..", "probability": 0.8037109375}], "temperature": 1.0}, {"id": 62, "seek": 177426, "start": 1746.36, "end": 1774.26, "text": "هي عندي xn و لو بدك تكتب xm ف xm هيكون بساوي واحد اول حد زائد نص زائد تلت و هكذا زائد واحد على n", "tokens": [3224, 1829, 18871, 16254, 2031, 77, 4032, 45164, 47525, 4117, 6055, 4117, 2655, 3555, 2031, 76, 6156, 2031, 76, 39896, 30544, 4724, 3794, 995, 45865, 36764, 24401, 1975, 12610, 11331, 3215, 30767, 16373, 3215, 8717, 9381, 30767, 16373, 3215, 6055, 1211, 2655, 4032, 8032, 4117, 15730, 30767, 16373, 3215, 36764, 24401, 15844, 297], "avg_logprob": -0.16001156800323063, "compression_ratio": 1.4770642201834863, "no_speech_prob": 0.0, "words": [{"start": 1746.36, "end": 1746.68, "word": "هي", "probability": 0.792236328125}, {"start": 1746.68, "end": 1746.96, "word": " عندي", "probability": 0.63330078125}, 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0.69677734375}, {"start": 2017.26, "end": 2017.76, "word": " على", "probability": 0.91162109375}, {"start": 2017.76, "end": 2018.1, "word": " M", "probability": 0.9765625}, {"start": 2018.1, "end": 2020.52, "word": " طيب", "probability": 0.9684244791666666}, {"start": 2020.52, "end": 2021.14, "word": " انا", "probability": 0.73974609375}, {"start": 2021.14, "end": 2021.58, "word": " في", "probability": 0.94189453125}, {"start": 2021.58, "end": 2022.26, "word": " التحليل", "probability": 0.9151611328125}, {"start": 2022.26, "end": 2022.66, "word": " هذا", "probability": 0.75146484375}, {"start": 2022.66, "end": 2023.4, "word": " ماخد", "probability": 0.8260091145833334}], "temperature": 1.0}, {"id": 73, "seek": 204273, "start": 2024.49, "end": 2042.73, "text": "ال M أكبر من N يعني هذا الكلام صحيح إذا كان M أكبر من N طيب الآن take M بساوة 2N بالتأكيد 2N أي عدد طبيعي N لأي عدد طبيعي N 2N أكبر من N", "tokens": [6027, 376, 5551, 4117, 26890, 9154, 426, 37495, 22653, 23758, 2423, 28820, 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"end": 2028.05, "word": " هذا", "probability": 0.93505859375}, {"start": 2028.05, "end": 2028.41, "word": " الكلام", "probability": 0.9309895833333334}, {"start": 2028.41, "end": 2028.91, "word": " صحيح", "probability": 0.995361328125}, {"start": 2028.91, "end": 2029.15, "word": " إذا", "probability": 0.868896484375}, {"start": 2029.15, "end": 2029.49, "word": " كان", "probability": 0.99169921875}, {"start": 2029.49, "end": 2029.81, "word": " M", "probability": 0.9755859375}, {"start": 2029.81, "end": 2030.25, "word": " أكبر", "probability": 0.978515625}, {"start": 2030.25, "end": 2030.45, "word": " من", "probability": 0.99560546875}, {"start": 2030.45, "end": 2030.69, "word": " N", "probability": 0.98681640625}, {"start": 2030.69, "end": 2031.57, "word": " طيب", "probability": 0.9798177083333334}, {"start": 2031.57, "end": 2032.05, "word": " الآن", "probability": 0.6522216796875}, {"start": 2032.05, "end": 2032.67, "word": " take", "probability": 0.6640625}, {"start": 2032.67, "end": 2034.67, "word": " M", "probability": 0.93798828125}, {"start": 2034.67, "end": 2035.39, "word": " بساوة", "probability": 0.77421875}, {"start": 2035.39, "end": 2036.05, "word": " 2N", "probability": 0.660888671875}, {"start": 2036.05, "end": 2036.85, "word": " بالتأكيد", "probability": 0.93779296875}, {"start": 2036.85, "end": 2037.57, "word": " 2N", "probability": 0.824951171875}, {"start": 2037.57, "end": 2037.89, "word": " أي", "probability": 0.66748046875}, {"start": 2037.89, "end": 2038.29, "word": " عدد", "probability": 0.9910481770833334}, {"start": 2038.29, "end": 2038.95, "word": " طبيعي", "probability": 0.9801025390625}, {"start": 2038.95, "end": 2039.21, "word": " N", "probability": 0.5830078125}, {"start": 2039.21, "end": 2039.55, "word": " لأي", "probability": 0.9306640625}, {"start": 2039.55, "end": 2039.81, "word": " عدد", "probability": 0.9951171875}, {"start": 2039.81, "end": 2040.23, "word": " طبيعي", "probability": 0.9896240234375}, {"start": 2040.23, "end": 2040.45, "word": " N", "probability": 0.88525390625}, {"start": 2040.45, "end": 2041.09, "word": " 2N", "probability": 0.6943359375}, {"start": 2041.09, "end": 2042.17, "word": " أكبر", "probability": 0.9812825520833334}, {"start": 2042.17, "end": 2042.41, "word": " من", "probability": 0.99462890625}, {"start": 2042.41, "end": 2042.73, "word": " N", "probability": 0.51171875}], "temperature": 1.0}, {"id": 74, "seek": 207089, "start": 2044.69, "end": 2070.89, "text": "إذا لو عوضت عن M بتنين N في المتباين الأخير هذه هيطلع عندي XM أو X to N negative XN بتطلع أكبر من أو يساوي واحد negative two N على اتنين N صح؟", "tokens": [28814, 15730, 45164, 6225, 2407, 11242, 2655, 18871, 376, 39894, 1863, 9957, 426, 8978, 9673, 2655, 3555, 995, 9957, 16247, 9778, 13546, 29538, 8032, 1829, 9566, 1211, 3615, 18871, 16254, 1783, 44, 34051, 1783, 281, 426, 3671, 1783, 45, 39894, 9566, 1211, 3615, 5551, 4117, 26890, 9154, 34051, 7251, 3794, 995, 45865, 36764, 24401, 3671, 732, 426, 15844, 1975, 2655, 1863, 9957, 426, 20328, 5016, 22807], "avg_logprob": -0.24440298685387, "compression_ratio": 1.3987341772151898, "no_speech_prob": 0.0, "words": [{"start": 2044.69, "end": 2045.07, "word": "إذا", "probability": 0.738525390625}, {"start": 2045.07, "end": 2045.33, "word": " لو", "probability": 0.9892578125}, {"start": 2045.33, "end": 2045.83, "word": " عوضت", "probability": 0.97509765625}, {"start": 2045.83, "end": 2046.05, "word": " عن", "probability": 0.98095703125}, {"start": 2046.05, "end": 2046.39, "word": " M", "probability": 0.6123046875}, {"start": 2046.39, "end": 2047.17, "word": " بتنين", "probability": 0.7277018229166666}, {"start": 2047.17, "end": 2047.55, "word": " N", "probability": 0.91552734375}, {"start": 2047.55, "end": 2048.23, "word": " في", "probability": 0.95166015625}, {"start": 2048.23, "end": 2048.89, "word": " المتباين", "probability": 0.9615234375}, {"start": 2048.89, "end": 2049.39, "word": " الأخير", "probability": 0.83349609375}, {"start": 2049.39, "end": 2049.81, "word": " هذه", "probability": 0.260498046875}, {"start": 2049.81, "end": 2050.87, "word": " هيطلع", "probability": 0.809912109375}, {"start": 2050.87, "end": 2051.21, "word": " عندي", "probability": 0.7017822265625}, {"start": 2051.21, "end": 2052.13, "word": " XM", "probability": 0.647705078125}, {"start": 2052.13, "end": 2053.91, "word": " أو", "probability": 0.5673828125}, {"start": 2053.91, "end": 2054.45, "word": " X", "probability": 0.962890625}, {"start": 2054.45, "end": 2055.49, "word": " to", "probability": 0.335205078125}, {"start": 2055.49, "end": 2055.91, "word": " N", "probability": 0.97607421875}, {"start": 2055.91, "end": 2057.65, "word": " negative", "probability": 0.49267578125}, {"start": 2057.65, "end": 2058.61, "word": " XN", "probability": 0.752197265625}, {"start": 2058.61, "end": 2060.05, "word": " بتطلع", "probability": 0.76666259765625}, {"start": 2060.05, "end": 2060.95, "word": " أكبر", "probability": 0.9493815104166666}, {"start": 2060.95, "end": 2061.21, "word": " من", "probability": 0.9580078125}, {"start": 2061.21, "end": 2061.41, "word": " أو", "probability": 0.9609375}, {"start": 2061.41, "end": 2062.13, "word": " يساوي", "probability": 0.9005126953125}, {"start": 2062.13, "end": 2064.63, "word": " واحد", "probability": 0.929443359375}, {"start": 2064.63, "end": 2065.35, "word": " negative", "probability": 0.9609375}, {"start": 2065.35, "end": 2067.29, "word": " two", "probability": 0.1458740234375}, {"start": 2067.29, "end": 2068.29, "word": " N", "probability": 0.88037109375}, {"start": 2068.29, "end": 2068.67, "word": " على", "probability": 0.8369140625}, {"start": 2068.67, "end": 2069.49, "word": " اتنين", "probability": 0.9268798828125}, {"start": 2069.49, "end": 2069.87, "word": " N", "probability": 0.96826171875}, {"start": 2069.87, "end": 2070.89, "word": " صح؟", "probability": 0.9259440104166666}], "temperature": 1.0}, {"id": 75, "seek": 209046, "start": 2072.15, "end": 2090.47, "text": "اللي هو واحد negative one-half بطلع one-half نص الكلام هذا صحيح لكل n ينتمي إلى n تمام؟", "tokens": [6027, 20292, 31439, 36764, 24401, 3671, 472, 12, 25461, 4724, 9566, 1211, 3615, 472, 12, 25461, 8717, 9381, 2423, 28820, 10943, 23758, 20328, 5016, 1829, 5016, 5296, 28820, 297, 7251, 29399, 2304, 1829, 30731, 297, 46811, 10943, 22807], "avg_logprob": -0.2013221169129396, "compression_ratio": 1.1478260869565218, "no_speech_prob": 0.0, "words": [{"start": 2072.15, "end": 2072.53, "word": "اللي", "probability": 0.96435546875}, {"start": 2072.53, "end": 2072.79, "word": " هو", "probability": 0.9892578125}, {"start": 2072.79, "end": 2073.25, "word": " واحد", "probability": 0.968017578125}, {"start": 2073.25, "end": 2073.87, "word": " negative", "probability": 0.86083984375}, {"start": 2073.87, "end": 2074.25, "word": " one", "probability": 0.72705078125}, {"start": 2074.25, "end": 2074.67, "word": "-half", "probability": 0.6064453125}, {"start": 2074.67, "end": 2075.19, "word": " بطلع", "probability": 0.7901611328125}, {"start": 2075.19, "end": 2075.45, "word": " one", "probability": 0.9443359375}, {"start": 2075.45, "end": 2075.75, "word": "-half", "probability": 0.80029296875}, {"start": 2075.75, "end": 2076.17, "word": " نص", "probability": 0.739990234375}, {"start": 2076.17, "end": 2078.83, "word": " الكلام", "probability": 0.8470052083333334}, {"start": 2078.83, "end": 2079.27, "word": " هذا", "probability": 0.84521484375}, {"start": 2079.27, "end": 2080.99, "word": " صحيح", "probability": 0.995361328125}, {"start": 2080.99, "end": 2082.05, "word": " لكل", "probability": 0.9814453125}, {"start": 2082.05, "end": 2082.53, "word": " n", "probability": 0.625}, {"start": 2082.53, "end": 2085.93, "word": " ينتمي", "probability": 0.9805908203125}, {"start": 2085.93, "end": 2086.33, "word": " إلى", "probability": 0.89404296875}, {"start": 2086.33, "end": 2086.91, "word": " n", "probability": 0.403076171875}, {"start": 2086.91, "end": 2090.47, "word": " تمام؟", "probability": 0.76171875}], "temperature": 1.0}, {"id": 76, "seek": 211471, "start": 2092.31, "end": 2114.71, "text": "إذا أنا أصبح في عندي المتباينة x to n negative xn أكبر من أو ساوي one half for all n belong to a now you can easily show", "tokens": [28814, 15730, 41850, 5551, 9381, 49628, 8978, 18871, 16254, 9673, 2655, 3555, 995, 9957, 3660, 2031, 281, 297, 3671, 2031, 77, 5551, 4117, 26890, 9154, 34051, 8608, 995, 45865, 472, 1922, 337, 439, 297, 5784, 281, 257, 586, 291, 393, 3612, 855], "avg_logprob": -0.22765262113061063, "compression_ratio": 1.162962962962963, "no_speech_prob": 0.0, "words": [{"start": 2092.31, "end": 2092.73, "word": "إذا", "probability": 0.7216796875}, {"start": 2092.73, "end": 2093.01, "word": " أنا", "probability": 0.755859375}, {"start": 2093.01, "end": 2093.53, "word": " أصبح", "probability": 0.9195963541666666}, {"start": 2093.53, "end": 2093.75, "word": " في", "probability": 0.90380859375}, {"start": 2093.75, "end": 2094.07, "word": " عندي", "probability": 0.643310546875}, {"start": 2094.07, "end": 2095.09, "word": " المتباينة", "probability": 0.905517578125}, {"start": 2095.09, "end": 2095.79, "word": " x", "probability": 0.54541015625}, {"start": 2095.79, "end": 2096.21, "word": " to", "probability": 0.5927734375}, {"start": 2096.21, "end": 2096.57, "word": " n", "probability": 0.8837890625}, {"start": 2096.57, "end": 2097.07, "word": " negative", "probability": 0.84912109375}, {"start": 2097.07, "end": 2098.91, "word": " xn", "probability": 0.584716796875}, {"start": 2098.91, "end": 2100.19, "word": " أكبر", "probability": 0.9635416666666666}, {"start": 2100.19, "end": 2100.39, "word": " من", "probability": 0.95751953125}, {"start": 2100.39, "end": 2100.59, "word": " أو", "probability": 0.958984375}, {"start": 2100.59, "end": 2101.31, "word": " ساوي", "probability": 0.9412434895833334}, {"start": 2101.31, "end": 2101.79, "word": " one", "probability": 0.78466796875}, {"start": 2101.79, "end": 2102.33, "word": " half", "probability": 0.59375}, {"start": 2102.33, "end": 2102.77, "word": " for", "probability": 0.85205078125}, {"start": 2102.77, "end": 2103.33, "word": " all", "probability": 0.95166015625}, {"start": 2103.33, "end": 2103.77, "word": " n", "probability": 0.94140625}, {"start": 2103.77, "end": 2104.29, "word": " belong", "probability": 0.463134765625}, {"start": 2104.29, "end": 2104.59, "word": " to", "probability": 0.984375}, {"start": 2104.59, "end": 2104.85, "word": " a", "probability": 0.46044921875}, {"start": 2104.85, "end": 2106.75, "word": " now", "probability": 0.634765625}, {"start": 2106.75, "end": 2109.23, "word": " you", "probability": 0.94775390625}, {"start": 2109.23, "end": 2110.75, "word": " can", "probability": 0.96630859375}, {"start": 2110.75, "end": 2113.25, "word": " easily", "probability": 0.91650390625}, {"start": 2113.25, "end": 2114.71, "word": " show", "probability": 0.97216796875}], "temperature": 1.0}, {"id": 77, "seek": 214362, "start": 2119.5, "end": 2143.62, "text": "ممكن بسهولة اثبات انه this implies المتباينة هذه الأخيرة بتقدي this implies that هذا بيقدي ان ال sequence xn is not Cauchy", "tokens": [2304, 43020, 4724, 3794, 3224, 12610, 3660, 1975, 12984, 3555, 9307, 16472, 3224, 341, 18779, 9673, 2655, 3555, 995, 9957, 3660, 29538, 16247, 9778, 48923, 39894, 4587, 16254, 341, 18779, 300, 23758, 4724, 1829, 4587, 16254, 16472, 2423, 8310, 2031, 77, 307, 406, 7544, 625, 88], "avg_logprob": -0.22440159511058888, "compression_ratio": 1.2753623188405796, "no_speech_prob": 0.0, "words": [{"start": 2119.5, "end": 2119.98, "word": "ممكن", "probability": 0.887939453125}, {"start": 2119.98, "end": 2120.58, "word": " بسهولة", "probability": 0.98505859375}, {"start": 2120.58, "end": 2121.1, "word": " اثبات", "probability": 0.8397216796875}, {"start": 2121.1, "end": 2121.68, "word": " انه", "probability": 0.5125732421875}, {"start": 2121.68, "end": 2122.92, "word": " this", "probability": 0.55322265625}, {"start": 2122.92, "end": 2125.86, "word": " implies", "probability": 0.87255859375}, {"start": 2125.86, "end": 2128.58, "word": " المتباينة", "probability": 0.8960774739583334}, {"start": 2128.58, "end": 2128.86, "word": " هذه", "probability": 0.884765625}, {"start": 2128.86, "end": 2129.5, "word": " الأخيرة", "probability": 0.9241536458333334}, {"start": 2129.5, "end": 2131.54, "word": " بتقدي", "probability": 0.6698404947916666}, {"start": 2131.54, "end": 2133.34, "word": " this", "probability": 0.61572265625}, {"start": 2133.34, "end": 2134.2, "word": " implies", "probability": 0.96484375}, {"start": 2134.2, "end": 2137.54, "word": " that", "probability": 0.83203125}, {"start": 2137.54, "end": 2139.12, "word": " هذا", "probability": 0.75048828125}, {"start": 2139.12, "end": 2139.58, "word": " بيقدي", "probability": 0.685302734375}, {"start": 2139.58, "end": 2139.72, "word": " ان", "probability": 0.8447265625}, {"start": 2139.72, "end": 2139.88, "word": " ال", "probability": 0.88134765625}, {"start": 2139.88, "end": 2140.24, "word": " sequence", "probability": 0.94873046875}, {"start": 2140.24, "end": 2141.26, "word": " xn", "probability": 0.5535888671875}, {"start": 2141.26, "end": 2142.62, "word": " is", "probability": 0.9345703125}, {"start": 2142.62, "end": 2143.02, "word": " not", "probability": 0.96923828125}, {"start": 2143.02, "end": 2143.62, "word": " Cauchy", "probability": 0.825927734375}], "temperature": 1.0}, {"id": 78, "seek": 216816, "start": 2146.72, "end": 2168.16, "text": "is not Cauchy as desired كما هو مطلوب تمام؟ من المتباين هذا ممكن نثبت أن ال sequence تبعتنا ليست Cauchy", "tokens": [271, 406, 7544, 625, 88, 382, 14721, 9122, 15042, 31439, 3714, 9566, 1211, 37746, 46811, 10943, 22807, 9154, 9673, 2655, 3555, 995, 9957, 23758, 3714, 43020, 8717, 12984, 3555, 2655, 14739, 2423, 8310, 6055, 3555, 34268, 8315, 32239, 14851, 7544, 625, 88], "avg_logprob": -0.22674417980881625, "compression_ratio": 1.204724409448819, "no_speech_prob": 0.0, "words": [{"start": 2146.72, "end": 2147.08, "word": "is", "probability": 0.8515625}, 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"compression_ratio": 0.7666666666666667, "no_speech_prob": 0.0, "words": [{"start": 2200.18, "end": 2200.64, "word": "Okay", "probability": 0.95703125}, {"start": 2200.64, "end": 2201.2, "word": " تمام", "probability": 0.8603515625}, {"start": 2201.2, "end": 2203.0, "word": " فال", "probability": 0.908447265625}, {"start": 2203.0, "end": 2203.36, "word": " ..", "probability": 0.98095703125}], "temperature": 1.0}, {"id": 81, "seek": 224371, "start": 2215.7, "end": 2243.72, "text": "إذن باقي إثبات إن عشان البرهان يكون كامل بإنه يثبت إن المتباينة الأخيرة هذه بتأدى إن ال sequence بتاعتنا لا تكون شيء فأنا بقول إذا حابين ممكن إنكم تكتبوا البرهان على ورقة خارجية وتعطوني إذا برهانكم بيكون صح بعطيه لكم علامتين يضافوا إلى امتحان النصف الأول", "tokens": [28814, 8848, 1863, 4724, 995, 38436, 11933, 12984, 3555, 9307, 36145, 6225, 8592, 7649, 2423, 26890, 3224, 7649, 7251, 30544, 9122, 10943, 1211, 4724, 28814, 1863, 3224, 7251, 12984, 3555, 2655, 36145, 9673, 2655, 3555, 995, 9957, 3660, 16247, 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1.05, "no_speech_prob": 0.0, "words": [{"start": 2246.11, "end": 2246.53, "word": "Okay", "probability": 0.95703125}, {"start": 2246.53, "end": 2247.81, "word": " تمام", "probability": 0.9326171875}, {"start": 2247.81, "end": 2248.67, "word": " اتفقنا", "probability": 0.764892578125}, {"start": 2248.67, "end": 2249.97, "word": " ولا", "probability": 0.34619140625}, {"start": 2249.97, "end": 2250.33, "word": " اعطيكم", "probability": 0.8451171875}, {"start": 2250.33, "end": 2250.79, "word": " البرهان", "probability": 0.841064453125}, {"start": 2250.79, "end": 2250.85, "word": " و", "probability": 0.82373046875}, {"start": 2250.85, "end": 2251.15, "word": " بلاش", "probability": 0.93359375}, {"start": 2251.15, "end": 2251.65, "word": " خلاص؟", "probability": 0.93505859375}], "temperature": 1.0}, {"id": 83, "seek": 227883, "start": 2253.91, "end": 2278.83, "text": "Okay إذا بنوقف هنا نكتفي بالأمثلة هذه وإن شاء الله المرة الجاية هناخد .. ندخل في الموضوع ال contractive sequences و بعدين نبدأ section جديد إن شاء الله فشكرا لكم و ال .. نكمل إن شاء الله في اللقاء القادم", "tokens": [8297, 11933, 15730, 44945, 30543, 5172, 34105, 8717, 4117, 2655, 41185, 20666, 10721, 2304, 12984, 37977, 29538, 4032, 28814, 1863, 13412, 16606, 21984, 9673, 25720, 25724, 995, 10632, 8032, 1863, 47283, 3215, 4386, 8717, 3215, 9778, 1211, 8978, 9673, 2407, 11242, 45367, 2423, 4364, 488, 22978, 4032, 39182, 9957, 8717, 44510, 10721, 3541, 10874, 16254, 3215, 36145, 13412, 16606, 21984, 6156, 8592, 4117, 23557, 5296, 24793, 4032, 2423, 4386, 8717, 24793, 1211, 36145, 13412, 16606, 21984, 8978, 13672, 4587, 16606, 25062, 18513, 2304], "avg_logprob": -0.17196801145161902, "compression_ratio": 1.6435643564356435, "no_speech_prob": 1.1920928955078125e-07, "words": [{"start": 2253.91, "end": 2254.37, "word": "Okay", "probability": 0.8994140625}, {"start": 2254.37, "end": 2254.71, "word": " إذا", "probability": 0.707275390625}, {"start": 2254.71, "end": 2256.81, "word": " بنوقف", "probability": 0.741943359375}, {"start": 2256.81, "end": 2257.41, "word": " هنا", "probability": 0.99462890625}, {"start": 2257.41, "end": 2259.49, "word": " نكتفي", "probability": 0.8570556640625}, {"start": 2259.49, "end": 2260.97, "word": " بالأمثلة", "probability": 0.89716796875}, {"start": 2260.97, "end": 2261.43, "word": " هذه", "probability": 0.72900390625}, {"start": 2261.43, "end": 2262.81, "word": " وإن", "probability": 0.78662109375}, {"start": 2262.81, "end": 2262.99, "word": " شاء", "probability": 0.9853515625}, {"start": 2262.99, "end": 2263.15, "word": " الله", "probability": 0.95361328125}, {"start": 2263.15, "end": 2263.57, "word": " المرة", "probability": 0.95703125}, {"start": 2263.57, "end": 2263.93, "word": " الجاية", "probability": 0.8972981770833334}, {"start": 2263.93, "end": 2264.73, "word": " هناخد", "probability": 0.8333740234375}, {"start": 2264.73, "end": 2264.87, "word": " ..", "probability": 0.44970703125}, {"start": 2264.87, "end": 2265.35, "word": " ندخل", "probability": 0.9715576171875}, {"start": 2265.35, "end": 2265.47, "word": " في", "probability": 0.8642578125}, {"start": 2265.47, "end": 2265.95, "word": " الموضوع", "probability": 0.90234375}, {"start": 2265.95, "end": 2266.87, "word": " ال", "probability": 0.85009765625}, {"start": 2266.87, "end": 2267.59, "word": " contractive", "probability": 0.6209716796875}, {"start": 2267.59, "end": 2268.19, "word": " sequences", "probability": 0.94140625}, {"start": 2268.19, "end": 2268.41, "word": " و", "probability": 0.77880859375}, {"start": 2268.41, "end": 2268.95, "word": " بعدين", "probability": 0.888427734375}, {"start": 2268.95, "end": 2270.19, "word": " نبدأ", "probability": 0.97998046875}, {"start": 2270.19, "end": 2270.69, "word": " section", "probability": 0.876953125}, {"start": 2270.69, "end": 2271.39, "word": " جديد", "probability": 0.7674967447916666}, {"start": 2271.39, "end": 2271.57, "word": " إن", "probability": 0.84375}, {"start": 2271.57, "end": 2271.69, "word": " شاء", "probability": 0.991455078125}, {"start": 2271.69, "end": 2271.87, "word": " الله", "probability": 0.96142578125}, {"start": 2271.87, "end": 2274.21, "word": " فشكرا", "probability": 0.8748779296875}, {"start": 2274.21, "end": 2274.69, "word": " لكم", "probability": 0.6898193359375}, {"start": 2274.69, "end": 2275.11, "word": " و", "probability": 0.986328125}, {"start": 2275.11, "end": 2275.69, "word": " ال", "probability": 0.40087890625}, {"start": 2275.69, "end": 2276.71, "word": " ..", "probability": 0.96630859375}, {"start": 2276.71, "end": 2277.33, "word": " نكمل", "probability": 0.9427083333333334}, {"start": 2277.33, "end": 2277.49, "word": " إن", "probability": 0.96240234375}, {"start": 2277.49, "end": 2277.65, "word": " شاء", "probability": 0.994140625}, {"start": 2277.65, "end": 2277.79, "word": " الله", "probability": 0.96533203125}, {"start": 2277.79, "end": 2278.01, "word": " في", "probability": 0.95166015625}, {"start": 2278.01, "end": 2278.37, "word": " اللقاء", "probability": 0.7845052083333334}, {"start": 2278.37, "end": 2278.83, "word": " القادم", "probability": 0.9905598958333334}], "temperature": 1.0}], "language": "ar", "language_probability": 1.0, "duration": 2280.1995, "duration_after_vad": 2050.9693749999924}
\ No newline at end of file
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/aQ184E7DSME.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/aQ184E7DSME.srt
new file mode 100644
index 0000000000000000000000000000000000000000..b068286fa2090f129eba86b3b2bba74e6bc3d29c
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/aQ184E7DSME.srt
@@ -0,0 +1,1883 @@
+1
+00:00:20,960 --> 00:00:27,560
+Okay إذا هنواصل إن شاء الله اللي بدنا فيه المحاضرة
+
+2
+00:00:27,560 --> 00:00:33,340
+السابقة المرة اللي فاتت خلينا بسرعة بس هيك نمر على
+
+3
+00:00:33,340 --> 00:00:40,220
+الحاجات اللي أخذناها أخذنا اللي هو ال algebraic
+
+4
+00:00:40,220 --> 00:00:43,060
+properties of the real number system اللي هو
+
+5
+00:00:43,060 --> 00:00:48,970
+الخواص الجبرية و عرفنا اللي هو ال real number
+
+6
+00:00:48,970 --> 00:00:52,930
+system فقلنا إن ال real number system عبارة عن
+
+7
+00:00:52,930 --> 00:00:58,430
+المجموعة R مع عمليتين جبريتين أو ثنائيتين عملية
+
+8
+00:00:58,430 --> 00:01:03,110
+جمع و عملية ضرب و هذول العمليات بتحققوا خمس خواص
+
+9
+00:01:03,110 --> 00:01:08,290
+خاصية الإبدال commutative law خاصية الدمج ال
+
+10
+00:01:08,290 --> 00:01:14,180
+associative laws خواص التوزيع distributive laws
+
+11
+00:01:14,180 --> 00:01:19,120
+خاصية الرابعة وجود ال identity elements العناصر
+
+12
+00:01:19,120 --> 00:01:25,280
+المحايدة اللي هي 0 و 1 ووجود ال inverse
+
+13
+00:01:25,280 --> 00:01:38,580
+elements أو العناصر النظائر أو المعكسات فلكل عدد 
+
+14
+00:01:38,580 --> 00:01:48,850
+حقيقي فيه له معكوس جمعي اللي هو سالب x  و لكل عدد
+
+15
+00:01:48,850 --> 00:01:55,330
+حقيقي غير مختلف عن الصفر له نظير ضربي أو
+
+16
+00:01:55,330 --> 00:01:59,030
+multiplicative inverse يُرمز له بالرمز x<sup>-1</sup>
+
+17
+00:01:59,030 --> 00:02:03,050
+negative واحد بحيث لو ضربتهم في بعض بيعطوني ال
+
+18
+00:02:03,050 --> 00:02:10,890
+identity element واحد هذول الخمس خواص اللي بتحققهم 
+
+19
+00:02:10,890 --> 00:02:15,760
+مجموعة الأعداد الحقيقية مع العملياتين ضرب وجمع
+
+20
+00:02:15,760 --> 00:02:21,600
+اللي عرفناها سابقاً في أول خواص أخذناها اللي هي
+
+21
+00:02:21,600 --> 00:02:27,680
+cancellation laws موجودة في نظرية 1.1 فعملية 
+
+22
+00:02:27,680 --> 00:02:33,220
+الجمع بتحقق cancellation law يعني أنا لو كان عندي
+
+23
+00:02:33,220 --> 00:02:38,940
+x plus z بساوي y plus z فممكن أشطب z من الطرفين
+
+24
+00:02:38,940 --> 00:02:44,680
+بيطلع عندي x بساوي y كذلك عملية الضرب بتحقق ال
+
+25
+00:02:44,680 --> 00:02:51,000
+cancellation law فلو في عندي حاصل ضرب زي هذا بساوي
+
+26
+00:02:51,000 --> 00:02:56,140
+حاصل الضرب هذا و ال answer w هذا العدد w ما بيساوي صفر
+
+27
+00:02:56,140 --> 00:03:01,470
+فمقدر أقسم الطرفين على w  بيطلع عندي x بساوي y هذه
+
+28
+00:03:01,470 --> 00:03:05,430
+الخواص مهمة و برهنت لكم المرة اللي فاتت الجزء 
+
+29
+00:03:05,430 --> 00:03:11,810
+الثاني و برهان الجزء الأول مشابه و بالتالي قلنا 
+
+30
+00:03:11,810 --> 00:03:17,650
+لكم حاولوا تثبتوا بنفس الطريقة بالمثل كمان أخذنا
+
+31
+00:03:17,650 --> 00:03:22,810
+نظرية ثانية اللي هي النظرية هذه ذكرناها المرة اللي
+
+32
+00:03:22,810 --> 00:03:30,050
+فاتت فيها حوالي عشر خواص للأعداد الحقيقية فأول
+
+33
+00:03:30,050 --> 00:03:35,350
+خاصية لو ضربت أي عدد حقيقي بالصفر سواء من اليمين
+
+34
+00:03:35,350 --> 00:03:39,370
+أو اليسار فالنتيجة العدد الصفري اللي هو ال additive
+
+35
+00:03:39,370 --> 00:03:47,090
+identity صفر فبرهان هذا الجزء هو برهان الجزء الأول 
+
+36
+00:03:47,090 --> 00:03:55,210
+يعني باين واحد هنا فكيف يتم البرهان أنا عايز أثبت 
+
+37
+00:03:55,210 --> 00:04:02,370
+أن x ضرب صفر بساوي صفر طيب أنا عندي لو جمعت x ضارب 
+
+38
+00:04:02,370 --> 00:04:07,470
+0 زائد x ضارب 0 بقدر باستخدام ال distributive law
+
+39
+00:04:07,470 --> 00:04:13,430
+أخذ x عامل مشترك كأني بضرب x في 0 زائد 0 هذا صحيح 
+
+40
+00:04:13,430 --> 00:04:17,550
+باستخدام ال distributive law الآن لما أجمع الصفر
+
+41
+00:04:17,550 --> 00:04:22,450
+على أي عدد حقيقي حتى لو نفسه الناتج بيطلع 0 هذا من 
+
+42
+00:04:22,450 --> 00:04:33,170
+خواص المحايد الجمعي و ال X ضرب صفر هو نفسه لو جمعت 
+
+43
+00:04:33,170 --> 00:04:37,330
+على العدد هذا صفر فيبقى زي ما هو من خواص الصفر
+
+44
+00:04:37,330 --> 00:04:43,950
+الآن أنا ممكن أشطب باستخدام cancellation law ممكن
+
+45
+00:04:43,950 --> 00:04:49,590
+أشطب هذا و أشطب هذا فبيطلع عندي X ضرب صفر بساوي صفر 
+
+46
+00:04:49,590 --> 00:04:55,490
+okay تمام الخاصية الثانية الخاصية الثانية عايزين
+
+47
+00:04:55,490 --> 00:05:01,230
+نثبت أنه لو أخذت أي عدد حقيقي و أخذت سالبه مرتين 
+
+48
+00:05:01,230 --> 00:05:06,530
+فهذا هو نفس ال X البرهان برضه بتم كالتالي هاي ال X
+
+49
+00:05:06,530 --> 00:05:12,370
+و بجمع عليه negative X اللي هو المعكوس الجمعي
+
+50
+00:05:12,370 --> 00:05:20,390
+طبعاً فهذا بساوي صفر هذا بساوي صفر من خواص المعكوس
+
+51
+00:05:20,390 --> 00:05:26,570
+الجمعي و ممكن أن احنا نبدل هذول مع بعض أو لأ، الآن
+
+52
+00:05:26,570 --> 00:05:31,610
+برضه لو أخذت هذا، هذا عدد حقيقي، و هذا المعكوس 
+
+53
+00:05:31,610 --> 00:05:36,970
+الجمعي تبعه، عدد حقيقي و المعكوس الجمعي تبعه دائماً
+
+54
+00:05:36,970 --> 00:05:41,830
+بساوي صفر، الآن ممكن نبدل عملية الجمع إبدالية،
+
+55
+00:05:41,830 --> 00:05:46,970
+commutative، فنبدل الحاجات هذه مع بعض الآن عملية 
+
+56
+00:05:46,970 --> 00:05:50,970
+الجمع بتحقق قانون الحدث cancellation law إذا
+
+57
+00:05:50,970 --> 00:05:55,390
+ممكن أشطب أنا العنصر هذا مع هذا بيبقى عندي على 
+
+58
+00:05:55,390 --> 00:05:59,810
+الشمال X وعلى اليمين بيبقى negative negative X
+
+59
+00:05:59,810 --> 00:06:06,070
+فبالتالي هي كمان أثبتنا صحة الخاصية هذه تمام؟
+
+60
+00:06:07,410 --> 00:06:11,250
+الخاصية الثالثة في النظرية اللي شوفناها قبل شوية
+
+61
+00:06:11,250 --> 00:06:17,770
+برهانها مشابه لخاصية الثانية وبالتالي هأسيبه تمرين، 
+
+62
+00:06:17,770 --> 00:06:21,770
+إذا بعض الحاجات اللي برهانها مشابه دائماً هنسيبها
+
+63
+00:06:21,770 --> 00:06:26,030
+كتمرين للطالب لأن الفكرة نفسها و رياضيات مجرد أفكار
+
+64
+00:06:26,030 --> 00:06:33,620
+فإذا عرفنا الفكرة انتهى الحل اللغز الخاصية الرابعة،
+
+65
+00:06:33,620 --> 00:06:37,120
+في الخاصية الرابعة عايزين نثبت أنه لو ضربت سالب
+
+66
+00:06:37,120 --> 00:06:42,880
+واحد في x بيطلع عندي ال negative x أو المحايد
+
+67
+00:06:42,880 --> 00:06:48,640
+الجمعي ل x فلبرهان ذلك بأخذ negative واحد في x و
+
+68
+00:06:48,640 --> 00:06:53,820
+بجمعها على x فهذا هو نفسه هي negative واحد في x و
+
+69
+00:06:53,820 --> 00:06:59,970
+ال x هذه عبارة عن واحد في x الآن ممكن هنا أستخدم
+
+70
+00:06:59,970 --> 00:07:03,070
+ال distributive law عملية الضرب تتوزع على عملية 
+
+71
+00:07:03,070 --> 00:07:07,670
+الجمع فممكن أكتب هذا سالب واحد زائد واحد مضروب من
+
+72
+00:07:07,670 --> 00:07:13,370
+اليمين في X و عملية الضرب إبدالية فهي نفس كما لو 
+
+73
+00:07:13,370 --> 00:07:19,040
+ضربت X من اليمين في سالب واحد زائد واحد Okay تمام 
+
+74
+00:07:19,040 --> 00:07:24,700
+الآن لما أجمع سالب واحد هذا سالب واحد على واحد
+
+75
+00:07:24,700 --> 00:07:31,440
+عنصر و نظير الجمع تبعهم مجموعهم صفر و صفر في أي
+
+76
+00:07:31,440 --> 00:07:37,500
+عنصر أثبتنا أنه بيساوي صفر و الصفر هو نفسه سالب x
+
+77
+00:07:37,500 --> 00:07:42,450
+زائد x وبالتالي بقدر أستخدم ال cancellation اللي هو
+
+78
+00:07:42,450 --> 00:07:47,250
+عامل cancelling ل X على الطرف الشمال و cancelling
+
+79
+00:07:47,250 --> 00:07:50,990
+ل X على الطرف اليمين يبقى في الشمال سالب واحد في X
+
+80
+00:07:50,990 --> 00:07:55,850
+في اليمين سالب X وبالتالي هيك ممكن أثبتنا الخاصية
+
+81
+00:07:55,850 --> 00:08:01,850
+هذه واضحة؟ أي حد عنده أي استفسار؟ خواص سهلة و براهين 
+
+82
+00:08:01,850 --> 00:08:06,150
+سهلة جداً أنتم دارسين مبادئ و اللي دارسين مبادئ 
+
+83
+00:08:06,150 --> 00:08:11,530
+رياضيات و فاهمينها كويس هذه حاجات يعني براهين أمثلة 
+
+84
+00:08:11,530 --> 00:08:16,170
+كلها على برهان مباشر بسيط باستخدام خواص ذكرناها
+
+85
+00:08:16,170 --> 00:08:22,990
+سابقاً فهذا مجرد يعني مبادئ رياضيات الخاصية الخامسة
+
+86
+00:08:22,990 --> 00:08:29,590
+احنا عايزين نثبت لو ضربت x في negative y تطلع ..
+
+87
+00:08:29,590 --> 00:08:34,960
+هي نفسها كم لو ضربت negative x في y و هذه يعني
+
+88
+00:08:34,960 --> 00:08:39,800
+البرهان مش صعب هاي ال X وهي negative Y الخاصية 
+
+89
+00:08:39,800 --> 00:08:43,560
+الرابعة أثبتنا فيها إنه negative Y بيساوي negative
+
+90
+00:08:43,560 --> 00:08:48,780
+واحد في Y هذا من الخاصية أربعة ال associative law
+
+91
+00:08:48,780 --> 00:08:52,120
+بيسمح لإن أنا الأقواس هذه أرتبها بالطريقة هذه
+
+92
+00:08:52,120 --> 00:08:56,180
+عملية الضرب associative و كمان عملية الضرب
+
+93
+00:08:56,180 --> 00:09:00,620
+commutative إذا ممكن أبدل ال X مع ال negative واحد
+
+94
+00:09:04,010 --> 00:09:07,470
+الخاصية الرابعة بتقول سالب واحد ضرب X عبارة عن
+
+95
+00:09:07,470 --> 00:09:14,030
+negative X إذاً هذا هو إيه هذا جزء من الجزء الخامس
+
+96
+00:09:14,030 --> 00:09:20,750
+في كمان جزء ثاني اللي هو عايزين نثبت إن X ضرب
+
+97
+00:09:20,750 --> 00:09:24,990
+negative Y هو نفس الحاجة كما لو ضربت X في Y الأول
+
+98
+00:09:24,990 --> 00:09:30,950
+وضربت الكل في negative و لبرهان ذلك هي X في
+
+99
+00:09:30,950 --> 00:09:34,950
+negative Y X في negative Y أثبتنا إنها طلعت بتساوي
+
+100
+00:09:34,950 --> 00:09:41,090
+سالب واحد في X في Y هذا هو نزلناها من هنا الآن 
+
+101
+00:09:41,090 --> 00:09:44,470
+باستخدام ال associative law ممكن إيه أبدل الأقواس 
+
+102
+00:09:44,470 --> 00:09:50,450
+يعني أخد X و Y مع بعض و أضربهم في سالب واحد وطبعاً
+
+103
+00:09:50,450 --> 00:09:54,490
+الآن أثبتنا أن سالب واحد لما أضربه في حاجة زي هذه
+
+104
+00:09:54,490 --> 00:09:56,910
+بيطلع سالب X في Y
+
+105
+00:10:01,430 --> 00:10:08,850
+بالنسبة لخاصية رقم 6 ممكن نستخدم الخاصية رقم 4 و
+
+106
+00:10:08,850 --> 00:10:12,510
+ال distributive law في إثباتها فطبعاً أنا هنا كاتب
+
+107
+00:10:12,510 --> 00:10:18,030
+لكم use الخاصية أو الجزء الرابع و ال distributive
+
+108
+00:10:18,030 --> 00:10:23,210
+law لبرهان مين؟ الخاصية رقم 6 هذه، هذه أرقام 
+
+109
+00:10:23,210 --> 00:10:29,920
+لاتينية فبتشوفوها أنتم طبعاً و بتحاولوا تثبتوها و
+
+110
+00:10:29,920 --> 00:10:35,660
+إذا ما عرفتوهاش ممكن تتواصلوا معايا نحاول نثبتها لكم
+
+111
+00:10:35,660 --> 00:10:42,780
+بالنسبة لخاصية السابعة إيه هي الخاصية السابعة لو
+
+112
+00:10:42,780 --> 00:10:46,700
+ضربت negative x في negative y المفروض يطلع نفس
+
+113
+00:10:46,700 --> 00:10:54,530
+الحاجة x ضرب y و هي البرهان بسيط احنا أخذنا أن لو 
+
+114
+00:10:54,530 --> 00:10:59,830
+ضربت negative x في عنصر negative y هو نفسه كما لو
+
+115
+00:10:59,830 --> 00:11:07,130
+أنا ضربت x في العنصر الثاني و أخذت السالب برا و 
+
+116
+00:11:07,130 --> 00:11:07,750
+بعدين
+
+117
+00:11:10,830 --> 00:11:17,230
+نفس الحاجة هنا x في negative y هي
+
+118
+00:11:17,230 --> 00:11:24,890
+نفسها negative x في y بدل x في negative y ب 
+
+119
+00:11:24,890 --> 00:11:30,290
+negative ضرب x y و أثبتنا قبل هيك أن negative ضرب
+
+120
+00:11:30,290 --> 00:11:34,930
+negative العنصر بساوي العنصر إذا هذا برضه
+
+121
+00:11:34,930 --> 00:11:45,630
+برهان هذا الجزء الجزء الثامن أو التاسع are left as
+
+122
+00:11:45,630 --> 00:11:48,410
+exercises برضه أنا سايب لكم إياهم تمرين لأن مش
+
+123
+00:11:48,410 --> 00:11:52,310
+ما جون نبرهنه كل شيء أنتم يعني كبار لأن بدأتوا
+
+124
+00:11:52,310 --> 00:11:58,050
+تُساوبوُا بدأتوا تفهموا فلازم برضه تشاركوا شوية مش
+
+125
+00:11:58,050 --> 00:12:02,030
+معقول زي اللي .. احنا مش في مدرسة ثانوية، بنأشي
+
+126
+00:12:02,030 --> 00:12:06,310
+نقوُل، ابتدائية، لازم يشرح لكم كل شيء و لازم يبرهن لكم 
+
+127
+00:12:06,310 --> 00:12:10,050
+كل شيء، لازم الطالب يشارك شوية، خاصة الحاجات اللي
+
+128
+00:12:10,050 --> 00:12:14,290
+براهينها مشابهة فبتزعلوش و حالكم أنتم تاخدوا
+
+129
+00:12:14,290 --> 00:12:19,430
+الأمور هذه بصدر رحب و كمان مرة بكرر لو أي شيء من
+
+130
+00:12:19,430 --> 00:12:23,570
+الحاجات اللي بنسيبها ما عرفتوهاش تحلوها أو تبرهنوها
+
+131
+00:12:23,570 --> 00:12:28,350
+فأنا على استعداد أن أساعدكم في برهانها الجزء الآخر
+
+132
+00:12:28,350 --> 00:12:32,490
+جزء العاشر إيه هو الجزء العاشر؟ الجزء العاشر بيقول
+
+133
+00:12:32,490 --> 00:12:40,530
+دائماً بندرسها في مبادئ الرياضيات مثال على برهان غير
+
+134
+00:12:40,530 --> 00:12:44,490
+مباشر في مبادئ رياضيات، لو كان X ضرب Y أعداد
+
+135
+00:12:44,490 --> 00:12:50,990
+حقيقية، حاصل ضربهم صفر، فإيه بيطلع؟ إما X بساوي صفر
+
+136
+00:12:50,990 --> 00:12:55,650
+أو Y بساوي صفر، بظبط؟
+
+137
+00:13:01,000 --> 00:13:05,060
+فهذا نعطي مثال في المبادئ الرياضية طيب البرهان هو 
+
+138
+00:13:05,060 --> 00:13:12,880
+نفسه البرهان هو نفسه عشان أثبت إنه x لو كان x ضرب
+
+139
+00:13:12,880 --> 00:13:19,760
+y بساوي 0 فبيطلع x بساوي 0 أو y بساوي 0 فبأفرض أن x
+
+140
+00:13:19,760 --> 00:13:26,220
+ما يساويش 0 و بأثبت أن y بيساوي 0 أو by symmetry 
+
+141
+00:13:26,220 --> 00:13:32,080
+بالتماثل ممكن أفرض أن y بيساوي 0 وأصل إلى أن x 
+
+142
+00:13:32,080 --> 00:13:41,640
+بيساوي 0 okay تمام فاللي عملناه هنا هي لبرهان أن
+
+143
+00:13:41,640 --> 00:13:47,400
+x بيساوي 0 أو y بيساوي 0 it suffices يعني يكفي أن
+
+144
+00:13:47,400 --> 00:13:54,070
+أفرض to assume أن x لا يساوي 0 وأثبت وأثبت أن y
+
+145
+00:13:54,070 --> 00:14:07,690
+بيساوي 0 طيب أنا عندي من الفرض x y بيساوي 0 ف
+
+146
+00:14:07,690 --> 00:14:14,610
+ال x y بيساوي 0  تُهيّئ لي
+
+147
+00:14:14,610 --> 00:14:21,110
+فيه شيء هنا مش مبين ده هو خلينا نبينه نعم هاي أنا
+
+148
+00:14:21,110 --> 00:14:26,470
+عندي x y من الفرض x y بيساوي صفر والصفر هذا ممكن
+
+149
+00:14:26,470 --> 00:14:30,970
+أكتبه على صورة x ضرب صفر برضه هذا بيساوي صفر تمام
+
+150
+00:14:30,970 --> 00:14:36,470
+الآن أنا أفرض أن x لا يساوي صفر فعملية الضرب بتحقق
+
+151
+00:14:36,470 --> 00:14:40,210
+cancellation law إذا من cancellation law تبع عملية 
+
+152
+00:14:40,210 --> 00:14:44,470
+الضرب مدام x لا يساوي صفر فبقدر أجزم عليها فبيطلع
+
+153
+00:14:44,470 --> 00:14:50,590
+عندي y بيساوي صفر وهذا هو المطلوب Okay تمام إذا هيك
+
+154
+00:14:50,590 --> 00:15:01,290
+بنكون برهنا النظرية الثانية كويس
+
+155
+00:15:01,290 --> 00:15:07,770
+تمام هيك طيب
+
+156
+00:15:07,770 --> 00:15:11,690
+نأخذ تعريفات أو تعريف مهم
+
+157
+00:15:21,020 --> 00:15:27,480
+في تعريف هنا الخواص
+
+158
+00:15:27,480 --> 00:15:30,660
+الخامسة هذه اللي حكينا عنها ال commutative law ال
+
+159
+00:15:30,660 --> 00:15:37,140
+associative law وال distributive law existence of
+
+160
+00:15:37,140 --> 00:15:42,780
+identity elements الخاصية الخامسة existence of
+
+161
+00:15:42,780 --> 00:15:47,240
+inverses خمس خواص هذه اللي بتحققها عمليات الجمع و
+
+162
+00:15:47,240 --> 00:15:52,770
+الضرب على الأعداد الحقيقية هذه الخواص بتشكل تعريف
+
+163
+00:15:52,770 --> 00:15:57,750
+ما يسمى في الجبر في الجبر الحديث في تركيبة جبرية
+
+164
+00:15:57,750 --> 00:16:03,870
+اسمها field أو حقل فما
+
+165
+00:16:03,870 --> 00:16:09,130
+هو الحقل لو أنتم هتدرسوا جبر حديث واحد أو .. أو
+
+166
+00:16:09,130 --> 00:16:14,850
+درستموه فيمكن مر عليكم ال field أو الحقل هو عبارة
+
+167
+00:16:14,850 --> 00:16:23,080
+عن set مع عمليتين ثنائيتين عملية جمع وعملية ضرب
+
+168
+00:16:23,080 --> 00:16:29,300
+معرفين على ال set F بحيث أن العمليتين هدول بيحققوا
+
+169
+00:16:29,300 --> 00:16:33,640
+الخواص الخمسة اللي هي حققتها مجموعة الأعداد
+
+170
+00:16:33,640 --> 00:16:40,250
+الحقيقية، إذا أي مجموعة F مع عمليتين ثنائيتين بتحقق
+
+171
+00:16:40,250 --> 00:16:46,630
+الخواص الخمسة بنسميها في الجبر field إذا ال .. ال
+
+172
+00:16:46,630 --> 00:16:51,510
+.. ال R ال R أو الأعداد الحقيقية مع عمليات الجمع
+
+173
+00:16:51,510 --> 00:16:56,650
+والضرب اللي عرفناها سابقا شفنا أنها بتحقق الخواص
+
+174
+00:16:56,650 --> 00:17:01,570
+الخمسة وبالتالي بتشكل field فبنسميها it the field of
+
+175
+00:17:01,570 --> 00:17:08,790
+real numbers أو المجال الأعداد الحقيقية هي
+
+176
+00:17:08,790 --> 00:17:14,430
+مثال فيه أمثلة كثيرة على fields على حقول فهي لو
+
+177
+00:17:14,430 --> 00:17:18,030
+أخذت المجموعة هي أبسط field أصغر وأبسط field
+
+178
+00:17:18,030 --> 00:17:24,190
+موجود في الرياضيات هو ال 6F اللي بتتكون من عنصرين
+
+179
+00:17:24,190 --> 00:17:33,050
+عددين حقيقيين 0, 1 الآن عشان أكون field على المجموعة
+
+180
+00:17:33,050 --> 00:17:37,590
+F اللي بتكون من عنصرين لازم أعرف عملية جمع وضرب
+
+181
+00:17:37,590 --> 00:17:43,690
+فممكن أعرف عملية جمع وضرب على F كالتالي ها يعني
+
+182
+00:17:43,690 --> 00:17:50,570
+عرفت لو ضربت 0 في نفسه أو 1 في 0 أو 0 في 1 أو
+
+183
+00:17:50,570 --> 00:17:57,780
+جمعت 0 على 0 أو 1 على 1 هذا بأعرفه أنه بيساوي 0 إذا
+
+184
+00:17:57,780 --> 00:18:02,020
+أنا عرفت حاصل الضرب والجمع هذا بيساوي صفر كذلك
+
+185
+00:18:02,020 --> 00:18:05,800
+أنا بأعرف أنه لو جمعت الصفر على الواحد أو الواحد
+
+186
+00:18:05,800 --> 00:18:10,000
+على الصفر أو الواحد على الواحد بيطلع واحد الآن إذا
+
+187
+00:18:10,000 --> 00:18:16,410
+أنا عرفت عمليات جمع وضرب على كل عناصر المجموعة الآن
+
+188
+00:18:16,410 --> 00:18:23,610
+من السهل التحقق أن ال خواص الخمسة كلها بتتحقق تابعة
+
+189
+00:18:23,610 --> 00:18:28,530
+ال field وبالتالي هذا بيكون field وهذا ال field في
+
+190
+00:18:28,530 --> 00:18:36,770
+الجبر برمز له بالرمز Z2 وفي
+
+191
+00:18:36,770 --> 00:18:40,890
+نفس الوقت هو cyclic group of order two إذا في حد
+
+192
+00:18:40,890 --> 00:18:45,120
+فيكم درس الجبر الحديث على أي حال احنا هذا مش موضوعنا
+
+193
+00:18:45,120 --> 00:18:49,400
+هذا موضوع جبر فبس يعني هذا مجرد مثال بسيط على
+
+194
+00:18:49,400 --> 00:18:53,600
+field واحنا أهم حاجة أنه احنا يعني اللي بدنا احنا
+
+195
+00:18:53,600 --> 00:18:58,420
+نصله أنه مجموعة الأعداد الحقيقية تبعتنا مجموعة
+
+196
+00:18:58,420 --> 00:19:04,980
+الأعداد الحقيقية R مع عملية الجمع والضرب بتشكل
+
+197
+00:19:04,980 --> 00:19:10,360
+ما يسمى في الجبر بال field تمام؟ هذا اللي احنا
+
+198
+00:19:10,360 --> 00:19:11,980
+عايزين نصله، نعم فضل
+
+199
+00:19:15,080 --> 00:19:18,860
+هذا تعريف احنا .. احنا .. احنا بنعرف أنه لو جمع
+
+200
+00:19:18,860 --> 00:19:23,700
+واحد على واحد يطلع صفر هذا definition تعريف وليس
+
+201
+00:19:23,700 --> 00:19:27,320
+آه يعني هاي مجموعة آه بدي أعرف عليها عملية جمع
+
+202
+00:19:27,320 --> 00:19:31,060
+عملية .. عملية الجمع هانا بأعرفها لو جمعت واحد على
+
+203
+00:19:31,060 --> 00:19:39,480
+واحد بيطلع صفر لو جمعت واحد آه
+
+204
+00:19:43,240 --> 00:19:52,920
+آه في هنا شيء مش مظبوط هنا هذه المفروض ضرب هذه هذه
+
+205
+00:19:52,920 --> 00:19:58,760
+المفروض ضرب هذه فهذا في خطأ مطبعي صحيح هذا كله
+
+206
+00:19:58,760 --> 00:20:02,200
+تعريف الآن لما هيك أنا بكون عرفت عملية الجمع
+
+207
+00:20:02,200 --> 00:20:07,240
+والضرب هنا في خطأ مطبعي دي المفروض تكون ضرب لأن
+
+208
+00:20:07,240 --> 00:20:10,240
+واحد زائد واحد عرفناها تساوي صفر ف
+
+209
+00:20:13,680 --> 00:20:19,300
+حسب التعريف هذا ممكن التحقق أن خمس خواص تبعت ال
+
+210
+00:20:19,300 --> 00:20:22,760
+field بتتحقق فممكن تحققوها إذا بدكم تحققوها إذا
+
+211
+00:20:22,760 --> 00:20:25,700
+بدكم تحققوها إذا بدكم تحققوها إذا بدكم تحققوها
+
+212
+00:20:25,700 --> 00:20:27,620
+إذا بدكم تحققوها إذا بدكم تحققوها إذا بدكم
+
+213
+00:20:27,620 --> 00:20:27,740
+تحققوها إذا بدكم تحققوها إذا بدكم تحققوها إذا
+
+214
+00:20:27,740 --> 00:20:28,060
+بدكم تحققوها إذا بدكم تحققوها إذا بدكم تحققوها
+
+215
+00:20:28,060 --> 00:20:28,700
+إذا بدكم تحققوها إذا بدكم تحققوها إذا بدكم
+
+216
+00:20:28,700 --> 00:20:36,280
+تحققوها إذا بدكم تحققوها إذا بدكم تحققوها إذا
+
+217
+00:20:36,280 --> 00:20:43,660
+بدكم تحققوها إذا بدكم توعملية القسمة هي عملية
+
+218
+00:20:43,660 --> 00:20:47,340
+ضرب، ضرب، ضرب، ضرب، ضرب، ضرب، ضرب، ضرب، ضرب، ضرب،
+
+219
+00:20:47,340 --> 00:20:49,400
+ضرب، ضرب، ضرب، ضرب، ضرب، ضرب، ضرب، ضرب، ضرب، ضرب،
+
+220
+00:20:49,400 --> 00:20:53,020
+ضرب، ضرب، ضرب، ضرب، ضرب، ضرب، ضرب، ضرب، ضرب، ضرب،
+
+221
+00:20:53,020 --> 00:20:54,300
+ضرب، ضرب، ضرب، ضرب، ضرب، ضرب، ضرب، ضرب، ضرب، ضرب،
+
+222
+00:20:54,300 --> 00:20:56,080
+ضرب، ضرب، ضرب، ضرب، ضرب، ضرب، ضرب، ضرب، ضرب، ضرب،
+
+223
+00:20:56,080 --> 00:20:56,480
+ضرب، ضرب، ضرب، ضرب، ضرب، ضرب، ضرب، ضرب، ضرب، ضرب،
+
+224
+00:20:56,480 --> 00:21:01,560
+ضرب، ضرب، ضرب، ضرب، ضرب، ضرب، ضرب، ضرب، ضرب، ضرب،
+
+225
+00:21:01,560 --> 00:21:06,040
+ضرب، إذن عملية الطرح هي عملية جمع كذلك عملية
+
+226
+00:21:06,040 --> 00:21:10,320
+القسمة division on R is defined by  هاي X على Y
+
+227
+00:21:10,320 --> 00:21:16,740
+بيساوي عملية ضرب X في ال multiplicative inverse ل Y
+
+228
+00:21:16,740 --> 00:21:22,320
+أو 1 على Y كذلك بنعرف هذا definition دائما لما
+
+229
+00:21:22,320 --> 00:21:25,660
+يكون فيه نقطتين ووراهم علامة تساوي معناه إن طرف
+
+230
+00:21:25,660 --> 00:21:30,040
+الشمال by definition بيساوي الطرف اليمين إذن هذا
+
+231
+00:21:30,040 --> 00:21:36,630
+تعريف لأن أنا بأعرف هذا التعريف أن a أس negative one
+
+232
+00:21:36,630 --> 00:21:43,470
+معناها واحد على a a to zero بيساوي واحد حسب التعريف
+
+233
+00:21:43,470 --> 00:21:48,970
+a to negative n و n عدد طبيعي هي عبارة عن مقلوب ال
+
+234
+00:21:48,970 --> 00:21:54,630
+a الكل to n فكل هذه تعريفات بناء على التعريفات هذه
+
+235
+00:21:54,630 --> 00:21:58,730
+ممكن أن نبرهن خواص كثيرة
+
+236
+00:22:00,150 --> 00:22:03,810
+وهتشوفوا بعضها في التمرين اللي موجودة في نهاية ال
+
+237
+00:22:03,810 --> 00:22:09,970
+section نتطرق
+
+238
+00:22:09,970 --> 00:22:14,330
+لحاجة اسمها rational numbers الأعداد النسبية
+
+239
+00:22:14,330 --> 00:22:20,210
+الأعداد النسبية أو rational numbers بنعرفها على
+
+240
+00:22:20,210 --> 00:22:24,130
+أنها مجموعة من الأعداد الحقيقية أو هي مجموعة جزئية
+
+241
+00:22:24,130 --> 00:22:30,010
+من الأعداد الحقيقية نموذجها بالرمز boldface q هذه
+
+242
+00:22:30,010 --> 00:22:34,370
+الرموز الأحرف هذه أو ال letters هذه نسميها
+
+243
+00:22:34,370 --> 00:22:41,790
+boldface يعني حرف مغمق هذه طبعا بتدل على مجموعات
+
+244
+00:22:43,170 --> 00:22:47,190
+فال rational numbers هي كل الأعداد الحقيقية اللي
+
+245
+00:22:47,190 --> 00:22:52,210
+ممكن كتبتها على صورة rational a على b حيث a وb 
+
+246
+00:22:52,210 --> 00:22:56,370
+أعداد صحيحة هذه مجموعة الأعداد الصحيحة bold في ال
+
+247
+00:22:56,370 --> 00:23:00,650
+z والمقام لازم ما يساويش صفر لأن القسمة على صفر مش
+
+248
+00:23:00,650 --> 00:23:05,840
+معرفة الآن لو أخذت الأعداد الحقيقية وشلت منها
+
+249
+00:23:05,840 --> 00:23:09,720
+الأعداد النسبية طرحت منها الأعداد النسبية فالأعداد
+
+250
+00:23:09,720 --> 00:23:13,160
+الحقيقية المتبقية بنسميها irrational numbers
+
+251
+00:23:13,160 --> 00:23:18,480
+irrational numbers الأعداد غير النسبية إذا الأعداد
+
+252
+00:23:18,480 --> 00:23:21,960
+النسبية هي كل الأعداد الحقيقية التي لا يمكن
+
+253
+00:23:21,960 --> 00:23:26,740
+كتابتها على صورة rational a على b حيث a وb أعداد
+
+254
+00:23:26,740 --> 00:23:32,910
+صحيحة والمقام لا يساوي صفر تمام؟ طبعا لو أخذت اتحاد ال
+
+255
+00:23:32,910 --> 00:23:35,590
+rational numbers مع ال irrational numbers بيعطوني
+
+256
+00:23:35,590 --> 00:23:38,770
+كل الأعداد الحقيقية يعني المعنى الآخر الأعداد
+
+257
+00:23:38,770 --> 00:23:43,390
+الحقيقية احنا جزأناها إلى مجموعتين irrational
+
+258
+00:23:43,390 --> 00:23:53,270
+numbers اتحاد ال irrational numbers طبعا
+
+259
+00:23:53,270 --> 00:23:57,830
+المجموعتين هدول disjoint يعني منفصلتين ما فيش بينهم
+
+260
+00:23:57,830 --> 00:23:58,850
+عناصر مشتركة
+
+261
+00:24:01,750 --> 00:24:08,710
+طيب ال .. النظرية التالية ممكن من السهل أن احنا
+
+262
+00:24:08,710 --> 00:24:14,950
+نثبتها باستخدام خواص الأعداد الصحيحة يعني
+
+263
+00:24:14,950 --> 00:24:25,150
+معروف احنا عندنا أن ال ..
+
+264
+00:24:25,150 --> 00:24:30,090
+معروف أن ال ..
+
+265
+00:24:33,390 --> 00:24:37,450
+لو في عندي عددين صحيحين فمجموعهم بيطلع عدد صحيح
+
+266
+00:24:37,450 --> 00:24:42,570
+وحاصل ضربهم عدد صحيح بمعنى آخر عملية مجموعة
+
+267
+00:24:42,570 --> 00:24:49,220
+الأعداد الصحيحة مغلقة تحت عمليات الجمع والضرب بنفس
+
+268
+00:24:49,220 --> 00:24:53,860
+.. باستخدام الحقيقة هذه أو ال fact هذه ممكن إثبات
+
+269
+00:24:53,860 --> 00:24:58,900
+أن مجموعة الأعداد النسبية مغلقة تحت عمليات الضرب
+
+270
+00:24:58,900 --> 00:25:06,270
+والجمع وأخذ ال additive inverse يعني لو كان XY
+
+271
+00:25:06,270 --> 00:25:10,810
+أعداد نسبية فمجموعهم بيطلع عدد نسبي وحاصل ضربهم
+
+272
+00:25:10,810 --> 00:25:15,330
+عدد نسبي وسالب عدد نسبي بيطلع عدد نسبي ولو في
+
+273
+00:25:15,330 --> 00:25:21,330
+عندي عدد نسبي مختلف عن الصفر ف ال multiplicative
+
+274
+00:25:21,330 --> 00:25:28,550
+inverse بيطلع عدد نسبي بمعنى آخر، مجموعة الأعداد
+
+275
+00:25:28,550 --> 00:25:33,950
+النسبية مغلقة تحت عملية الجمع والضرب، وأخذ الـ
+
+276
+00:25:33,950 --> 00:25:39,950
+Additive Inverse وأخذ مغلقة تحت الـ Multiplicative
+
+277
+00:25:39,950 --> 00:25:44,470
+Inverse وبالتالي ممكن
+
+278
+00:25:46,690 --> 00:25:51,850
+نستنتج أن مجموعة الأعداد النسبية بتحقق الخواص
+
+279
+00:25:51,850 --> 00:25:56,430
+الخامسة تبع ال field وبالتالي هي بتشكل field بحد
+
+280
+00:25:56,430 --> 00:26:05,690
+ذاتها الآن ال Q subset من R وR field وQ subset من R
+
+281
+00:26:05,690 --> 00:26:11,830
+R وQ field فبنسمي Q sub field زي لما يكون في عندي
+
+282
+00:26:11,830 --> 00:26:19,160
+set وفي داخلها setفبنقول sub set لو في عندي group
+
+283
+00:26:19,160 --> 00:26:25,240
+وفي عندي مجموعة جزئية من ال group فبنسمي المجموعة
+
+284
+00:26:25,240 --> 00:26:31,940
+جزئية نفسها group أيضا فبنسميها sub group إذا ال Q
+
+285
+00:26:31,940 --> 00:26:38,340
+is a sub field of the field R بناء على النظرية هذه
+
+286
+00:26:38,340 --> 00:26:44,770
+إذا هذه كلها حقائق معروفة يعني وهتركزوا عليها 
+
+287
+00:26:44,770 --> 00:26:48,410
+أنتم في الجبر يعني ما حنخوض فيها كثير لأن هذا
+
+288
+00:26:48,410 --> 00:26:53,370
+مواضيع جبرية احنا هنركز على ال analysis في كمان
+
+289
+00:26:53,370 --> 00:26:57,990
+نظرية هنا مهمة بتخص الأعداد النسبية والغير نسبية
+
+290
+00:26:57,990 --> 00:27:03,570
+وهذه طبعا برضه بنعطيها دائما مثال في مبادئ
+
+291
+00:27:03,570 --> 00:27:04,430
+الرياضيات
+
+292
+00:27:07,090 --> 00:27:14,250
+برهان على مثال على برهان غير مباشر النظرية هذه
+
+293
+00:27:14,250 --> 00:27:17,930
+بتقول there does not exist R ينتمي ل Q بحيث R
+
+294
+00:27:17,930 --> 00:27:22,530
+تربيعه بيساوي 2 ما فيش لا يوجد عدد نسبي مربعه
+
+295
+00:27:22,530 --> 00:27:27,450
+بيساوي 2 يعني بمعنى آخر النظرية هذه باختصار
+
+296
+00:27:27,450 --> 00:27:36,590
+بتقول أن جذر الـ 2 ليس عدد نسبي ومالوش عدد غير
+
+297
+00:27:36,590 --> 00:27:41,830
+نسبي يعني جذر 2 عدد غير نسبي فهي برهان
+
+298
+00:27:41,830 --> 00:27:45,490
+بالتناقض برهان غير مباشر بالتناقض by contradiction
+
+299
+00:27:45,490 --> 00:27:53,510
+احنا عايزين نثبت أنه ما فيش عدد أو جذر 2 ليس
+
+300
+00:27:53,510 --> 00:28:01,010
+عدد نسبي فنفرض النقيض نفرض النفي هو الصح يعني نفرض
+
+301
+00:28:01,010 --> 00:28:11,490
+أن جذر الـ 2 عدد نسبي يعني بيساوي P على Q حيث P
+
+302
+00:28:11,490 --> 00:28:17,310
+وQ أعداد صحيحة و
+
+303
+00:28:17,310 --> 00:28:23,990
+Q لا يساوي صفر تمام؟ وبدنا نصل إلى تناقض إذا
+
+304
+00:28:23,990 --> 00:28:27,870
+وصلنا إلى تناقض معناته فرضنا هذا غلط والصح أن
+
+305
+00:28:27,870 --> 00:28:32,410
+النظرية تكون صحيحة، بصبوت؟ طيب نشوف مع بعض
+
+306
+00:28:36,080 --> 00:28:46,520
+هي فرضنا أن الـ 2 عدد هي 2 بيساوي هي
+
+307
+00:28:46,520 --> 00:28:52,320
+فرضنا أن الـ 2 بيساوي P على Q أو جدر 2
+
+308
+00:28:52,320 --> 00:28:57,100
+بيساوي P على Q وبالتالي P على Q تربيع فبيطلع عندي P
+
+309
+00:28:57,100 --> 00:29:02,900
+على Q تربيع بيساوي 2 صح؟ هايطيب الآن لما يكون
+
+310
+00:29:02,900 --> 00:29:12,820
+في عندي أعداد هذا عدد نسبي فممكن نختصر ونفرض أنه
+
+311
+00:29:12,820 --> 00:29:19,480
+ما فيش عامل مشترك بين الـ P والـ Q إلا الـ 1 الصحيح
+
+312
+00:29:19,480 --> 00:29:26,660
+يعني مثلا هي عندي مثلا 4 على مثلا 10 هذا عدد
+
+313
+00:29:26,660 --> 00:29:32,130
+نسبي فممكن أكتبه أختصر أقسم على 2 وأقسم على
+
+314
+00:29:32,130 --> 00:29:36,530
+2 بيطلع 2 على 5 لاحظوا الآن ما فيش عامل
+
+315
+00:29:36,530 --> 00:29:40,370
+مشترك بين 2 والـ 5 إلا الـ 1 فهذا في الجبر
+
+316
+00:29:40,370 --> 00:29:44,410
+بيسموه الـ greatest common divisor لـ 2 و 5
+
+317
+00:29:44,410 --> 00:29:53,130
+بيساوي 1 هي مثلا سالب 10 على 30 هي هذا
+
+318
+00:29:53,130 --> 00:29:59,940
+عدد نسبي فممكن نختصر هذا بيصير 2 على والا ايش
+
+319
+00:29:59,940 --> 00:30:06,760
+هذا أو سالب 1 على 3 فالـ  bus سالب 1 عدد
+
+320
+00:30:06,760 --> 00:30:10,460
+صحيح والمقام 3 والـ greatest common divisor
+
+321
+00:30:10,460 --> 00:30:17,420
+لسالب 1 و3 بيساوي 1 إذا أي عدد نسبي ممكن
+
+322
+00:30:17,420 --> 00:30:23,260
+أختصره أو أبسط وأكتبه بأبسط صورة يعني أخلي الـ
+
+323
+00:30:23,260 --> 00:30:29,640
+greatest common divisor لـ P وQ هنا بيساوي 1 طيب
+
+324
+00:30:29,640 --> 00:30:35,580
+الآن تعالوا نربع هنا هذه المعادلة P على Q تربيع
+
+325
+00:30:35,580 --> 00:30:40,500
+بيساوي 2 فمنها بنستنتج أن P تربيع بيساوي 2 و
+
+326
+00:30:40,500 --> 00:30:47,040
+Q تربيع طيب هي عندي P تربيع بيساوي 2 في Q تربيع الـ Q
+
+327
+00:30:47,040 --> 00:30:50,820
+هذا عدد صحيح فمربع العدد الصحيح اللي هو Q تربيع
+
+328
+00:30:50,820 --> 00:30:54,720
+بيطلع عدد صحيح عدد صحيح مضروب في 2 بيطلع even
+
+329
+00:30:54,720 --> 00:30:59,400
+number عشان المبادئ صح؟ إذا P تربيع بيطلع even
+
+330
+00:30:59,400 --> 00:31:06,520
+number تمام؟ هذا بيؤدي أنه ممكن إثبات أن P بيطلع
+
+331
+00:31:06,520 --> 00:31:10,960
+even لو كان مربع عدد صحيح even فممكن إثبات أن
+
+332
+00:31:10,960 --> 00:31:14,860
+العدد الصحيح نفسه لازم يطلع even وهذا ممكن نعمل
+
+333
+00:31:14,860 --> 00:31:21,940
+برهان صغير بالتناقض افرضه أن P مش even يعني odd
+
+334
+00:31:21,940 --> 00:31:28,360
+تربيعه فبيطلع مربع odd تناقض هي البرهان إذا هذا ايه
+
+335
+00:31:28,360 --> 00:31:33,220
+الإجابة على الـ Y كمان مرة إذا كان عندي P تربيع أنا
+
+336
+00:31:33,220 --> 00:31:39,900
+عندي P تربيع even هذا بيعني أن P even كيف نبرهن هذا
+
+337
+00:31:39,900 --> 00:31:46,420
+برهان بالتناقض افرض أنه P odd ايه يعني P odd يعني
+
+338
+00:31:46,420 --> 00:31:51,600
+P هي
+
+339
+00:31:51,600 --> 00:31:58,500
+P بيساوي 2K زي 1 هذا معناه odd وطبعا الـ K هي عدد
+
+340
+00:31:58,500 --> 00:32:04,620
+صحيح تربيع إذا P تربيع بيساوي 4 K تربيع زائد 4
+
+341
+00:32:04,620 --> 00:32:14,080
+K زائد 1 وهذا بيساوي 2 في 2 K تربيع زائد
+
+342
+00:32:14,080 --> 00:32:19,900
+2 K مع بعض زائد 1 وهذا بيساوي 2 في M زائد
+
+343
+00:32:19,900 --> 00:32:26,350
+1 حيث M عدد صحيح إذاً P تربيع طلع بيساوي 2 في
+
+344
+00:32:26,350 --> 00:32:30,410
+عدد صحيح زائد 1 وبالتالي هذا  Contradiction
+
+345
+00:32:30,410 --> 00:32:35,590
+تناقض لأن احنا عندنا P تربيع P تربيع is even okay
+
+346
+00:32:35,590 --> 00:32:42,670
+إذاً هذا برهان الـ why طيب إذا احنا وصلنا إلى أن P
+
+347
+00:32:42,670 --> 00:32:48,750
+تربيع even بقدر أن P even الآن الـ P والـ Q have no
+
+348
+00:32:48,750 --> 00:32:51,410
+common factor other than 1 ما فيش بينهم عامل
+
+349
+00:32:51,410 --> 00:32:57,460
+مشترك إلا الـ 1 والـ P even إذا لازم الـ Q يكون odd
+
+350
+00:32:57,460 --> 00:33:03,320
+لأن لو كان الـ Q even والـ P even فيه عامل مشترك
+
+351
+00:33:03,320 --> 00:33:09,560
+بينهم 2 على الأقل أو 4 وهذا بيتناقض مع ايه
+
+352
+00:33:09,560 --> 00:33:13,100
+أن احنا فرضنا أنه ما فيش common factor بين الـ P و
+
+353
+00:33:13,100 --> 00:33:18,560
+الـ Q إلا الـ 1 تمام إذا أنا عندي هنا نستنتج أن الـ
+
+354
+00:33:18,560 --> 00:33:25,030
+Q لازم يكون odd الآن أنا عندي P even يعني معناته
+
+355
+00:33:25,030 --> 00:33:33,020
+بيساوي 2M for some M عدد صحيح وبالتالي لو ربعت له
+
+356
+00:33:33,020 --> 00:33:37,780
+نرجع نعوض في المعادلة هذه P تربيع بيساوي 2Q تربيع
+
+357
+00:33:37,780 --> 00:33:43,120
+عوض عن P بيساوي 2M فبيصير 4M تربيع بيساوي 2Q تربيع
+
+358
+00:33:43,120 --> 00:33:48,760
+فبختصر 2 من الطرفين بيطلع 2M تربيع بيساوي Q تربيع
+
+359
+00:33:48,760 --> 00:33:54,620
+إذن Q تربيع بيساوي 2 ضرب عدد صحيح وبالتالي Q تربيع
+
+360
+00:33:54,620 --> 00:34:00,020
+is even وبالتالي منها بنستنتج أن الـ Q نفسها is
+
+361
+00:34:00,020 --> 00:34:08,720
+even إذا الآن أنا عندي الـ Q is even وهي نفس الـ Q
+
+362
+00:34:08,720 --> 00:34:14,260
+استنتجنا أنها odd فهذا
+
+363
+00:34:14,260 --> 00:34:20,680
+تناقض .. هذا تناقض صح؟ الـ Q هنا odd وهنا طلعت even
+
+364
+00:34:20,680 --> 00:34:24,240
+فهذا يعطيني contradiction which is a contradiction
+
+365
+00:34:24,240 --> 00:34:30,620
+فهذا التناقض أن احنا هنا وصلنا بدينا بالبرهان احنا
+
+366
+00:34:30,620 --> 00:34:35,860
+عايزين نثبت جدر 2 لا تنتمي لـ Q فرضنا النقيض الـ
+
+367
+00:34:35,860 --> 00:34:40,660
+contrary أن جدر 2 تنتمي لـ Q يعني ممكن كتبتها
+
+368
+00:34:40,660 --> 00:34:46,240
+على صورة P على Q P وQ أعداد صحيحة وصلنا لتناقض
+
+369
+00:34:46,240 --> 00:34:50,020
+معناته أن هذا فرضنا غلط الصح أن جدر 2 لاتن
+
+370
+00:34:50,020 --> 00:34:54,640
+تمي لـ Q as required كما هو مطلوب okay تمام هذا
+
+371
+00:34:54,640 --> 00:34:59,660
+برهان بالتناقض تمام إذا هنا شوية راجعنا شوية
+
+372
+00:34:59,660 --> 00:35:04,820
+براهين تعلمتوها في مبادئ رياضيات طبعا الناس اللي
+
+373
+00:35:04,820 --> 00:35:09,180
+اتعلموا مبادئ رياضيات وأخوياء الحاجات هذه
+
+374
+00:35:09,180 --> 00:35:14,870
+بالنسبة لهم يعني صارت مجرد تسلية والناس اللي عندهم
+
+375
+00:35:14,870 --> 00:35:20,790
+مشاكل في المبادئ نجحوا بالعافية فممكن يعني يجد أن
+
+376
+00:35:20,790 --> 00:35:28,670
+هذا مش كثير مثير طيب هيك بنكون خلصنا الـ
+
+377
+00:35:28,670 --> 00:35:33,410
+algebraic properties of R الخواص الجبرية لنظام
+
+378
+00:35:33,410 --> 00:35:37,650
+الأعداد الحقيقية أو الـ real number system ننتقل لـ
+
+379
+00:35:37,650 --> 00:35:40,550
+section ثاني داخل الـ chapter الأول هيك خلصنا
+
+380
+00:35:40,550 --> 00:35:46,110
+section الـ section الثاني عنوانه الـ order
+
+381
+00:35:46,110 --> 00:35:52,370
+properties of R خواص الترتيب على R order properties
+
+382
+00:35:52,370 --> 00:35:58,470
+خواص الترتيب احنا شفنا أو قلنا لما عرفنا نظام
+
+383
+00:35:58,470 --> 00:36:02,070
+الأعداد الحقيقية قلنا أن نظام الأعداد الحقيقية
+
+384
+00:36:02,070 --> 00:36:07,670
+مجرد مجموعة R boldface R مع عمليتين ثنائيتين two
+
+385
+00:36:07,670 --> 00:36:11,710
+binary operations بيحققوا الخمس خواص تبعت الـ field
+
+386
+00:36:11,710 --> 00:36:16,730
+اليوم هنفترض أيضا أن نظام الأعداد الحقيقية بيحقق
+
+387
+00:36:17,890 --> 00:36:24,850
+order properties الخاصية رقم 6 هذه الخاصية رقم
+
+388
+00:36:24,850 --> 00:36:30,710
+6 هذه الخاصية رقم 6 تتجزأ إلى 3 خواص 3
+
+389
+00:36:30,710 --> 00:36:41,410
+خواص نسميهم order properties أو أول خاصية نفترض
+
+390
+00:36:41,410 --> 00:36:48,400
+ماهي order property نفترض وجود مجموعة جزئية من R
+
+391
+00:36:48,400 --> 00:36:53,500
+وغير خالية، ليست خالية، non-empty subset of R
+
+392
+00:36:53,500 --> 00:36:58,220
+نفترض أن يوجد مجموعة P subset of R غير خالية
+
+393
+00:36:58,220 --> 00:37:04,620
+وبتحقق الثلاث خواص هذه الـ set P is closed under
+
+394
+00:37:04,620 --> 00:37:09,920
+addition يعني لو أخذت أي عنصرين في P فمجموعهم بيطلع
+
+395
+00:37:09,920 --> 00:37:15,040
+عنصر ثالث فيها كذلك المجموعة P closed under
+
+396
+00:37:15,040 --> 00:37:20,620
+multiplication يعني لو أخذت أي عنصرين وحصل ضربهم
+
+397
+00:37:20,620 --> 00:37:26,980
+بيطلع عنصر ثالث الخاصية الثالثة من خاصية الترتيب
+
+398
+00:37:26,980 --> 00:37:32,340
+اللي لها اسم بنسميها trichotomy property الخاصية
+
+399
+00:37:32,340 --> 00:37:37,940
+الثلاثية الخاصية الثلاثية ايه هي؟ لو أخذت أي عدد
+
+400
+00:37:37,940 --> 00:37:39,180
+حقيقي R
+
+401
+00:37:41,750 --> 00:37:51,250
+فواحد من الثلاث احتمالات هذه لازم يكون صحيح وهو أن
+
+402
+00:37:51,250 --> 00:37:57,290
+إما a تنتمي للمجموعة P هذه أو a بيساوي صفر أو
+
+403
+00:37:57,290 --> 00:38:01,410
+negative a ينتمي للمجموعة P هذه هي الخاصية
+
+404
+00:38:01,410 --> 00:38:07,670
+الثالثة أي عدد حقيقي إما يكون عنصر في P أو بيساوي
+
+405
+00:38:07,670 --> 00:38:10,630
+صفر أو الـ negative تبقى عنصر في P
+
+406
+00:38:16,020 --> 00:38:21,780
+الآن بنعرف المجموعة remark ملاحظة بنعرف المجموعة
+
+407
+00:38:21,780 --> 00:38:25,820
+negative P المجموعة negative P هي مجموعة كل
+
+408
+00:38:25,820 --> 00:38:33,050
+العناصر negative A حيث A ينتمي لـ P الآن الخاصية C
+
+409
+00:38:33,050 --> 00:38:36,310
+من الـ order property اللي هي trichotomy property
+
+410
+00:38:36,310 --> 00:38:43,610
+الخاصية C says تقول أو بتقول أن الـ sets المجموعات
+
+411
+00:38:43,610 --> 00:38:52,110
+اللي هي المجموعة الأحادية صفر والمجموعة P و
+
+412
+00:38:52,110 --> 00:38:57,570
+المجموعة negative P الثلاث 
+
+413
+00:38:57,570 --> 00:39:02,860
+هدول are pairwise disjoint منفصلة مثنى مثنى
+
+414
+00:39:02,860 --> 00:39:06,840
+pairwise disjoint يعني منفصلة مثنى مثنى ايه يعني؟
+
+415
+00:39:06,840 --> 00:39:11,360
+لو أخذت أي اثنتين من الثلاث مجموعات هدول وقاطعتهم
+
+416
+00:39:11,360 --> 00:39:15,200
+مع بعض فبتقعوا اعتبارهم five ما فيش بينهم عناصر
+
+417
+00:39:15,200 --> 00:39:19,060
+مشتركة فبنقول إن المجموعات هذه pairwise disjoint
+
+418
+00:39:20,610 --> 00:39:26,410
+ومش هيكوا بس واتحادهم بيساوي كل الأعداد الحقيقية هذا
+
+419
+00:39:26,410 --> 00:39:33,430
+صحيح من الخاصية C لأن C بتقول لأي عدد حقيقي أي A
+
+420
+00:39:33,430 --> 00:39:41,310
+ينتمي إلى R أي A ينتمي إلى R إما ينتمي إلى P أو 
+
+421
+00:39:41,310 --> 00:39:45,370
+بساوي 0 وبالتالي ينتمي إلى المجموعة هذه أو ينتمي
+
+422
+00:39:45,370 --> 00:39:52,030
+إلى negative P صح؟ وبالتالي كل a هنا موجود في واحدة
+
+423
+00:39:52,030 --> 00:39:54,810
+من هذول التلاتة وبالتالي موجود في اتحادهم مش هيك
+
+424
+00:39:54,810 --> 00:39:59,990
+تعريف اتحاد؟ إذاً الـ R الآن أصبحت مجموعة جزئية من
+
+425
+00:39:59,990 --> 00:40:06,130
+الاتحاد، مظبوط؟ طب الـ P مجموعة جزئية من R و
+
+426
+00:40:06,130 --> 00:40:10,850
+negative P مجموعة جزئية من R و singleton 0 برضه 
+
+427
+00:40:10,850 --> 00:40:15,130
+مجموعة جزئية من R إذا اتحادهم بيطلع مجموعة جزئية
+
+428
+00:40:15,130 --> 00:40:20,060
+من R وبالتالي أنا عندي الاحتواء من الناحيتين
+
+429
+00:40:20,060 --> 00:40:24,280
+وبالتالي عندي تساوي إذاً هنا برهنت لكم أن الاتحاد
+
+430
+00:40:24,280 --> 00:40:26,740
+هذا بساوي R تمام؟
+
+431
+00:40:33,090 --> 00:40:37,470
+طب ليش هدول disjoint؟ لأنه لو .. لو فرضت أنه مثلاً
+
+432
+00:40:37,470 --> 00:40:42,130
+في عنصر بيقع ينتمي لتقاطع المجموعتين هدول، معناته 
+
+433
+00:40:42,130 --> 00:40:45,990
+هذا العنصر بيساوي صفر وفي نفس الوقت ينتمي لـ P
+
+434
+00:40:45,990 --> 00:40:49,830
+وهذا بتناقض مع الخاصية الثلاثية، الخاصية الثلاثية
+
+435
+00:40:49,830 --> 00:40:53,950
+بتقول لازم و exactly one، واحد من الاحتمالات
+
+436
+00:40:53,950 --> 00:41:00,090
+الثلاثة هذه صح، أما اثنين مش صح، تمام؟ okay
+
+437
+00:41:02,970 --> 00:41:07,390
+العنى المجموعة P هذه اللي عرفناها هنا بنسميها
+
+438
+00:41:07,390 --> 00:41:10,610
+مجموعة الأعداد الحقيقية الموجبة the set of
+
+439
+00:41:10,610 --> 00:41:15,830
+positive real number مجموعة الأعداد الحقيقية
+
+440
+00:41:15,830 --> 00:41:19,750
+الموجبة و الـ set negative P هذه بنسميها مجموعة
+
+441
+00:41:19,750 --> 00:41:24,560
+الأعداد الحقيقية السالبة، ولما نضيف عليهم الصفر هيك
+
+442
+00:41:24,560 --> 00:41:28,980
+بنكون غطينا كل الأعداد الحقيقية صح؟ okay تمام إذاً
+
+443
+00:41:28,980 --> 00:41:33,000
+P بنسميها مجموعة الأعداد الحقيقية الموجبة negative
+
+444
+00:41:33,000 --> 00:41:42,440
+P مجموعة الأعداد الحقيقية السالبة وهكذا طيب
+
+445
+00:41:42,440 --> 00:41:47,980
+لحد الآن احنا ما عرفناش علاقة أصغر أو أكبر أو أصغر
+
+446
+00:41:47,980 --> 00:41:52,380
+من أو يساوي أو أكبر من أو يساوي اللي هو الترتيب في
+
+447
+00:41:52,380 --> 00:41:56,100
+عندي definition هنا لو أخدت أي عددين حقيقيين a و b
+
+448
+00:41:56,100 --> 00:42:04,620
+فهنكتب أو هنعلن أن a أصغر من b أو ما يكافئها b
+
+449
+00:42:04,620 --> 00:42:12,640
+أكبر من a هذا معناه نقصد نقصد بذلك أن الفرق بين b
+
+450
+00:42:12,640 --> 00:42:18,920
+و a ينتمي لـ p يعني الفرق هذا موجب يعني هذا عدد
+
+451
+00:42:18,920 --> 00:42:26,080
+موجب، إذا لو كان الفرق بين b و a عدد موجب فبنكتب a
+
+452
+00:42:26,080 --> 00:42:32,780
+أصغر من b أو b أكبر من a طيب طب
+
+453
+00:42:32,780 --> 00:42:36,840
+متى بكتب a أصغر من أو يساوي b أو b أكبر من أو
+
+454
+00:42:36,840 --> 00:42:45,760
+يساوي a هذا معناه يعني هذا مثلاً معناه أن a أصغر من
+
+455
+00:42:45,760 --> 00:42:54,840
+b يعني الفرق بين B و A ينتمي لـ P أو A بيساوي B لما
+
+456
+00:42:54,840 --> 00:42:58,860
+يكون A بيساوي B لاحتمال الثاني هذا معناه أن الفرق
+
+457
+00:42:58,860 --> 00:43:03,620
+بيساوي صفر يعني ينتمي للمجموعة Singleton Zero okay
+
+458
+00:43:03,620 --> 00:43:07,500
+تمام؟ إذن أصغر من أو يساوي أو أكبر من أو يساوي
+
+459
+00:43:07,500 --> 00:43:13,410
+معناته الفرق ينتمي لـ P اتحاد Singleton Zero الآن في
+
+460
+00:43:13,410 --> 00:43:20,170
+عندي نظرية والنظرية هذه بتعطيني خواص لـ الـ
+
+461
+00:43:20,170 --> 00:43:28,450
+الـ خليني بس نبص عليها بسرعة النظرية
+
+462
+00:43:28,450 --> 00:43:33,410
+هذه بتعطيني خواص للـ order properties يعني خواص
+
+463
+00:43:33,410 --> 00:43:40,210
+أخرى نقدر نشتقها من الـ order properties وكل هذه
+
+464
+00:43:40,210 --> 00:43:44,990
+خواص معروفة وسهلة وبسيطة وكلها .. كلها عارفين لكن
+
+465
+00:43:44,990 --> 00:43:48,230
+بدها برهان .. بدها برهان ما حدش عمره برهان لنا إياها
+
+466
+00:43:49,460 --> 00:43:54,600
+Okay فحنوقف عند النظرية هذه وإن شاء الله المرة
+
+467
+00:43:54,600 --> 00:44:01,120
+الجاية بنحاول نبرهن النظرية okay حاولوا أنتم
+
+468
+00:44:01,120 --> 00:44:05,260
+meanwhile في نفس الوقت كتحضير للمحاضرة الجاية
+
+469
+00:44:05,260 --> 00:44:10,140
+حاولوا أنكم تقرأوا البرهان تبع النظرية وشوفوا هل
+
+470
+00:44:10,140 --> 00:44:16,360
+تفهموه ولا لأ okay تمام في أي سؤال okay شكراً لكم
+
+471
+00:44:16,360 --> 00:44:17,540
+ومبارك الله فيكم
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/aQ184E7DSME_raw.json b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/aQ184E7DSME_raw.json
new file mode 100644
index 0000000000000000000000000000000000000000..49f3aa0b8b107d27f824692ca632289b0f73cc37
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/aQ184E7DSME_raw.json
@@ -0,0 +1 @@
+{"segments": [{"id": 1, "seek": 4440, "start": 20.96, "end": 44.4, "text": "Okay إذا هنواصل ان شاء الله اللي بدنا في المحاضرة السادقة المرة اللي فاتت خلينا بسرعة بس هيك نمر على الحاجات اللي أخدناها أخدنا اللي هو ال algebraic properties of the real number system اللي هو القواصة الجبرية", "tokens": [8297, 11933, 15730, 8032, 1863, 2407, 33546, 1211, 16472, 13412, 16606, 21984, 13672, 1829, 47525, 8315, 8978, 9673, 5016, 46958, 25720, 21136, 18513, 28671, 9673, 25720, 13672, 1829, 6156, 9307, 2655, 16490, 1211, 9957, 995, 4724, 3794, 2288, 27884, 4724, 3794, 39896, 4117, 8717, 29973, 15844, 21542, 26108, 9307, 13672, 1829, 5551, 9778, 3215, 8315, 11296, 5551, 9778, 3215, 8315, 13672, 1829, 31439, 2423, 21989, 299, 7221, 295, 264, 957, 1230, 1185, 13672, 1829, 31439, 25062, 2407, 33546, 3660, 25724, 26890, 10632], "avg_logprob": -0.14664909638554216, "compression_ratio": 1.6280193236714975, "no_speech_prob": 0.0, "words": [{"start": 20.96, "end": 21.46, "word": "Okay", "probability": 0.9140625}, {"start": 21.46, "end": 21.82, "word": " إذا", "probability": 0.643798828125}, {"start": 21.82, "end": 24.1, "word": " هنواصل", "probability": 0.8673828125}, {"start": 24.1, "end": 24.3, "word": " ان", "probability": 0.53857421875}, {"start": 24.3, "end": 24.56, "word": " شاء", "probability": 0.988037109375}, {"start": 24.56, "end": 24.76, "word": " الله", "probability": 0.96484375}, {"start": 24.76, "end": 25.42, "word": " اللي", "probability": 0.8017578125}, {"start": 25.42, "end": 25.94, "word": " بدنا", "probability": 0.751708984375}, {"start": 25.94, "end": 27.04, "word": " في", "probability": 0.77734375}, {"start": 27.04, "end": 27.56, "word": " المحاضرة", "probability": 0.9791259765625}, {"start": 27.56, "end": 28.2, "word": " السادقة", "probability": 0.8572591145833334}, {"start": 28.2, "end": 29.56, "word": " المرة", "probability": 0.90673828125}, {"start": 29.56, "end": 29.74, "word": " اللي", "probability": 0.908935546875}, {"start": 29.74, "end": 30.12, "word": " فاتت", "probability": 0.9609375}, {"start": 30.12, "end": 30.52, "word": " خلينا", "probability": 0.79931640625}, {"start": 30.52, "end": 30.96, "word": " بسرعة", "probability": 0.975830078125}, {"start": 30.96, "end": 31.28, "word": " بس", "probability": 0.8818359375}, {"start": 31.28, "end": 31.64, "word": " هيك", "probability": 0.683837890625}, {"start": 31.64, "end": 33.14, "word": " نمر", "probability": 0.842529296875}, {"start": 33.14, "end": 33.34, "word": " على", "probability": 0.74072265625}, {"start": 33.34, "end": 33.98, "word": " الحاجات", "probability": 0.9954427083333334}, {"start": 33.98, "end": 35.5, "word": " اللي", "probability": 0.98486328125}, {"start": 35.5, "end": 36.94, "word": " أخدناها", "probability": 0.96279296875}, {"start": 36.94, "end": 38.64, "word": " أخدنا", "probability": 0.9459228515625}, {"start": 38.64, "end": 38.9, "word": " اللي", "probability": 0.976318359375}, {"start": 38.9, "end": 39.22, "word": " هو", "probability": 0.98876953125}, {"start": 39.22, "end": 39.52, "word": " ال", "probability": 0.93408203125}, {"start": 39.52, "end": 40.22, "word": " algebraic", "probability": 0.798095703125}, {"start": 40.22, "end": 40.84, "word": " properties", "probability": 0.884765625}, {"start": 40.84, "end": 41.3, "word": " of", "probability": 0.984375}, {"start": 41.3, "end": 41.5, "word": " the", "probability": 0.7861328125}, {"start": 41.5, "end": 41.72, "word": " real", "probability": 0.939453125}, {"start": 41.72, "end": 42.1, "word": " number", "probability": 0.9052734375}, {"start": 42.1, "end": 42.64, "word": " system", "probability": 0.96484375}, {"start": 42.64, "end": 42.86, "word": " اللي", "probability": 0.971435546875}, {"start": 42.86, "end": 43.06, "word": " هو", "probability": 0.9873046875}, {"start": 43.06, "end": 43.78, "word": " القواصة", "probability": 0.68035888671875}, {"start": 43.78, "end": 44.4, "word": " الجبرية", "probability": 0.8072102864583334}], "temperature": 1.0}, {"id": 2, "seek": 6937, "start": 45.53, "end": 69.37, "text": "و أعرفنا اللي هو ال real number system فقلنا إن ال real number system عبارة عن المجموعة R مع عمليتين جبريتين أو ثنائيتين عملية جمع و عملية ضرب و هدول العمليات بحققوا خمس خواص خاصية الإبدال competitive law خاصية الدمج ال associative laws", "tokens": [2407, 5551, 3615, 28480, 8315, 13672, 1829, 31439, 2423, 957, 1230, 1185, 6156, 4587, 1211, 8315, 36145, 2423, 957, 1230, 1185, 6225, 3555, 9640, 3660, 18871, 9673, 7435, 2304, 2407, 27884, 497, 20449, 6225, 2304, 20292, 2655, 9957, 10874, 3555, 16572, 2655, 9957, 34051, 38637, 1863, 16373, 36081, 9957, 6225, 42213, 10632, 10874, 2304, 3615, 4032, 6225, 42213, 10632, 48812, 25513, 4032, 8032, 3215, 12610, 18863, 2304, 20292, 9307, 4724, 5016, 4587, 4587, 14407, 16490, 2304, 3794, 16490, 2407, 33546, 16490, 33546, 10632, 33688, 44510, 6027, 10043, 2101, 16490, 33546, 10632, 32748, 2304, 7435, 2423, 4180, 1166, 6064], "avg_logprob": -0.1429924224362229, "compression_ratio": 1.7440758293838863, 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"word": " العناصر", "probability": 0.83984375}, {"start": 89.14, "end": 91.76, "word": " النظائر", "probability": 0.9478759765625}, {"start": 91.76, "end": 93.54, "word": " أو", "probability": 0.67138671875}, {"start": 93.54, "end": 94.46, "word": " المعكسات", "probability": 0.91298828125}, {"start": 94.46, "end": 97.46, "word": " فلكل", "probability": 0.8361002604166666}, {"start": 97.46, "end": 98.58, "word": " عدد", "probability": 0.9860026041666666}, {"start": 98.58, "end": 99.58, "word": " حقيقي", "probability": 0.9894205729166666}], "temperature": 1.0}, {"id": 4, "seek": 10543, "start": 100.69, "end": 105.43, "text": "فيه إله معكوس جمعي اللي هو Salvo", "tokens": [41185, 3224, 11933, 43761, 20449, 4117, 41779, 10874, 2304, 3615, 1829, 13672, 1829, 31439, 5996, 3080], "avg_logprob": -0.29940257352941174, "compression_ratio": 0.9464285714285714, "no_speech_prob": 0.0, "words": [{"start": 100.69, "end": 101.77, "word": "فيه", "probability": 0.66796875}, {"start": 101.77, "end": 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واحد", "probability": 0.98193359375}, {"start": 125.23, "end": 127.25, "word": " هدول", "probability": 0.6897786458333334}, {"start": 127.25, "end": 127.87, "word": " الخمس", "probability": 0.9853515625}, {"start": 127.87, "end": 128.63, "word": " خواص", "probability": 0.9039713541666666}, {"start": 128.63, "end": 129.65, "word": " اللي", "probability": 0.84228515625}, {"start": 129.65, "end": 130.89, "word": " بتحققهم", "probability": 0.94951171875}, {"start": 130.89, "end": 131.35, "word": " مجموعة", "probability": 0.9744140625}, {"start": 131.35, "end": 131.77, "word": " العداد", "probability": 0.8154296875}, {"start": 131.77, "end": 132.49, "word": " الحقيقية", "probability": 0.984619140625}, {"start": 132.49, "end": 132.87, "word": " مع", "probability": 0.98583984375}], "temperature": 1.0}, {"id": 6, "seek": 15730, "start": 133.36, "end": 157.3, "text": "العمليات الدرب والجامع اللي عرفناها سابقا في أول خواص أخدناها اللي هي cancellation laws موجودة في نظرية 1 1 فعملية الجامع بتحقق cancellation law يعني انا لو كان عندي x plus z بساوي y plus z فممكن اشطب z", "tokens": [6027, 25957, 20292, 9307, 32748, 25513, 16070, 7435, 10943, 3615, 13672, 1829, 6225, 28480, 8315, 11296, 8608, 16758, 4587, 995, 8978, 5551, 12610, 16490, 2407, 33546, 5551, 9778, 3215, 8315, 11296, 13672, 1829, 39896, 45867, 6064, 3714, 29245, 23328, 3660, 8978, 8717, 19913, 2288, 10632, 502, 502, 6156, 25957, 1211, 10632, 25724, 10943, 3615, 39894, 5016, 4587, 4587, 45867, 2101, 37495, 22653, 1975, 8315, 45164, 25961, 18871, 16254, 2031, 1804, 710, 4724, 3794, 995, 45865, 288, 1804, 710, 6156, 2304, 43020, 1975, 8592, 9566, 3555, 710], "avg_logprob": -0.16073994184362478, "compression_ratio": 1.6, "no_speech_prob": 0.0, "words": [{"start": 133.36, "end": 134.48, "word": "العمليات", "probability": 0.84375}, {"start": 134.48, "end": 135.0, "word": " الدرب", "probability": 0.75732421875}, {"start": 135.0, "end": 135.76, "word": " والجامع", "probability": 0.764404296875}, {"start": 135.76, "end": 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X", "probability": 0.92138671875}, {"start": 560.41, "end": 560.75, "word": " ضرب", "probability": 0.9853515625}, {"start": 560.75, "end": 561.13, "word": " negative", "probability": 0.8173828125}, {"start": 561.13, "end": 561.59, "word": " Y", "probability": 0.9765625}, {"start": 561.59, "end": 562.01, "word": " هو", "probability": 0.9755859375}, {"start": 562.01, "end": 562.41, "word": " نفس", "probability": 0.9970703125}, {"start": 562.41, "end": 562.93, "word": " الحاجة", "probability": 0.9895833333333334}, {"start": 562.93, "end": 563.25, "word": " كما", "probability": 0.900634765625}, {"start": 563.25, "end": 563.41, "word": " لو", "probability": 0.99267578125}, {"start": 563.41, "end": 563.91, "word": " ضربت", "probability": 0.9842122395833334}, {"start": 563.91, "end": 564.19, "word": " X", "probability": 0.98681640625}, {"start": 564.19, "end": 564.35, "word": " في", "probability": 0.98583984375}, {"start": 564.35, "end": 564.55, "word": " Y", "probability": 0.9658203125}, 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كان X ضرب Y أعداد حقيقية، حصل ضربهم سفر، فإيه بيطلع؟ إما X بساوي سفر أو Y بساوي سفر، بظبط؟", "tokens": [2304, 12984, 6027, 15844, 4724, 2288, 3224, 7649, 32771, 13546, 3714, 3555, 33599, 2288, 8978, 3714, 3555, 18513, 2288, 12602, 1829, 46958, 1829, 9307, 12399, 45164, 25961, 1783, 48812, 25513, 398, 5551, 22488, 18513, 11331, 38436, 4587, 10632, 12399, 11331, 36520, 48812, 25513, 16095, 8608, 5172, 2288, 12399, 6156, 28814, 1829, 3224, 4724, 1829, 9566, 1211, 3615, 22807, 11933, 15042, 1783, 4724, 3794, 995, 45865, 8608, 5172, 2288, 34051, 398, 4724, 3794, 995, 45865, 8608, 5172, 2288, 12399, 4724, 19913, 3555, 9566, 22807], "avg_logprob": -0.14918155471483865, "compression_ratio": 1.5584415584415585, "no_speech_prob": 0.0, "words": [{"start": 758.61, "end": 759.31, "word": "مثال", "probability": 0.9591471354166666}, {"start": 759.31, "end": 759.83, "word": " على", "probability": 0.85302734375}, {"start": 759.83, "end": 760.27, "word": " برهان", "probability": 0.9090576171875}, 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"word": " سفر،", "probability": 0.980712890625}, {"start": 775.01, "end": 775.65, "word": " بظبط؟", "probability": 0.86015625}], "temperature": 1.0}, {"id": 32, "seek": 80074, "start": 781.0, "end": 800.74, "text": "فهذا نعطي مثال في المبادئ الرياضية طيب البرهان هو نفسه البرهان هو نفسه عشان اثبت انه x لو كان x ضرب y بساوة 0 فبطلع x بساوة 0 او y بساوة 0 فبأفرض ان x مايسويش 0", "tokens": [5172, 3224, 15730, 8717, 3615, 9566, 1829, 50113, 6027, 8978, 9673, 3555, 18513, 19986, 2423, 16572, 46958, 10632, 23032, 1829, 3555, 2423, 26890, 3224, 7649, 31439, 8717, 36178, 3224, 2423, 26890, 3224, 7649, 31439, 8717, 36178, 3224, 6225, 8592, 7649, 1975, 12984, 3555, 2655, 16472, 3224, 2031, 45164, 25961, 2031, 48812, 25513, 288, 4724, 3794, 995, 2407, 3660, 1958, 6156, 3555, 9566, 1211, 3615, 2031, 4724, 3794, 995, 2407, 3660, 1958, 1975, 2407, 288, 4724, 3794, 995, 2407, 3660, 1958, 6156, 3555, 10721, 5172, 43042, 16472, 2031, 19446, 1829, 3794, 45865, 8592, 1958], "avg_logprob": 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91169, "start": 888.75, "end": 911.69, "text": "Okay تمام إذا هيك بنكون برهننا النظرية التانية كويس تمام هيك طيب ناخد تعريفات أو تعريف مهم", "tokens": [8297, 46811, 10943, 11933, 15730, 39896, 4117, 44945, 30544, 4724, 2288, 3224, 1863, 8315, 28239, 19913, 2288, 10632, 16712, 7649, 10632, 9122, 45865, 3794, 46811, 10943, 39896, 4117, 23032, 1829, 3555, 8717, 47283, 3215, 37279, 16572, 5172, 9307, 34051, 37279, 16572, 5172, 3714, 16095], "avg_logprob": -0.1506076455116272, "compression_ratio": 1.391304347826087, "no_speech_prob": 0.0, "words": [{"start": 888.75, "end": 889.15, "word": "Okay", "probability": 0.92919921875}, {"start": 889.15, "end": 889.69, "word": " تمام", "probability": 0.84619140625}, {"start": 889.69, "end": 890.37, "word": " إذا", "probability": 0.614013671875}, {"start": 890.37, "end": 890.59, "word": " هيك", "probability": 0.6787109375}, {"start": 890.59, "end": 890.91, "word": " بنكون", "probability": 0.85498046875}, {"start": 890.91, "end": 891.95, "word": " 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{"start": 1266.04, "end": 1266.54, "word": " القسمة", "probability": 0.9189453125}, {"start": 1266.54, "end": 1266.84, "word": " division", "probability": 0.7666015625}, {"start": 1266.84, "end": 1267.26, "word": " on", "probability": 0.8662109375}, {"start": 1267.26, "end": 1267.54, "word": " R", "probability": 0.8193359375}, {"start": 1267.54, "end": 1267.72, "word": " is", "probability": 0.85400390625}, {"start": 1267.72, "end": 1268.14, "word": " defined", "probability": 0.92578125}, {"start": 1268.14, "end": 1268.58, "word": " by", "probability": 0.94677734375}, {"start": 1268.58, "end": 1269.36, "word": " هاي", "probability": 0.4813232421875}, {"start": 1269.36, "end": 1269.72, "word": " X", "probability": 0.7041015625}, {"start": 1269.72, "end": 1269.96, "word": " على", "probability": 0.74365234375}, {"start": 1269.96, "end": 1270.32, "word": " Y", "probability": 0.98876953125}, {"start": 1270.32, "end": 1271.5, "word": " بساوي", "probability": 0.72900390625}, {"start": 1271.5, 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"end": 1284.24, "word": " علامة", "probability": 0.9724934895833334}, {"start": 1284.24, "end": 1284.66, "word": " تساوي", "probability": 0.9345703125}, {"start": 1284.66, "end": 1285.12, "word": " معناه", "probability": 0.9482421875}, {"start": 1285.12, "end": 1285.36, "word": " إن", "probability": 0.421875}, {"start": 1285.36, "end": 1285.66, "word": " طرف", "probability": 0.916748046875}, {"start": 1285.66, "end": 1286.28, "word": " الشمال", "probability": 0.9671223958333334}, {"start": 1286.28, "end": 1287.02, "word": " by", "probability": 0.90283203125}, {"start": 1287.02, "end": 1287.6, "word": " definition", "probability": 0.8486328125}, {"start": 1287.6, "end": 1288.2, "word": " بساوي", "probability": 0.825927734375}, {"start": 1288.2, "end": 1288.54, "word": " الطرف", "probability": 0.9755859375}, {"start": 1288.54, "end": 1289.0, "word": " اليمين", "probability": 0.8487955729166666}, {"start": 1289.0, "end": 1289.86, "word": " إذن", "probability": 0.73779296875}, {"start": 1289.86, "end": 1290.04, "word": " هذا", "probability": 0.908203125}, {"start": 1290.04, "end": 1290.6, "word": " تعريف", "probability": 0.9762369791666666}], "temperature": 1.0}, {"id": 51, "seek": 131873, "start": 1291.09, "end": 1318.73, "text": "لأن أنا بعرف هذا التعريف أن a أس negative one معناها واحد على a a to zero بساوي واحد حسب التعريف a to negative n و n عدد طبيعي هي عبارة عن مقلوب ال a الكل to n فكل هذه تعريفات بناء على التعريفات هذه ممكن أنبرهن خواص كثيرة", "tokens": [1211, 33456, 41850, 45030, 28480, 23758, 16712, 3615, 16572, 5172, 14739, 257, 5551, 3794, 3671, 472, 20449, 8315, 11296, 36764, 24401, 15844, 257, 257, 281, 4018, 4724, 3794, 995, 45865, 36764, 24401, 11331, 35457, 16712, 3615, 16572, 5172, 257, 281, 3671, 297, 4032, 297, 6225, 3215, 3215, 23032, 21292, 3615, 1829, 39896, 6225, 3555, 9640, 3660, 18871, 3714, 4587, 1211, 37746, 2423, 257, 2423, 28820, 281, 297, 6156, 28820, 29538, 37279, 16572, 5172, 9307, 44945, 16606, 15844, 16712, 3615, 16572, 5172, 9307, 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{"start": 1308.97, "end": 1309.25, "word": " a", "probability": 0.7490234375}, {"start": 1309.25, "end": 1309.85, "word": " الكل", "probability": 0.669921875}, {"start": 1309.85, "end": 1310.07, "word": " to", "probability": 0.63330078125}, {"start": 1310.07, "end": 1310.33, "word": " n", "probability": 0.9287109375}, {"start": 1310.33, "end": 1310.85, "word": " فكل", "probability": 0.38287353515625}, {"start": 1310.85, "end": 1311.09, "word": " هذه", "probability": 0.9072265625}, {"start": 1311.09, "end": 1311.83, "word": " تعريفات", "probability": 0.970703125}, {"start": 1311.83, "end": 1313.27, "word": " بناء", "probability": 0.863037109375}, {"start": 1313.27, "end": 1313.49, "word": " على", "probability": 0.8505859375}, {"start": 1313.49, "end": 1314.25, "word": " التعريفات", "probability": 0.92958984375}, {"start": 1314.25, "end": 1314.63, "word": " هذه", "probability": 0.92236328125}, {"start": 1314.63, "end": 1315.19, "word": " ممكن", "probability": 0.95947265625}, {"start": 1315.19, "end": 1316.09, "word": " أنبرهن", "probability": 0.81640625}, {"start": 1316.09, "end": 1316.89, "word": " خواص", "probability": 0.86962890625}, {"start": 1316.89, "end": 1318.73, "word": " كثيرة", "probability": 0.6901041666666666}], "temperature": 1.0}, {"id": 52, "seek": 134535, "start": 1320.15, "end": 1345.35, "text": "و هتشوفوا بعضها في التمرين اللي موجودة في نهاية ال section نتطرق لحاجة اسمها rational numbers الاعداد النسبية الاعداد النسبية او rational numbers بنعرفها على انها مجموعة من الاعداد الحقيقية او هي مجموعة جزئية من الاعداد الحقيقية", "tokens": [2407, 8032, 2655, 8592, 38688, 14407, 45030, 11242, 11296, 8978, 16712, 29973, 9957, 13672, 1829, 3714, 29245, 23328, 3660, 8978, 8717, 11296, 10632, 2423, 3541, 8717, 2655, 9566, 2288, 4587, 5296, 5016, 26108, 3660, 24525, 2304, 11296, 15090, 3547, 42963, 22488, 18513, 28239, 35457, 10632, 42963, 22488, 18513, 28239, 35457, 10632, 1975, 2407, 15090, 3547, 44945, 3615, 28480, 11296, 15844, 16472, 11296, 3714, 7435, 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"end": 1323.65, "word": " نهاية", "probability": 0.9554036458333334}, {"start": 1323.65, "end": 1323.81, "word": " ال", "probability": 0.9072265625}, {"start": 1323.81, "end": 1324.13, "word": " section", "probability": 0.87744140625}, {"start": 1324.13, "end": 1329.97, "word": " نتطرق", "probability": 0.8869140625}, {"start": 1329.97, "end": 1330.55, "word": " لحاجة", "probability": 0.9681396484375}, {"start": 1330.55, "end": 1331.25, "word": " اسمها", "probability": 0.94482421875}, {"start": 1331.25, "end": 1332.47, "word": " rational", "probability": 0.45751953125}, {"start": 1332.47, "end": 1333.07, "word": " numbers", "probability": 0.77978515625}, {"start": 1333.07, "end": 1333.65, "word": " الاعداد", "probability": 0.75537109375}, {"start": 1333.65, "end": 1334.33, "word": " النسبية", "probability": 0.9620768229166666}, {"start": 1334.33, "end": 1335.05, "word": " الاعداد", "probability": 0.8527018229166666}, {"start": 1335.05, "end": 1335.79, "word": " النسبية", "probability": 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"probability": 0.66943359375}, {"start": 1343.21, "end": 1343.63, "word": " مجموعة", "probability": 0.961328125}, {"start": 1343.63, "end": 1344.13, "word": " جزئية", "probability": 0.9600830078125}, {"start": 1344.13, "end": 1344.25, "word": " من", "probability": 0.99365234375}, {"start": 1344.25, "end": 1344.65, "word": " الاعداد", "probability": 0.9710286458333334}, {"start": 1344.65, "end": 1345.35, "word": " الحقيقية", "probability": 0.98681640625}], "temperature": 1.0}, {"id": 53, "seek": 136179, "start": 1346.37, "end": 1361.79, "text": "نموذجها بالرمز boldfaceq هذه الرموز الأحرف هذه أو ال letters هذه نسميها boldface يعني حرف مغمق هذه طبعا بتدل على مجموعات", "tokens": [1863, 2304, 2407, 8848, 7435, 11296, 20666, 2288, 2304, 11622, 11928, 2868, 80, 29538, 34892, 2304, 2407, 11622, 16247, 5016, 28480, 29538, 34051, 2423, 7825, 29538, 8717, 38251, 1829, 11296, 11928, 2868, 37495, 22653, 11331, 28480, 3714, 17082, 2304, 4587, 29538, 23032, 3555, 3615, 995, 39894, 3215, 1211, 15844, 3714, 7435, 2304, 45367, 9307], "avg_logprob": -0.24218749783255838, "compression_ratio": 1.462686567164179, "no_speech_prob": 0.0, "words": [{"start": 1346.37, "end": 1346.95, "word": "نموذجها", "probability": 0.6444498697916666}, {"start": 1346.95, "end": 1347.51, "word": " بالرمز", "probability": 0.93310546875}, {"start": 1347.51, "end": 1349.15, "word": " boldfaceq", "probability": 0.6451009114583334}, {"start": 1349.15, "end": 1350.01, "word": " هذه", "probability": 0.4599609375}, {"start": 1350.01, "end": 1350.63, "word": " الرموز", "probability": 0.774169921875}, {"start": 1350.63, "end": 1351.55, "word": " الأحرف", "probability": 0.876953125}, {"start": 1351.55, "end": 1352.13, "word": " هذه", "probability": 0.6162109375}, {"start": 1352.13, "end": 1353.05, "word": " أو", "probability": 0.6982421875}, {"start": 1353.05, "end": 1353.17, "word": " ال", "probability": 0.7236328125}, {"start": 1353.17, "end": 1353.49, "word": " letters", "probability": 0.818359375}, {"start": 1353.49, "end": 1353.79, "word": " هذه", "probability": 0.65185546875}, {"start": 1353.79, "end": 1354.37, "word": " نسميها", "probability": 0.83966064453125}, {"start": 1354.37, "end": 1355.15, "word": " boldface", "probability": 0.83984375}, {"start": 1355.15, "end": 1355.63, "word": " يعني", "probability": 0.7314453125}, {"start": 1355.63, "end": 1356.35, "word": " حرف", "probability": 0.996337890625}, {"start": 1356.35, "end": 1357.45, "word": " مغمق", "probability": 0.9833984375}, {"start": 1357.45, "end": 1360.15, "word": " هذه", "probability": 0.5625}, {"start": 1360.15, "end": 1360.45, "word": " طبعا", "probability": 0.85235595703125}, {"start": 1360.45, "end": 1360.93, "word": " بتدل", "probability": 0.87841796875}, {"start": 1360.93, "end": 1361.11, "word": " على", "probability": 0.83203125}, {"start": 1361.11, "end": 1361.79, "word": " مجموعات", "probability": 0.9759765625}], "temperature": 1.0}, {"id": 54, "seek": 138135, "start": 1363.17, "end": 1381.35, "text": "فال rational numbers هي كل الأعداد الحقيقية اللي ممكن كتبتها على صورة rational a على b حيث a و b أعداد صحيحة هذه مجموعة الأعداد الصحيحة bold في ال z و المقام لازم مايسويش سفر لأن القسم على سفر مش معرفة", "tokens": [5172, 6027, 15090, 3547, 39896, 28242, 16247, 22488, 18513, 21542, 38436, 4587, 10632, 13672, 1829, 3714, 43020, 9122, 2655, 3555, 2655, 11296, 15844, 20328, 13063, 3660, 15090, 257, 15844, 272, 11331, 1829, 12984, 257, 4032, 272, 5551, 22488, 18513, 20328, 5016, 1829, 5016, 3660, 29538, 3714, 7435, 2304, 2407, 27884, 16247, 22488, 18513, 31767, 5016, 1829, 5016, 3660, 11928, 8978, 2423, 710, 4032, 9673, 4587, 10943, 5296, 31377, 2304, 19446, 1829, 3794, 45865, 8592, 8608, 5172, 2288, 5296, 33456, 25062, 38251, 15844, 8608, 5172, 2288, 37893, 20449, 28480, 3660], "avg_logprob": -0.1828125043047799, "compression_ratio": 1.6923076923076923, "no_speech_prob": 0.0, "words": [{"start": 1363.17, "end": 1364.01, "word": "فال", "probability": 0.447479248046875}, {"start": 1364.01, "end": 1364.69, "word": " rational", "probability": 0.80517578125}, {"start": 1364.69, "end": 1365.33, "word": " numbers", "probability": 0.677734375}, {"start": 1365.33, "end": 1365.59, "word": " هي", "probability": 0.89208984375}, {"start": 1365.59, "end": 1365.91, "word": " كل", "probability": 0.83642578125}, {"start": 1365.91, "end": 1366.33, "word": " الأعداد", "probability": 0.841796875}, {"start": 1366.33, "end": 1366.97, "word": " الحقيقية", "probability": 0.9859619140625}, {"start": 1366.97, "end": 1367.19, "word": " اللي", "probability": 0.711181640625}, {"start": 1367.19, "end": 1367.81, "word": " ممكن", "probability": 0.986083984375}, {"start": 1367.81, "end": 1368.45, "word": " كتبتها", "probability": 0.97255859375}, {"start": 1368.45, "end": 1368.65, "word": " على", "probability": 0.9462890625}, {"start": 1368.65, "end": 1369.13, "word": " صورة", "probability": 0.9939778645833334}, {"start": 1369.13, "end": 1370.17, "word": " rational", "probability": 0.8359375}, {"start": 1370.17, "end": 1370.75, "word": " a", "probability": 0.408447265625}, {"start": 1370.75, "end": 1371.05, "word": " على", "probability": 0.79443359375}, {"start": 1371.05, "end": 1371.35, "word": " b", "probability": 0.93701171875}, {"start": 1371.35, "end": 1371.75, "word": " حيث", "probability": 0.6990559895833334}, {"start": 1371.75, "end": 1371.95, "word": " a", "probability": 0.95556640625}, {"start": 1371.95, "end": 1372.07, "word": " و", "probability": 0.921875}, {"start": 1372.07, "end": 1372.21, "word": " b", "probability": 0.392822265625}, {"start": 1372.21, "end": 1372.81, "word": " أعداد", "probability": 0.8932291666666666}, {"start": 1372.81, "end": 1373.65, "word": " صحيحة", "probability": 0.98984375}, {"start": 1373.65, "end": 1374.19, "word": " هذه", "probability": 0.26025390625}, {"start": 1374.19, "end": 1374.67, "word": " مجموعة", "probability": 0.97021484375}, {"start": 1374.67, "end": 1375.13, "word": " الأعداد", "probability": 0.9754231770833334}, {"start": 1375.13, "end": 1375.73, "word": " الصحيحة", "probability": 0.99248046875}, {"start": 1375.73, "end": 1376.01, "word": " bold", "probability": 0.439697265625}, {"start": 1376.01, "end": 1376.23, "word": " في", "probability": 0.96044921875}, {"start": 1376.23, "end": 1376.37, "word": " ال", "probability": 0.86083984375}, {"start": 1376.37, "end": 1376.57, "word": " z", "probability": 0.5634765625}, {"start": 1376.57, "end": 1377.43, "word": " و", "probability": 0.62060546875}, {"start": 1377.43, "end": 1378.05, "word": " المقام", "probability": 0.9200846354166666}, {"start": 1378.05, "end": 1378.43, "word": " لازم", "probability": 0.9348958333333334}, {"start": 1378.43, "end": 1378.93, "word": " مايسويش", "probability": 0.8734375}, {"start": 1378.93, "end": 1379.17, "word": " سفر", "probability": 0.8064778645833334}, {"start": 1379.17, "end": 1379.41, "word": " لأن", "probability": 0.781982421875}, {"start": 1379.41, "end": 1379.75, "word": " القسم", "probability": 0.984619140625}, {"start": 1379.75, "end": 1379.93, "word": " على", "probability": 0.94287109375}, {"start": 1379.93, "end": 1380.35, "word": " سفر", "probability": 0.98974609375}, {"start": 1380.35, "end": 1380.65, "word": " مش", "probability": 0.90869140625}, {"start": 1380.65, "end": 1381.35, "word": " معرفة", "probability": 0.9265950520833334}], "temperature": 1.0}, {"id": 55, "seek": 141012, "start": 1382.36, "end": 1410.12, "text": "الان لو أخدت الأعداد الحقيقية وشلت منها الأعداد النسبية طرحت منها الأعداد النسبية فالأعداد الحقيقية المتبقية بنسميها irrational numbers irrational numbers الأعداد غير النسبية إذا الأعداد النسبية هي كل الأعداد الحقيقية التي لا يمكن كتابتها على صورة rational a على b حيث a وb أعداد صحيحة والمقان بسويش سفر تمام؟", "tokens": [6027, 7649, 45164, 5551, 9778, 3215, 2655, 16247, 22488, 18513, 21542, 38436, 4587, 10632, 4032, 8592, 1211, 2655, 9154, 11296, 16247, 22488, 18513, 28239, 35457, 10632, 23032, 2288, 33753, 9154, 11296, 16247, 22488, 18513, 28239, 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1404.68, "end": 1405.02, "word": " a", "probability": 0.533203125}, {"start": 1405.02, "end": 1405.22, "word": " على", "probability": 0.78955078125}, {"start": 1405.22, "end": 1405.44, "word": " b", "probability": 0.94580078125}, {"start": 1405.44, "end": 1405.8, "word": " حيث", "probability": 0.9781901041666666}, {"start": 1405.8, "end": 1405.94, "word": " a", "probability": 0.9619140625}, {"start": 1405.94, "end": 1406.22, "word": " وb", "probability": 0.769775390625}, {"start": 1406.22, "end": 1406.74, "word": " أعداد", "probability": 0.9658203125}, {"start": 1406.74, "end": 1407.48, "word": " صحيحة", "probability": 0.98642578125}, {"start": 1407.48, "end": 1408.38, "word": " والمقان", "probability": 0.69378662109375}, {"start": 1408.38, "end": 1409.0, "word": " بسويش", "probability": 0.7274169921875}, {"start": 1409.0, "end": 1409.36, "word": " سفر", "probability": 0.87158203125}, {"start": 1409.36, "end": 1410.12, "word": " تمام؟", "probability": 0.82568359375}], "temperature": 1.0}, {"id": 56, "seek": 143885, "start": 1411.19, "end": 1438.85, "text": "طبعا لو أخدت اتحاد ال rational numbers مع ال irrational numbers بيعطوني كل الأعداد الحقيقية يعني المعنى أخر الأعداد الحقيقية احنا جزقناها إلى مجموعتين irrational numbers اتحاد ال irrational numbers طبعا المجموعتين هدول disjoint يعني منفصلتين مافيش بينهم عناصر مشتركة", "tokens": [9566, 3555, 3615, 995, 45164, 5551, 9778, 3215, 2655, 1975, 2655, 5016, 18513, 2423, 15090, 3547, 20449, 2423, 39914, 3547, 4724, 40228, 9566, 11536, 1829, 28242, 16247, 22488, 18513, 21542, 38436, 4587, 10632, 37495, 22653, 9673, 3615, 1863, 7578, 5551, 34740, 16247, 22488, 18513, 21542, 38436, 4587, 10632, 1975, 5016, 8315, 10874, 11622, 4587, 8315, 11296, 30731, 3714, 7435, 2304, 2407, 34268, 9957, 39914, 3547, 1975, 2655, 5016, 18513, 2423, 39914, 3547, 23032, 3555, 3615, 995, 9673, 7435, 2304, 2407, 34268, 9957, 8032, 3215, 12610, 717, 48613, 37495, 22653, 9154, 5172, 36520, 2655, 9957, 19446, 41185, 8592, 49374, 16095, 18871, 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حقائق معروفة يعني", "tokens": [5172, 3555, 1863, 39648, 1422, 992, 45164, 8978, 18871, 16254, 1594, 4032, 8978, 18871, 16254, 3714, 7435, 2304, 2407, 27884, 10874, 11622, 19986, 10632, 9154, 2423, 1594, 6156, 3555, 1863, 38251, 1829, 9673, 7435, 2304, 2407, 27884, 10874, 11622, 19986, 10632, 8717, 36178, 11296, 1594, 36632, 11242, 995, 6156, 3555, 1863, 38251, 1829, 11296, 1422, 1594, 1975, 15730, 2423, 1249, 307, 257, 1422, 2519, 295, 264, 2519, 497, 44945, 16606, 15844, 28239, 19913, 2288, 10632, 29538, 1975, 15730, 29538, 28242, 11296, 11331, 4587, 16373, 4587, 20449, 32887, 5172, 3660, 37495, 22653], "avg_logprob": -0.25611412622358487, "compression_ratio": 1.7305699481865284, "no_speech_prob": 0.0, "words": [{"start": 1574.94, "end": 1575.56, "word": "فبنقول", "probability": 0.6136474609375}, {"start": 1575.56, "end": 1576.04, "word": " sub", "probability": 0.59130859375}, {"start": 1576.04, "end": 1576.58, "word": " set", "probability": 0.63623046875}, {"start": 1576.58, "end": 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"word": " odd", "probability": 0.91943359375}], "temperature": 1.0}, {"id": 76, "seek": 194284, "start": 1917.1, "end": 1942.84, "text": "وطبعا الـ K هي عدد صحيح ربي إذا P تربية بساوي أربعة K تربية زاد أربعة K زاد واحد وهذا بساوي اتنين في اتنين K تربية زاد اتنين K مع بعض زاد واحد وهذا بساوي اتنين في M زاد واحد حيث M عدد صحيح", "tokens": [2407, 9566, 3555, 3615, 995, 2423, 39184, 591, 39896, 6225, 3215, 3215, 20328, 5016, 1829, 5016, 12602, 21292, 11933, 15730, 430, 6055, 25513, 10632, 4724, 3794, 995, 45865, 5551, 25513, 27884, 591, 6055, 25513, 10632, 30767, 18513, 5551, 25513, 27884, 591, 30767, 18513, 36764, 24401, 37037, 15730, 4724, 3794, 995, 45865, 1975, 2655, 1863, 9957, 8978, 1975, 2655, 1863, 9957, 591, 6055, 25513, 10632, 30767, 18513, 1975, 2655, 1863, 9957, 591, 20449, 45030, 11242, 30767, 18513, 36764, 24401, 37037, 15730, 4724, 3794, 995, 45865, 1975, 2655, 1863, 9957, 8978, 376, 30767, 18513, 36764, 24401, 11331, 1829, 12984, 376, 6225, 3215, 3215, 20328, 5016, 1829, 5016], "avg_logprob": -0.1544811302479708, "compression_ratio": 2.3214285714285716, "no_speech_prob": 0.0, "words": [{"start": 1917.1, "end": 1917.68, "word": "وطبعا", "probability": 0.775244140625}, {"start": 1917.68, "end": 1917.88, "word": " الـ", "probability": 0.406982421875}, {"start": 1917.88, "end": 1918.0, "word": " K", "probability": 0.3583984375}, {"start": 1918.0, "end": 1918.18, "word": " هي", "probability": 0.794921875}, {"start": 1918.18, "end": 1918.5, "word": " عدد", "probability": 0.96435546875}, {"start": 1918.5, "end": 1918.96, "word": " صحيح", "probability": 0.9906005859375}, {"start": 1918.96, "end": 1919.64, "word": " ربي", "probability": 0.80615234375}, {"start": 1919.64, "end": 1920.84, "word": " إذا", "probability": 0.5}, {"start": 1920.84, "end": 1921.06, "word": " P", "probability": 0.82861328125}, {"start": 1921.06, "end": 1921.52, "word": " تربية", "probability": 0.806640625}, {"start": 1921.52, "end": 1922.02, "word": " بساوي", "probability": 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0.5771484375}, {"start": 1930.18, "end": 1930.68, "word": " اتنين", "probability": 0.964111328125}, {"start": 1930.68, "end": 1930.94, "word": " K", "probability": 0.93115234375}, {"start": 1930.94, "end": 1931.68, "word": " تربية", "probability": 0.9661458333333334}, {"start": 1931.68, "end": 1934.08, "word": " زاد", "probability": 0.91064453125}, {"start": 1934.08, "end": 1934.62, "word": " اتنين", "probability": 0.97998046875}, {"start": 1934.62, "end": 1935.08, "word": " K", "probability": 0.96533203125}, {"start": 1935.08, "end": 1935.34, "word": " مع", "probability": 0.986328125}, {"start": 1935.34, "end": 1935.8, "word": " بعض", "probability": 0.990234375}, {"start": 1935.8, "end": 1936.22, "word": " زاد", "probability": 0.963623046875}, {"start": 1936.22, "end": 1936.66, "word": " واحد", "probability": 0.98486328125}, {"start": 1936.66, "end": 1937.58, "word": " وهذا", "probability": 0.95947265625}, {"start": 1937.58, "end": 1938.16, "word": " بساوي", "probability": 0.94970703125}, {"start": 1938.16, "end": 1938.76, "word": " اتنين", "probability": 0.988525390625}, {"start": 1938.76, "end": 1939.02, "word": " في", "probability": 0.95166015625}, {"start": 1939.02, "end": 1939.42, "word": " M", "probability": 0.91796875}, {"start": 1939.42, "end": 1939.9, "word": " زاد", "probability": 0.955810546875}, {"start": 1939.9, "end": 1940.44, "word": " واحد", "probability": 0.988525390625}, {"start": 1940.44, "end": 1941.5, "word": " حيث", "probability": 0.9864908854166666}, {"start": 1941.5, "end": 1941.84, "word": " M", "probability": 0.85107421875}, {"start": 1941.84, "end": 1942.32, "word": " عدد", "probability": 0.9856770833333334}, {"start": 1942.32, "end": 1942.84, "word": " صحيح", "probability": 0.9964599609375}], "temperature": 1.0}, {"id": 77, "seek": 197251, "start": 1944.29, "end": 1972.51, "text": "إذاً P تربية طلع بساوة اتنين في عدد صحيح زائد واحد وبالتالي هذا اقض Contradiction تناقض لأن احنا عندنا P تربية P تربية is even okay إذاً هذا برهان ال why طيب إذا احنا وصلنا إلى أن P تربية even بقدر أن P even الآن ال P و ال Q have no common factor other than واحد مافيش بينهم عامل مشترك إلا الواحد", "tokens": [28814, 15730, 14111, 430, 6055, 25513, 10632, 23032, 1211, 3615, 4724, 3794, 995, 2407, 3660, 1975, 2655, 1863, 9957, 8978, 6225, 3215, 3215, 20328, 5016, 1829, 5016, 30767, 16373, 3215, 36764, 24401, 46599, 6027, 2655, 6027, 1829, 23758, 1975, 4587, 11242, 4839, 6206, 4105, 6055, 8315, 4587, 11242, 5296, 33456, 1975, 5016, 8315, 43242, 8315, 430, 6055, 25513, 10632, 430, 6055, 25513, 10632, 307, 754, 1392, 11933, 15730, 14111, 23758, 4724, 2288, 3224, 7649, 2423, 983, 23032, 1829, 3555, 11933, 15730, 1975, 5016, 8315, 4032, 36520, 8315, 30731, 14739, 430, 6055, 25513, 10632, 754, 4724, 28543, 2288, 14739, 430, 754, 6024, 48506, 2423, 430, 4032, 2423, 1249, 362, 572, 2689, 5952, 661, 813, 36764, 24401, 19446, 41185, 8592, 49374, 16095, 6225, 10943, 1211, 37893, 2655, 31747, 11933, 15040, 2423, 14407, 24401], "avg_logprob": 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0.984375}, {"start": 1947.81, "end": 1948.95, "word": " وبالتالي", "probability": 0.936328125}, {"start": 1948.95, "end": 1949.27, "word": " هذا", "probability": 0.8349609375}, {"start": 1949.27, "end": 1949.59, "word": " اقض", "probability": 0.3402506510416667}, {"start": 1949.59, "end": 1950.41, "word": " Contradiction", "probability": 0.81396484375}, {"start": 1950.41, "end": 1951.07, "word": " تناقض", "probability": 0.82427978515625}, {"start": 1951.07, "end": 1952.03, "word": " لأن", "probability": 0.7626953125}, {"start": 1952.03, "end": 1952.25, "word": " احنا", "probability": 0.8111979166666666}, {"start": 1952.25, "end": 1952.55, "word": " عندنا", "probability": 0.71435546875}, {"start": 1952.55, "end": 1952.75, "word": " P", "probability": 0.94677734375}, {"start": 1952.75, "end": 1953.27, "word": " تربية", "probability": 0.9659830729166666}, {"start": 1953.27, "end": 1953.61, "word": " P", "probability": 0.595703125}, {"start": 1953.61, "end": 1954.05, "word": " تربية", 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" وصلنا", "probability": 0.9754231770833334}, {"start": 1962.01, "end": 1962.17, "word": " إلى", "probability": 0.91943359375}, {"start": 1962.17, "end": 1962.39, "word": " أن", "probability": 0.36279296875}, {"start": 1962.39, "end": 1962.67, "word": " P", "probability": 0.92919921875}, {"start": 1962.67, "end": 1963.19, "word": " تربية", "probability": 0.98046875}, {"start": 1963.19, "end": 1963.57, "word": " even", "probability": 0.83544921875}, {"start": 1963.57, "end": 1964.01, "word": " بقدر", "probability": 0.79345703125}, {"start": 1964.01, "end": 1964.15, "word": " أن", "probability": 0.47607421875}, {"start": 1964.15, "end": 1964.45, "word": " P", "probability": 0.92333984375}, {"start": 1964.45, "end": 1964.97, "word": " even", "probability": 0.6474609375}, {"start": 1964.97, "end": 1966.61, "word": " الآن", "probability": 0.6871337890625}, {"start": 1966.61, "end": 1967.31, "word": " ال", "probability": 0.80712890625}, {"start": 1967.31, "end": 1967.61, "word": " P", "probability": 0.9521484375}, {"start": 1967.61, "end": 1967.85, "word": " و", "probability": 0.7822265625}, {"start": 1967.85, "end": 1967.89, "word": " ال", "probability": 0.7216796875}, {"start": 1967.89, "end": 1968.13, "word": " Q", "probability": 0.92919921875}, {"start": 1968.13, "end": 1968.47, "word": " have", "probability": 0.50390625}, {"start": 1968.47, "end": 1968.75, "word": " no", "probability": 0.93408203125}, {"start": 1968.75, "end": 1969.15, "word": " common", "probability": 0.919921875}, {"start": 1969.15, "end": 1969.71, "word": " factor", "probability": 0.91650390625}, {"start": 1969.71, "end": 1969.99, "word": " other", "probability": 0.84619140625}, {"start": 1969.99, "end": 1970.31, "word": " than", "probability": 0.96337890625}, {"start": 1970.31, "end": 1970.63, "word": " واحد", "probability": 0.93115234375}, {"start": 1970.63, "end": 1970.95, "word": " مافيش", "probability": 0.9427083333333334}, {"start": 1970.95, "end": 1971.21, "word": " بينهم", "probability": 0.813232421875}, {"start": 1971.21, "end": 1971.41, "word": " عامل", "probability": 0.9855143229166666}, {"start": 1971.41, "end": 1971.77, "word": " مشترك", "probability": 0.97216796875}, {"start": 1971.77, "end": 1971.99, "word": " إلا", "probability": 0.802978515625}, {"start": 1971.99, "end": 1972.51, "word": " الواحد", "probability": 0.97412109375}], "temperature": 1.0}, {"id": 78, "seek": 200164, "start": 1974.22, "end": 2001.64, "text": "و ال P even إذا لازم ال Q يكون odd لأن لو كان ال Q even و ال P even ففيه عامل مشترك بينهم اتنين على الأقل او أربعة وهذا بتناقض مع ايه أنه احنا فرضين أنه مافيش common factor بين ال P و ال Q إلا الواحد تمام إذا أنا عندي هنا سنتجة أن ال Q لازم يكون odd الآن", "tokens": [2407, 2423, 430, 754, 11933, 15730, 5296, 31377, 2304, 2423, 1249, 7251, 30544, 7401, 5296, 33456, 45164, 25961, 2423, 1249, 754, 4032, 2423, 430, 754, 6156, 41185, 3224, 6225, 10943, 1211, 37893, 2655, 31747, 49374, 16095, 1975, 2655, 1863, 9957, 15844, 16247, 4587, 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"probability": 0.95458984375}, {"start": 1976.08, "end": 1976.22, "word": " ال", "probability": 0.7998046875}, {"start": 1976.22, "end": 1976.42, "word": " Q", "probability": 0.8994140625}, {"start": 1976.42, "end": 1976.96, "word": " يكون", "probability": 0.939208984375}, {"start": 1976.96, "end": 1977.46, "word": " odd", "probability": 0.369140625}, {"start": 1977.46, "end": 1977.94, "word": " لأن", "probability": 0.797119140625}, {"start": 1977.94, "end": 1978.14, "word": " لو", "probability": 0.8603515625}, {"start": 1978.14, "end": 1978.5, "word": " كان", "probability": 0.962890625}, {"start": 1978.5, "end": 1979.28, "word": " ال", "probability": 0.9306640625}, {"start": 1979.28, "end": 1979.66, "word": " Q", "probability": 0.7265625}, {"start": 1979.66, "end": 1980.34, "word": " even", "probability": 0.85693359375}, {"start": 1980.34, "end": 1980.98, "word": " و", "probability": 0.876953125}, {"start": 1980.98, "end": 1981.1, "word": " ال", "probability": 0.56640625}, {"start": 1981.1, "end": 1981.24, "word": " P", "probability": 0.99462890625}, {"start": 1981.24, "end": 1981.58, "word": " even", "probability": 0.92431640625}, {"start": 1981.58, "end": 1982.52, "word": " ففيه", "probability": 0.6649576822916666}, {"start": 1982.52, "end": 1982.92, "word": " عامل", "probability": 0.9881184895833334}, {"start": 1982.92, "end": 1983.32, "word": " مشترك", "probability": 0.9703776041666666}, {"start": 1983.32, "end": 1983.78, "word": " بينهم", "probability": 0.959716796875}, {"start": 1983.78, "end": 1984.68, "word": " اتنين", "probability": 0.83203125}, {"start": 1984.68, "end": 1984.82, "word": " على", "probability": 0.93994140625}, {"start": 1984.82, "end": 1985.24, "word": " الأقل", "probability": 0.890625}, {"start": 1985.24, "end": 1986.16, "word": " او", "probability": 0.798095703125}, {"start": 1986.16, "end": 1986.68, "word": " أربعة", "probability": 0.7132161458333334}, {"start": 1986.68, "end": 1987.96, "word": " وهذا", "probability": 0.75}, {"start": 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R خواص الترتيب على R", "tokens": [6027, 3541, 16712, 7649, 1829, 18871, 2407, 7649, 3224, 2423, 1668, 7221, 295, 497, 16490, 2407, 33546, 16712, 2288, 31371, 3555, 15844, 497], "avg_logprob": -0.1725260398040215, "compression_ratio": 1.1136363636363635, "no_speech_prob": 0.0, "words": [{"start": 2142.47, "end": 2142.91, "word": "ال", "probability": 0.955078125}, {"start": 2142.91, "end": 2143.23, "word": " section", "probability": 0.231201171875}, {"start": 2143.23, "end": 2143.95, "word": " التاني", "probability": 0.9085286458333334}, {"start": 2143.95, "end": 2145.65, "word": " عنوانه", "probability": 0.9290771484375}, {"start": 2145.65, "end": 2145.77, "word": " ال", "probability": 0.70068359375}, {"start": 2145.77, "end": 2146.11, "word": " order", "probability": 0.96142578125}, {"start": 2146.11, "end": 2146.87, "word": " properties", "probability": 0.693359375}, {"start": 2146.87, "end": 2147.13, "word": " of", "probability": 0.96240234375}, {"start": 2147.13, "end": 2147.45, 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\ No newline at end of file
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/be-bepplyPs_raw.json b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/be-bepplyPs_raw.json
new file mode 100644
index 0000000000000000000000000000000000000000..6050db6e76f6b7a3e1afb183ed2cb5b70fa8b03d
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/be-bepplyPs_raw.json
@@ -0,0 +1 @@
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0.42822265625}, {"start": 74.25, "end": 74.51, "word": " عندي", "probability": 0.9326171875}, {"start": 74.51, "end": 74.63, "word": " ال", "probability": 0.92138671875}, {"start": 74.63, "end": 75.05, "word": " function", "probability": 0.83203125}, {"start": 75.05, "end": 76.13, "word": " g", "probability": 0.6123046875}, {"start": 76.13, "end": 76.33, "word": " of", "probability": 0.92236328125}, {"start": 76.33, "end": 76.57, "word": " x", "probability": 0.9794921875}, {"start": 76.57, "end": 77.07, "word": " بيساوي", "probability": 0.96474609375}, {"start": 77.07, "end": 77.41, "word": " إيص", "probability": 0.5298258463541666}, {"start": 77.41, "end": 77.79, "word": " واحد", "probability": 0.983154296875}, {"start": 77.79, "end": 77.95, "word": " على", "probability": 0.7470703125}, {"start": 77.95, "end": 78.29, "word": " x", "probability": 0.611328125}, {"start": 78.29, "end": 79.45, "word": " ده", "probability": 0.6719970703125}, {"start": 79.45, "end": 79.89, "word": " قيمها", "probability": 0.9539388020833334}, {"start": 79.89, "end": 80.49, "word": " موجبة", "probability": 0.9532877604166666}, {"start": 80.49, "end": 81.99, "word": " أصغر", "probability": 0.9388427734375}, {"start": 81.99, "end": 82.39, "word": " من", "probability": 0.9892578125}, {"start": 82.39, "end": 84.79, "word": " أو", "probability": 0.720703125}, {"start": 84.79, "end": 85.63, "word": " يساوي", "probability": 0.9473876953125}, {"start": 85.63, "end": 86.15, "word": " m", "probability": 0.666015625}, {"start": 86.15, "end": 88.07, "word": " لكل", "probability": 0.981689453125}, {"start": 88.07, "end": 88.61, "word": " x", "probability": 0.89794921875}, {"start": 88.61, "end": 90.37, "word": " في", "probability": 0.974609375}, {"start": 90.37, "end": 92.67, "word": " الجوار", "probability": 0.9334309895833334}], "temperature": 1.0}, {"id": 4, "seek": 12206, "start": 94.58, "end": 122.06, "text": "أو الـ delta الجوار من اليمين للصفر طيب by Archimedean property يوجد ان عدد طبيعي او ان واحد", "tokens": [10721, 2407, 2423, 39184, 8289, 25724, 2407, 9640, 9154, 2423, 32640, 9957, 24976, 9381, 5172, 2288, 23032, 1829, 3555, 538, 10984, 332, 4858, 282, 4707, 7251, 29245, 3215, 16472, 6225, 3215, 3215, 23032, 21292, 3615, 1829, 1975, 2407, 16472, 36764, 24401], "avg_logprob": -0.2209821357613518, "compression_ratio": 1.1557377049180328, "no_speech_prob": 0.0, "words": [{"start": 94.58, "end": 94.96, "word": "أو", "probability": 0.6727294921875}, {"start": 94.96, "end": 95.14, "word": " الـ", "probability": 0.673095703125}, {"start": 95.14, "end": 95.46, "word": " delta", "probability": 0.3486328125}, {"start": 95.46, "end": 96.9, "word": " الجوار", "probability": 0.78564453125}, {"start": 96.9, "end": 97.08, "word": " من", "probability": 0.9814453125}, {"start": 97.08, "end": 97.7, "word": " اليمين", "probability": 0.8306477864583334}, {"start": 97.7, "end": 100.86, "word": " للصفر", "probability": 0.7432861328125}, {"start": 100.86, "end": 108.68, "word": " طيب", "probability": 0.8483072916666666}, {"start": 108.68, "end": 109.14, "word": " by", "probability": 0.57568359375}, {"start": 109.14, "end": 112.14, "word": " Archimedean", "probability": 0.908203125}, {"start": 112.14, "end": 114.14, "word": " property", "probability": 0.91845703125}, {"start": 114.14, "end": 116.4, "word": " يوجد", "probability": 0.9708658854166666}, {"start": 116.4, "end": 118.4, "word": " ان", "probability": 0.57958984375}, {"start": 118.4, "end": 118.88, "word": " عدد", "probability": 0.9505208333333334}, {"start": 118.88, "end": 119.64, "word": " طبيعي", "probability": 0.9677734375}, {"start": 119.64, "end": 121.16, "word": " او", "probability": 0.7457275390625}, {"start": 121.16, "end": 121.46, "word": " ان", "probability": 0.9052734375}, {"start": 121.46, "end": 122.06, "word": " واحد", "probability": 0.98291015625}], "temperature": 1.0}, {"id": 5, "seek": 14996, "start": 123.12, "end": 149.96, "text": "عدد طبيعي بحيث ان ان واحد اكبر من ان طيب also by Archimedean property انا عندي delta zero هذه عدد موجة", "tokens": [3615, 3215, 3215, 23032, 21292, 3615, 1829, 4724, 5016, 1829, 12984, 16472, 16472, 36764, 24401, 1975, 4117, 26890, 9154, 16472, 23032, 1829, 3555, 611, 538, 10984, 332, 4858, 282, 4707, 1975, 8315, 18871, 16254, 8289, 4018, 29538, 6225, 3215, 3215, 3714, 29245, 3660], "avg_logprob": -0.30415481803092087, "compression_ratio": 1.1627906976744187, "no_speech_prob": 0.0, "words": [{"start": 123.12, "end": 123.76, "word": "عدد", "probability": 0.7242838541666666}, {"start": 123.76, "end": 124.6, "word": " طبيعي", "probability": 0.884033203125}, {"start": 124.6, "end": 126.18, "word": " بحيث", "probability": 0.8985595703125}, {"start": 126.18, "end": 126.42, "word": " ان", "probability": 0.681640625}, {"start": 126.42, "end": 128.4, "word": " ان", "probability": 0.08746337890625}, {"start": 128.4, "end": 129.1, "word": " واحد", "probability": 0.948974609375}, {"start": 129.1, "end": 132.7, "word": " اكبر", "probability": 0.90087890625}, {"start": 132.7, "end": 132.94, "word": " من", "probability": 0.9599609375}, {"start": 132.94, "end": 133.28, "word": " ان", "probability": 0.2164306640625}, {"start": 133.28, "end": 138.22, "word": " طيب", "probability": 0.80615234375}, {"start": 138.22, "end": 138.88, "word": " also", "probability": 0.41748046875}, {"start": 138.88, "end": 142.14, "word": " by", "probability": 0.83837890625}, {"start": 142.14, "end": 143.64, "word": " Archimedean", "probability": 0.861083984375}, {"start": 143.64, "end": 144.62, "word": " property", "probability": 0.8505859375}, {"start": 144.62, "end": 147.72, "word": " انا", "probability": 0.81396484375}, {"start": 147.72, "end": 148.08, "word": " عندي", "probability": 0.75634765625}, {"start": 148.08, "end": 148.42, "word": " delta", "probability": 0.5166015625}, {"start": 148.42, "end": 148.86, "word": " zero", "probability": 0.7919921875}, {"start": 148.86, "end": 149.14, "word": " هذه", "probability": 0.26806640625}, {"start": 149.14, "end": 149.46, "word": " عدد", "probability": 0.9650065104166666}, {"start": 149.46, "end": 149.96, "word": " موجة", "probability": 0.9825846354166666}], "temperature": 1.0}, {"id": 6, "seek": 17643, "start": 151.37, "end": 176.43, "text": "أنا عندي delta zero عدد موجد بيقدّي أن يوجد عدد طبيعي M2 عدد طبيعي بحيث أنه واحد على M2 أصغر من delta zero left end", "tokens": [10721, 8315, 18871, 16254, 8289, 4018, 6225, 3215, 3215, 3714, 29245, 3215, 4724, 1829, 28543, 11703, 1829, 14739, 7251, 29245, 3215, 6225, 3215, 3215, 23032, 21292, 3615, 1829, 376, 17, 6225, 3215, 3215, 23032, 21292, 3615, 1829, 4724, 5016, 1829, 12984, 14739, 3224, 36764, 24401, 15844, 376, 17, 5551, 9381, 17082, 2288, 9154, 8289, 4018, 1411, 917], "avg_logprob": -0.19490840851232924, "compression_ratio": 1.3615384615384616, "no_speech_prob": 0.0, "words": [{"start": 151.37, "end": 151.65, "word": "أنا", "probability": 0.57391357421875}, {"start": 151.65, "end": 151.97, "word": " عندي", "probability": 0.788818359375}, {"start": 151.97, "end": 152.25, "word": " delta", "probability": 0.5751953125}, {"start": 152.25, "end": 152.65, "word": " zero", "probability": 0.55517578125}, {"start": 152.65, "end": 153.01, "word": " عدد", "probability": 0.9446614583333334}, {"start": 153.01, "end": 153.49, "word": " موجد", "probability": 0.8541666666666666}, {"start": 153.49, "end": 154.11, "word": " بيقدّي", "probability": 0.7123046875}, {"start": 154.11, "end": 154.29, "word": " أن", "probability": 0.380126953125}, {"start": 154.29, "end": 155.05, "word": " يوجد", "probability": 0.9381510416666666}, {"start": 155.05, "end": 156.59, "word": " عدد", "probability": 0.98291015625}, {"start": 156.59, "end": 157.35, "word": " طبيعي", "probability": 0.9752197265625}, {"start": 157.35, "end": 158.31, "word": " M2", "probability": 0.662841796875}, {"start": 158.31, "end": 159.21, "word": " عدد", "probability": 0.9451497395833334}, {"start": 159.21, "end": 160.27, "word": " طبيعي", "probability": 0.9827880859375}, {"start": 160.27, "end": 162.09, "word": " بحيث", "probability": 0.9158935546875}, {"start": 162.09, "end": 162.71, "word": " أنه", "probability": 0.72900390625}, {"start": 162.71, "end": 164.69, "word": " واحد", "probability": 0.8603515625}, {"start": 164.69, "end": 165.07, "word": " على", "probability": 0.7861328125}, {"start": 165.07, "end": 166.27, "word": " M2", "probability": 0.948486328125}, {"start": 166.27, "end": 168.03, "word": " أصغر", "probability": 0.9747314453125}, {"start": 168.03, "end": 168.31, "word": " من", "probability": 0.98876953125}, {"start": 168.31, "end": 168.75, "word": " delta", "probability": 0.75341796875}, {"start": 168.75, "end": 169.45, "word": " zero", "probability": 0.87255859375}, {"start": 169.45, "end": 175.93, "word": " left", "probability": 0.7978515625}, {"start": 175.93, "end": 176.43, "word": " end", "probability": 0.76416015625}], "temperature": 1.0}, {"id": 7, "seek": 20412, "start": 179.36, "end": 204.12, "text": "let n بساوي ال maximum الأكبر بين n واحد و n اتنين واضح لأن n أكبر من نهو يساوي n واحد و أكبر من نهو يساوي n اتنين وبالتالي x", "tokens": [2631, 297, 4724, 3794, 995, 45865, 2423, 6674, 16247, 4117, 26890, 49374, 297, 36764, 24401, 4032, 297, 1975, 2655, 1863, 9957, 4032, 46958, 5016, 5296, 33456, 297, 5551, 4117, 26890, 9154, 8717, 3224, 2407, 7251, 3794, 995, 45865, 297, 36764, 24401, 4032, 5551, 4117, 26890, 9154, 8717, 3224, 2407, 7251, 3794, 995, 45865, 297, 1975, 2655, 1863, 9957, 46599, 6027, 2655, 6027, 1829, 2031], "avg_logprob": -0.2800480769230769, "compression_ratio": 1.7894736842105263, "no_speech_prob": 0.0, "words": [{"start": 179.36, "end": 179.9, "word": "let", "probability": 0.1494140625}, {"start": 179.9, "end": 180.38, "word": " n", "probability": 0.6591796875}, {"start": 180.38, "end": 181.2, "word": " بساوي", "probability": 0.6174545288085938}, {"start": 181.2, "end": 181.34, "word": " ال", "probability": 0.638671875}, {"start": 181.34, "end": 181.92, "word": " maximum", "probability": 0.92529296875}, {"start": 181.92, "end": 182.86, "word": " الأكبر", "probability": 0.8658854166666666}, {"start": 182.86, "end": 183.28, "word": " بين", "probability": 0.8623046875}, {"start": 183.28, "end": 184.86, "word": " n", "probability": 0.56396484375}, {"start": 184.86, "end": 185.5, "word": " واحد", "probability": 0.822265625}, {"start": 185.5, "end": 185.68, "word": " و", "probability": 0.970703125}, {"start": 185.68, "end": 186.16, "word": " n", "probability": 0.52099609375}, {"start": 186.16, "end": 187.82, "word": " اتنين", "probability": 0.926513671875}, {"start": 187.82, "end": 192.74, "word": " واضح", "probability": 0.761962890625}, {"start": 192.74, "end": 193.16, "word": " لأن", "probability": 0.4742431640625}, {"start": 193.16, "end": 193.56, "word": " n", "probability": 0.6982421875}, {"start": 193.56, "end": 194.16, "word": " أكبر", "probability": 0.9471028645833334}, {"start": 194.16, "end": 194.34, "word": " من", "probability": 0.87841796875}, {"start": 194.34, "end": 194.54, "word": " نهو", "probability": 0.380126953125}, {"start": 194.54, "end": 195.08, "word": " يساوي", "probability": 0.9241943359375}, {"start": 195.08, "end": 195.42, "word": " n", "probability": 0.80908203125}, {"start": 195.42, "end": 196.06, "word": " واحد", "probability": 0.9814453125}, {"start": 196.06, "end": 197.0, "word": " و", "probability": 0.576171875}, {"start": 197.0, "end": 197.44, "word": " أكبر", "probability": 0.8907877604166666}, {"start": 197.44, "end": 197.64, "word": " من", "probability": 0.9873046875}, {"start": 197.64, "end": 197.86, "word": " نهو", "probability": 0.8899739583333334}, {"start": 197.86, "end": 198.52, "word": " يساوي", "probability": 0.9893798828125}, {"start": 198.52, "end": 198.78, "word": " n", "probability": 0.94873046875}, {"start": 198.78, "end": 200.48, "word": " اتنين", "probability": 0.9447021484375}, {"start": 200.48, "end": 203.08, "word": " وبالتالي", "probability": 0.85087890625}, {"start": 203.08, "end": 204.12, "word": " x", "probability": 0.259521484375}], "temperature": 1.0}, {"id": 8, "seek": 23764, "start": 211.26, "end": 237.64, "text": "لو أخدت xn بيساوي واحد على n بيساوي واحد على n فهذا دفلع ينتمي إلى ال delta zero neighborhood للسفر ال right neighborhood للسفر اللي هو هذا", "tokens": [1211, 2407, 5551, 9778, 3215, 2655, 2031, 77, 4724, 1829, 3794, 995, 45865, 36764, 24401, 15844, 297, 4724, 1829, 3794, 995, 45865, 36764, 24401, 15844, 297, 6156, 3224, 15730, 11778, 5172, 1211, 3615, 7251, 29399, 2304, 1829, 30731, 2423, 8289, 4018, 7630, 24976, 3794, 5172, 2288, 2423, 558, 7630, 24976, 3794, 5172, 2288, 13672, 1829, 31439, 23758], "avg_logprob": -0.3246228366062559, "compression_ratio": 1.5789473684210527, "no_speech_prob": 0.0, "words": [{"start": 211.26000000000002, "end": 212.46, "word": "لو", "probability": 0.5068817138671875}, {"start": 212.46, "end": 212.94, "word": " أخدت", "probability": 0.82122802734375}, {"start": 212.94, "end": 213.78, "word": " xn", "probability": 0.3353271484375}, {"start": 213.78, "end": 215.4, "word": " بيساوي", "probability": 0.54449462890625}, {"start": 215.4, "end": 215.84, "word": " واحد", "probability": 0.8603515625}, {"start": 215.84, "end": 216.04, "word": " على", "probability": 0.68212890625}, {"start": 216.04, "end": 216.44, "word": " n", "probability": 0.49169921875}, {"start": 216.44, "end": 220.48, "word": " بيساوي", "probability": 0.876611328125}, {"start": 220.48, "end": 221.08, "word": " واحد", "probability": 0.985595703125}, {"start": 221.08, "end": 221.28, "word": " على", "probability": 0.89453125}, {"start": 221.28, "end": 221.74, "word": " n", "probability": 0.67138671875}, {"start": 221.74, "end": 226.08, "word": " فهذا", "probability": 0.931640625}, {"start": 226.08, "end": 226.76, "word": " دفلع", "probability": 0.6464385986328125}, {"start": 226.76, "end": 228.24, "word": " ينتمي", "probability": 0.9669189453125}, {"start": 228.24, "end": 229.14, "word": " إلى", "probability": 0.60498046875}, {"start": 229.14, "end": 230.48, "word": " ال", "probability": 0.90869140625}, {"start": 230.48, "end": 230.84, "word": " delta", "probability": 0.40771484375}, {"start": 230.84, "end": 231.48, "word": " zero", "probability": 0.74560546875}, {"start": 231.48, "end": 232.08, "word": " neighborhood", "probability": 0.75927734375}, {"start": 232.08, "end": 233.1, "word": " للسفر", "probability": 0.822021484375}, {"start": 233.1, "end": 234.8, "word": " ال", "probability": 0.45458984375}, {"start": 234.8, "end": 235.24, "word": " right", "probability": 0.7607421875}, {"start": 235.24, "end": 235.76, "word": " neighborhood", "probability": 0.9287109375}, {"start": 235.76, "end": 236.84, "word": " للسفر", "probability": 0.983154296875}, {"start": 236.84, "end": 237.06, "word": " اللي", "probability": 0.87890625}, {"start": 237.06, "end": 237.26, "word": " هو", "probability": 0.97509765625}, {"start": 237.26, "end": 237.64, "word": " هذا", "probability": 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0.501708984375}, {"start": 685.38, "end": 685.64, "word": " X", "probability": 0.94775390625}, {"start": 685.64, "end": 686.04, "word": " تقول", "probability": 0.975341796875}, {"start": 686.04, "end": 686.26, "word": " إذا", "probability": 0.83544921875}, {"start": 686.26, "end": 686.74, "word": " سفر", "probability": 0.6765950520833334}, {"start": 686.74, "end": 686.94, "word": " من", "probability": 0.99560546875}, {"start": 686.94, "end": 687.72, "word": " اليسار", "probability": 0.994140625}, {"start": 687.72, "end": 688.74, "word": " بيساوي", "probability": 0.728369140625}, {"start": 688.74, "end": 689.22, "word": " سفر", "probability": 0.9591471354166666}, {"start": 689.22, "end": 691.6, "word": " تمام؟", "probability": 0.713134765625}, {"start": 691.6, "end": 692.2, "word": " وهيك", "probability": 0.7139485677083334}, {"start": 692.2, "end": 692.7, "word": " بتكون", "probability": 0.7403564453125}, {"start": 692.7, "end": 693.7, "word": " فرهنة", "probability": 0.679833984375}], "temperature": 1.0}, {"id": 25, "seek": 72410, "start": 694.54, "end": 724.1, "text": "الجزر التاني إذا هذا مثال على function النهاية تبعتها من اليمين غير موجودة بينما النهاية من اليسار على نفس النقطة موجودة كذلك مثال تالت لو أخدنا let H of X بسوء واحد على واحد زائد E أس واحد على X", "tokens": [6027, 7435, 11622, 2288, 16712, 7649, 1829, 11933, 15730, 23758, 50113, 6027, 15844, 2445, 28239, 11296, 10632, 6055, 3555, 34268, 11296, 9154, 2423, 32640, 9957, 32771, 13546, 3714, 29245, 23328, 3660, 49374, 15042, 28239, 11296, 10632, 9154, 45595, 3794, 9640, 15844, 8717, 36178, 28239, 47432, 3660, 3714, 29245, 23328, 3660, 9122, 8848, 23275, 50113, 6027, 6055, 6027, 2655, 45164, 5551, 9778, 3215, 8315, 718, 389, 295, 1783, 4724, 3794, 2407, 38207, 36764, 24401, 15844, 36764, 24401, 30767, 16373, 3215, 462, 5551, 3794, 36764, 24401, 15844, 1783], "avg_logprob": -0.19055316708553796, "compression_ratio": 1.7671957671957672, "no_speech_prob": 3.5762786865234375e-07, "words": [{"start": 694.54, "end": 695.24, "word": "الجزر", "probability": 0.7686767578125}, {"start": 695.24, "end": 695.82, "word": " التاني", "probability": 0.8984375}, {"start": 695.82, "end": 696.0, "word": " إذا", "probability": 0.471710205078125}, {"start": 696.0, "end": 696.32, "word": " هذا", "probability": 0.60791015625}, {"start": 696.32, "end": 696.9, "word": " مثال", "probability": 0.94775390625}, {"start": 696.9, "end": 698.14, "word": " على", "probability": 0.8388671875}, {"start": 698.14, "end": 698.86, "word": " function", "probability": 0.8642578125}, {"start": 698.86, "end": 700.48, "word": " النهاية", "probability": 0.8821614583333334}, {"start": 700.48, "end": 700.96, "word": " تبعتها", "probability": 0.80364990234375}, {"start": 700.96, "end": 701.1, "word": " من", "probability": 0.98681640625}, {"start": 701.1, "end": 701.54, "word": " اليمين", "probability": 0.8414713541666666}, {"start": 701.54, "end": 701.92, "word": " غير", "probability": 0.96240234375}, {"start": 701.92, "end": 702.56, "word": " موجودة", "probability": 0.9892578125}, {"start": 702.56, "end": 703.0, "word": " بينما", "probability": 0.94091796875}, {"start": 703.0, "end": 703.4, "word": " النهاية", "probability": 0.8898111979166666}, {"start": 703.4, "end": 703.62, "word": " من", "probability": 0.9921875}, {"start": 703.62, "end": 704.24, "word": " اليسار", "probability": 0.9934895833333334}, {"start": 704.24, "end": 705.08, "word": " على", "probability": 0.5966796875}, {"start": 705.08, "end": 705.38, "word": " نفس", "probability": 0.98974609375}, {"start": 705.38, "end": 705.9, "word": " النقطة", "probability": 0.95751953125}, {"start": 705.9, "end": 706.74, "word": " موجودة", "probability": 0.98583984375}, {"start": 706.74, "end": 711.62, "word": " كذلك", "probability": 0.9671223958333334}, {"start": 711.62, "end": 712.4, "word": " مثال", "probability": 0.891357421875}, {"start": 712.4, "end": 713.02, "word": " تالت", "probability": 0.9524739583333334}, {"start": 713.02, "end": 713.36, "word": " لو", "probability": 0.890625}, {"start": 713.36, "end": 714.26, "word": " أخدنا", "probability": 0.9566650390625}, {"start": 714.26, "end": 714.8, "word": " let", "probability": 0.52294921875}, {"start": 714.8, "end": 716.36, "word": " H", "probability": 0.32958984375}, {"start": 716.36, "end": 716.6, "word": " of", "probability": 0.389892578125}, {"start": 716.6, "end": 717.06, "word": " X", "probability": 0.97998046875}, {"start": 717.06, "end": 718.26, "word": " بسوء", "probability": 0.606201171875}, {"start": 718.26, "end": 721.1, "word": " واحد", "probability": 0.88427734375}, {"start": 721.1, "end": 721.34, "word": " على", "probability": 0.69140625}, {"start": 721.34, "end": 721.88, "word": " واحد", "probability": 0.986572265625}, {"start": 721.88, "end": 722.52, "word": " زائد", "probability": 0.86669921875}, {"start": 722.52, "end": 722.84, "word": " E", "probability": 0.63671875}, {"start": 722.84, "end": 723.12, "word": " أس", "probability": 0.5244140625}, {"start": 723.12, "end": 723.56, "word": " واحد", "probability": 0.980712890625}, {"start": 723.56, "end": 723.66, "word": " على", "probability": 0.69384765625}, {"start": 723.66, "end": 724.1, "word": " X", "probability": 0.89892578125}], "temperature": 1.0}, {"id": 26, "seek": 75463, "start": 726.33, "end": 754.63, "text": "لو أخدنا الدالة هذه طبعا و X هنا لا يساوي سفر فالممكن اثبات ان ال limit للدالة هذه لما X تقول اللي هي سفر من اليمين موجودة و بتساوي سفر", "tokens": [1211, 2407, 5551, 9778, 3215, 8315, 32748, 6027, 3660, 29538, 23032, 3555, 3615, 995, 4032, 1783, 34105, 20193, 7251, 3794, 995, 45865, 8608, 5172, 2288, 6156, 45340, 43020, 1975, 12984, 3555, 9307, 16472, 2423, 4948, 24976, 3215, 6027, 3660, 29538, 5296, 15042, 1783, 6055, 39648, 13672, 1829, 39896, 8608, 5172, 2288, 9154, 2423, 32640, 9957, 3714, 29245, 23328, 3660, 4032, 39894, 3794, 995, 45865, 8608, 5172, 2288], "avg_logprob": -0.20795036479830742, "compression_ratio": 1.6137931034482758, "no_speech_prob": 0.0, "words": [{"start": 726.33, "end": 726.65, "word": "لو", "probability": 0.740234375}, {"start": 726.65, "end": 727.03, "word": " أخدنا", "probability": 0.867919921875}, {"start": 727.03, "end": 727.47, "word": " الدالة", "probability": 0.9615885416666666}, {"start": 727.47, "end": 727.93, "word": " هذه", "probability": 0.78369140625}, {"start": 727.93, "end": 731.55, "word": " طبعا", "probability": 0.803497314453125}, {"start": 731.55, "end": 731.67, "word": " و", "probability": 0.30224609375}, {"start": 731.67, "end": 732.03, "word": " X", "probability": 0.62060546875}, {"start": 732.03, "end": 732.33, "word": " هنا", "probability": 0.90087890625}, {"start": 732.33, "end": 732.53, "word": " لا", "probability": 0.85888671875}, {"start": 732.53, "end": 733.07, "word": " يساوي", "probability": 0.8284912109375}, {"start": 733.07, "end": 733.43, "word": " سفر", "probability": 0.83251953125}, {"start": 733.43, "end": 739.53, "word": " فالممكن", "probability": 0.8583984375}, {"start": 739.53, "end": 740.27, "word": " اثبات", "probability": 0.888916015625}, {"start": 740.27, "end": 741.67, "word": " ان", "probability": 0.63134765625}, {"start": 741.67, "end": 741.83, "word": " ال", "probability": 0.55224609375}, {"start": 741.83, "end": 742.23, "word": " limit", "probability": 0.8955078125}, {"start": 742.23, "end": 744.23, "word": " للدالة", "probability": 0.960205078125}, {"start": 744.23, "end": 744.75, "word": " هذه", "probability": 0.87255859375}, {"start": 744.75, "end": 745.21, "word": " لما", "probability": 0.90087890625}, {"start": 745.21, "end": 745.51, "word": " X", "probability": 0.9384765625}, {"start": 745.51, "end": 746.03, "word": " تقول", "probability": 0.92724609375}, {"start": 746.03, "end": 747.13, "word": " اللي", "probability": 0.5467529296875}, {"start": 747.13, "end": 747.27, "word": " هي", "probability": 0.22021484375}, {"start": 747.27, "end": 747.71, "word": " سفر", "probability": 0.9485677083333334}, {"start": 747.71, "end": 747.97, "word": " من", "probability": 0.98779296875}, {"start": 747.97, "end": 748.61, "word": " اليمين", "probability": 0.833984375}, {"start": 748.61, "end": 750.57, "word": " موجودة", "probability": 0.9842529296875}, {"start": 750.57, "end": 751.57, "word": " و", "probability": 0.65576171875}, {"start": 751.57, "end": 752.57, "word": " بتساوي", "probability": 0.9603271484375}, {"start": 752.57, "end": 754.63, "word": " سفر", "probability": 0.9825846354166666}], "temperature": 1.0}, {"id": 27, "seek": 78101, "start": 758.39, "end": 781.01, "text": "و ال limit لنفس الدالة لما x تقول إلى سفر من اليسار أيضا موجودة لكن بالساوي واحد وبالتالي هذا طبعا هذا الإمثال محلول", "tokens": [2407, 2423, 4948, 5296, 1863, 36178, 32748, 6027, 3660, 5296, 15042, 2031, 6055, 39648, 30731, 8608, 5172, 2288, 9154, 45595, 3794, 9640, 36632, 11242, 995, 3714, 29245, 23328, 3660, 44381, 20666, 3794, 995, 45865, 36764, 24401, 46599, 6027, 2655, 6027, 1829, 23758, 23032, 3555, 3615, 995, 23758, 33688, 2304, 12984, 6027, 3714, 5016, 1211, 12610], "avg_logprob": -0.22642298734613828, "compression_ratio": 1.4676258992805755, "no_speech_prob": 5.960464477539063e-08, "words": [{"start": 758.39, "end": 758.79, "word": "و", "probability": 0.6533203125}, {"start": 758.79, "end": 760.37, "word": " ال", "probability": 0.1368408203125}, {"start": 760.37, "end": 760.81, "word": " limit", "probability": 0.83935546875}, {"start": 760.81, "end": 763.53, "word": " لنفس", "probability": 0.8780924479166666}, {"start": 763.53, "end": 764.33, "word": " الدالة", "probability": 0.9200846354166666}, {"start": 764.33, "end": 764.77, "word": " لما", "probability": 0.84521484375}, {"start": 764.77, "end": 765.35, "word": " x", "probability": 0.59130859375}, {"start": 765.35, "end": 765.79, "word": " تقول", "probability": 0.797119140625}, {"start": 765.79, "end": 766.09, "word": " إلى", "probability": 0.56201171875}, {"start": 766.09, "end": 766.61, "word": " سفر", "probability": 0.7811686197916666}, {"start": 766.61, "end": 767.61, "word": " من", "probability": 0.94677734375}, {"start": 767.61, "end": 768.27, "word": " اليسار", "probability": 0.9939778645833334}, {"start": 768.27, "end": 768.85, "word": " أيضا", "probability": 0.8797200520833334}, {"start": 768.85, "end": 770.99, "word": " موجودة", "probability": 0.9775390625}, {"start": 770.99, "end": 772.11, "word": " لكن", "probability": 0.78662109375}, {"start": 772.11, "end": 773.05, "word": " بالساوي", "probability": 0.72882080078125}, {"start": 773.05, "end": 773.83, "word": " واحد", "probability": 0.950439453125}, {"start": 773.83, "end": 778.57, "word": " وبالتالي", "probability": 0.93125}, {"start": 778.57, "end": 778.85, "word": " هذا", "probability": 0.81494140625}, {"start": 778.85, "end": 779.31, "word": " طبعا", "probability": 0.8160400390625}, {"start": 779.31, "end": 779.57, "word": " هذا", "probability": 0.72998046875}, {"start": 779.57, "end": 780.23, "word": " الإمثال", "probability": 0.737060546875}, {"start": 780.23, "end": 781.01, "word": " محلول", "probability": 0.9857177734375}], "temperature": 1.0}, {"id": 28, "seek": 81175, "start": 782.95, "end": 811.75, "text": "بالتفصيل في الكتاب وبرهانه أسهل بكتير من المثال اللي فات ويعتمد برضه على المثال السابق اللي هو المثال رقم اتنين see the text انظروا في الكتاب للتفاصيل الحالية لأن هذا المثال الآخر زي ال signal function", "tokens": [3555, 6027, 2655, 5172, 9381, 26895, 8978, 33251, 2655, 16758, 4032, 26890, 3224, 7649, 3224, 5551, 3794, 3224, 1211, 4724, 4117, 2655, 13546, 9154, 9673, 12984, 6027, 13672, 1829, 6156, 9307, 4032, 1829, 34268, 2304, 3215, 4724, 43042, 3224, 15844, 9673, 12984, 6027, 21136, 16758, 4587, 13672, 1829, 31439, 9673, 12984, 6027, 12602, 4587, 2304, 1975, 2655, 1863, 9957, 536, 264, 2487, 16472, 19913, 2288, 14407, 8978, 33251, 2655, 16758, 24976, 2655, 5172, 33546, 26895, 21542, 6027, 10632, 5296, 33456, 23758, 9673, 12984, 6027, 6024, 95, 34740, 30767, 1829, 2423, 6358, 2445], "avg_logprob": -0.1518817210710177, "compression_ratio": 1.758974358974359, "no_speech_prob": 5.960464477539063e-08, "words": [{"start": 782.95, "end": 783.81, "word": "بالتفصيل", "probability": 0.927001953125}, {"start": 783.81, "end": 783.99, "word": " في", "probability": 0.87353515625}, {"start": 783.99, "end": 784.51, "word": " الكتاب", "probability": 0.98388671875}, {"start": 784.51, "end": 785.33, "word": " وبرهانه", "probability": 0.79912109375}, {"start": 785.33, "end": 785.93, "word": " أسهل", "probability": 0.9246826171875}, {"start": 785.93, "end": 787.15, "word": " بكتير", "probability": 0.9451904296875}, {"start": 787.15, "end": 787.43, "word": " من", "probability": 0.990234375}, {"start": 787.43, "end": 788.09, "word": " المثال", "probability": 0.95263671875}, {"start": 788.09, "end": 789.01, "word": " اللي", "probability": 0.837890625}, {"start": 789.01, "end": 789.41, "word": " فات", "probability": 0.89013671875}, {"start": 789.41, "end": 792.67, "word": " ويعتمد", "probability": 0.8435546875}, {"start": 792.67, "end": 793.03, "word": " برضه", "probability": 0.9723307291666666}, {"start": 793.03, "end": 793.19, "word": " على", "probability": 0.90185546875}, {"start": 793.19, "end": 793.63, "word": " المثال", "probability": 0.9876302083333334}, {"start": 793.63, "end": 794.23, "word": " السابق", "probability": 0.9884440104166666}, {"start": 794.23, "end": 794.49, "word": " اللي", "probability": 0.933349609375}, {"start": 794.49, "end": 794.61, "word": " هو", "probability": 0.99365234375}, {"start": 794.61, "end": 795.23, "word": " المثال", "probability": 0.8447265625}, {"start": 795.23, "end": 796.75, "word": " رقم", "probability": 0.8849283854166666}, {"start": 796.75, "end": 797.13, "word": " اتنين", "probability": 0.83056640625}, {"start": 797.13, "end": 797.83, "word": " see", "probability": 0.072509765625}, {"start": 797.83, "end": 798.21, "word": " the", "probability": 0.88818359375}, {"start": 798.21, "end": 798.77, "word": " text", "probability": 0.96728515625}, {"start": 798.77, "end": 801.53, "word": " انظروا", 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"probability": 0.90625}], "temperature": 1.0}, {"id": 29, "seek": 83871, "start": 812.83, "end": 838.71, "text": "الـ one-sided limits both exist لكن مش متساويات وبالتالي ال limit عند السفر للدالة هذه غير موجودة okay تمام لان هذه بعض الأمثلة على ال one-sided limits خلينا ننتقل إلى موضوع ال infinite limits", "tokens": [6027, 39184, 472, 12, 30941, 10406, 1293, 2514, 44381, 37893, 44650, 3794, 995, 45865, 9307, 46599, 6027, 2655, 6027, 1829, 2423, 4948, 43242, 21136, 5172, 2288, 24976, 3215, 6027, 3660, 29538, 32771, 13546, 3714, 29245, 23328, 3660, 1392, 46811, 10943, 5296, 7649, 29538, 45030, 11242, 16247, 2304, 12984, 37977, 15844, 2423, 472, 12, 30941, 10406, 16490, 1211, 9957, 995, 8717, 29399, 4587, 1211, 30731, 3714, 2407, 11242, 45367, 2423, 13785, 10406], "avg_logprob": -0.20399304831193554, "compression_ratio": 1.4974093264248705, "no_speech_prob": 0.0, "words": [{"start": 812.83, "end": 813.95, "word": "الـ", "probability": 0.47174072265625}, {"start": 813.95, "end": 814.15, "word": " one", "probability": 0.8359375}, {"start": 814.15, "end": 814.49, "word": "-sided", "probability": 0.67919921875}, {"start": 814.49, "end": 814.97, "word": " limits", "probability": 0.9755859375}, {"start": 814.97, "end": 815.29, "word": " both", "probability": 0.78564453125}, {"start": 815.29, "end": 815.87, "word": " exist", "probability": 0.931640625}, {"start": 815.87, "end": 816.15, "word": " لكن", "probability": 0.8681640625}, {"start": 816.15, "end": 816.41, "word": " مش", "probability": 0.96728515625}, {"start": 816.41, "end": 817.31, "word": " متساويات", "probability": 0.92109375}, {"start": 817.31, "end": 818.39, "word": " وبالتالي", "probability": 0.92333984375}, {"start": 818.39, "end": 818.53, "word": " ال", "probability": 0.63720703125}, {"start": 818.53, "end": 818.85, "word": " limit", "probability": 0.88330078125}, {"start": 818.85, "end": 819.93, "word": " عند", "probability": 0.77880859375}, {"start": 819.93, "end": 820.73, "word": " السفر", "probability": 0.8401692708333334}, {"start": 820.73, "end": 821.29, "word": " للدالة", "probability": 0.7872314453125}, {"start": 821.29, "end": 821.57, "word": " هذه", "probability": 0.428955078125}, {"start": 821.57, "end": 821.89, "word": " غير", "probability": 0.988037109375}, {"start": 821.89, "end": 823.07, "word": " موجودة", "probability": 0.9912109375}, {"start": 823.07, "end": 823.87, "word": " okay", "probability": 0.51904296875}, {"start": 823.87, "end": 824.43, "word": " تمام", "probability": 0.970458984375}, {"start": 824.43, "end": 825.91, "word": " لان", "probability": 0.26690673828125}, {"start": 825.91, "end": 826.11, "word": " هذه", "probability": 0.505859375}, {"start": 826.11, "end": 826.45, "word": " بعض", "probability": 0.970947265625}, {"start": 826.45, "end": 827.13, "word": " الأمثلة", "probability": 0.9293212890625}, {"start": 827.13, "end": 827.51, "word": " على", "probability": 0.78369140625}, {"start": 827.51, "end": 828.59, "word": " ال", "probability": 0.95751953125}, {"start": 828.59, "end": 829.63, "word": " one", "probability": 0.90234375}, {"start": 829.63, "end": 830.13, "word": "-sided", "probability": 0.94970703125}, {"start": 830.13, "end": 831.67, "word": " limits", "probability": 0.9873046875}, {"start": 831.67, "end": 834.33, "word": " خلينا", "probability": 0.839111328125}, {"start": 834.33, "end": 835.53, "word": " ننتقل", "probability": 0.9100341796875}, {"start": 835.53, "end": 836.63, "word": " إلى", "probability": 0.63134765625}, {"start": 836.63, "end": 837.39, "word": " موضوع", "probability": 0.981689453125}, {"start": 837.39, "end": 837.59, "word": " ال", "probability": 0.98974609375}, {"start": 837.59, "end": 838.01, "word": " infinite", "probability": 0.904296875}, {"start": 838.01, "end": 838.71, "word": " limits", "probability": 0.96826171875}], "temperature": 1.0}, {"id": 30, "seek": 87955, "start": 859.75, "end": 879.55, "text": "فداخل definition let F be function from A to R وC be cluster point of set A", "tokens": [5172, 3215, 47283, 1211, 7123, 718, 479, 312, 2445, 490, 316, 281, 497, 4032, 34, 312, 13630, 935, 295, 992, 316], "avg_logprob": -0.41015625, "compression_ratio": 0.9642857142857143, "no_speech_prob": 0.0, "words": [{"start": 859.75, "end": 860.51, "word": "فداخل", "probability": 0.623779296875}, {"start": 860.51, "end": 861.31, "word": " definition", "probability": 0.32275390625}, {"start": 861.31, "end": 866.41, "word": " let", "probability": 0.331787109375}, {"start": 866.41, "end": 866.97, "word": " F", "probability": 0.35546875}, {"start": 866.97, "end": 867.33, "word": " be", "probability": 0.7431640625}, {"start": 867.33, "end": 868.05, "word": " function", "probability": 0.71142578125}, {"start": 868.05, "end": 868.71, "word": " from", "probability": 0.876953125}, {"start": 868.71, "end": 869.11, "word": " A", "probability": 0.90966796875}, {"start": 869.11, "end": 869.35, "word": " to", "probability": 0.97998046875}, {"start": 869.35, "end": 869.95, "word": " R", "probability": 0.99365234375}, {"start": 869.95, "end": 872.49, "word": " وC", "probability": 0.56689453125}, {"start": 872.49, "end": 873.05, "word": " be", "probability": 0.37890625}, {"start": 873.05, "end": 873.83, "word": " cluster", "probability": 0.84326171875}, {"start": 873.83, "end": 876.89, "word": " point", "probability": 0.9521484375}, {"start": 876.89, "end": 878.43, "word": " of", "probability": 0.9609375}, {"start": 878.43, "end": 879.09, "word": " set", "probability": 0.91552734375}, {"start": 879.09, "end": 879.55, "word": " A", "probability": 0.98388671875}], "temperature": 1.0}, {"id": 31, "seek": 91429, "start": 887.05, "end": 914.29, "text": "نقول إن قيمة f of x كما أن x هو c بساوي plus infinity إذا تحقق الشرط التالي for any alpha real number", "tokens": [1863, 39648, 36145, 12174, 32640, 3660, 283, 295, 2031, 9122, 15042, 14739, 2031, 31439, 269, 4724, 3794, 995, 45865, 1804, 13202, 11933, 15730, 6055, 5016, 4587, 4587, 25124, 2288, 9566, 16712, 6027, 1829, 337, 604, 8961, 957, 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"temperature": 1.0}, {"id": 32, "seek": 94417, "start": 915.75, "end": 944.17, "text": "There exists delta تعتمد على alpha على موجب يعني هذا شبه بتعريف بتعريف أن ال sequence xn ال limit بتاعتها تكون plus infinity فقلنا هذا معناه أن xn أكبر من أي real alpha", "tokens": [51, 511, 68, 8198, 8289, 6055, 34268, 2304, 3215, 15844, 8961, 15844, 3714, 29245, 3555, 37495, 22653, 23758, 13412, 3555, 3224, 39894, 3615, 16572, 5172, 39894, 3615, 16572, 5172, 14739, 2423, 8310, 2031, 77, 2423, 4948, 39894, 995, 34268, 11296, 6055, 30544, 1804, 13202, 6156, 4587, 1211, 8315, 23758, 20449, 8315, 3224, 14739, 2031, 77, 5551, 4117, 26890, 9154, 36632, 957, 8961], "avg_logprob": -0.34002976758139475, "compression_ratio": 1.4, "no_speech_prob": 0.0, "words": [{"start": 915.75, "end": 916.13, "word": "There", "probability": 0.5078328450520834}, {"start": 916.13, "end": 916.65, "word": " exists", "probability": 0.67578125}, {"start": 916.65, "end": 917.21, "word": " delta", "probability": 0.416259765625}, 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"end": 932.25, "word": " sequence", "probability": 0.4951171875}, {"start": 932.25, "end": 933.17, "word": " xn", "probability": 0.5440673828125}, {"start": 933.17, "end": 934.07, "word": " ال", "probability": 0.2158203125}, {"start": 934.07, "end": 934.33, "word": " limit", "probability": 0.8623046875}, {"start": 934.33, "end": 934.81, "word": " بتاعتها", "probability": 0.8985595703125}, {"start": 934.81, "end": 935.23, "word": " تكون", "probability": 0.960693359375}, {"start": 935.23, "end": 935.69, "word": " plus", "probability": 0.9130859375}, {"start": 935.69, "end": 936.37, "word": " infinity", "probability": 0.845703125}, {"start": 936.37, "end": 939.33, "word": " فقلنا", "probability": 0.7569580078125}, {"start": 939.33, "end": 939.91, "word": " هذا", "probability": 0.748046875}, {"start": 939.91, "end": 940.45, "word": " معناه", "probability": 0.9794921875}, {"start": 940.45, "end": 940.61, "word": " أن", "probability": 0.603515625}, {"start": 940.61, "end": 941.27, "word": " xn", "probability": 0.955810546875}, {"start": 941.27, "end": 941.81, "word": " أكبر", "probability": 0.9586588541666666}, {"start": 941.81, "end": 942.03, "word": " من", "probability": 0.98779296875}, {"start": 942.03, "end": 942.41, "word": " أي", "probability": 0.802734375}, {"start": 942.41, "end": 943.61, "word": " real", "probability": 0.8310546875}, {"start": 943.61, "end": 944.17, "word": " alpha", "probability": 0.81640625}], "temperature": 1.0}, {"id": 33, "seek": 97515, "start": 945.81, "end": 975.15, "text": "لكل n أكبر من أو ساوي capital N حيث capital N عدد طبيعي يعتمد على Alpha مش هيك التعريف تقريبا وهذا نفس الحاجة ما معناه ان ال limit لل function نقطة لساوي infinity هذا معناه ان اخلي ال function أكبر من أي given Alpha لكل X قريبة من C", "tokens": [1211, 28820, 297, 5551, 4117, 26890, 9154, 34051, 8608, 995, 45865, 4238, 426, 11331, 1829, 12984, 4238, 426, 6225, 3215, 3215, 23032, 21292, 3615, 1829, 7251, 34268, 2304, 3215, 15844, 20588, 37893, 39896, 4117, 16712, 3615, 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{"start": 1197.69, "end": 1197.89, "word": " على", "probability": 0.6806640625}, {"start": 1197.89, "end": 1198.41, "word": " الجذر", "probability": 0.8502604166666666}, {"start": 1198.41, "end": 1199.55, "word": " التربيهي", "probability": 0.85244140625}, {"start": 1199.55, "end": 1201.03, "word": " ل", "probability": 0.88720703125}, {"start": 1201.03, "end": 1201.41, "word": " alpha", "probability": 0.78662109375}, {"start": 1201.41, "end": 1206.69, "word": " او", "probability": 0.6810302734375}, {"start": 1206.69, "end": 1206.97, "word": " ممكن", "probability": 0.972412109375}, {"start": 1206.97, "end": 1207.35, "word": " تقول", "probability": 0.991455078125}, {"start": 1207.35, "end": 1207.53, "word": " ان", "probability": 0.66455078125}, {"start": 1207.53, "end": 1207.71, "word": " انا", "probability": 0.883544921875}, {"start": 1207.71, "end": 1208.17, "word": " بدي", "probability": 0.80517578125}, {"start": 1208.17, "end": 1208.71, "word": " f", "probability": 0.7216796875}, 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طيب لما يكون هذا الكلام صح معناه absolute x أصغر من delta و هذا معناه ان x تربية أصغر من delta تربية لأن absolute x بيساوي جدر x تربية", "tokens": [3555, 1863, 8592, 38688, 1975, 1829, 8592, 11331, 25528, 3660, 11778, 9957, 7578, 23032, 1829, 3555, 5296, 15042, 7251, 30544, 23758, 2423, 28820, 10943, 20328, 5016, 20449, 8315, 3224, 8236, 2031, 5551, 9381, 17082, 2288, 9154, 8289, 4032, 23758, 20449, 8315, 3224, 16472, 2031, 6055, 25513, 10632, 5551, 9381, 17082, 2288, 9154, 8289, 6055, 25513, 10632, 5296, 33456, 8236, 2031, 4724, 1829, 3794, 995, 45865, 10874, 3215, 2288, 2031, 6055, 25513, 10632], "avg_logprob": -0.17990153456387453, "compression_ratio": 1.6291390728476822, "no_speech_prob": 0.0, "words": [{"start": 1345.02, "end": 1346.02, "word": "بنشوف", "probability": 0.82000732421875}, {"start": 1346.02, "end": 1347.02, "word": " ايش", "probability": 0.7157796223958334}, {"start": 1347.02, "end": 1347.32, "word": " حياة", "probability": 0.689208984375}, {"start": 1347.32, "end": 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واحد أكبر من absolute alpha صح؟ طب و absolute alpha أكبر من أو ساوي alpha لأي real number دا من ال absolute value لل number أكبر من أو ساوي ال number إذن هي اللي أثبتت أن ال F of X أكبر من ال given alpha", "tokens": [2304, 4587, 1211, 37746, 33251, 3794, 2288, 23758, 13672, 1829, 31439, 8236, 8961, 30767, 995, 25708, 36764, 24401, 23032, 3555, 8236, 8961, 30767, 995, 25708, 36764, 24401, 5551, 4117, 26890, 9154, 8236, 8961, 20328, 5016, 22807, 23032, 3555, 4032, 8236, 8961, 5551, 4117, 26890, 9154, 34051, 8608, 995, 45865, 8961, 5296, 10721, 1829, 957, 1230, 11778, 995, 9154, 2423, 8236, 2158, 24976, 1230, 5551, 4117, 26890, 9154, 34051, 8608, 995, 45865, 2423, 1230, 11933, 8848, 1863, 39896, 13672, 1829, 5551, 12984, 3555, 2655, 2655, 14739, 2423, 479, 295, 1783, 5551, 4117, 26890, 9154, 2423, 2212, 8961], "avg_logprob": -0.19684278842100164, "compression_ratio": 2.026178010471204, "no_speech_prob": 0.0, "words": [{"start": 1400.98, "end": 1401.82, "word": "مقلوب", "probability": 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"end": 2323.69, "word": " of", "probability": 0.9619140625}, {"start": 2323.69, "end": 2324.23, "word": " x", "probability": 0.97265625}, {"start": 2324.23, "end": 2325.59, "word": " أصغر", "probability": 0.9652099609375}, {"start": 2325.59, "end": 2325.79, "word": " من", "probability": 0.99462890625}, {"start": 2325.79, "end": 2326.27, "word": " beta", "probability": 0.56396484375}, {"start": 2326.27, "end": 2326.75, "word": " وهنا", "probability": 0.70166015625}, {"start": 2326.75, "end": 2327.13, "word": " طبعا", "probability": 0.9437255859375}, {"start": 2327.13, "end": 2327.65, "word": " beta", "probability": 0.4951171875}], "temperature": 1.0}, {"id": 81, "seek": 234234, "start": 2334.04, "end": 2342.34, "text": "هنا سنستبدل F of X أصغر من أو يساوي G of X أصغر من D", "tokens": [3224, 8315, 8608, 1863, 14851, 44510, 1211, 479, 295, 1783, 5551, 9381, 17082, 2288, 9154, 34051, 7251, 3794, 995, 45865, 460, 295, 1783, 5551, 9381, 17082, 2288, 9154, 413], "avg_logprob": -0.4278645972410838, "compression_ratio": 1.1408450704225352, "no_speech_prob": 0.0, "words": [{"start": 2334.0399999999995, "end": 2334.7999999999997, "word": "هنا", "probability": 0.373779296875}, {"start": 2334.7999999999997, "end": 2335.56, "word": " سنستبدل", "probability": 0.730029296875}, {"start": 2335.56, "end": 2337.1, "word": " F", "probability": 0.06884765625}, {"start": 2337.1, "end": 2337.48, "word": " of", "probability": 0.1326904296875}, {"start": 2337.48, "end": 2338.0, "word": " X", "probability": 0.9814453125}, {"start": 2338.0, "end": 2339.24, "word": " أصغر", "probability": 0.78399658203125}, {"start": 2339.24, "end": 2339.42, "word": " من", "probability": 0.93212890625}, {"start": 2339.42, "end": 2339.6, "word": " أو", "probability": 0.50146484375}, {"start": 2339.6, "end": 2340.28, "word": " يساوي", "probability": 0.8585205078125}, {"start": 2340.28, "end": 2340.56, "word": " G", "probability": 0.61572265625}, {"start": 2340.56, "end": 2340.82, "word": " of", "probability": 0.87646484375}, {"start": 2340.82, "end": 2341.18, "word": " X", "probability": 0.9951171875}, {"start": 2341.18, "end": 2341.82, "word": " أصغر", "probability": 0.9681396484375}, {"start": 2341.82, "end": 2342.02, "word": " من", "probability": 0.8291015625}, {"start": 2342.02, "end": 2342.34, "word": " D", "probability": 0.348876953125}], "temperature": 1.0}, {"id": 82, "seek": 237429, "start": 2348.31, "end": 2374.29, "text": "أصغر من beta و بالتالي هذا معناه حسب التعريف انه limit f of x لما x تقول يا c تساوي ثالث من beta دكتور بس there is this delta تتمد على alpha مش إيه؟ اه هاد المفروض يكون alpha كانت و الأن في الجزء التاني هصير beta مظبوط كرامك صحيح", "tokens": [10721, 9381, 17082, 2288, 9154, 9861, 4032, 20666, 2655, 6027, 1829, 23758, 20449, 8315, 3224, 11331, 35457, 16712, 3615, 16572, 5172, 16472, 3224, 4948, 283, 295, 2031, 5296, 15042, 2031, 6055, 39648, 35186, 269, 6055, 3794, 995, 45865, 38637, 6027, 12984, 9154, 9861, 11778, 4117, 2655, 13063, 4724, 3794, 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"probability": 0.958984375}, {"start": 2351.41, "end": 2351.79, "word": " حسب", "probability": 0.91943359375}, {"start": 2351.79, "end": 2353.03, "word": " التعريف", "probability": 0.7620849609375}, {"start": 2353.03, "end": 2353.47, "word": " انه", "probability": 0.5966796875}, {"start": 2353.47, "end": 2353.83, "word": " limit", "probability": 0.9765625}, {"start": 2353.83, "end": 2354.27, "word": " f", "probability": 0.6552734375}, {"start": 2354.27, "end": 2354.53, "word": " of", "probability": 0.841796875}, {"start": 2354.53, "end": 2354.97, "word": " x", "probability": 0.92041015625}, {"start": 2354.97, "end": 2355.99, "word": " لما", "probability": 0.41217041015625}, {"start": 2355.99, "end": 2356.27, "word": " x", "probability": 0.87548828125}, {"start": 2356.27, "end": 2356.59, "word": " تقول", "probability": 0.90869140625}, {"start": 2356.59, "end": 2356.81, "word": " يا", "probability": 0.2578125}, {"start": 2356.81, "end": 2357.13, "word": " c", "probability": 0.73681640625}, {"start": 2357.13, "end": 2357.85, "word": " تساوي", "probability": 0.776153564453125}, {"start": 2357.85, "end": 2358.51, "word": " ثالث", "probability": 0.7666015625}, {"start": 2358.51, "end": 2358.71, "word": " من", "probability": 0.47509765625}, {"start": 2358.71, "end": 2359.15, "word": " beta", "probability": 0.423583984375}, {"start": 2359.15, "end": 2359.91, "word": " دكتور", "probability": 0.8822021484375}, {"start": 2359.91, "end": 2360.49, "word": " بس", "probability": 0.96728515625}, {"start": 2360.49, "end": 2360.87, "word": " there", "probability": 0.8291015625}, {"start": 2360.87, "end": 2361.07, "word": " is", "probability": 0.7548828125}, {"start": 2361.07, "end": 2361.21, "word": " this", "probability": 0.55517578125}, {"start": 2361.21, "end": 2361.47, "word": " delta", "probability": 0.6591796875}, {"start": 2361.47, "end": 2361.97, "word": " تتمد", "probability": 0.87109375}, {"start": 2361.97, "end": 2362.21, "word": " على", "probability": 0.86083984375}, {"start": 2362.21, "end": 2362.51, "word": " alpha", "probability": 0.64599609375}, {"start": 2362.51, "end": 2362.75, "word": " مش", "probability": 0.96826171875}, {"start": 2362.75, "end": 2365.21, "word": " إيه؟", "probability": 0.4625244140625}, {"start": 2365.21, "end": 2365.49, "word": " اه", "probability": 0.7052001953125}, {"start": 2365.49, "end": 2365.73, "word": " هاد", "probability": 0.5438232421875}, {"start": 2365.73, "end": 2366.11, "word": " المفروض", "probability": 0.843017578125}, {"start": 2366.11, "end": 2366.35, "word": " يكون", "probability": 0.7568359375}, {"start": 2366.35, "end": 2366.77, "word": " alpha", "probability": 0.75830078125}, {"start": 2366.77, "end": 2368.05, "word": " كانت", "probability": 0.97802734375}, {"start": 2368.05, "end": 2369.93, "word": " و", "probability": 0.62255859375}, {"start": 2369.93, "end": 2370.49, "word": " الأن", "probability": 0.771484375}, {"start": 2370.49, "end": 2370.73, "word": " في", "probability": 0.76513671875}, {"start": 2370.73, "end": 2371.21, "word": " الجزء", "probability": 0.984375}, {"start": 2371.21, "end": 2371.71, "word": " التاني", "probability": 0.9396158854166666}, {"start": 2371.71, "end": 2372.29, "word": " هصير", "probability": 0.8411458333333334}, {"start": 2372.29, "end": 2372.77, "word": " beta", "probability": 0.896484375}, {"start": 2372.77, "end": 2373.45, "word": " مظبوط", "probability": 0.8958984375}, {"start": 2373.45, "end": 2373.91, "word": " كرامك", "probability": 0.75830078125}, {"start": 2373.91, "end": 2374.29, "word": " صحيح", "probability": 0.9476318359375}], "temperature": 1.0}, {"id": 83, "seek": 239856, "start": 2375.12, "end": 2398.56, "text": "إذا هذا بيكون إيه هكذا بيكون برهان الجزء التاني المرة الجاية هنشوف ناخد تطبيقات على النظرية هذه وناخد مزيد من النظريات على ال infinite limits okay شكرا لصراعكم ونشوفكم ان شاء الله المرة الجاية", "tokens": [28814, 15730, 23758, 4724, 1829, 30544, 11933, 1829, 3224, 8032, 4117, 15730, 4724, 1829, 30544, 4724, 2288, 3224, 7649, 25724, 11622, 38207, 16712, 7649, 1829, 9673, 25720, 25724, 995, 10632, 8032, 1863, 8592, 38688, 8717, 47283, 3215, 6055, 9566, 21292, 4587, 9307, 15844, 28239, 19913, 2288, 10632, 29538, 4032, 1863, 47283, 3215, 3714, 11622, 25708, 9154, 28239, 19913, 16572, 9307, 15844, 2423, 13785, 10406, 1392, 13412, 4117, 23557, 5296, 9381, 23557, 3615, 24793, 4032, 1863, 8592, 38688, 24793, 16472, 13412, 16606, 21984, 9673, 25720, 25724, 995, 10632], "avg_logprob": -0.16281960091807626, "compression_ratio": 1.8296703296703296, "no_speech_prob": 2.4437904357910156e-06, "words": [{"start": 2375.12, "end": 2375.42, "word": "إذا", "probability": 0.5592041015625}, {"start": 2375.42, "end": 2375.7, "word": " هذا", "probability": 0.68408203125}, {"start": 2375.7, "end": 2376.24, "word": " بيكون", "probability": 0.80859375}, {"start": 2376.24, "end": 2376.98, "word": " إيه", "probability": 0.83056640625}, {"start": 2376.98, "end": 2377.62, "word": " هكذا", "probability": 0.739990234375}, {"start": 2377.62, "end": 2378.04, "word": " بيكون", "probability": 0.9588216145833334}, {"start": 2378.04, "end": 2378.64, "word": " برهان", "probability": 0.885498046875}, {"start": 2378.64, "end": 2379.88, "word": " الجزء", "probability": 0.96630859375}, {"start": 2379.88, "end": 2380.46, "word": " التاني", "probability": 0.7848307291666666}, {"start": 2380.46, "end": 2380.78, "word": " المرة", "probability": 0.869873046875}, {"start": 2380.78, "end": 2381.18, "word": " الجاية", "probability": 0.92919921875}, {"start": 2381.18, "end": 2382.52, "word": " هنشوف", "probability": 0.9288330078125}, {"start": 2382.52, "end": 2383.08, "word": " ناخد", "probability": 0.9866536458333334}, {"start": 2383.08, "end": 2383.96, "word": " تطبيقات", "probability": 0.95712890625}, {"start": 2383.96, "end": 2384.82, "word": " على", "probability": 0.92333984375}, {"start": 2384.82, "end": 2385.6, "word": " النظرية", "probability": 0.8388671875}, {"start": 2385.6, "end": 2386.08, "word": " هذه", "probability": 0.876953125}, {"start": 2386.08, "end": 2387.66, "word": " وناخد", "probability": 0.8726806640625}, {"start": 2387.66, "end": 2389.08, "word": " مزيد", "probability": 0.9322916666666666}, {"start": 2389.08, "end": 2389.52, "word": " من", "probability": 0.98291015625}, {"start": 2389.52, "end": 2391.1, "word": " النظريات", "probability": 0.9149169921875}, {"start": 2391.1, "end": 2391.46, "word": " على", "probability": 0.9599609375}, {"start": 2391.46, "end": 2391.66, "word": " ال", "probability": 0.794921875}, {"start": 2391.66, "end": 2392.02, "word": " infinite", "probability": 0.8447265625}, {"start": 2392.02, "end": 2392.58, "word": " limits", "probability": 0.94091796875}, {"start": 2392.58, "end": 2393.86, "word": " okay", "probability": 0.434326171875}, {"start": 2393.86, "end": 2395.32, "word": " شكرا", "probability": 0.9226888020833334}, {"start": 2395.32, "end": 2396.5, "word": " لصراعكم", "probability": 0.75654296875}, {"start": 2396.5, "end": 2397.42, "word": " ونشوفكم", "probability": 0.91328125}, {"start": 2397.42, "end": 2397.56, "word": " ان", "probability": 0.5107421875}, {"start": 2397.56, "end": 2397.7, "word": " شاء", "probability": 0.978759765625}, {"start": 2397.7, "end": 2397.78, "word": " الله", "probability": 0.9677734375}, {"start": 2397.78, "end": 2398.12, "word": " المرة", "probability": 0.949462890625}, {"start": 2398.12, "end": 2398.56, "word": " الجاية", "probability": 0.9375}], "temperature": 1.0}], "language": "ar", "language_probability": 1.0, "duration": 2400.15375, "duration_after_vad": 2113.4393749999927}
\ No newline at end of file
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/dw89EvC63CE_postprocess.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/dw89EvC63CE_postprocess.srt
new file mode 100644
index 0000000000000000000000000000000000000000..da9efc7b273879b8afb1a282babfee6c32a729ac
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/dw89EvC63CE_postprocess.srt
@@ -0,0 +1,1236 @@
+1
+00:00:20,910 --> 00:00:26,430
+بسم الله الرحمن الرحيم في المحاضرة اليوم هناخد
+
+2
+00:00:26,430 --> 00:00:32,890
+انتباه المقدمة البسيطة عن ال infinite series اللي
+
+3
+00:00:32,890 --> 00:00:37,890
+بدأناها في المحاضرة السابقةففي المحاضرة السابقة
+
+4
+00:00:37,890 --> 00:00:41,610
+عرفنا ما معنى ان infinite series of real numbers
+
+5
+00:00:41,610 --> 00:00:46,050
+converge معناه هذا بكافئ ان ال sequence of partial
+
+6
+00:00:46,050 --> 00:00:51,970
+sums S اللي سمناها Sn حيث ال inf partial sum هو
+
+7
+00:00:51,970 --> 00:00:56,170
+مجموع أول n من حدود ال series إذا كانت ال sequence
+
+8
+00:00:56,170 --> 00:01:00,550
+هذه convergentإذا ال sequence of partial sums
+
+9
+00:01:00,550 --> 00:01:03,990
+convergent بتكون ال series convergent، إذا ال
+
+10
+00:01:03,990 --> 00:01:06,770
+sequence of partial sums divergent بتكون ال series
+
+11
+00:01:06,770 --> 00:01:11,760
+divergentطيب لو كانت ال sequence of partial sums
+
+12
+00:01:11,760 --> 00:01:15,920
+convergent و ال limit تبعتها عدد S ففي الحالة هذه
+
+13
+00:01:15,920 --> 00:01:19,420
+بتقول ان ال series converged طبعا و ال sum تبع ال
+
+14
+00:01:19,420 --> 00:01:22,680
+series مجموعة ال series بساوي ال limit لل sequence
+
+15
+00:01:22,680 --> 00:01:26,800
+of partial sums اللي هو S اذا في الحالة هذه S is
+
+16
+00:01:26,800 --> 00:01:32,860
+the sum of the infinite series الان في عنديالنظرية
+
+17
+00:01:32,860 --> 00:01:38,220
+بتعطيني test for divergence بتسميه الانث term test
+
+18
+00:01:38,220 --> 00:01:43,360
+زي ما درسته في calculus B فالنظرية هذه بتقول لو
+
+19
+00:01:43,360 --> 00:01:46,120
+كان في عندي series و ال series كانت convergent
+
+20
+00:01:46,120 --> 00:01:50,440
+فضروري it is necessary that the limit of the inf
+
+21
+00:01:50,440 --> 00:01:57,250
+term بساوي سفروالبرهان سهل هاي ال Nth partial sum
+
+22
+00:01:57,250 --> 00:02:01,830
+مجموعة أول N من حدود ال series وهي ال N minus
+
+23
+00:02:01,830 --> 00:02:06,110
+first partial sum اللي هو مجموعة N سالب واحد من
+
+24
+00:02:06,110 --> 00:02:12,310
+الحدود الأولانية لما نيجي نطرح لما نيجي نطرح فكل
+
+25
+00:02:12,310 --> 00:02:19,910
+الحدود هذه بتروح مع بعضها يبقى عندى الفرق XNأنا
+
+26
+00:02:19,910 --> 00:02:23,650
+فارض أن ال series converge إذا ال limit لل
+
+27
+00:02:23,650 --> 00:02:27,270
+sequence of partial sums exist و بالساوية sum real
+
+28
+00:02:27,270 --> 00:02:32,570
+number S الآن باخد ال limit للطرفين في ال star هنا
+
+29
+00:02:33,630 --> 00:02:38,130
+باخد limit of both sides we get انه limit x in
+
+30
+00:02:38,130 --> 00:02:43,590
+بيساوي limit s in minus limit s in minus واحد ال
+
+31
+00:02:43,590 --> 00:02:46,930
+limit ال sequence of partial sums s وهذه هي نفس ال
+
+32
+00:02:46,930 --> 00:02:51,070
+sequence of partial sums بس حدفنا أو زودنا حد برضه
+
+33
+00:02:51,070 --> 00:02:55,170
+ال limit تبعتها s إذا ال limit ل x in بيطلع بيساوي
+
+34
+00:02:55,170 --> 00:03:01,710
+سفر كما هو مظلوم تمام؟ واضح البرهان؟ الآن في عندي
+
+35
+00:03:01,710 --> 00:03:02,310
+remark
+
+36
+00:03:10,040 --> 00:03:15,900
+the converse of
+
+37
+00:03:15,900 --> 00:03:23,140
+above theorem is
+
+38
+00:03:23,140 --> 00:03:27,860
+false عكس
+
+39
+00:03:27,860 --> 00:03:36,120
+النظرية السابقة ليس صحيحا بمعنى لو كانت x in
+
+40
+00:03:38,350 --> 00:03:45,490
+converge to zero هذا لا يؤدي بالضرورة .. لا يؤدي
+
+41
+00:03:45,490 --> 00:03:56,310
+مش شرط يؤدي ان ال series sigma x in converge وهي
+
+42
+00:03:56,310 --> 00:03:58,310
+مثال على ذلك ناخد مثال
+
+43
+00:04:17,840 --> 00:04:25,520
+example the harmonic series satisfies
+
+44
+00:04:39,920 --> 00:04:48,080
+إن ال limit ل xn عبارة عن ال limit ل 1 على n يسوى
+
+45
+00:04:48,080 --> 00:04:59,420
+سفر لكن however .. however
+
+46
+00:04:59,420 --> 00:05:04,560
+ال series ال harmonic series diverges
+
+47
+00:05:09,690 --> 00:05:14,990
+ال harmonic series is divergent مش convergent ليش؟
+
+48
+00:05:14,990 --> 00:05:22,770
+why؟
+
+49
+00:05:22,770 --> 00:05:29,330
+احنا برهننا الكلام هذا في chapter تلاتة بس كش في
+
+50
+00:05:29,330 --> 00:05:31,410
+section تلاتة خمسة
+
+51
+00:05:36,620 --> 00:05:44,980
+فاكرين ان ال sequence هذه كانت طيب نجاوب ليه ال
+
+52
+00:05:44,980 --> 00:05:52,720
+harmonic series diverges في عندي جوابين answer one
+
+53
+00:05:52,720 --> 00:06:00,660
+الإجابة الأولى في
+
+54
+00:06:00,660 --> 00:06:03,740
+عندي مثال أخدنا
+
+55
+00:06:08,530 --> 00:06:14,430
+by example لسه ماخدينه اللي كان هذا الجزء اللي
+
+56
+00:06:14,430 --> 00:06:18,590
+بتعلق فيه اللي هو ال sequence of partial sums لل
+
+57
+00:06:18,590 --> 00:06:25,470
+series هذه اثبتنا انها not cauchy صح؟ وكان هذا جزء
+
+58
+00:06:25,470 --> 00:06:31,610
+من ايه؟ من تكملة البرهان صح؟ اذا نشوف الرقم تبعها
+
+59
+00:06:31,610 --> 00:06:34,830
+بالظبط section
+
+60
+00:06:37,490 --> 00:06:50,870
+مثال تلاتة خمسة ستة الجزء C by this example the
+
+61
+00:06:50,870 --> 00:07:02,430
+sequence of partial sums Sn
+
+62
+00:07:02,430 --> 00:07:03,370
+where
+
+63
+00:07:05,750 --> 00:07:13,890
+SN بيساوي سيجما من K بيساوي واحد إلى N لواحد على K
+
+64
+00:07:13,890 --> 00:07:21,150
+اللي هو واحد زاد نص زاد واحد على N أثبتنا إن ال
+
+65
+00:07:21,150 --> 00:07:30,630
+sequence هذه is not Cauchy is not Cauchy and
+
+66
+00:07:30,630 --> 00:07:34,850
+therefore not convergent
+
+67
+00:07:40,330 --> 00:07:49,510
+حسب by cushy by cushy criterion cushy criterion
+
+68
+00:07:49,510 --> 00:07:52,950
+بتقول أي sequence of real numbers is convergent if
+
+69
+00:07:52,950 --> 00:07:56,190
+and only if it is cushy اللي إحنا أثبتنا إن ال
+
+70
+00:07:56,190 --> 00:08:01,180
+sequence هذه في المثال هذاماهياش كوشي و بالتالي
+
+71
+00:08:01,180 --> 00:08:04,460
+not convergent طب ما هذه هي ال sequence of partial
+
+72
+00:08:04,460 --> 00:08:07,680
+.. هذه هي ال sequence of partial sums لل harmonic
+
+73
+00:08:07,680 --> 00:08:11,020
+series اذا by above definition
+
+74
+00:08:15,530 --> 00:08:19,870
+طبعا ال definition المسعة اللى كتبناها أول واحد
+
+75
+00:08:19,870 --> 00:08:23,830
+مادام ال series of partial sums is not convergent
+
+76
+00:08:23,830 --> 00:08:30,150
+اذا ال series نفسها is divergent
+
+77
+00:08:33,300 --> 00:08:38,940
+Okay إذا هذا كان إحدى الإجابات و ليه .. ليه ال
+
+78
+00:08:38,940 --> 00:08:42,960
+series هي divergent هاي الإثبات تبعها هنا جاي من
+
+79
+00:08:42,960 --> 00:08:47,360
+المثال هذا إن ال sequence of partial sums طلعت not
+
+80
+00:08:47,360 --> 00:08:51,340
+Cauchy وبالتالي not convergent هذا السبب في كمان
+
+81
+00:08:51,340 --> 00:09:02,140
+سبب تاني أو حل تاني أو إجابة أخرى answer
+
+82
+00:09:02,140 --> 00:09:13,540
+twoإنه by .. في مثال آخر أثبتنا فيه by example
+
+83
+00:09:13,540 --> 00:09:17,900
+تلاتة
+
+84
+00:09:17,900 --> 00:09:24,800
+تلاتة تلاتة بي .. إذا في section تلاتة تلاتة
+
+85
+00:09:24,800 --> 00:09:31,540
+المثال تلاتة بي أثبتنا فيه إن ال sequence و
+
+86
+00:09:31,540 --> 00:09:32,480
+partial sums
+
+87
+00:09:34,910 --> 00:09:40,910
+the sequence of partial sums
+
+88
+00:09:40,910 --> 00:09:44,190
+Sn
+
+89
+00:09:44,190 --> 00:09:50,370
+is unbounded
+
+90
+00:09:50,370 --> 00:09:55,910
+أثبتنا بالمثال هذا أن ال sequence Sn of partial
+
+91
+00:09:55,910 --> 00:10:04,760
+sums اللي هي هذه is unbounded and thereforeإذا
+
+92
+00:10:04,760 --> 00:10:08,380
+كانت الـ sequence unbounded إذا بتطلع divergent أو
+
+93
+00:10:08,380 --> 00:10:16,060
+not convergent لأن sn is divergent لأن لو كانت
+
+94
+00:10:16,060 --> 00:10:20,800
+convergent بتكون bounded وبالتالي therefore ال
+
+95
+00:10:20,800 --> 00:10:28,090
+series أو ال harmonic seriesdivergent إذا هذا برضه
+
+96
+00:10:28,090 --> 00:10:33,350
+إجابة تانية بتخلي ال harmonic series divergent
+
+97
+00:10:33,350 --> 00:10:40,830
+تمام طيب نرجع ناخد أمثلة أخرى
+
+98
+00:11:08,590 --> 00:11:18,590
+أول شيء الـ geometric series الـ
+
+99
+00:11:18,590 --> 00:11:26,210
+geometric series اللي هي summation من k بساوي سفر
+
+100
+00:11:26,210 --> 00:11:32,090
+to infinity لR to K اللي الحد الأول تبعها واحد
+
+101
+00:11:32,090 --> 00:11:37,250
+والأساس تبعها R اشمالها واحد
+
+102
+00:11:40,350 --> 00:11:49,570
+converges if absolute r أصغر من one and
+
+103
+00:11:49,570 --> 00:12:06,550
+diverges and its sum its sum بيساوي واحد على واحد
+
+104
+00:12:06,550 --> 00:12:08,250
+minus r
+
+105
+00:12:14,230 --> 00:12:20,410
+and diverges if
+
+106
+00:12:20,410 --> 00:12:27,790
+absolute are أكبر من أوسى واحد أثبتنا
+
+107
+00:12:27,790 --> 00:12:31,250
+هذا الجزء المرة اللي فاتت في الأول ولا ما أثبتناه
+
+108
+00:12:31,250 --> 00:12:34,350
+مش؟
+
+109
+00:12:34,350 --> 00:12:38,010
+أثبتنا صحيح؟ okay إذا
+
+110
+00:12:44,740 --> 00:12:52,060
+proof part one was
+
+111
+00:12:52,060 --> 00:12:55,940
+proved last
+
+112
+00:12:55,940 --> 00:13:01,400
+time .. last time
+
+113
+00:13:01,400 --> 00:13:09,160
+to
+
+114
+00:13:09,160 --> 00:13:12,760
+prove two assume
+
+115
+00:13:15,240 --> 00:13:29,520
+absolute are أكبر من أو ساوي الواحد افترض
+
+116
+00:13:29,520 --> 00:13:36,400
+ان absolute are أكبر من أو ساوي الواحد طيب then
+
+117
+00:13:44,600 --> 00:13:53,300
+limit xn as n tends to infinity بيساوي limit r to
+
+118
+00:13:53,300 --> 00:13:59,220
+n as n tends to infinity لحظة
+
+119
+00:13:59,220 --> 00:14:06,940
+n to absolute r أكبر من واحد بيقدر ان ر أكبر من ..
+
+120
+00:14:06,940 --> 00:14:08,580
+أصغر من واحد
+
+121
+00:14:11,700 --> 00:14:22,700
+or R أكبر من سالب واحد او
+
+122
+00:14:22,700 --> 00:14:30,200
+R أكبر من واحد او R أصغر من سالب
+
+123
+00:14:30,200 --> 00:14:34,640
+واحد وبالتالي
+
+124
+00:14:34,640 --> 00:14:42,080
+limit R أُس N مش ممكن تساوي سفرlimit ل R أسن شفنا
+
+125
+00:14:42,080 --> 00:14:50,420
+أن هذه بساوية سفر فقط لما ال R يكون أصغر من واحد و
+
+126
+00:14:50,420 --> 00:14:57,940
+أكبر من أو بساوية سفر وبالتالي اذا by .. اذا by
+
+127
+00:14:57,940 --> 00:15:08,300
+nth by nth term test اذا الحد العام الحد النوني لل
+
+128
+00:15:08,300 --> 00:15:12,700
+series هذهلا يقول لا سفر ال limit تبعته ما بيسويش
+
+129
+00:15:12,700 --> 00:15:20,400
+سفر وبالتالي by the nth term test المسح اه اذا كان
+
+130
+00:15:20,400 --> 00:15:25,480
+limit الحد النوني بيسويش سفر فال test بتقول لي ان
+
+131
+00:15:25,480 --> 00:15:31,500
+ال series is
+
+132
+00:15:31,500 --> 00:15:37,480
+divergent ال series sigma ل R to N من N equal 0 to
+
+133
+00:15:37,480 --> 00:15:39,000
+infinity diverges
+
+134
+00:15:45,290 --> 00:15:48,990
+لأنه لو كانت convergent فالمفروض limit الحد اللوني
+
+135
+00:15:48,990 --> 00:15:55,090
+يساوي سفر وهذا مش موجود تمام؟ ان انا شوفت هنا كيف
+
+136
+00:15:55,090 --> 00:16:00,710
+أخدنا ال .. او استخدمنا ال in term test لإثبات ال
+
+137
+00:16:00,710 --> 00:16:05,150
+divergence مثال
+
+138
+00:16:05,150 --> 00:16:05,630
+تاني
+
+139
+00:16:14,400 --> 00:16:23,600
+show that ال series اللي متولدة من ال sequence
+
+140
+00:16:23,600 --> 00:16:29,840
+سالب واحد to end اللي
+
+141
+00:16:29,840 --> 00:16:37,100
+هي أول حد هيكون واحد بعدين سالب واحد بعدين واحد
+
+142
+00:16:37,100 --> 00:16:43,030
+بعدين سالب واحد و هكذاshow that this series
+
+143
+00:16:43,030 --> 00:16:55,610
+diverges تعالى
+
+144
+00:16:55,610 --> 00:16:59,510
+نشوف ال sequence of partial sums ناخد ال sequence
+
+145
+00:16:59,510 --> 00:17:03,070
+of partial sums والله ده طلعت ال sequence of
+
+146
+00:17:03,070 --> 00:17:06,810
+partial sums convergent بيكون ال series convergent
+
+147
+00:17:06,810 --> 00:17:10,730
+وال sum تبعها بيساوي ال limitللـ sequence of
+
+148
+00:17:10,730 --> 00:17:14,290
+partial sums مظبوط وإذا طلعت ال sequence of
+
+149
+00:17:14,290 --> 00:17:19,570
+partial sums divergent فال series اللي تابعة إلىها
+
+150
+00:17:19,570 --> 00:17:24,650
+بتكون divergent لذا هنا هنفحص هل ال sequence of
+
+151
+00:17:24,650 --> 00:17:29,410
+partial sums divergent ولا convergent طيب تعالوا
+
+152
+00:17:29,410 --> 00:17:34,750
+نفحص Sn ال N في partial sums هذا summation من K
+
+153
+00:17:34,750 --> 00:17:47,900
+equal 0 to Nلا سالب واحد plus k هذا xk فهذا هيطلع
+
+154
+00:17:47,900 --> 00:17:57,300
+الو قيمتين الو هيطلع قيمتين
+
+155
+00:17:57,300 --> 00:18:05,300
+لو
+
+156
+00:18:05,300 --> 00:18:14,590
+كان in evenأو n odd لو كانت n even يعني نفر بأن n
+
+157
+00:18:14,590 --> 00:18:22,470
+بساوي سفر إذن هيطلع واحد، n بساوي اتنين، هيطلع
+
+158
+00:18:22,470 --> 00:18:29,470
+اندي تلت حدود، مجموعهم برضه واحد، و هكذا، إذن
+
+159
+00:18:29,470 --> 00:18:35,160
+المجموع هذا هيطلع واحدإذا كانت ال in even أما لو
+
+160
+00:18:35,160 --> 00:18:40,680
+كانت ال in odd يعني لو أخدت in بالساوية واحدلو
+
+161
+00:18:40,680 --> 00:18:46,040
+أخدت in بالساوية واحد فهيطلع عندي حدين مجموعهم سفر
+
+162
+00:18:46,040 --> 00:18:50,960
+لو أخدت in بالساوية تلاتة هيطلع مجموع أول أربع
+
+163
+00:18:50,960 --> 00:18:56,220
+حدود برضه بيطلع مجموعهم سفر و هكذا اذا سفر ال in
+
+164
+00:18:56,220 --> 00:19:00,160
+partial sums بساوية سفر اذا كانت in odd واحد اذا
+
+165
+00:19:00,160 --> 00:19:05,650
+كانت in even وبالتالي thereforetherefore a
+
+166
+00:19:05,650 --> 00:19:11,110
+sequence of partial sums is an alternating series
+
+167
+00:19:11,110 --> 00:19:16,170
+يعني حدودها متدبدة بين صفر و واحد أو أول حد واحد
+
+168
+00:19:16,170 --> 00:19:23,710
+التاني صفر واحد صفر و هكذا which
+
+169
+00:19:23,710 --> 00:19:29,710
+is divergent هذه أمرها ما كانت convergent
+
+170
+00:19:33,470 --> 00:19:39,270
+الـ sequence هذه divergent عارفين ليه؟ لأن هي ال
+
+171
+00:19:39,270 --> 00:19:54,130
+proof الـ subsequence الحدود
+
+172
+00:19:54,130 --> 00:19:58,330
+مثلا الزوجية S2K
+
+173
+00:20:03,520 --> 00:20:08,760
+هي أول حد وهي التاني لإن الحدود الزوجية أصفار كلهم
+
+174
+00:20:08,760 --> 00:20:17,120
+صح؟ تكون بيرجل سفر and ال subsequence التانية اللي
+
+175
+00:20:17,120 --> 00:20:23,400
+حدودها فردية كلهم
+
+176
+00:20:23,400 --> 00:20:30,960
+واحد، ثابت واحد وهدي تكون بيرجل واحد نعم؟
+
+177
+00:20:31,600 --> 00:20:36,180
+وبالتالي إذا مادام في عندي two subsequences واحدة
+
+178
+00:20:36,180 --> 00:20:40,120
+converge لصفر وواحدة converge لواحد وصفر بيسويش
+
+179
+00:20:40,120 --> 00:20:46,380
+الواحد وصفر لا يساوي الواحد معناته by the
+
+180
+00:20:46,380 --> 00:20:50,400
+divergence criterion ال sequence هذه is divergent
+
+181
+00:20:51,890 --> 00:20:56,830
+لأن لو كانت ال sequence هذي convergent فمفروض أي
+
+182
+00:20:56,830 --> 00:21:00,250
+subsequence تكون convergent لنفس ال limit يعني
+
+183
+00:21:00,250 --> 00:21:03,630
+المفروض دول ال limits تكون متساويين و هذا مستحيل
+
+184
+00:21:03,630 --> 00:21:07,130
+إذا
+
+185
+00:21:07,130 --> 00:21:11,490
+ال sequence of partial sums هنا divergent وبالتالي
+
+186
+00:21:11,490 --> 00:21:17,130
+نكمل الحل هنا و بالتالي therefore the series
+
+187
+00:21:21,790 --> 00:21:27,290
+المتولدة من الـ sequence سالب واحد أس N اللي هي ال
+
+188
+00:21:27,290 --> 00:21:31,970
+series هذه is
+
+189
+00:21:31,970 --> 00:21:38,510
+divergent okay
+
+190
+00:21:38,510 --> 00:21:43,210
+تمام وبالتالي هيك ممكن أثبتنا هذا مثال على
+
+191
+00:21:43,210 --> 00:21:46,430
+divergence series إذن عشان أثبت ال series
+
+192
+00:21:46,430 --> 00:21:51,260
+divergent أو convergent بحاول أفحصالـ sequence of
+
+193
+00:21:51,260 --> 00:21:55,120
+partial sums و أفحص هل الـ sequence of partial
+
+194
+00:21:55,120 --> 00:22:01,880
+sums convergent ولا divergent ناخد
+
+195
+00:22:01,880 --> 00:22:06,760
+كمان مثال مختلف من نوع آخر مثال رقم 3
+
+196
+00:22:21,550 --> 00:22:29,550
+Discuss the convergence of
+
+197
+00:22:29,550 --> 00:22:33,110
+the
+
+198
+00:22:33,110 --> 00:22:43,010
+series sigma from N equals one to infinity لواحد
+
+199
+00:22:43,010 --> 00:22:45,130
+على N في N زايد واحد
+
+200
+00:22:53,320 --> 00:22:59,340
+لما يكون الحد العام لل series xn عبارة عن كسر زي
+
+201
+00:22:59,340 --> 00:23:04,260
+هذا و الكسر اللي زي هذا ممكن تجزئته إلى كسور جزئية
+
+202
+00:23:04,260 --> 00:23:10,220
+فبنعمل كسور جزئية الأول و بنجزئ الكسر هذا إلى
+
+203
+00:23:10,220 --> 00:23:16,600
+مجموع كسرين جزئيين اذا by partial fractions او use
+
+204
+00:23:16,600 --> 00:23:22,280
+partial fractions و هذا اتعلمته في تفاضل بقى
+
+205
+00:23:25,060 --> 00:23:30,740
+by partial fractions هي
+
+206
+00:23:30,740 --> 00:23:40,560
+عندي واحد على k في k زائد واحد بساوي a على k زائد
+
+207
+00:23:40,560 --> 00:23:42,920
+b over k plus one
+
+208
+00:23:48,420 --> 00:23:53,360
+الان لتحديد ثوابت A وB نتخلص من الكسور في المعادلة
+
+209
+00:23:53,360 --> 00:23:58,940
+هذه و ذلك بضرب طرفي المعادلة في المقام تبع الطرف
+
+210
+00:23:58,940 --> 00:24:08,530
+اليسار اذا بضرب في K في K زائد واحديطلع عندي واحد
+
+211
+00:24:08,530 --> 00:24:16,450
+بساوي a في k plus one زايد b في k الآن خد k بساوي
+
+212
+00:24:16,450 --> 00:24:22,610
+سفر هذا بيقدر ان a بساوي واحد وخد k بساوي سالب
+
+213
+00:24:22,610 --> 00:24:29,650
+واحد بيقدر ان السالب b بساوي واحد بيقدر ان b بساوي
+
+214
+00:24:29,650 --> 00:24:38,450
+سالب واحدإذا طلع أنا عندي واحد على ك في ك زائد
+
+215
+00:24:38,450 --> 00:24:45,170
+واحد بساوي واحد على ك سالب واحد على ك زائد واحد
+
+216
+00:24:45,170 --> 00:24:51,550
+الان now تعالوا نشوف ال inf partial sum أو نبحث ال
+
+217
+00:24:51,550 --> 00:24:55,430
+sequence of partial sums ونبحث هل هي convergent
+
+218
+00:24:55,430 --> 00:25:03,610
+ولا divergentفهي عندي sm بساوي summation من k
+
+219
+00:25:03,610 --> 00:25:13,170
+بساوي واحد إلى n ل xk اللي
+
+220
+00:25:13,170 --> 00:25:19,670
+هو summation من k بساوي واحد إلى n لواحد على k في
+
+221
+00:25:19,670 --> 00:25:27,210
+k زايد واحدوهذا الان باستخدام ال partial fractions
+
+222
+00:25:27,210 --> 00:25:33,810
+summation من k بساوي واحد الى n الى واحد على k
+
+223
+00:25:33,810 --> 00:25:42,790
+سالب واحد على k موجب واحد الان
+
+224
+00:25:42,790 --> 00:25:47,450
+هذا المجموع تعالوا نفرد تعالوا نفرفته let's expand
+
+225
+00:25:47,450 --> 00:25:49,490
+it يعني نفرفته
+
+226
+00:25:53,610 --> 00:25:58,830
+هذا مجموع منتهي في n من الحدود خد k بالساوي واحد
+
+227
+00:25:58,830 --> 00:26:06,930
+بطلع أول حد واحد سالب نص خد k بساوي اتنين بطلع
+
+228
+00:26:06,930 --> 00:26:16,010
+الحد البعد و نص سالب تلت و هكذا إلى الحد الأخير
+
+229
+00:26:16,010 --> 00:26:23,560
+هيكون واحد على n minus واحد على n زاد واحدبنلاحظ
+
+230
+00:26:23,560 --> 00:26:28,920
+أن المجموع هذا telescoping يعني في حدود فيها إزاي
+
+231
+00:26:28,920 --> 00:26:33,280
+تتلاشى مع الحدود اللي بعديها يعني سالب نص لاحظوا
+
+232
+00:26:33,280 --> 00:26:38,640
+بيروح مع موجب نص وسالب تلت هيروح مع موجب تلت في
+
+233
+00:26:38,640 --> 00:26:43,340
+الحد اللي يليه مباشرة وواحد على ان هذا بيروح مع
+
+234
+00:26:43,340 --> 00:26:50,510
+سالب واحد على ان في الحد اللي يصدقه مباشرةفيبقى كل
+
+235
+00:26:50,510 --> 00:26:55,650
+شي بيروح بيظل يبقى عندي واحد negative one over n
+
+236
+00:26:55,650 --> 00:27:01,410
+زايد واحد اذا انا طلع عيني اثبتت ان sn بساوي واحد
+
+237
+00:27:01,410 --> 00:27:06,410
+negative واحد على n زايد واحد الكلام هذا صحيح for
+
+238
+00:27:06,410 --> 00:27:10,480
+every natural number nأخذ قيمة كما نسميها limit of
+
+239
+00:27:10,480 --> 00:27:16,580
+both sides as n tends to infinity نحصل على قيمة Sn
+
+240
+00:27:16,580 --> 00:27:17,980
+كما نسميها limit of both sides as n tends to
+
+241
+00:27:17,980 --> 00:27:18,500
+infinity نحصل على قيمة Sn كما نسميها limit of both
+
+242
+00:27:18,500 --> 00:27:19,140
+sides as n tends to infinity نحصل على قيمة Sn كما
+
+243
+00:27:19,140 --> 00:27:20,040
+نحصل على قيمة Sn كما نسميها limit of both sides as
+
+244
+00:27:20,040 --> 00:27:20,180
+n tends to infinity نحصل على قيمة Sn كما نسميها
+
+245
+00:27:20,180 --> 00:27:22,340
+limit of both sides as n tends to infinity نحصل
+
+246
+00:27:22,340 --> 00:27:24,580
+على قيمة Sn كما نسميها limit of both sides as n
+
+247
+00:27:24,580 --> 00:27:29,920
+tends to infinity نحصل على قيمة Sn كما نسميها
+
+248
+00:27:29,920 --> 00:27:36,040
+limit of both sides as
+
+249
+00:27:36,380 --> 00:27:41,720
+is convergent و ال limit تبعتها بساوي واحد تمام
+
+250
+00:27:41,720 --> 00:27:46,480
+وبالتالي حسب التعريف تبع ال convergence لل
+
+251
+00:27:46,480 --> 00:27:49,680
+infinite series the infinite series اللي احنا
+
+252
+00:27:49,680 --> 00:27:54,320
+عايزين نفحصها اللي هي summation from n equals one
+
+253
+00:27:54,320 --> 00:28:00,200
+to infinity ل one over n في n plus one اشملها
+
+254
+00:28:00,200 --> 00:28:02,260
+converges
+
+255
+00:28:03,720 --> 00:28:10,920
+and ال sum تبعها يعني مجموعها بساوي واحد اللي هو
+
+256
+00:28:10,920 --> 00:28:17,360
+limit لل sequence of partial sums تمام؟
+
+257
+00:28:17,360 --> 00:28:20,800
+إذا هذا النوع من ال series بنسميه telescoping ..
+
+258
+00:28:20,800 --> 00:28:28,140
+telescoping .. telescoping series
+
+259
+00:28:31,710 --> 00:28:36,990
+such type of series is called telescoping series
+
+260
+00:28:36,990 --> 00:28:41,110
+okay تمام واضح
+
+261
+00:28:54,430 --> 00:29:00,810
+طيب هناخد بس نظرية سريعة و يعني مش .. البرهانة مش
+
+262
+00:29:00,810 --> 00:29:10,810
+هياخد الا دقيقة واحدة theorem Cauchy
+
+263
+00:29:10,810 --> 00:29:17,930
+.. Cauchy criterion for
+
+264
+00:29:17,930 --> 00:29:20,950
+series
+
+265
+00:29:25,660 --> 00:29:31,060
+أحنا أخدنا قبلها كوشي كريتيريا for sequences لأن
+
+266
+00:29:31,060 --> 00:29:37,560
+في كوشي كريتيريا for series فال .. الكوشي كريتيريا
+
+267
+00:29:37,560 --> 00:29:45,840
+for series بتنص على أنه ال series sigma
+
+268
+00:29:45,840 --> 00:29:53,100
+x in converges if
+
+269
+00:29:53,100 --> 00:29:54,500
+and only if
+
+270
+00:29:58,620 --> 00:30:05,760
+لكل ابسلون اكبر من صفر يوجد capital N يعتمد على
+
+271
+00:30:05,760 --> 00:30:13,500
+ابسلون عدد طبيعي بحيث انه لكل M اكبر من N اكبر من
+
+272
+00:30:13,500 --> 00:30:21,360
+او ساوي capital N بطلع absolute SM minus SN اللي
+
+273
+00:30:21,360 --> 00:30:28,830
+هو عبارة عن absolute Xm زائد واحد زائد xn زائد
+
+274
+00:30:28,830 --> 00:30:35,030
+اتنين زائد xm اصغر من اكسمن
+
+275
+00:30:49,160 --> 00:30:55,400
+كمان مرة ال series هذه converges if and only if
+
+276
+00:30:55,400 --> 00:31:00,440
+for every epsilon فيه capital N بحيث لكل M و N
+
+277
+00:31:00,440 --> 00:31:05,140
+أكبر من أو ساوي capital N و M بتكون أكبر من N ال
+
+278
+00:31:05,140 --> 00:31:10,520
+absolute value للفرق هذا أصغر من epsilon طب هذا هو
+
+279
+00:31:10,520 --> 00:31:11,440
+شرط كوشي
+
+280
+00:31:17,000 --> 00:31:24,920
+that is .. that is .. يعني هذا يعني أنه sigma x in
+
+281
+00:31:24,920 --> 00:31:30,640
+converges if and only if ال sequence of partial
+
+282
+00:31:30,640 --> 00:31:34,860
+sums is Cauchy
+
+283
+00:31:34,860 --> 00:31:39,880
+إذا كانت ال sequence of partial sums is Cauchy هذا
+
+284
+00:31:39,880 --> 00:31:45,280
+شرط Cauchy البرهان ينتج من Cauchy criterion for
+
+285
+00:31:45,280 --> 00:31:51,890
+sequencesطيب احنا أخدنا by definition من ال
+
+286
+00:31:51,890 --> 00:31:56,050
+definition تبع ال convergence لل series by
+
+287
+00:31:56,050 --> 00:32:00,910
+definition the series sigma
+
+288
+00:32:00,910 --> 00:32:07,510
+xn converges if and only if the sequence of
+
+289
+00:32:07,510 --> 00:32:13,870
+partial sums اللي
+
+290
+00:32:13,870 --> 00:32:14,950
+هي Sn
+
+291
+00:32:22,440 --> 00:32:33,300
+converges مظبوط؟ by Cauchy criterion
+
+292
+00:32:33,300 --> 00:32:41,990
+for sequences .. for sequences من المتالياتالـ
+
+293
+00:32:41,990 --> 00:32:52,750
+sequence sn converges if and only if sn is cauchy
+
+294
+00:32:56,430 --> 00:33:02,310
+هو بالتالي ال series sigma xn converges if and
+
+295
+00:33:02,310 --> 00:33:06,150
+only if ال sequence Sn converges طيب ما هذي
+
+296
+00:33:06,150 --> 00:33:09,690
+converges if and only if it is Cauchy، إذا ال
+
+297
+00:33:09,690 --> 00:33:14,870
+series sigma xn converges if and only if ال
+
+298
+00:33:14,870 --> 00:33:20,590
+sequence of partial sums is Cauchy، و هذا هو
+
+299
+00:33:20,590 --> 00:33:21,150
+المطلوب
+
+300
+00:33:24,120 --> 00:33:28,920
+طبعا هذا هو شرط كوشي لحظوا SM ال M هنا أكبر من N
+
+301
+00:33:28,920 --> 00:33:39,020
+فلو اتكتبت الحدود فتبعت SM هيكون X1 إلى XM SN ال N
+
+302
+00:33:39,020 --> 00:33:45,740
+أصغر من M فالحدود من X1 إلى XN لما أطرح بيضل عندي
+
+303
+00:33:45,740 --> 00:33:50,260
+كل الحدود المتشابهة بيضل عندي XN زياد واحد XN زياد
+
+304
+00:33:50,260 --> 00:33:51,840
+اتنين إلى XM
+
+305
+00:33:55,430 --> 00:34:02,690
+تمام؟ اذا بنوقف هنا ال .. هذا برهان كوشي criterion
+
+306
+00:34:02,690 --> 00:34:07,190
+for series بنوقف عند النظرية هذه و المرة الجاية ان
+
+307
+00:34:07,190 --> 00:34:14,310
+شاء الله بنكمل هناخد limit comparison test
+
+308
+00:34:21,040 --> 00:34:25,600
+هناك تجارب تجارب تجارب تجارب تجارب تجارب تجارب
+
+309
+00:34:25,600 --> 00:34:27,540
+تجارب تجارب تجارب
+
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/eX4Dw2M3cKU_postprocess.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/eX4Dw2M3cKU_postprocess.srt
new file mode 100644
index 0000000000000000000000000000000000000000..58a60d03535a710a5a391a8e6d1ed793a489cad0
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/eX4Dw2M3cKU_postprocess.srt
@@ -0,0 +1,1328 @@
+1
+00:00:23,030 --> 00:00:31,610
+في exercise هنحاول ان شاء الله نحله وهذا ال
+
+2
+00:00:31,610 --> 00:00:37,190
+exercise مهم رقم عشرة section تلاتة اربعة في ال
+
+3
+00:00:37,190 --> 00:00:41,470
+textbook او الكتاب المقرر ال exercise هذا بيقول
+
+4
+00:00:41,470 --> 00:00:48,370
+let xn be bounded sequence of real numbers let Sn
+
+5
+00:00:48,370 --> 00:00:53,550
+be the supremumلكل حدود ال sequence من N و M تطالع
+
+6
+00:00:53,550 --> 00:01:05,030
+يعني هذا عبارة عن ال supremum ل X N X رقم N زياد
+
+7
+00:01:05,030 --> 00:01:09,950
+واحد و هكذا فمثلا
+
+8
+00:01:09,950 --> 00:01:15,390
+S1 هيسوي ال supremum لكل ال sequence لما N بسوا
+
+9
+00:01:15,390 --> 00:01:23,160
+واحد صح؟S2 هيساوي ال supremum ل X2 و X3 إلى آخرها
+
+10
+00:01:23,160 --> 00:01:30,260
+لأن هذا ال supremum ل ال one tail ل ال two tail S3
+
+11
+00:01:30,260 --> 00:01:35,060
+هو ال supremum ل X3 و X4 إلى آخرها لأن هذا ال
+
+12
+00:01:35,060 --> 00:01:39,600
+supremum لل تلاتة tail of the sequence M tail
+
+13
+00:01:42,530 --> 00:01:47,570
+لاحظوا إن الـ Sn هذه كل ما N كبرت كل ما ال 6 هذه
+
+14
+00:01:47,570 --> 00:01:54,130
+زغرت و ال suprem زغرت فبتطلع decreasing sequence و
+
+15
+00:01:54,130 --> 00:02:00,050
+طبعا bounded إذا الإنفمام تبعها موجود بسبب عدد S
+
+16
+00:02:00,050 --> 00:02:07,790
+إذن هنا بنعرف ال S على إن الإنفمام لل sequence Sn
+
+17
+00:02:12,100 --> 00:02:16,480
+الان بدنا نثبت ان يوجد subsequence XKN من سيكوانس
+
+18
+00:02:16,480 --> 00:02:21,720
+XN وهذه السبسيكوانس convergence للعدد S اللي هو
+
+19
+00:02:21,720 --> 00:02:27,720
+الانثمام لكل S لبرهان ذلك انا عندي S بالساوية
+
+20
+00:02:27,720 --> 00:02:31,620
+الانثمام من الفرض لسيكوانس SN
+
+21
+00:02:34,540 --> 00:02:41,960
+أخدنا جابل هيك لمبة بتقول أنه w بساول infimum ل
+
+22
+00:02:41,960 --> 00:02:47,960
+set S if and only if لكل epsilon أكبر من الصفر
+
+23
+00:02:47,960 --> 00:02:57,560
+يوجد S epsilon ينتمي ل S بحيث أنه S
+
+24
+00:02:57,560 --> 00:02:58,360
+epsilon
+
+25
+00:03:01,030 --> 00:03:09,790
+فأصغر من W زاد Epsilon، مظبوط هكذا، وفي كمان حقيقة
+
+26
+00:03:09,790 --> 00:03:14,890
+تانية أو لمبة تانية بتقول أن U بساوي ال suprem ل
+
+27
+00:03:14,890 --> 00:03:21,050
+S، F and only F لكل Epsilon أكبر من السفر يوجد S
+
+28
+00:03:21,050 --> 00:03:24,670
+Epsilon ينتمي ل S بحيث أنه
+
+29
+00:03:27,490 --> 00:03:35,870
+u minus epsilon أصغر من s epsilon نظبط يعنى فهنطبخ
+
+30
+00:03:35,870 --> 00:03:40,750
+الكلام هذا إذا هي عندي s بساوة n في وملة ال set
+
+31
+00:03:40,750 --> 00:03:53,330
+هذه اعتبر هذه ال set S فهذا هو ال W فلو أخدت for
+
+32
+00:03:53,330 --> 00:04:04,110
+eachin عدد طبيعي لكل عدد طبيعي in there
+
+33
+00:04:04,110 --> 00:04:07,910
+exist sn
+
+34
+00:04:07,910 --> 00:04:17,090
+ينتمي لل set هذه ال set of all sn hyphen عدد طبيعي
+
+35
+00:04:17,090 --> 00:04:28,190
+بحيث انه ال هذا ال snأصغر من الانفرام اللي هو S زي
+
+36
+00:04:28,190 --> 00:04:34,830
+واحد على N اللي هو الابسلون هذا عبارة عن الابسلون
+
+37
+00:04:34,830 --> 00:04:42,940
+ممكن اخد ابسلون بساوي واحد على N هذا عدد موجةفلأي
+
+38
+00:04:42,940 --> 00:04:47,760
+epsilon عدد موجة يوجد عنصر في ال set اللي باخدلها
+
+39
+00:04:47,760 --> 00:04:51,680
+ال infimum بحيث أن هذا العنصر أصغر من ال infimum
+
+40
+00:04:51,680 --> 00:05:05,200
+زاد epsilon حسب الحقيقة الأولى طيب كذلك since ايضا
+
+41
+00:05:05,200 --> 00:05:12,890
+من الفرض احنا فرضين ان sn بساوي supremumللset of
+
+42
+00:05:12,890 --> 00:05:21,530
+all xk حيث k أكبر من أو ساوي ال n بما أن هذا صحيح
+
+43
+00:05:21,530 --> 00:05:28,210
+إذا أنا باستخدام الحقيقة التانية then
+
+44
+00:05:28,210 --> 00:05:33,410
+for epsilon لو أخدت epsilon برضه واحد على n هذا
+
+45
+00:05:33,410 --> 00:05:42,830
+عدد موجب فبنقدر نلاقي يوجدx عنصر هنا في ال
+
+46
+00:05:42,830 --> 00:05:52,250
+sequence هذه أو في ال set هذه نسمي xkn ينتمي لset
+
+47
+00:05:52,250 --> 00:06:04,570
+of all xk حيث k أكبر من أو يساوي ال n such
+
+48
+00:06:04,570 --> 00:06:05,070
+that
+
+49
+00:06:08,240 --> 00:06:13,400
+الـ supremum اللي هو SN minus الـ Y اللي هو واحد
+
+50
+00:06:13,400 --> 00:06:28,220
+على N أصغر من الـ SY اللي هو العنصر X KN طيب
+
+51
+00:06:28,220 --> 00:06:30,200
+إذا من هنا
+
+52
+00:06:36,260 --> 00:06:45,240
+من المتباينة هذه والمتباينة هذه هينتج
+
+53
+00:06:45,240 --> 00:06:49,700
+it follows that
+
+54
+00:06:49,700 --> 00:07:00,020
+من المتباينتين هذول أنا عندي ال sn
+
+55
+00:07:00,020 --> 00:07:07,730
+سالب واحد على n أكبر من أو يساويcapital S minus
+
+56
+00:07:07,730 --> 00:07:13,890
+واحد على N لحظة
+
+57
+00:07:13,890 --> 00:07:20,330
+من هنا S N ال capital S lower bound للست هذه
+
+58
+00:07:20,330 --> 00:07:25,270
+وبالتالي أي أنصر هنا أكبر من أو يساوي ال S لأن أنا
+
+59
+00:07:25,270 --> 00:07:32,230
+عندي S N أكبر من أو يساوي capital S اطرحي واحد على
+
+60
+00:07:32,230 --> 00:07:37,590
+N من الطرفين ونحصل على الكلام هذاهذا بيقدّي إن هذا
+
+61
+00:07:37,590 --> 00:07:45,330
+الكلام صحيح طيب و من هنا من المتباين الأخير هذا
+
+62
+00:07:45,330 --> 00:07:49,650
+أصغر من xkn
+
+63
+00:07:49,650 --> 00:07:52,750
+و
+
+64
+00:07:52,750 --> 00:08:01,030
+xkn هذا عنصر في ال set هذه عنصر في ال set هذه و sn
+
+65
+00:08:01,030 --> 00:08:05,450
+upper bound لل setإذن هذا ال upper bound أكبر من
+
+66
+00:08:05,450 --> 00:08:11,790
+أو ساوي كل عناصر ال set إذن هذا أصغر من أو ساوي Sn
+
+67
+00:08:11,790 --> 00:08:16,630
+ومن
+
+68
+00:08:16,630 --> 00:08:23,290
+المتباينة هذه Sn أصغر من capital S زائد واحد على N
+
+69
+00:08:23,290 --> 00:08:26,490
+مظبوط؟
+
+70
+00:08:39,470 --> 00:08:50,830
+thus we obtain a subsequence الكلام هذا صحيح لكل
+
+71
+00:08:50,830 --> 00:08:51,070
+ان
+
+72
+00:08:54,210 --> 00:08:59,590
+لأن هذا الكلام صحيح لكل n و هذا الكلام صحيح لكل n
+
+73
+00:08:59,590 --> 00:09:05,790
+لكل epsilon بالساعة واحد على n و لكل .. لكل n ممكن
+
+74
+00:09:05,790 --> 00:09:10,450
+أكون epsilon بالساعة واحد على n وهي هنا لكل n for
+
+75
+00:09:10,450 --> 00:09:14,670
+each n كلام هذا صحيحأذا هذا الكلام المتباين هذه
+
+76
+00:09:14,670 --> 00:09:21,130
+صحيحة لكل n وبالتالي أنا لكل n بقدر ألاقي xkn
+
+77
+00:09:21,130 --> 00:09:25,590
+موجود هنا و بيحقق المتباين هذه
+
+78
+00:09:28,570 --> 00:09:34,390
+عناصرها xkn من n بالساوية واحد to infinity of ال
+
+79
+00:09:34,390 --> 00:09:43,430
+sequence xn such that بحيث أنه xkn
+
+80
+00:09:43,430 --> 00:09:51,310
+أصغر من s زايد واحد على n أكبر من s سالب واحد على
+
+81
+00:09:51,310 --> 00:09:57,610
+n الكلام هذا صحيح لكل n في nطب ماذا معناه أن
+
+82
+00:09:57,610 --> 00:10:06,510
+absolute x k n minus s أصغر من واحد على n لكل n في
+
+83
+00:10:06,510 --> 00:10:14,690
+n؟ so
+
+84
+00:10:14,690 --> 00:10:17,930
+by theorem فاكرين نظرية اتنين اربعة؟
+
+85
+00:10:25,000 --> 00:10:33,100
+with c بساوي واحد أكبر من سفر and ال sequence an
+
+86
+00:10:33,100 --> 00:10:38,820
+بساوي واحد على n tensor zero بتخلي
+
+87
+00:10:38,820 --> 00:10:44,820
+.. so by this theorem بطلع عندي limit ال sequence
+
+88
+00:10:44,820 --> 00:10:54,430
+xkn as n tensor infinity بساوي ال S وهو المطلوبإذن
+
+89
+00:10:54,430 --> 00:11:00,690
+هي اللي أثبتت إنه يوجد subsequence من ال sequence
+
+90
+00:11:00,690 --> 00:11:06,310
+XN وال subsequence هذه convergent إلى العدد S
+
+91
+00:11:22,860 --> 00:11:27,900
+بالمناسبة في تعريف هنا في الكتب لو شفتوا طلعتوا
+
+92
+00:11:27,900 --> 00:11:33,380
+على الكتب في الأول واضح البرهان هنا؟ واضح البرهان؟
+
+93
+00:11:33,380 --> 00:11:42,560
+في أي شيء مش واضح؟ في
+
+94
+00:11:42,560 --> 00:11:46,860
+أي شيء مش واضح؟ اللي عندها سؤال تسألني أنا هنا مش
+
+95
+00:11:46,860 --> 00:11:51,040
+تتكلم مع اللي جنبها في أي شيء في البرهان مش واضح؟
+
+96
+00:11:52,600 --> 00:12:03,620
+طيب إذا المرهان معناته واضح في
+
+97
+00:12:03,620 --> 00:12:18,340
+definition هنا definition تعريف the
+
+98
+00:12:18,340 --> 00:12:28,700
+number isin above example is
+
+99
+00:12:28,700 --> 00:12:36,260
+called limit
+
+100
+00:12:36,260 --> 00:12:40,620
+superior
+
+101
+00:12:40,620 --> 00:12:47,060
+of sequence x in
+
+102
+00:12:50,000 --> 00:12:57,160
+and we write limit
+
+103
+00:12:57,160 --> 00:13:06,800
+superior ل
+
+104
+00:13:06,800 --> 00:13:10,520
+Xn بساوي S
+
+105
+00:13:13,780 --> 00:13:19,800
+إذا العدد S هذا في التمرين أو في ال exercise أو في
+
+106
+00:13:19,800 --> 00:13:25,560
+المثال هذا بنسميه
+
+107
+00:13:25,560 --> 00:13:30,780
+limit superior ل sequence xn limit superior of the
+
+108
+00:13:30,780 --> 00:13:33,760
+sequence xn بيسموها بالعربي النهاية العليا
+
+109
+00:13:33,760 --> 00:13:41,640
+للمتتالية في نهاية عالية و في نهاية سفلة وبالتالي
+
+110
+00:13:46,810 --> 00:13:52,670
+هذه عبارة عن الـ limit يعني
+
+111
+00:13:52,670 --> 00:13:55,750
+هذه بتطلع .. يعني ممكن أثبات باستخدام الـ monotone
+
+112
+00:13:55,750 --> 00:14:01,290
+convergence theorem أن هذه بالساوي هي نفسها limit
+
+113
+00:14:01,290 --> 00:14:11,530
+S as N tends to infinity limit
+
+114
+00:14:11,530 --> 00:14:17,940
+ال supremum لل sequence هذه أو limit superiorلأن
+
+115
+00:14:17,940 --> 00:14:22,060
+الـ sequence هذي decrease in وبما أن X in bounded
+
+116
+00:14:22,060 --> 00:14:27,400
+إذا هذي bounded وبالتالي
+
+117
+00:14:27,400 --> 00:14:32,360
+limit S in بساوي ال infimum لل sequence S in by
+
+118
+00:14:32,360 --> 00:14:39,480
+monotone convergence theorem اللي هو S تمام؟
+
+119
+00:14:39,480 --> 00:14:43,500
+بالمثل في حاجة اسمها limit inferior أو النهاية
+
+120
+00:14:43,500 --> 00:14:44,120
+السفلة
+
+121
+00:14:48,820 --> 00:14:59,140
+similarly أو remark ..similarly
+
+122
+00:14:59,140 --> 00:15:02,700
+أو using
+
+123
+00:15:02,700 --> 00:15:12,320
+this .. using a similar .. a similar argument
+
+124
+00:15:14,860 --> 00:15:20,100
+يعني باستخدام برهان مشابه لبرهان التمرين اللي فات
+
+125
+00:15:20,100 --> 00:15:27,760
+one can easily easily
+
+126
+00:15:27,760 --> 00:15:37,380
+show that if x
+
+127
+00:15:37,380 --> 00:15:39,880
+in contained in R is bounded
+
+128
+00:15:43,230 --> 00:15:52,710
+و S N بساوي المرهد الانفمام ل X K حيث K أكبر من أو
+
+129
+00:15:52,710 --> 00:16:03,670
+ساوي N و S بساوي Supremum بدل الانفمام لل sequence
+
+130
+00:16:03,670 --> 00:16:10,450
+S N حيث N تنتمي إلى N
+
+131
+00:16:15,800 --> 00:16:21,340
+ثم يوجد تجارب تجارب
+
+132
+00:16:21,340 --> 00:16:24,700
+XKN
+
+133
+00:16:24,700 --> 00:16:29,100
+تجارب XN تجارب تجارب XN تجارب XN تجارب XN تجارب XN
+
+134
+00:16:29,100 --> 00:16:32,500
+تجارب XN تجارب XN تجارب XN تجارب XN تجارب XN تجارب
+
+135
+00:16:32,500 --> 00:16:35,800
+XN تجارب XN تجارب XN تجارب XN تجارب XN تجارب XN
+
+136
+00:16:35,800 --> 00:16:35,900
+تجارب XN تجارب XN تجارب XN تجارب XN تجارب XN تجارب
+
+137
+00:16:35,900 --> 00:16:36,120
+XN تجارب XN تجارب XN تجارب XN تجارب XN تجارب XN
+
+138
+00:16:36,120 --> 00:16:36,560
+تجارب XN تجارب XN تجارب XN تجارب XN تجارب XN تجارب
+
+139
+00:16:36,560 --> 00:16:38,000
+XN تجارب XN
+
+140
+00:16:46,190 --> 00:16:50,570
+برضه العدد S في الحالة هذه بنسميه limit inferior
+
+141
+00:16:50,570 --> 00:17:05,330
+لل sequence XM S in above remark is
+
+142
+00:17:05,330 --> 00:17:11,850
+called is called limit
+
+143
+00:17:14,350 --> 00:17:26,290
+limit inferior limit inferior of ال sequence x in
+
+144
+00:17:26,290 --> 00:17:29,730
+and
+
+145
+00:17:29,730 --> 00:17:35,490
+we write limit
+
+146
+00:17:35,490 --> 00:17:40,910
+inferior ل x in بساوي s
+
+147
+00:17:45,400 --> 00:17:49,760
+وكمان مرة هذه limit inferior بتساوي .. ممكن اثبات
+
+148
+00:17:49,760 --> 00:17:54,080
+باستخدام الـ monotone convergence theorem ممكن
+
+149
+00:17:54,080 --> 00:18:00,200
+اثبات ان هذا بساوي limit sequence
+
+150
+00:18:00,200 --> 00:18:07,240
+S as N tends to infinityلأن هذه الـ sequence Sn
+
+151
+00:18:07,240 --> 00:18:12,280
+ممكن اثبات إنها increasing متزايدة و bounded
+
+152
+00:18:12,280 --> 00:18:15,960
+therefore by monotone convergence theorem it
+
+153
+00:18:15,960 --> 00:18:20,400
+converges و ال limit تبعتها هتساوي ال supremum إلى
+
+154
+00:18:20,400 --> 00:18:27,840
+اللي هو Sتمام؟ إذا هذه بنتسميها limit inferior of
+
+155
+00:18:27,840 --> 00:18:32,840
+the sequence xn وهذه limit superior هي النهاية
+
+156
+00:18:32,840 --> 00:18:37,880
+العليا الصفلة النهاية العليا طبعا إذا كانت ال
+
+157
+00:18:37,880 --> 00:18:42,780
+sequence bounded ف limit superior لها exist
+
+158
+00:18:42,780 --> 00:18:48,240
+وبالساوي العدد S وكذا لك limit inferior لها exist
+
+159
+00:18:48,240 --> 00:18:54,990
+وبالساوي العدد Sبس العدد S هذا غير عن العدد S دا
+
+160
+00:18:54,990 --> 00:18:59,610
+يعني هذا خلينا نسميه S star عشان نميزه عن العدد
+
+161
+00:18:59,610 --> 00:19:07,530
+اللي فات ده خلينا نسميه S star S star وS star هنا
+
+162
+00:19:07,530 --> 00:19:13,090
+غير مختلف مش شرط يكون هو نفس ال S في ال exercise
+
+163
+00:19:13,090 --> 00:19:20,430
+اللي فات إذا نحاولوا تثبتهحاولوا تثبتوا التمرين
+
+164
+00:19:20,430 --> 00:19:25,530
+هذا اللي هو الخاص ب limit inferior بنفس ال
+
+165
+00:19:25,530 --> 00:19:30,990
+argument by similar argument باستخدام استنتاج أو
+
+166
+00:19:30,990 --> 00:19:36,470
+برهان مشابه للبرهان اللي عملناها للجزء اللي فات
+
+167
+00:19:36,470 --> 00:19:41,840
+اللي هو تمرين عشرة okay تمام؟واضح؟ اذا هنا اليوم
+
+168
+00:19:41,840 --> 00:19:45,360
+اتعلمنا ان في حاجة اسمها النهاية العليا والنهاية
+
+169
+00:19:45,360 --> 00:19:49,360
+السفلة لل sequence إذا كانت ال sequence bounded
+
+170
+00:19:49,360 --> 00:19:53,720
+فأثبتنا هنا أن النهاية العليا تبقاتها موجودة
+
+171
+00:19:53,720 --> 00:19:56,960
+وبالساوي العدد S اللي هو ال infimum لل sequence
+
+172
+00:19:56,960 --> 00:20:01,050
+هذهإذا كانت برضه ال sequence bounded فالنهاية
+
+173
+00:20:01,050 --> 00:20:06,670
+السفلة لها أو limit inferior بتطلع exist و بتساوي
+
+174
+00:20:06,670 --> 00:20:10,210
+العدد s star اللي هو ال supremum لل sequence هذه
+
+175
+00:20:10,210 --> 00:20:17,810
+أو بساوي limit ال sequence s هذه تمام؟ فحاولوا
+
+176
+00:20:17,810 --> 00:20:22,310
+تبرهنوا الجزء هذا المشابه للجزء الأول يعني في أي
+
+177
+00:20:22,310 --> 00:20:23,650
+سؤال أو استفسار؟
+
+178
+00:20:26,730 --> 00:20:30,430
+طيب الآن إذا ننتقل ل section جديد هيك احنا متكون
+
+179
+00:20:30,430 --> 00:20:36,110
+خلصنا section تلاتة اربعة في الكتاب المخرج و هنبدأ
+
+180
+00:20:36,110 --> 00:20:42,730
+section جديد اللي
+
+181
+00:20:42,730 --> 00:20:51,010
+هو section تلاتة خمسة الانوان
+
+182
+00:20:51,010 --> 00:20:54,990
+تبع koshi sequences
+
+183
+00:20:57,810 --> 00:21:03,410
+متتاليات كوشي طبعا كوشي هاد عالم ألماني German
+
+184
+00:21:03,410 --> 00:21:09,050
+mathematician كان مهتم بدراسة نوع خاص من ال
+
+185
+00:21:09,050 --> 00:21:14,570
+sequences وكان بحاول يدرس التقارب و التباعت تباعهم
+
+186
+00:21:14,570 --> 00:21:18,740
+و علاقتهم بالconvergent و divergence sequences
+
+187
+00:21:18,740 --> 00:21:24,820
+العادية فهذا يعني من كان جزء من اهتماماته طبعا له
+
+188
+00:21:24,820 --> 00:21:27,320
+حاجات كتيرة في ال analysis
+
+189
+00:21:31,200 --> 00:21:35,700
+فالجماعة الرياضيات او الناس اللي جوا بعده عشان
+
+190
+00:21:35,700 --> 00:21:39,060
+يعني تقديرا للجهد اللي قامله وللدراسة اللي قاملها
+
+191
+00:21:39,060 --> 00:21:44,380
+لهذا النوع الخاص من المتتاليات سموا المتتاليات هذه
+
+192
+00:21:44,380 --> 00:21:50,780
+باسمه متتاليات كوشي او كوشي sequences فما هي
+
+193
+00:21:50,780 --> 00:21:55,800
+متتاليات كوشي؟ متى بنقول ان ال sequence كوشي؟ فهذه
+
+194
+00:21:55,800 --> 00:21:56,360
+ال definition
+
+195
+00:22:01,970 --> 00:22:08,870
+a sequence of real numbers a sequence xn contained
+
+196
+00:22:08,870 --> 00:22:21,230
+in R is Cauchy is Cauchy if الشرط التالي بتحقق لكل
+
+197
+00:22:21,230 --> 00:22:27,970
+إبسلون أكبر من السفر يوجد capital N يعتمد على
+
+198
+00:22:27,970 --> 00:22:37,300
+إبسلون عدد طبيعيبحيث أنه لو كان n و m أكبر من أو
+
+199
+00:22:37,300 --> 00:22:42,920
+ساوي capital N فهذا لازم يقدي أن المسافة بين xn و
+
+200
+00:22:42,920 --> 00:22:52,600
+xm أصغر من إبسلون هذه هي كوشي sequence sequence of
+
+201
+00:22:52,600 --> 00:22:57,360
+real number بنسميها كوشي متتالية كوشي إذا كان لأي
+
+202
+00:22:57,360 --> 00:23:02,360
+given إبسلونبقدر ألاقي عدد طبيعي يعتمد على epsilon
+
+203
+00:23:02,360 --> 00:23:08,400
+بحيث لكل N و M لكل المؤشرات اللي من capital N و N
+
+204
+00:23:08,400 --> 00:23:14,620
+طالع المسافة تبع الحدود الفرق بين حدودهم أصغر من
+
+205
+00:23:14,620 --> 00:23:19,800
+epsilon في
+
+206
+00:23:19,800 --> 00:23:23,040
+هنا أول لمة
+
+207
+00:23:33,720 --> 00:23:42,080
+لمّة واحدة عشرين اش لمّة بتقول every ..
+
+208
+00:23:42,080 --> 00:23:52,420
+every convergent .. every convergent sequence of
+
+209
+00:23:52,420 --> 00:23:58,120
+real numbers is
+
+210
+00:23:58,120 --> 00:24:08,020
+Cauchy كل متتاليةمتقاربة بتكون كشية وهي البرهان
+
+211
+00:24:08,020 --> 00:24:14,180
+prove assume
+
+212
+00:24:14,180 --> 00:24:20,160
+let x
+
+213
+00:24:20,160 --> 00:24:26,920
+in contained in R be
+
+214
+00:24:26,920 --> 00:24:32,000
+such that limit x in بساوي x
+
+215
+00:24:36,590 --> 00:24:41,730
+يعني افرض ان في عندي sequence xn وconvergent ل x
+
+216
+00:24:41,730 --> 00:24:45,110
+من
+
+217
+00:24:45,110 --> 00:24:50,190
+ان اثبت ان ال sequence هذه كوشي او كوشية ف let
+
+218
+00:24:50,190 --> 00:24:54,170
+epsilon لاثبات ان ال sequence كوشي لازم نبدأ
+
+219
+00:24:54,170 --> 00:24:58,210
+بepsilon ونرد عليها بcapital N تعطين ال
+
+220
+00:24:58,210 --> 00:25:02,070
+implication هذه صح؟ مش هيك التعريب يقول ان نبدأ
+
+221
+00:25:02,070 --> 00:25:06,090
+let epsilon أكبر من الصفر be given
+
+222
+00:25:11,220 --> 00:25:15,720
+طيب من تعريف epsilon capital N للكنفرجنس ال
+
+223
+00:25:15,720 --> 00:25:20,840
+sequence xn converge لx إذا there exist لأي given
+
+224
+00:25:20,840 --> 00:25:26,020
+epsilon بما أن xn converge لx لأي epsilon يوجد
+
+225
+00:25:26,020 --> 00:25:35,540
+capital N يعتمد على epsilon عدد طبيعي بحيث أنه
+
+226
+00:25:45,980 --> 00:25:53,820
+بحيث انه لو كان n أكبر من أو ساوي capital N هذا
+
+227
+00:25:53,820 --> 00:26:00,460
+بيقدي ان absolute xn minus x أصغر من epsilon ع 2
+
+228
+00:26:00,460 --> 00:26:04,680
+نسمي ال implication هذه start
+
+229
+00:26:09,880 --> 00:26:16,500
+الان لو أخدت N و M كلا هما أعداد طبيعية أكبر من أو
+
+230
+00:26:16,500 --> 00:26:24,680
+يساوي capital N هذا بيقدي ان absolute X N minus X
+
+231
+00:26:24,680 --> 00:26:34,020
+M هذا بيساوي absolute X N minus X زائد X minus X M
+
+232
+00:26:34,020 --> 00:26:39,320
+هاي المجموع هذا وهي المجموع هذا
+
+233
+00:26:49,790 --> 00:26:53,850
+إذا أنا طرحت x ورجعت x باستخدام ال triangle
+
+234
+00:26:53,850 --> 00:27:00,510
+inequality ال absolute value للمجموع أصلا أو ساوي
+
+235
+00:27:00,510 --> 00:27:04,410
+مجموع ال absolute values إذا أنا عندي absolute xn
+
+236
+00:27:04,410 --> 00:27:14,010
+minus x زائد absolute x minus xm طيب باستخدام ال
+
+237
+00:27:14,010 --> 00:27:20,260
+implication starأنا عندي ال N، ال N هذه أكبر من أو
+
+238
+00:27:20,260 --> 00:27:24,760
+ساوي capital N، إذن حسب ال star، absolute الفرخ
+
+239
+00:27:24,760 --> 00:27:29,740
+هذا أصغر من epsilon ع اتنين كذلك أنا عندي ال M
+
+240
+00:27:29,740 --> 00:27:36,800
+أكبر من أو ساوي N، إذن by star بدل N ب M، المسافة
+
+241
+00:27:36,800 --> 00:27:42,630
+هذه تطلع أصغر من epsilon ع اتنينالمجموعة epsilon
+
+242
+00:27:42,630 --> 00:27:49,350
+إذا هاني أثبتت أنه لأي epsilon أكبر من السفر في
+
+243
+00:27:49,350 --> 00:27:54,890
+capital N يوجد capital N تعتمد على epsilon بحيث
+
+244
+00:27:54,890 --> 00:27:59,910
+لكل N و M أكبر من أو ساوي capital N بطلع المسافة
+
+245
+00:27:59,910 --> 00:28:05,690
+بين الحد رقم N والحد رقم M أصغر من epsilon إذا
+
+246
+00:28:05,690 --> 00:28:08,290
+since
+
+247
+00:28:10,030 --> 00:28:17,650
+إبسلون أكبر من السفر was arbitrary إن
+
+248
+00:28:17,650 --> 00:28:22,330
+الكلام هذا صحيح أثبتنا هذا الكلام صحيح لكل إبسلون
+
+249
+00:28:22,330 --> 00:28:28,030
+الإبسلون هذي was arbitrary عشوائية إن هيك بتكون
+
+250
+00:28:28,030 --> 00:28:34,610
+حققنا الشرط تبقى cushy sequences وبالتالي we have
+
+251
+00:28:34,610 --> 00:28:38,370
+we get that من التعريف
+
+252
+00:28:56,870 --> 00:29:04,070
+العكس تبع النظرية هذه صحيحهنشوفه بعد شوية لكن في
+
+253
+00:29:04,070 --> 00:29:11,770
+كمان نظرية او لمّة بسيطة زي هذه هنثبتها
+
+254
+00:29:11,770 --> 00:29:20,450
+الآن بتقول ان كل كوشي sequence is bounded واضح
+
+255
+00:29:20,450 --> 00:29:23,470
+البرهان؟ في اي استفسار على البرهان؟ اعتقد البرهان
+
+256
+00:29:23,470 --> 00:29:28,480
+سهل وبسيطهذا ال .. النوع من البرهان بنسميه epsilon
+
+257
+00:29:28,480 --> 00:29:37,360
+بنسميه epsilon over two argument epsilon
+
+258
+00:29:37,360 --> 00:29:44,380
+over two argument استخدامنا epsilon over two برهان
+
+259
+00:29:44,380 --> 00:29:51,140
+باستخدام epsilon over two إذن نثبت اللمة التانية
+
+260
+00:29:51,140 --> 00:29:54,320
+قبل ما نثبت عكس النظر اللي فاتت صحيح
+
+261
+00:30:00,800 --> 00:30:10,180
+لمّا 22 every Cauchy sequence
+
+262
+00:30:10,180 --> 00:30:20,600
+in R is bounded والبرهان
+
+263
+00:30:20,600 --> 00:30:27,760
+تبعها شبيه إلى حد كبير ببرهان أن كل convergence
+
+264
+00:30:27,760 --> 00:30:32,110
+sequence is boundedأثبتنا قبل هيك أن كل conversion
+
+265
+00:30:32,110 --> 00:30:35,390
+sequence is bounded اليوم هأثبت أن كل Cauchy
+
+266
+00:30:35,390 --> 00:30:40,090
+sequence is bounded والبرهان مشابه للبرهان السابق
+
+267
+00:30:40,090 --> 00:30:44,950
+proof let
+
+268
+00:30:44,950 --> 00:30:54,590
+xn contained in R be Cauchy be Cauchy sequence
+
+269
+00:30:57,010 --> 00:31:03,350
+و خلّينا ناخد epsilon then
+
+270
+00:31:03,350 --> 00:31:09,990
+من تعريف الكوشي sequence for epsilon بساوي واحد
+
+271
+00:31:09,990 --> 00:31:16,130
+أكبر من السفر يوجد capital N يعتمد على ال epsilon
+
+272
+00:31:16,130 --> 00:31:26,330
+اللي هو الواحد عدد طبيعي بحيث أنه لabsolute xn
+
+273
+00:31:26,330 --> 00:31:34,150
+minus xm أصغر من إبسلون اللي هو واحد وهذا الكلام
+
+274
+00:31:34,150 --> 00:31:40,930
+صحيح لكل N و M أكبر من أو ساوي capital N هذا من
+
+275
+00:31:40,930 --> 00:31:49,190
+تعريف الكوشي sequence تمام؟ طيب hence
+
+276
+00:31:50,870 --> 00:31:59,410
+من المتباينة هذه بنحصل على absolute xn minus x
+
+277
+00:31:59,410 --> 00:32:06,670
+capital N أصغر من واحد وهذا الكلام صحيحلكل N أكبر
+
+278
+00:32:06,670 --> 00:32:11,130
+منه ساوي capital N اذا انا اشعلنت هنا المتباين هي
+
+279
+00:32:11,130 --> 00:32:18,910
+دي حصلت عليها من اللي فوق و ذلك بياخذ take M بساوي
+
+280
+00:32:18,910 --> 00:32:24,110
+capital N انا أخدت ال M هنا بساوي capital N وهذا
+
+281
+00:32:24,110 --> 00:32:30,430
+مسموح ليه فصارت هيكطب ليش أنا عملت هيك لحاجة في
+
+282
+00:32:30,430 --> 00:32:36,930
+نفسي عقوب؟ عشان فيه غرض للاستفاء البرهان أو
+
+283
+00:32:36,930 --> 00:32:45,150
+المطلوب في البرهان فمن هنا by triangle inequality
+
+284
+00:32:45,150 --> 00:32:52,690
+من متباينة المثلثأنا عندي لو كان n أكبر من أو ساوي
+
+285
+00:32:52,690 --> 00:32:58,150
+capital N هذا بيقدي ان absolute x in بالظبط زي ما
+
+286
+00:32:58,150 --> 00:33:02,330
+عملنا في برهان كل convergence sequence is bounded
+
+287
+00:33:02,330 --> 00:33:10,010
+فهذا بنكتب على صورة x in minus x capital N زائد x
+
+288
+00:33:10,010 --> 00:33:16,920
+capital N إذا أنا طرحت x capital N و رجعتهاالان
+
+289
+00:33:16,920 --> 00:33:21,740
+هذا اصغر من او ساوي absolute x n سالب x capital n
+
+290
+00:33:21,740 --> 00:33:28,340
+زاد absolute x capital n وهذا
+
+291
+00:33:28,340 --> 00:33:33,660
+هذا هو هذا المقدار لكل n اكبر من او ساوي capital n
+
+292
+00:33:33,660 --> 00:33:37,960
+اصغر من واحد اذا هذا اصغر من واحد زاد absolute x
+
+293
+00:33:37,960 --> 00:33:43,540
+capital n تمام الان
+
+294
+00:33:43,540 --> 00:33:52,020
+nowlet خلّينا ناخد m .. نعرف عدد m على أنه ال
+
+295
+00:33:52,020 --> 00:33:57,480
+supremum أو ال maximum للأعداد الحقيقية غير
+
+296
+00:33:57,480 --> 00:34:05,720
+السالبة اللي هي absolute x1 absolute x2 إلى
+
+297
+00:34:05,720 --> 00:34:16,000
+absolute x رقم capital M minus 1واخر حد absolute
+
+298
+00:34:16,000 --> 00:34:20,420
+واحد
+
+299
+00:34:20,420 --> 00:34:38,180
+زائد واحد زائد absolute x رقم capital N you
+
+300
+00:34:38,180 --> 00:34:39,340
+can verify
+
+301
+00:34:42,590 --> 00:34:51,350
+بإمكانكم التحقق إنه absolute xn أصغر طبعاً العدد m
+
+302
+00:34:51,350 --> 00:34:58,210
+هذا موجب بالتأكيد هذا موجب لأن
+
+303
+00:34:58,210 --> 00:35:05,420
+العداد هذه كلها موجبةو بعدين absolute xn ممكن تحقق
+
+304
+00:35:05,420 --> 00:35:10,380
+أنه أخدت أي عنصر في ال sequence xn و أخدت القيمة
+
+305
+00:35:10,380 --> 00:35:17,100
+المطلقة إليه فهذا هيطلع أصغر من أو ساوي m لكل n في
+
+306
+00:35:17,100 --> 00:35:17,460
+n
+
+307
+00:35:20,190 --> 00:35:25,570
+هل إيش هذا صحيح؟ لأنه لو كانت ال N هذه لو كانت ال
+
+308
+00:35:25,570 --> 00:35:31,750
+N هي index من واحد لcapital N لcapital N سالب واحد
+
+309
+00:35:31,750 --> 00:35:36,130
+لو كانت ال N هذه مؤشر N ده واحد من المؤشرات واحد
+
+310
+00:35:36,130 --> 00:35:40,310
+إلى capital N سالب واحد فالقيمة المطلقة اللي هو
+
+311
+00:35:40,310 --> 00:35:45,530
+واحد من الحدود هدول وكل حد من هدول أصلا من أو ساوي
+
+312
+00:35:45,530 --> 00:35:49,490
+ال upper bound ال M هذا upper bound لل set هذه
+
+313
+00:35:52,340 --> 00:35:57,880
+طب لو كانت ال n small n هذه بالساوي capital N او
+
+314
+00:35:57,880 --> 00:36:04,980
+اكبر capital N او اكبر هيا لو كانت small n اكبر من
+
+315
+00:36:04,980 --> 00:36:10,320
+او ساوي capital N فبطلع absolute xn اصغر من واحد
+
+316
+00:36:10,320 --> 00:36:14,900
+زاد absolute xn اذا القيمة المطلقة هذه اصغر من
+
+317
+00:36:14,900 --> 00:36:22,500
+واحد زاد absolute xn وهذا العدد اصغر منأو يساوي ال
+
+318
+00:36:22,500 --> 00:36:26,340
+upper bound لل set اللي بتحتويه و بتحتوي عناصر
+
+319
+00:36:26,340 --> 00:36:31,240
+تانية وبالتالي absolute xm أصغر من أو ساوي okay
+
+320
+00:36:31,240 --> 00:36:39,900
+تمام؟ إذن هذا بكمل البرهجة المرة الجاية طبعا هنشوف
+
+321
+00:36:39,900 --> 00:36:45,730
+أنه عكس النمة هذه صحيحوها دي بنسميها Cauchy
+
+322
+00:36:45,730 --> 00:36:50,510
+criterion أو معيار Cauchy ومعيار Cauchy بيقول إن
+
+323
+00:36:50,510 --> 00:36:54,510
+لو كانت ال sequence Cauchy بتكون أيضا convergent
+
+324
+00:36:54,510 --> 00:36:58,430
+وبالتالي بنحصل على نظرية Cauchy أو معيار Cauchy
+
+325
+00:36:58,430 --> 00:37:04,010
+Cauchy criterion وهو إن ال sequence is convergent
+
+326
+00:37:04,010 --> 00:37:09,690
+if and only if it is Cauchyإذا هذا الجزء التالي من
+
+327
+00:37:09,690 --> 00:37:14,050
+النظرية أو عكس اللمحة دي هنشوفه المرة الجاية وناخد
+
+328
+00:37:14,050 --> 00:37:20,350
+أمثلة كيف نثبت أن ال sequence معينة is Cauchy فهذا
+
+329
+00:37:20,350 --> 00:37:22,830
+موجود في section تلاتة خمسة في الكتاب حاولوا
+
+330
+00:37:22,830 --> 00:37:27,930
+تقرؤوا باقى ال section و تحضروه للمرة الجاية تمام؟
+
+331
+00:37:27,930 --> 00:37:30,730
+إذا انتهت هيك المحاضرة نشوفكم ان شاء الله يوم
+
+332
+00:37:30,730 --> 00:37:31,190
+اتنين
+
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/eX4Dw2M3cKU_raw.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/eX4Dw2M3cKU_raw.srt
new file mode 100644
index 0000000000000000000000000000000000000000..58a60d03535a710a5a391a8e6d1ed793a489cad0
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/eX4Dw2M3cKU_raw.srt
@@ -0,0 +1,1328 @@
+1
+00:00:23,030 --> 00:00:31,610
+في exercise هنحاول ان شاء الله نحله وهذا ال
+
+2
+00:00:31,610 --> 00:00:37,190
+exercise مهم رقم عشرة section تلاتة اربعة في ال
+
+3
+00:00:37,190 --> 00:00:41,470
+textbook او الكتاب المقرر ال exercise هذا بيقول
+
+4
+00:00:41,470 --> 00:00:48,370
+let xn be bounded sequence of real numbers let Sn
+
+5
+00:00:48,370 --> 00:00:53,550
+be the supremumلكل حدود ال sequence من N و M تطالع
+
+6
+00:00:53,550 --> 00:01:05,030
+يعني هذا عبارة عن ال supremum ل X N X رقم N زياد
+
+7
+00:01:05,030 --> 00:01:09,950
+واحد و هكذا فمثلا
+
+8
+00:01:09,950 --> 00:01:15,390
+S1 هيسوي ال supremum لكل ال sequence لما N بسوا
+
+9
+00:01:15,390 --> 00:01:23,160
+واحد صح؟S2 هيساوي ال supremum ل X2 و X3 إلى آخرها
+
+10
+00:01:23,160 --> 00:01:30,260
+لأن هذا ال supremum ل ال one tail ل ال two tail S3
+
+11
+00:01:30,260 --> 00:01:35,060
+هو ال supremum ل X3 و X4 إلى آخرها لأن هذا ال
+
+12
+00:01:35,060 --> 00:01:39,600
+supremum لل تلاتة tail of the sequence M tail
+
+13
+00:01:42,530 --> 00:01:47,570
+لاحظوا إن الـ Sn هذه كل ما N كبرت كل ما ال 6 هذه
+
+14
+00:01:47,570 --> 00:01:54,130
+زغرت و ال suprem زغرت فبتطلع decreasing sequence و
+
+15
+00:01:54,130 --> 00:02:00,050
+طبعا bounded إذا الإنفمام تبعها موجود بسبب عدد S
+
+16
+00:02:00,050 --> 00:02:07,790
+إذن هنا بنعرف ال S على إن الإنفمام لل sequence Sn
+
+17
+00:02:12,100 --> 00:02:16,480
+الان بدنا نثبت ان يوجد subsequence XKN من سيكوانس
+
+18
+00:02:16,480 --> 00:02:21,720
+XN وهذه السبسيكوانس convergence للعدد S اللي هو
+
+19
+00:02:21,720 --> 00:02:27,720
+الانثمام لكل S لبرهان ذلك انا عندي S بالساوية
+
+20
+00:02:27,720 --> 00:02:31,620
+الانثمام من الفرض لسيكوانس SN
+
+21
+00:02:34,540 --> 00:02:41,960
+أخدنا جابل هيك لمبة بتقول أنه w بساول infimum ل
+
+22
+00:02:41,960 --> 00:02:47,960
+set S if and only if لكل epsilon أكبر من الصفر
+
+23
+00:02:47,960 --> 00:02:57,560
+يوجد S epsilon ينتمي ل S بحيث أنه S
+
+24
+00:02:57,560 --> 00:02:58,360
+epsilon
+
+25
+00:03:01,030 --> 00:03:09,790
+فأصغر من W زاد Epsilon، مظبوط هكذا، وفي كمان حقيقة
+
+26
+00:03:09,790 --> 00:03:14,890
+تانية أو لمبة تانية بتقول أن U بساوي ال suprem ل
+
+27
+00:03:14,890 --> 00:03:21,050
+S، F and only F لكل Epsilon أكبر من السفر يوجد S
+
+28
+00:03:21,050 --> 00:03:24,670
+Epsilon ينتمي ل S بحيث أنه
+
+29
+00:03:27,490 --> 00:03:35,870
+u minus epsilon أصغر من s epsilon نظبط يعنى فهنطبخ
+
+30
+00:03:35,870 --> 00:03:40,750
+الكلام هذا إذا هي عندي s بساوة n في وملة ال set
+
+31
+00:03:40,750 --> 00:03:53,330
+هذه اعتبر هذه ال set S فهذا هو ال W فلو أخدت for
+
+32
+00:03:53,330 --> 00:04:04,110
+eachin عدد طبيعي لكل عدد طبيعي in there
+
+33
+00:04:04,110 --> 00:04:07,910
+exist sn
+
+34
+00:04:07,910 --> 00:04:17,090
+ينتمي لل set هذه ال set of all sn hyphen عدد طبيعي
+
+35
+00:04:17,090 --> 00:04:28,190
+بحيث انه ال هذا ال snأصغر من الانفرام اللي هو S زي
+
+36
+00:04:28,190 --> 00:04:34,830
+واحد على N اللي هو الابسلون هذا عبارة عن الابسلون
+
+37
+00:04:34,830 --> 00:04:42,940
+ممكن اخد ابسلون بساوي واحد على N هذا عدد موجةفلأي
+
+38
+00:04:42,940 --> 00:04:47,760
+epsilon عدد موجة يوجد عنصر في ال set اللي باخدلها
+
+39
+00:04:47,760 --> 00:04:51,680
+ال infimum بحيث أن هذا العنصر أصغر من ال infimum
+
+40
+00:04:51,680 --> 00:05:05,200
+زاد epsilon حسب الحقيقة الأولى طيب كذلك since ايضا
+
+41
+00:05:05,200 --> 00:05:12,890
+من الفرض احنا فرضين ان sn بساوي supremumللset of
+
+42
+00:05:12,890 --> 00:05:21,530
+all xk حيث k أكبر من أو ساوي ال n بما أن هذا صحيح
+
+43
+00:05:21,530 --> 00:05:28,210
+إذا أنا باستخدام الحقيقة التانية then
+
+44
+00:05:28,210 --> 00:05:33,410
+for epsilon لو أخدت epsilon برضه واحد على n هذا
+
+45
+00:05:33,410 --> 00:05:42,830
+عدد موجب فبنقدر نلاقي يوجدx عنصر هنا في ال
+
+46
+00:05:42,830 --> 00:05:52,250
+sequence هذه أو في ال set هذه نسمي xkn ينتمي لset
+
+47
+00:05:52,250 --> 00:06:04,570
+of all xk حيث k أكبر من أو يساوي ال n such
+
+48
+00:06:04,570 --> 00:06:05,070
+that
+
+49
+00:06:08,240 --> 00:06:13,400
+الـ supremum اللي هو SN minus الـ Y اللي هو واحد
+
+50
+00:06:13,400 --> 00:06:28,220
+على N أصغر من الـ SY اللي هو العنصر X KN طيب
+
+51
+00:06:28,220 --> 00:06:30,200
+إذا من هنا
+
+52
+00:06:36,260 --> 00:06:45,240
+من المتباينة هذه والمتباينة هذه هينتج
+
+53
+00:06:45,240 --> 00:06:49,700
+it follows that
+
+54
+00:06:49,700 --> 00:07:00,020
+من المتباينتين هذول أنا عندي ال sn
+
+55
+00:07:00,020 --> 00:07:07,730
+سالب واحد على n أكبر من أو يساويcapital S minus
+
+56
+00:07:07,730 --> 00:07:13,890
+واحد على N لحظة
+
+57
+00:07:13,890 --> 00:07:20,330
+من هنا S N ال capital S lower bound للست هذه
+
+58
+00:07:20,330 --> 00:07:25,270
+وبالتالي أي أنصر هنا أكبر من أو يساوي ال S لأن أنا
+
+59
+00:07:25,270 --> 00:07:32,230
+عندي S N أكبر من أو يساوي capital S اطرحي واحد على
+
+60
+00:07:32,230 --> 00:07:37,590
+N من الطرفين ونحصل على الكلام هذاهذا بيقدّي إن هذا
+
+61
+00:07:37,590 --> 00:07:45,330
+الكلام صحيح طيب و من هنا من المتباين الأخير هذا
+
+62
+00:07:45,330 --> 00:07:49,650
+أصغر من xkn
+
+63
+00:07:49,650 --> 00:07:52,750
+و
+
+64
+00:07:52,750 --> 00:08:01,030
+xkn هذا عنصر في ال set هذه عنصر في ال set هذه و sn
+
+65
+00:08:01,030 --> 00:08:05,450
+upper bound لل setإذن هذا ال upper bound أكبر من
+
+66
+00:08:05,450 --> 00:08:11,790
+أو ساوي كل عناصر ال set إذن هذا أصغر من أو ساوي Sn
+
+67
+00:08:11,790 --> 00:08:16,630
+ومن
+
+68
+00:08:16,630 --> 00:08:23,290
+المتباينة هذه Sn أصغر من capital S زائد واحد على N
+
+69
+00:08:23,290 --> 00:08:26,490
+مظبوط؟
+
+70
+00:08:39,470 --> 00:08:50,830
+thus we obtain a subsequence الكلام هذا صحيح لكل
+
+71
+00:08:50,830 --> 00:08:51,070
+ان
+
+72
+00:08:54,210 --> 00:08:59,590
+لأن هذا الكلام صحيح لكل n و هذا الكلام صحيح لكل n
+
+73
+00:08:59,590 --> 00:09:05,790
+لكل epsilon بالساعة واحد على n و لكل .. لكل n ممكن
+
+74
+00:09:05,790 --> 00:09:10,450
+أكون epsilon بالساعة واحد على n وهي هنا لكل n for
+
+75
+00:09:10,450 --> 00:09:14,670
+each n كلام هذا صحيحأذا هذا الكلام المتباين هذه
+
+76
+00:09:14,670 --> 00:09:21,130
+صحيحة لكل n وبالتالي أنا لكل n بقدر ألاقي xkn
+
+77
+00:09:21,130 --> 00:09:25,590
+موجود هنا و بيحقق المتباين هذه
+
+78
+00:09:28,570 --> 00:09:34,390
+عناصرها xkn من n بالساوية واحد to infinity of ال
+
+79
+00:09:34,390 --> 00:09:43,430
+sequence xn such that بحيث أنه xkn
+
+80
+00:09:43,430 --> 00:09:51,310
+أصغر من s زايد واحد على n أكبر من s سالب واحد على
+
+81
+00:09:51,310 --> 00:09:57,610
+n الكلام هذا صحيح لكل n في nطب ماذا معناه أن
+
+82
+00:09:57,610 --> 00:10:06,510
+absolute x k n minus s أصغر من واحد على n لكل n في
+
+83
+00:10:06,510 --> 00:10:14,690
+n؟ so
+
+84
+00:10:14,690 --> 00:10:17,930
+by theorem فاكرين نظرية اتنين اربعة؟
+
+85
+00:10:25,000 --> 00:10:33,100
+with c بساوي واحد أكبر من سفر and ال sequence an
+
+86
+00:10:33,100 --> 00:10:38,820
+بساوي واحد على n tensor zero بتخلي
+
+87
+00:10:38,820 --> 00:10:44,820
+.. so by this theorem بطلع عندي limit ال sequence
+
+88
+00:10:44,820 --> 00:10:54,430
+xkn as n tensor infinity بساوي ال S وهو المطلوبإذن
+
+89
+00:10:54,430 --> 00:11:00,690
+هي اللي أثبتت إنه يوجد subsequence من ال sequence
+
+90
+00:11:00,690 --> 00:11:06,310
+XN وال subsequence هذه convergent إلى العدد S
+
+91
+00:11:22,860 --> 00:11:27,900
+بالمناسبة في تعريف هنا في الكتب لو شفتوا طلعتوا
+
+92
+00:11:27,900 --> 00:11:33,380
+على الكتب في الأول واضح البرهان هنا؟ واضح البرهان؟
+
+93
+00:11:33,380 --> 00:11:42,560
+في أي شيء مش واضح؟ في
+
+94
+00:11:42,560 --> 00:11:46,860
+أي شيء مش واضح؟ اللي عندها سؤال تسألني أنا هنا مش
+
+95
+00:11:46,860 --> 00:11:51,040
+تتكلم مع اللي جنبها في أي شيء في البرهان مش واضح؟
+
+96
+00:11:52,600 --> 00:12:03,620
+طيب إذا المرهان معناته واضح في
+
+97
+00:12:03,620 --> 00:12:18,340
+definition هنا definition تعريف the
+
+98
+00:12:18,340 --> 00:12:28,700
+number isin above example is
+
+99
+00:12:28,700 --> 00:12:36,260
+called limit
+
+100
+00:12:36,260 --> 00:12:40,620
+superior
+
+101
+00:12:40,620 --> 00:12:47,060
+of sequence x in
+
+102
+00:12:50,000 --> 00:12:57,160
+and we write limit
+
+103
+00:12:57,160 --> 00:13:06,800
+superior ل
+
+104
+00:13:06,800 --> 00:13:10,520
+Xn بساوي S
+
+105
+00:13:13,780 --> 00:13:19,800
+إذا العدد S هذا في التمرين أو في ال exercise أو في
+
+106
+00:13:19,800 --> 00:13:25,560
+المثال هذا بنسميه
+
+107
+00:13:25,560 --> 00:13:30,780
+limit superior ل sequence xn limit superior of the
+
+108
+00:13:30,780 --> 00:13:33,760
+sequence xn بيسموها بالعربي النهاية العليا
+
+109
+00:13:33,760 --> 00:13:41,640
+للمتتالية في نهاية عالية و في نهاية سفلة وبالتالي
+
+110
+00:13:46,810 --> 00:13:52,670
+هذه عبارة عن الـ limit يعني
+
+111
+00:13:52,670 --> 00:13:55,750
+هذه بتطلع .. يعني ممكن أثبات باستخدام الـ monotone
+
+112
+00:13:55,750 --> 00:14:01,290
+convergence theorem أن هذه بالساوي هي نفسها limit
+
+113
+00:14:01,290 --> 00:14:11,530
+S as N tends to infinity limit
+
+114
+00:14:11,530 --> 00:14:17,940
+ال supremum لل sequence هذه أو limit superiorلأن
+
+115
+00:14:17,940 --> 00:14:22,060
+الـ sequence هذي decrease in وبما أن X in bounded
+
+116
+00:14:22,060 --> 00:14:27,400
+إذا هذي bounded وبالتالي
+
+117
+00:14:27,400 --> 00:14:32,360
+limit S in بساوي ال infimum لل sequence S in by
+
+118
+00:14:32,360 --> 00:14:39,480
+monotone convergence theorem اللي هو S تمام؟
+
+119
+00:14:39,480 --> 00:14:43,500
+بالمثل في حاجة اسمها limit inferior أو النهاية
+
+120
+00:14:43,500 --> 00:14:44,120
+السفلة
+
+121
+00:14:48,820 --> 00:14:59,140
+similarly أو remark ..similarly
+
+122
+00:14:59,140 --> 00:15:02,700
+أو using
+
+123
+00:15:02,700 --> 00:15:12,320
+this .. using a similar .. a similar argument
+
+124
+00:15:14,860 --> 00:15:20,100
+يعني باستخدام برهان مشابه لبرهان التمرين اللي فات
+
+125
+00:15:20,100 --> 00:15:27,760
+one can easily easily
+
+126
+00:15:27,760 --> 00:15:37,380
+show that if x
+
+127
+00:15:37,380 --> 00:15:39,880
+in contained in R is bounded
+
+128
+00:15:43,230 --> 00:15:52,710
+و S N بساوي المرهد الانفمام ل X K حيث K أكبر من أو
+
+129
+00:15:52,710 --> 00:16:03,670
+ساوي N و S بساوي Supremum بدل الانفمام لل sequence
+
+130
+00:16:03,670 --> 00:16:10,450
+S N حيث N تنتمي إلى N
+
+131
+00:16:15,800 --> 00:16:21,340
+ثم يوجد تجارب تجارب
+
+132
+00:16:21,340 --> 00:16:24,700
+XKN
+
+133
+00:16:24,700 --> 00:16:29,100
+تجارب XN تجارب تجارب XN تجارب XN تجارب XN تجارب XN
+
+134
+00:16:29,100 --> 00:16:32,500
+تجارب XN تجارب XN تجارب XN تجارب XN تجارب XN تجارب
+
+135
+00:16:32,500 --> 00:16:35,800
+XN تجارب XN تجارب XN تجارب XN تجارب XN تجارب XN
+
+136
+00:16:35,800 --> 00:16:35,900
+تجارب XN تجارب XN تجارب XN تجارب XN تجارب XN تجارب
+
+137
+00:16:35,900 --> 00:16:36,120
+XN تجارب XN تجارب XN تجارب XN تجارب XN تجارب XN
+
+138
+00:16:36,120 --> 00:16:36,560
+تجارب XN تجارب XN تجارب XN تجارب XN تجارب XN تجارب
+
+139
+00:16:36,560 --> 00:16:38,000
+XN تجارب XN
+
+140
+00:16:46,190 --> 00:16:50,570
+برضه العدد S في الحالة هذه بنسميه limit inferior
+
+141
+00:16:50,570 --> 00:17:05,330
+لل sequence XM S in above remark is
+
+142
+00:17:05,330 --> 00:17:11,850
+called is called limit
+
+143
+00:17:14,350 --> 00:17:26,290
+limit inferior limit inferior of ال sequence x in
+
+144
+00:17:26,290 --> 00:17:29,730
+and
+
+145
+00:17:29,730 --> 00:17:35,490
+we write limit
+
+146
+00:17:35,490 --> 00:17:40,910
+inferior ل x in بساوي s
+
+147
+00:17:45,400 --> 00:17:49,760
+وكمان مرة هذه limit inferior بتساوي .. ممكن اثبات
+
+148
+00:17:49,760 --> 00:17:54,080
+باستخدام الـ monotone convergence theorem ممكن
+
+149
+00:17:54,080 --> 00:18:00,200
+اثبات ان هذا بساوي limit sequence
+
+150
+00:18:00,200 --> 00:18:07,240
+S as N tends to infinityلأن هذه الـ sequence Sn
+
+151
+00:18:07,240 --> 00:18:12,280
+ممكن اثبات إنها increasing متزايدة و bounded
+
+152
+00:18:12,280 --> 00:18:15,960
+therefore by monotone convergence theorem it
+
+153
+00:18:15,960 --> 00:18:20,400
+converges و ال limit تبعتها هتساوي ال supremum إلى
+
+154
+00:18:20,400 --> 00:18:27,840
+اللي هو Sتمام؟ إذا هذه بنتسميها limit inferior of
+
+155
+00:18:27,840 --> 00:18:32,840
+the sequence xn وهذه limit superior هي النهاية
+
+156
+00:18:32,840 --> 00:18:37,880
+العليا الصفلة النهاية العليا طبعا إذا كانت ال
+
+157
+00:18:37,880 --> 00:18:42,780
+sequence bounded ف limit superior لها exist
+
+158
+00:18:42,780 --> 00:18:48,240
+وبالساوي العدد S وكذا لك limit inferior لها exist
+
+159
+00:18:48,240 --> 00:18:54,990
+وبالساوي العدد Sبس العدد S هذا غير عن العدد S دا
+
+160
+00:18:54,990 --> 00:18:59,610
+يعني هذا خلينا نسميه S star عشان نميزه عن العدد
+
+161
+00:18:59,610 --> 00:19:07,530
+اللي فات ده خلينا نسميه S star S star وS star هنا
+
+162
+00:19:07,530 --> 00:19:13,090
+غير مختلف مش شرط يكون هو نفس ال S في ال exercise
+
+163
+00:19:13,090 --> 00:19:20,430
+اللي فات إذا نحاولوا تثبتهحاولوا تثبتوا التمرين
+
+164
+00:19:20,430 --> 00:19:25,530
+هذا اللي هو الخاص ب limit inferior بنفس ال
+
+165
+00:19:25,530 --> 00:19:30,990
+argument by similar argument باستخدام استنتاج أو
+
+166
+00:19:30,990 --> 00:19:36,470
+برهان مشابه للبرهان اللي عملناها للجزء اللي فات
+
+167
+00:19:36,470 --> 00:19:41,840
+اللي هو تمرين عشرة okay تمام؟واضح؟ اذا هنا اليوم
+
+168
+00:19:41,840 --> 00:19:45,360
+اتعلمنا ان في حاجة اسمها النهاية العليا والنهاية
+
+169
+00:19:45,360 --> 00:19:49,360
+السفلة لل sequence إذا كانت ال sequence bounded
+
+170
+00:19:49,360 --> 00:19:53,720
+فأثبتنا هنا أن النهاية العليا تبقاتها موجودة
+
+171
+00:19:53,720 --> 00:19:56,960
+وبالساوي العدد S اللي هو ال infimum لل sequence
+
+172
+00:19:56,960 --> 00:20:01,050
+هذهإذا كانت برضه ال sequence bounded فالنهاية
+
+173
+00:20:01,050 --> 00:20:06,670
+السفلة لها أو limit inferior بتطلع exist و بتساوي
+
+174
+00:20:06,670 --> 00:20:10,210
+العدد s star اللي هو ال supremum لل sequence هذه
+
+175
+00:20:10,210 --> 00:20:17,810
+أو بساوي limit ال sequence s هذه تمام؟ فحاولوا
+
+176
+00:20:17,810 --> 00:20:22,310
+تبرهنوا الجزء هذا المشابه للجزء الأول يعني في أي
+
+177
+00:20:22,310 --> 00:20:23,650
+سؤال أو استفسار؟
+
+178
+00:20:26,730 --> 00:20:30,430
+طيب الآن إذا ننتقل ل section جديد هيك احنا متكون
+
+179
+00:20:30,430 --> 00:20:36,110
+خلصنا section تلاتة اربعة في الكتاب المخرج و هنبدأ
+
+180
+00:20:36,110 --> 00:20:42,730
+section جديد اللي
+
+181
+00:20:42,730 --> 00:20:51,010
+هو section تلاتة خمسة الانوان
+
+182
+00:20:51,010 --> 00:20:54,990
+تبع koshi sequences
+
+183
+00:20:57,810 --> 00:21:03,410
+متتاليات كوشي طبعا كوشي هاد عالم ألماني German
+
+184
+00:21:03,410 --> 00:21:09,050
+mathematician كان مهتم بدراسة نوع خاص من ال
+
+185
+00:21:09,050 --> 00:21:14,570
+sequences وكان بحاول يدرس التقارب و التباعت تباعهم
+
+186
+00:21:14,570 --> 00:21:18,740
+و علاقتهم بالconvergent و divergence sequences
+
+187
+00:21:18,740 --> 00:21:24,820
+العادية فهذا يعني من كان جزء من اهتماماته طبعا له
+
+188
+00:21:24,820 --> 00:21:27,320
+حاجات كتيرة في ال analysis
+
+189
+00:21:31,200 --> 00:21:35,700
+فالجماعة الرياضيات او الناس اللي جوا بعده عشان
+
+190
+00:21:35,700 --> 00:21:39,060
+يعني تقديرا للجهد اللي قامله وللدراسة اللي قاملها
+
+191
+00:21:39,060 --> 00:21:44,380
+لهذا النوع الخاص من المتتاليات سموا المتتاليات هذه
+
+192
+00:21:44,380 --> 00:21:50,780
+باسمه متتاليات كوشي او كوشي sequences فما هي
+
+193
+00:21:50,780 --> 00:21:55,800
+متتاليات كوشي؟ متى بنقول ان ال sequence كوشي؟ فهذه
+
+194
+00:21:55,800 --> 00:21:56,360
+ال definition
+
+195
+00:22:01,970 --> 00:22:08,870
+a sequence of real numbers a sequence xn contained
+
+196
+00:22:08,870 --> 00:22:21,230
+in R is Cauchy is Cauchy if الشرط التالي بتحقق لكل
+
+197
+00:22:21,230 --> 00:22:27,970
+إبسلون أكبر من السفر يوجد capital N يعتمد على
+
+198
+00:22:27,970 --> 00:22:37,300
+إبسلون عدد طبيعيبحيث أنه لو كان n و m أكبر من أو
+
+199
+00:22:37,300 --> 00:22:42,920
+ساوي capital N فهذا لازم يقدي أن المسافة بين xn و
+
+200
+00:22:42,920 --> 00:22:52,600
+xm أصغر من إبسلون هذه هي كوشي sequence sequence of
+
+201
+00:22:52,600 --> 00:22:57,360
+real number بنسميها كوشي متتالية كوشي إذا كان لأي
+
+202
+00:22:57,360 --> 00:23:02,360
+given إبسلونبقدر ألاقي عدد طبيعي يعتمد على epsilon
+
+203
+00:23:02,360 --> 00:23:08,400
+بحيث لكل N و M لكل المؤشرات اللي من capital N و N
+
+204
+00:23:08,400 --> 00:23:14,620
+طالع المسافة تبع الحدود الفرق بين حدودهم أصغر من
+
+205
+00:23:14,620 --> 00:23:19,800
+epsilon في
+
+206
+00:23:19,800 --> 00:23:23,040
+هنا أول لمة
+
+207
+00:23:33,720 --> 00:23:42,080
+لمّة واحدة عشرين اش لمّة بتقول every ..
+
+208
+00:23:42,080 --> 00:23:52,420
+every convergent .. every convergent sequence of
+
+209
+00:23:52,420 --> 00:23:58,120
+real numbers is
+
+210
+00:23:58,120 --> 00:24:08,020
+Cauchy كل متتاليةمتقاربة بتكون كشية وهي البرهان
+
+211
+00:24:08,020 --> 00:24:14,180
+prove assume
+
+212
+00:24:14,180 --> 00:24:20,160
+let x
+
+213
+00:24:20,160 --> 00:24:26,920
+in contained in R be
+
+214
+00:24:26,920 --> 00:24:32,000
+such that limit x in بساوي x
+
+215
+00:24:36,590 --> 00:24:41,730
+يعني افرض ان في عندي sequence xn وconvergent ل x
+
+216
+00:24:41,730 --> 00:24:45,110
+من
+
+217
+00:24:45,110 --> 00:24:50,190
+ان اثبت ان ال sequence هذه كوشي او كوشية ف let
+
+218
+00:24:50,190 --> 00:24:54,170
+epsilon لاثبات ان ال sequence كوشي لازم نبدأ
+
+219
+00:24:54,170 --> 00:24:58,210
+بepsilon ونرد عليها بcapital N تعطين ال
+
+220
+00:24:58,210 --> 00:25:02,070
+implication هذه صح؟ مش هيك التعريب يقول ان نبدأ
+
+221
+00:25:02,070 --> 00:25:06,090
+let epsilon أكبر من الصفر be given
+
+222
+00:25:11,220 --> 00:25:15,720
+طيب من تعريف epsilon capital N للكنفرجنس ال
+
+223
+00:25:15,720 --> 00:25:20,840
+sequence xn converge لx إذا there exist لأي given
+
+224
+00:25:20,840 --> 00:25:26,020
+epsilon بما أن xn converge لx لأي epsilon يوجد
+
+225
+00:25:26,020 --> 00:25:35,540
+capital N يعتمد على epsilon عدد طبيعي بحيث أنه
+
+226
+00:25:45,980 --> 00:25:53,820
+بحيث انه لو كان n أكبر من أو ساوي capital N هذا
+
+227
+00:25:53,820 --> 00:26:00,460
+بيقدي ان absolute xn minus x أصغر من epsilon ع 2
+
+228
+00:26:00,460 --> 00:26:04,680
+نسمي ال implication هذه start
+
+229
+00:26:09,880 --> 00:26:16,500
+الان لو أخدت N و M كلا هما أعداد طبيعية أكبر من أو
+
+230
+00:26:16,500 --> 00:26:24,680
+يساوي capital N هذا بيقدي ان absolute X N minus X
+
+231
+00:26:24,680 --> 00:26:34,020
+M هذا بيساوي absolute X N minus X زائد X minus X M
+
+232
+00:26:34,020 --> 00:26:39,320
+هاي المجموع هذا وهي المجموع هذا
+
+233
+00:26:49,790 --> 00:26:53,850
+إذا أنا طرحت x ورجعت x باستخدام ال triangle
+
+234
+00:26:53,850 --> 00:27:00,510
+inequality ال absolute value للمجموع أصلا أو ساوي
+
+235
+00:27:00,510 --> 00:27:04,410
+مجموع ال absolute values إذا أنا عندي absolute xn
+
+236
+00:27:04,410 --> 00:27:14,010
+minus x زائد absolute x minus xm طيب باستخدام ال
+
+237
+00:27:14,010 --> 00:27:20,260
+implication starأنا عندي ال N، ال N هذه أكبر من أو
+
+238
+00:27:20,260 --> 00:27:24,760
+ساوي capital N، إذن حسب ال star، absolute الفرخ
+
+239
+00:27:24,760 --> 00:27:29,740
+هذا أصغر من epsilon ع اتنين كذلك أنا عندي ال M
+
+240
+00:27:29,740 --> 00:27:36,800
+أكبر من أو ساوي N، إذن by star بدل N ب M، المسافة
+
+241
+00:27:36,800 --> 00:27:42,630
+هذه تطلع أصغر من epsilon ع اتنينالمجموعة epsilon
+
+242
+00:27:42,630 --> 00:27:49,350
+إذا هاني أثبتت أنه لأي epsilon أكبر من السفر في
+
+243
+00:27:49,350 --> 00:27:54,890
+capital N يوجد capital N تعتمد على epsilon بحيث
+
+244
+00:27:54,890 --> 00:27:59,910
+لكل N و M أكبر من أو ساوي capital N بطلع المسافة
+
+245
+00:27:59,910 --> 00:28:05,690
+بين الحد رقم N والحد رقم M أصغر من epsilon إذا
+
+246
+00:28:05,690 --> 00:28:08,290
+since
+
+247
+00:28:10,030 --> 00:28:17,650
+إبسلون أكبر من السفر was arbitrary إن
+
+248
+00:28:17,650 --> 00:28:22,330
+الكلام هذا صحيح أثبتنا هذا الكلام صحيح لكل إبسلون
+
+249
+00:28:22,330 --> 00:28:28,030
+الإبسلون هذي was arbitrary عشوائية إن هيك بتكون
+
+250
+00:28:28,030 --> 00:28:34,610
+حققنا الشرط تبقى cushy sequences وبالتالي we have
+
+251
+00:28:34,610 --> 00:28:38,370
+we get that من التعريف
+
+252
+00:28:56,870 --> 00:29:04,070
+العكس تبع النظرية هذه صحيحهنشوفه بعد شوية لكن في
+
+253
+00:29:04,070 --> 00:29:11,770
+كمان نظرية او لمّة بسيطة زي هذه هنثبتها
+
+254
+00:29:11,770 --> 00:29:20,450
+الآن بتقول ان كل كوشي sequence is bounded واضح
+
+255
+00:29:20,450 --> 00:29:23,470
+البرهان؟ في اي استفسار على البرهان؟ اعتقد البرهان
+
+256
+00:29:23,470 --> 00:29:28,480
+سهل وبسيطهذا ال .. النوع من البرهان بنسميه epsilon
+
+257
+00:29:28,480 --> 00:29:37,360
+بنسميه epsilon over two argument epsilon
+
+258
+00:29:37,360 --> 00:29:44,380
+over two argument استخدامنا epsilon over two برهان
+
+259
+00:29:44,380 --> 00:29:51,140
+باستخدام epsilon over two إذن نثبت اللمة التانية
+
+260
+00:29:51,140 --> 00:29:54,320
+قبل ما نثبت عكس النظر اللي فاتت صحيح
+
+261
+00:30:00,800 --> 00:30:10,180
+لمّا 22 every Cauchy sequence
+
+262
+00:30:10,180 --> 00:30:20,600
+in R is bounded والبرهان
+
+263
+00:30:20,600 --> 00:30:27,760
+تبعها شبيه إلى حد كبير ببرهان أن كل convergence
+
+264
+00:30:27,760 --> 00:30:32,110
+sequence is boundedأثبتنا قبل هيك أن كل conversion
+
+265
+00:30:32,110 --> 00:30:35,390
+sequence is bounded اليوم هأثبت أن كل Cauchy
+
+266
+00:30:35,390 --> 00:30:40,090
+sequence is bounded والبرهان مشابه للبرهان السابق
+
+267
+00:30:40,090 --> 00:30:44,950
+proof let
+
+268
+00:30:44,950 --> 00:30:54,590
+xn contained in R be Cauchy be Cauchy sequence
+
+269
+00:30:57,010 --> 00:31:03,350
+و خلّينا ناخد epsilon then
+
+270
+00:31:03,350 --> 00:31:09,990
+من تعريف الكوشي sequence for epsilon بساوي واحد
+
+271
+00:31:09,990 --> 00:31:16,130
+أكبر من السفر يوجد capital N يعتمد على ال epsilon
+
+272
+00:31:16,130 --> 00:31:26,330
+اللي هو الواحد عدد طبيعي بحيث أنه لabsolute xn
+
+273
+00:31:26,330 --> 00:31:34,150
+minus xm أصغر من إبسلون اللي هو واحد وهذا الكلام
+
+274
+00:31:34,150 --> 00:31:40,930
+صحيح لكل N و M أكبر من أو ساوي capital N هذا من
+
+275
+00:31:40,930 --> 00:31:49,190
+تعريف الكوشي sequence تمام؟ طيب hence
+
+276
+00:31:50,870 --> 00:31:59,410
+من المتباينة هذه بنحصل على absolute xn minus x
+
+277
+00:31:59,410 --> 00:32:06,670
+capital N أصغر من واحد وهذا الكلام صحيحلكل N أكبر
+
+278
+00:32:06,670 --> 00:32:11,130
+منه ساوي capital N اذا انا اشعلنت هنا المتباين هي
+
+279
+00:32:11,130 --> 00:32:18,910
+دي حصلت عليها من اللي فوق و ذلك بياخذ take M بساوي
+
+280
+00:32:18,910 --> 00:32:24,110
+capital N انا أخدت ال M هنا بساوي capital N وهذا
+
+281
+00:32:24,110 --> 00:32:30,430
+مسموح ليه فصارت هيكطب ليش أنا عملت هيك لحاجة في
+
+282
+00:32:30,430 --> 00:32:36,930
+نفسي عقوب؟ عشان فيه غرض للاستفاء البرهان أو
+
+283
+00:32:36,930 --> 00:32:45,150
+المطلوب في البرهان فمن هنا by triangle inequality
+
+284
+00:32:45,150 --> 00:32:52,690
+من متباينة المثلثأنا عندي لو كان n أكبر من أو ساوي
+
+285
+00:32:52,690 --> 00:32:58,150
+capital N هذا بيقدي ان absolute x in بالظبط زي ما
+
+286
+00:32:58,150 --> 00:33:02,330
+عملنا في برهان كل convergence sequence is bounded
+
+287
+00:33:02,330 --> 00:33:10,010
+فهذا بنكتب على صورة x in minus x capital N زائد x
+
+288
+00:33:10,010 --> 00:33:16,920
+capital N إذا أنا طرحت x capital N و رجعتهاالان
+
+289
+00:33:16,920 --> 00:33:21,740
+هذا اصغر من او ساوي absolute x n سالب x capital n
+
+290
+00:33:21,740 --> 00:33:28,340
+زاد absolute x capital n وهذا
+
+291
+00:33:28,340 --> 00:33:33,660
+هذا هو هذا المقدار لكل n اكبر من او ساوي capital n
+
+292
+00:33:33,660 --> 00:33:37,960
+اصغر من واحد اذا هذا اصغر من واحد زاد absolute x
+
+293
+00:33:37,960 --> 00:33:43,540
+capital n تمام الان
+
+294
+00:33:43,540 --> 00:33:52,020
+nowlet خلّينا ناخد m .. نعرف عدد m على أنه ال
+
+295
+00:33:52,020 --> 00:33:57,480
+supremum أو ال maximum للأعداد الحقيقية غير
+
+296
+00:33:57,480 --> 00:34:05,720
+السالبة اللي هي absolute x1 absolute x2 إلى
+
+297
+00:34:05,720 --> 00:34:16,000
+absolute x رقم capital M minus 1واخر حد absolute
+
+298
+00:34:16,000 --> 00:34:20,420
+واحد
+
+299
+00:34:20,420 --> 00:34:38,180
+زائد واحد زائد absolute x رقم capital N you
+
+300
+00:34:38,180 --> 00:34:39,340
+can verify
+
+301
+00:34:42,590 --> 00:34:51,350
+بإمكانكم التحقق إنه absolute xn أصغر طبعاً العدد m
+
+302
+00:34:51,350 --> 00:34:58,210
+هذا موجب بالتأكيد هذا موجب لأن
+
+303
+00:34:58,210 --> 00:35:05,420
+العداد هذه كلها موجبةو بعدين absolute xn ممكن تحقق
+
+304
+00:35:05,420 --> 00:35:10,380
+أنه أخدت أي عنصر في ال sequence xn و أخدت القيمة
+
+305
+00:35:10,380 --> 00:35:17,100
+المطلقة إليه فهذا هيطلع أصغر من أو ساوي m لكل n في
+
+306
+00:35:17,100 --> 00:35:17,460
+n
+
+307
+00:35:20,190 --> 00:35:25,570
+هل إيش هذا صحيح؟ لأنه لو كانت ال N هذه لو كانت ال
+
+308
+00:35:25,570 --> 00:35:31,750
+N هي index من واحد لcapital N لcapital N سالب واحد
+
+309
+00:35:31,750 --> 00:35:36,130
+لو كانت ال N هذه مؤشر N ده واحد من المؤشرات واحد
+
+310
+00:35:36,130 --> 00:35:40,310
+إلى capital N سالب واحد فالقيمة المطلقة اللي هو
+
+311
+00:35:40,310 --> 00:35:45,530
+واحد من الحدود هدول وكل حد من هدول أصلا من أو ساوي
+
+312
+00:35:45,530 --> 00:35:49,490
+ال upper bound ال M هذا upper bound لل set هذه
+
+313
+00:35:52,340 --> 00:35:57,880
+طب لو كانت ال n small n هذه بالساوي capital N او
+
+314
+00:35:57,880 --> 00:36:04,980
+اكبر capital N او اكبر هيا لو كانت small n اكبر من
+
+315
+00:36:04,980 --> 00:36:10,320
+او ساوي capital N فبطلع absolute xn اصغر من واحد
+
+316
+00:36:10,320 --> 00:36:14,900
+زاد absolute xn اذا القيمة المطلقة هذه اصغر من
+
+317
+00:36:14,900 --> 00:36:22,500
+واحد زاد absolute xn وهذا العدد اصغر منأو يساوي ال
+
+318
+00:36:22,500 --> 00:36:26,340
+upper bound لل set اللي بتحتويه و بتحتوي عناصر
+
+319
+00:36:26,340 --> 00:36:31,240
+تانية وبالتالي absolute xm أصغر من أو ساوي okay
+
+320
+00:36:31,240 --> 00:36:39,900
+تمام؟ إذن هذا بكمل البرهجة المرة الجاية طبعا هنشوف
+
+321
+00:36:39,900 --> 00:36:45,730
+أنه عكس النمة هذه صحيحوها دي بنسميها Cauchy
+
+322
+00:36:45,730 --> 00:36:50,510
+criterion أو معيار Cauchy ومعيار Cauchy بيقول إن
+
+323
+00:36:50,510 --> 00:36:54,510
+لو كانت ال sequence Cauchy بتكون أيضا convergent
+
+324
+00:36:54,510 --> 00:36:58,430
+وبالتالي بنحصل على نظرية Cauchy أو معيار Cauchy
+
+325
+00:36:58,430 --> 00:37:04,010
+Cauchy criterion وهو إن ال sequence is convergent
+
+326
+00:37:04,010 --> 00:37:09,690
+if and only if it is Cauchyإذا هذا الجزء التالي من
+
+327
+00:37:09,690 --> 00:37:14,050
+النظرية أو عكس اللمحة دي هنشوفه المرة الجاية وناخد
+
+328
+00:37:14,050 --> 00:37:20,350
+أمثلة كيف نثبت أن ال sequence معينة is Cauchy فهذا
+
+329
+00:37:20,350 --> 00:37:22,830
+موجود في section تلاتة خمسة في الكتاب حاولوا
+
+330
+00:37:22,830 --> 00:37:27,930
+تقرؤوا باقى ال section و تحضروه للمرة الجاية تمام؟
+
+331
+00:37:27,930 --> 00:37:30,730
+إذا انتهت هيك المحاضرة نشوفكم ان شاء الله يوم
+
+332
+00:37:30,730 --> 00:37:31,190
+اتنين
+
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/nokw77ZubUw_postprocess.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/nokw77ZubUw_postprocess.srt
new file mode 100644
index 0000000000000000000000000000000000000000..d3b3a2f372b2e3cc5302a93336097b21495d1084
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/nokw77ZubUw_postprocess.srt
@@ -0,0 +1,1764 @@
+1
+00:00:21,940 --> 00:00:28,180
+إذا في ال section هذا هن ان شاء الله هنتعلم ال
+
+2
+00:00:28,180 --> 00:00:34,860
+limited theorems و بعض النظريات الخاصة بالنهايات
+
+3
+00:00:34,860 --> 00:00:43,890
+فعشان نشوف النظريات هذه ناخد الأول تعريفsequence
+
+4
+00:00:43,890 --> 00:00:49,150
+of real numbers Xn نسميها bounded إذا قدرنا نلاقي
+
+5
+00:00:49,150 --> 00:00:56,110
+عدد موجة ب M بحيث إنه ال absolute value لكل عناصر
+
+6
+00:00:56,110 --> 00:01:00,510
+ال sequence أصغر من أو ساوي M هذا بكافئ
+
+7
+00:01:04,010 --> 00:01:10,550
+إن أن عناصر الـ sequence Xn تنتمي للفترة المغلقة
+
+8
+00:01:10,550 --> 00:01:19,030
+من سالب M إلى M لكل N عدد طبيعي إذن الـ sequence
+
+9
+00:01:19,030 --> 00:01:25,310
+بتكون bounded إذا قدرت أحصرها في closed interval
+
+10
+00:01:25,310 --> 00:01:29,730
+from سالب M إلى M حيث M عدد مجمع
+
+11
+00:01:32,190 --> 00:01:37,030
+النظرية هذه بتقول ان اي convergence sequence of
+
+12
+00:01:37,030 --> 00:01:41,710
+real numbers is bounded كل convergence sequence
+
+13
+00:01:41,710 --> 00:01:47,050
+بتكون ضروري انها تكون bounded ولبرهان ذلك هي افرض
+
+14
+00:01:47,050 --> 00:01:53,910
+ان sequence x in convergent to some x في R دعونا
+
+15
+00:01:53,910 --> 00:02:00,590
+ناخد y بساوي واحد هي هاد عدد موجب given من نظرية 2
+
+16
+00:02:00,590 --> 00:02:09,130
+اتنيناللي هو تعريف epsilon capital N لـ limit بقول
+
+17
+00:02:09,130 --> 00:02:13,390
+ال sequence هذه ال limit تبعتها X إذا كان لأي
+
+18
+00:02:13,390 --> 00:02:19,010
+epsilon زي هذه بقدر ألاقي عدد طبيعي يعتمد على
+
+19
+00:02:19,010 --> 00:02:23,410
+epsilon اللي هو واحد بحيث لكل N أكبر من أو ساوي
+
+20
+00:02:23,410 --> 00:02:28,570
+capital N المسافة بين XN و X أصغر من ال epsilon
+
+21
+00:02:28,570 --> 00:02:30,570
+اللي هي مخدينها واحد
+
+22
+00:02:34,370 --> 00:02:42,670
+الان absolute x in هذا عبارة عن ممكن ابدله هذا
+
+23
+00:02:42,670 --> 00:02:51,830
+بساوي absolute x in سالب x زاد x باخد
+
+24
+00:02:51,830 --> 00:02:58,840
+اتنين هدول مع بعض يعني اترحت x و رجعتby triangle
+
+25
+00:02:58,840 --> 00:03:03,520
+inequality less
+
+26
+00:03:03,520 --> 00:03:10,200
+than or equal absolute الحد الأول زاد absolute
+
+27
+00:03:10,200 --> 00:03:16,540
+الحد الثاني ومن هنا من هنا لكل n أكبر من أو ساوي
+
+28
+00:03:16,540 --> 00:03:20,680
+capital N المقدار هذا أو ال absolute value هذه
+
+29
+00:03:20,680 --> 00:03:26,790
+أصغر من واحدأذن absolute لكل n أكبر من لو ساوي
+
+30
+00:03:26,790 --> 00:03:36,730
+capital N أصغر من واحد واحد زائد absolute X تمام؟
+
+31
+00:03:36,730 --> 00:03:40,930
+الآن لو أخدت عرفت capital M على انها
+
+32
+00:03:46,510 --> 00:03:51,250
+لو عرفت capital N و capital M علي انها ال supremum
+
+33
+00:03:51,250 --> 00:03:55,890
+أو الأكبر بين الأعداد اللي هي absolute x1 و
+
+34
+00:03:55,890 --> 00:04:01,530
+absolute x2 و هكذا إلى absolute x رقم capital N
+
+35
+00:04:01,530 --> 00:04:07,170
+سالق واحد والعدد الأخير absolute x زائد واحد
+
+36
+00:04:10,460 --> 00:04:16,660
+ففي الحالة هذه ممكن نثبت أن absolute xn أصغر من أو
+
+37
+00:04:16,660 --> 00:04:27,980
+ساول عدد الموجب هذا لكل n لكل العداد الطبيعي هذا
+
+38
+00:04:27,980 --> 00:04:32,860
+بيجي من أربعة من المتبينة أربعة اللي هي هان ومن
+
+39
+00:04:32,860 --> 00:04:37,680
+التعريف capital M يعني لاحظوا أنتوا أن ال sequence
+
+40
+00:04:37,680 --> 00:04:50,680
+xnحدودها X1 X2 إلى X رقم capital N سالب واحد X رقم
+
+41
+00:04:50,680 --> 00:05:00,060
+capital N X رقم capital N زاد واحد و هكذا الان
+
+42
+00:05:00,060 --> 00:05:06,960
+لو ال N هذه كانت واحدة من الاعداد هدول المؤشر N
+
+43
+00:05:06,960 --> 00:05:14,380
+واحد من المؤشراتواحد اتنين الى capital N سالب واحد
+
+44
+00:05:14,380 --> 00:05:18,140
+ف
+
+45
+00:05:18,140 --> 00:05:23,860
+absolute ال XN هذا هيكون
+
+46
+00:05:23,860 --> 00:05:26,980
+اما absolute X واحد او absolute X اتنين او
+
+47
+00:05:26,980 --> 00:05:32,740
+absolute XN وكل واحد من هدولكل واحد من هذول أصغر
+
+48
+00:05:32,740 --> 00:05:37,020
+من أو ساوي ال supreme لهم كلهم وبالتالي هذا أكيد
+
+49
+00:05:37,020 --> 00:05:44,500
+أصغر من أو ساوي .. أصغر من أو ساوي ال N تمام؟ لو
+
+50
+00:05:44,500 --> 00:05:50,180
+كان ال N .. لو كان ال N أكبر من أو ساوي capital N
+
+51
+00:05:52,740 --> 00:05:57,480
+يعني المؤشر هذا تبع ال Xn هذي المؤشر تبع لو كان
+
+52
+00:05:57,480 --> 00:06:01,700
+أكبر يعني بسوء capital N أو capital N زائد واحد أو
+
+53
+00:06:01,700 --> 00:06:07,320
+أي عدد أكبر من capital N فمن
+
+54
+00:06:07,320 --> 00:06:11,880
+ال .. من ال implication أربع هاي و اذا كان ال N
+
+55
+00:06:11,880 --> 00:06:15,560
+هذي أكبر مما بسوء capital N فبطلع absolute Xn
+
+56
+00:06:15,560 --> 00:06:21,960
+فبطلع absolute Xn هذا أصغر من absolute X زائد واحد
+
+57
+00:06:22,790 --> 00:06:25,630
+و هذا العدد .. هذا العدد من هنا
+
+58
+00:06:28,510 --> 00:06:33,430
+أصغر من أو يساوي ال supremum له ولا باقي الأعداد
+
+59
+00:06:33,430 --> 00:06:37,870
+هذه وبالتالي أصغر من أو يساوي M إذن هذا العدد أصغر
+
+60
+00:06:37,870 --> 00:06:42,490
+من أو يساوي M إذن في كل الأحوال سواء ال N كانت من
+
+61
+00:06:42,490 --> 00:06:46,770
+واحد إلى capital N سالب واحد أو N أكبر من أو يساوي
+
+62
+00:06:46,770 --> 00:06:51,810
+capital N فطلع absolute xn أصغر من أو يساوي M حيث
+
+63
+00:06:51,810 --> 00:06:56,110
+M عدد موجب وبالتالي ال sequence xn is bounded حسب
+
+64
+00:06:56,110 --> 00:07:06,350
+التعريف تمام؟إذا هذا يبقى هنا نظرية و هذا طيب
+
+65
+00:07:06,350 --> 00:07:09,010
+في بعض الملاحظات عن نظرية هذه
+
+66
+00:07:20,320 --> 00:07:24,280
+في بعض الملاحظات الملاحظة الأولى بتقول ان ال
+
+67
+00:07:24,280 --> 00:07:29,240
+converts of theorem two six need not be true يعني
+
+68
+00:07:29,240 --> 00:07:34,060
+عكس النظرية السابقة مش شرط يكون صحيح طب ما هو عكس
+
+69
+00:07:34,060 --> 00:07:37,300
+النظرية النظرية بتقول كل convergence sequence is
+
+70
+00:07:37,300 --> 00:07:41,980
+bounded عكس ال statement هذا المفروض ان يكون كل
+
+71
+00:07:41,980 --> 00:07:46,990
+bounded sequence is convergent هذا مش صحو هاي في
+
+72
+00:07:46,990 --> 00:07:55,350
+counter example يثبت أو يوضح أن عكس نظرية 2.6 ليس
+
+73
+00:07:55,350 --> 00:08:00,930
+صحيحالسيكوانس ان الحد العام تبعها xn بساوي سالب
+
+74
+00:08:00,930 --> 00:08:06,550
+واحد قص ان السيكوانس هذي bounded ليه bounded؟ لأن
+
+75
+00:08:06,550 --> 00:08:11,130
+هاي في عدد موجب capital M بساوي واحد هذا عدد موجب
+
+76
+00:08:11,130 --> 00:08:15,670
+إذا يوجد عدد موجب اللي هو الواحد بحيث ان absolute
+
+77
+00:08:15,670 --> 00:08:20,950
+xn هذا ال absolute value العدد هذا واحد واحد أصغر
+
+78
+00:08:20,950 --> 00:08:25,490
+من اذا بساوي واحدإن ال absolute xn هي أصغر من أو
+
+79
+00:08:25,490 --> 00:08:28,590
+ساوي عدد موجة بواحد لكل n إذا ال sequence هي
+
+80
+00:08:28,590 --> 00:08:33,030
+bounded واضحة حسب التعريف الآن هذه ال sequence
+
+81
+00:08:33,030 --> 00:08:39,950
+بنثبت إنها ليست convergent إنها divergent فلبرهان
+
+82
+00:08:39,950 --> 00:08:44,430
+ذلك نعمل برهان بالتناقضأنا بدأ أثبت أن الـ
+
+83
+00:08:44,430 --> 00:08:48,070
+sequence هذه is not convergent ف assume ال
+
+84
+00:08:48,070 --> 00:08:52,530
+contrary يعني assume أنها convergent to some a
+
+85
+00:08:52,530 --> 00:08:56,490
+belongs to the real numbers افرض أنها convergent
+
+86
+00:08:56,490 --> 00:09:02,220
+وال limit تبعتها عدد aفلو أخدت epsilon بساوي واحد
+
+87
+00:09:02,220 --> 00:09:06,340
+عدد موجب إذا من تعريف ال convergence إذا يوجد
+
+88
+00:09:06,340 --> 00:09:09,220
+capital N يعتمد على ال epsilon اللي أنا مختاره
+
+89
+00:09:09,220 --> 00:09:13,500
+اللي هي الواحد بحيث لكل N أكبر من أو ساوي capital
+
+90
+00:09:13,500 --> 00:09:21,540
+N المسافة بين xn و a أصغر من epsilon الآن
+
+91
+00:09:21,540 --> 00:09:25,460
+ال N هذه اللي هي أكبر من أو ساوي capital N ممكن
+
+92
+00:09:25,460 --> 00:09:30,650
+يكون عدد فردي أو زوجيفلو كانت ال end أكبر من نفسه
+
+93
+00:09:30,650 --> 00:09:36,270
+capital N odd فردي فمن المتباينة خمسة بيصير هذا
+
+94
+00:09:36,270 --> 00:09:42,550
+العدد سالب واحد لأن N فردي فبصير absolute سالب
+
+95
+00:09:42,550 --> 00:09:50,250
+واحد سالب a أصغر من واحد و لو رفعت ال absolute
+
+96
+00:09:50,250 --> 00:09:55,010
+value فبينتج منها أن a أكبر من سالب اتنين أصغر من
+
+97
+00:09:55,010 --> 00:10:00,590
+سفرلو كرت ال N اللي هي أكبر ما يساوي capital N
+
+98
+00:10:00,590 --> 00:10:06,430
+even فهذا العدد بيصير واحد وبيصير المتباين هذا
+
+99
+00:10:06,430 --> 00:10:12,610
+absolute واحد سالب a أصغر من واحدلو رفعت .. فكيت
+
+100
+00:10:12,610 --> 00:10:15,990
+ال absolute value هنا أو حليت المتباينة هذه في a
+
+101
+00:10:15,990 --> 00:10:20,630
+فبطلع a أكبر من صفر أصغر من اتنين اذا الان
+
+102
+00:10:20,630 --> 00:10:25,110
+المتباينتين ستة و سبعة بيعطوني تناقض ليه؟ لأن
+
+103
+00:10:25,110 --> 00:10:31,010
+المتباينة الستة بتقوللي ان a ينتمي للفترة المفتوحة
+
+104
+00:10:31,010 --> 00:10:35,810
+من صفر من سالب اتنين لصفر والمتباينة سبعة بتقوللي
+
+105
+00:10:35,810 --> 00:10:41,680
+ان العدد a ينتمي للفترة المفتوحة من صفر لاتنينطب
+
+106
+00:10:41,680 --> 00:10:47,780
+هذه الفترة تقاطعها مع هذه بساوي five يعني إيه
+
+107
+00:10:47,780 --> 00:10:51,720
+ينتمي للمجموع الخالي، هذا contradiction، هذا
+
+108
+00:10:51,720 --> 00:10:59,580
+تناقض، okay تمام؟ إذا هذا التناقض بيقول إن فرضنا،
+
+109
+00:10:59,580 --> 00:11:04,540
+الفرض طبعنا إن ال sequence هذه convergent كان خطأ،
+
+110
+00:11:04,540 --> 00:11:08,160
+إن ال sequence هذه ليست convergent، ال sequence
+
+111
+00:11:08,160 --> 00:11:13,960
+هذه ليست convergentإذا هذا مثال على sequence
+
+112
+00:11:13,960 --> 00:11:16,880
+bounded but not convergent إذا مش كل bounded
+
+113
+00:11:16,880 --> 00:11:21,240
+sequence is convergent لكن العكس كمان في النظرية 2
+
+114
+00:11:21,240 --> 00:11:25,760
+-6 العكس أثبتنا أنه صحيح كل convergence sequence
+
+115
+00:11:25,760 --> 00:11:30,500
+ضروري أن تكون bounded النظرية
+
+116
+00:11:30,500 --> 00:11:35,660
+دي 2-6 الملاحظة التانية بتقول إن نظرية 2-6 هي دي
+
+117
+00:11:35,660 --> 00:11:40,040
+زي اختبار الدم اللي بيلف الأبوة ولا يثبتها
+
+118
+00:11:46,810 --> 00:11:52,550
+الملاحظة التانية بتقول إن نظرية 2.6 ممكن نستخدمها
+
+119
+00:11:52,550 --> 00:11:56,910
+تثبت إن certain sequences are divergent
+
+120
+00:12:02,100 --> 00:12:06,000
+يعني نظرية هذه ممكن نستخدمها لإثبات أن ال sequence
+
+121
+00:12:06,000 --> 00:12:11,880
+is معينة not convergent زي اختبار الدم بينفي
+
+122
+00:12:11,880 --> 00:12:16,020
+الأبوة و لا يثبتها و هنا هذه نظرية بتنفي ال
+
+123
+00:12:16,020 --> 00:12:22,060
+convergence و لا تثبته تمام مثلا
+
+124
+00:12:22,060 --> 00:12:27,040
+ال sequence in هذه ال sequence of national numbers
+
+125
+00:12:27,040 --> 00:12:32,930
+هذه ال sequence هذه bounded ولا مش boundednot
+
+126
+00:12:32,930 --> 00:12:36,210
+bounded, unbounded، الـ sequence هذه unbounded،
+
+127
+00:12:36,210 --> 00:12:43,030
+okay؟ إذا حسب نظرية 2.6، مدامها unbounded، إذا
+
+128
+00:12:43,030 --> 00:12:47,750
+لازم تكون divergent، لبرهان ذلك، افرضي ال
+
+129
+00:12:47,750 --> 00:12:52,330
+contrary، افرضي إنها convergent، إذا حسب نظرية 2
+
+130
+00:12:52,330 --> 00:12:56,250
+.6، المفروض تطلع bounded، لكنها ليست bounded
+
+131
+00:12:57,900 --> 00:13:03,860
+contradiction إذا ال sequence هذه كونها unbounded
+
+132
+00:13:03,860 --> 00:13:07,840
+حسب نظرية نين سيفتر بتطلع divergent و احنا طبعا
+
+133
+00:13:07,840 --> 00:13:12,260
+عارفين ان ال limit اللي عارفين انها divergent لأن
+
+134
+00:13:12,260 --> 00:13:17,380
+limit n as n tends to infinity بساوي infinity ليس
+
+135
+00:13:17,380 --> 00:13:23,820
+عدد حقيقي ال sequence هذه مالهاش limit okay تمام
+
+136
+00:13:30,200 --> 00:13:38,940
+تابع الـ sequence in ليش unbounded؟
+
+137
+00:13:38,940 --> 00:13:42,080
+ليش الـ sequence هذه unbounded؟ هاي ال proof
+
+138
+00:13:42,080 --> 00:13:52,280
+assume إنها bounded، in is bounded، بدنا نعمل
+
+139
+00:13:52,280 --> 00:13:59,460
+برهان بالتناقلإذا يوجد عدد مجد M أكبر من سفر بحيث
+
+140
+00:13:59,460 --> 00:14:07,280
+أنه absolute xn اللي هي M أصغر من M لكل M
+
+141
+00:14:07,280 --> 00:14:16,380
+belonging to N طيب
+
+142
+00:14:16,380 --> 00:14:18,280
+by Archimedean property
+
+143
+00:14:25,940 --> 00:14:33,220
+م أكبر من السفر بتأدي لأي عدد موجب يوجد عدد طبيعي
+
+144
+00:14:33,220 --> 00:14:42,300
+n0 ينتمي إلى n بحيث انه n0 أكبر من m طبعا
+
+145
+00:14:42,300 --> 00:14:53,400
+هذا بساوي n لأن لو سميت هذه واحد و هذه اتنين فمن
+
+146
+00:14:53,400 --> 00:15:02,440
+واحد و اتنين from واحدand اتنين بيطلع عندي N0 أكبر
+
+147
+00:15:02,440 --> 00:15:09,760
+من M و M أكبر من N0 يعني N0 أكبر من N0 هذا تناقض
+
+148
+00:15:09,760 --> 00:15:16,060
+فى عدد أكبر من نفسه مافيش لأن
+
+149
+00:15:16,060 --> 00:15:18,860
+التناقض هذا بيقول ان ال sequence هذه مش ممكن تكون
+
+150
+00:15:18,860 --> 00:15:24,740
+bounded تمام استخدمنا هنا ال Archimedean property
+
+151
+00:15:26,820 --> 00:15:32,140
+طيب نشوف الـ Limited theorems نظريات
+
+152
+00:15:32,140 --> 00:15:37,660
+النهايات أو قوانين النهايات النظرية هذه بتلخص
+
+153
+00:15:37,660 --> 00:15:44,920
+قوانين النهايات المعروفة لدينا من calculus بي وهي
+
+154
+00:15:44,920 --> 00:15:48,660
+إن لو في عندي sequence x in convergent ل x و
+
+155
+00:15:48,660 --> 00:15:52,300
+sequence ثانية y in convergent ل y و c أي عدد
+
+156
+00:15:52,300 --> 00:15:54,640
+حقيقي ف
+
+157
+00:15:56,610 --> 00:16:01,510
+الـ sequence اللي لحد الآن تبعها مجموعة الحد العام
+
+158
+00:16:01,510 --> 00:16:05,690
+تبع ال sequence xn و ال sequence yn ال sequence
+
+159
+00:16:05,690 --> 00:16:09,590
+هذا convergent و ال limit تبعتها بساوي مجموعة
+
+160
+00:16:09,590 --> 00:16:17,890
+النهايات كذلك نفس الشيء بالنسبة للفرق الآن
+
+161
+00:16:17,890 --> 00:16:20,830
+ال sequence اللي لحد الآن تبعها حاصل ضرب
+
+162
+00:16:23,930 --> 00:16:27,690
+الـ sequence xn مع الـ sequence yn هذه بتطلع
+
+163
+00:16:27,690 --> 00:16:34,290
+convergent ونهايتها بسبب حصر ضارب النهايات لو في
+
+164
+00:16:34,290 --> 00:16:38,370
+ثابت limit ثابت في ال sequence بسبب ثابت في نهاية
+
+165
+00:16:38,370 --> 00:16:45,310
+ال sequence لو كان ال zn sequence حدودها
+
+166
+00:16:45,310 --> 00:16:51,250
+كلها غير .. لا تساوي السفر وconvergent لعدد z لا
+
+167
+00:16:51,250 --> 00:16:59,260
+يساوي سفرالسيكوينس XM over ZM تظهر كونفرجنت
+
+168
+00:16:59,260 --> 00:17:05,540
+ونهايتها بساوي نهاية البسط على نهاية المقام طبعا
+
+169
+00:17:05,540 --> 00:17:12,180
+براهين الفروح كلها مطلوبة منكمفانا هاي اللي
+
+170
+00:17:12,180 --> 00:17:17,460
+برهنلكم الفرع بيه وبالمثل ممكن تجدوا الفروع الأخرى
+
+171
+00:17:17,460 --> 00:17:21,580
+موجودة برهنها في الكتاب المقرر وإذا فيش مش برهن
+
+172
+00:17:21,580 --> 00:17:29,460
+فبتبرهنوه لوحدكم فاعتبروه تمرين homework exercise
+
+173
+00:17:29,460 --> 00:17:36,480
+إذا نشوف برهان الجزء بيه في الجزء بيه عايزين نثبت
+
+174
+00:17:36,480 --> 00:17:43,250
+أنه limitيبقى ان في جزء بيه عايزين نثبت ان limit
+
+175
+00:17:43,250 --> 00:17:53,730
+xn في yn بساوي x في y فالبرهان
+
+176
+00:17:53,730 --> 00:18:01,640
+ذلكناخد المتباينة هذه أو الأعداد هذه في نهاية
+
+177
+00:18:01,640 --> 00:18:07,180
+الأمر حسب تعريف epsilon capital N للنهايات بتثبت
+
+178
+00:18:07,180 --> 00:18:11,920
+أن ال absolute value للحد العام لل sequence minus
+
+179
+00:18:11,920 --> 00:18:16,080
+ال limit المقترحة بتثبت أن ال absolute value هذه
+
+180
+00:18:16,080 --> 00:18:21,880
+أصغر من أي given epsilonفالبرهان ده عليك ببدأ هي
+
+181
+00:18:21,880 --> 00:18:29,380
+absolute xn yn minus xy هطرح من الحد هذا xn في y و
+
+182
+00:18:29,380 --> 00:18:33,940
+أرجعه فكأني ماعملتش عاجب أخد الحدين هذول مع بعض و
+
+183
+00:18:33,940 --> 00:18:37,260
+الحدين هذول مع بعض by ال triangle inequality هذا
+
+184
+00:18:37,260 --> 00:18:41,160
+أصغر من أو ساوي absolute الحد الأول زاد absolute
+
+185
+00:18:41,160 --> 00:18:47,240
+الحد التاني في الحد الأول هذا عامل مشترك xn بطلعه
+
+186
+00:18:47,750 --> 00:18:53,010
+فبصير absolute xn في absolute yn سالب y وفي هنا
+
+187
+00:18:53,010 --> 00:18:59,270
+اعمل مشترك y فبطلعه فبصير absolute الحد التاني
+
+188
+00:18:59,270 --> 00:19:06,850
+بيسوي absolute xn minus x في absolute ال y الان
+
+189
+00:19:06,850 --> 00:19:14,210
+بما ان xn converge ل xأذا حسب نظرية 2.6 الـ
+
+190
+00:19:14,210 --> 00:19:18,150
+sequence هذه بما أنها convergent فهي bounded فهي
+
+191
+00:19:18,150 --> 00:19:23,930
+bounded وبالتالي بنقدر نلاقي عدد موجب M1 بحيث أن
+
+192
+00:19:23,930 --> 00:19:26,810
+ال absolute value لحدود ال sequence أصغر من أو
+
+193
+00:19:26,810 --> 00:19:32,370
+ساوي M1 لو أخدت capital M المaximum الأكبر بين
+
+194
+00:19:32,370 --> 00:19:38,630
+العدد الموجب M1 والعدد الموجبabsolute y ف m بيطلع
+
+195
+00:19:38,630 --> 00:19:44,450
+عدد موجب لأنه أكبر من أو ساوي هذا وهذاأذا
+
+196
+00:19:44,450 --> 00:19:49,790
+المتباينة هذه بتصير absolute xny m minus xy أصغر
+
+197
+00:19:49,790 --> 00:19:55,970
+من أو ساوي absolute xn هذا هي أصغر من أو ساوي m
+
+198
+00:19:55,970 --> 00:20:01,110
+واحد و m واحد أصغر من أو ساوي capital M إذا بشيل
+
+199
+00:20:01,110 --> 00:20:06,990
+absolute xn بحط أصغر من أو ساوي capital Mفي
+
+200
+00:20:06,990 --> 00:20:12,770
+absolute y n minus y كذلك absolute y هذه هي أصغر
+
+201
+00:20:12,770 --> 00:20:18,290
+من أو يساوي capital M وبالتالي الحد هذا بيصير أصغر
+
+202
+00:20:18,290 --> 00:20:22,750
+من أو يساوي capital M بدل absolute y في absolute x
+
+203
+00:20:22,750 --> 00:20:28,350
+n minus x الآن احنا من الفرض فرضين ان ال sequence
+
+204
+00:20:28,350 --> 00:20:33,630
+x n converge ل x و ال sequence y n converge ل yإذا
+
+205
+00:20:33,630 --> 00:20:37,450
+لو أخدنا أي إبسلون لت إبسلون أكبر من السفر P given
+
+206
+00:20:37,450 --> 00:20:42,210
+بما أنه ال sequence XM converged ل X إذا يوجد N
+
+207
+00:20:42,210 --> 00:20:47,350
+واحد عدد طبيعي يعتمد على إبسلون بحيث لكل N أكبر من
+
+208
+00:20:47,350 --> 00:20:51,710
+أو ساوي capital N واحد بيطلع المسافة بين XM X أصغر
+
+209
+00:20:51,710 --> 00:20:55,790
+من إبسلون أو إبسلون على 2M يعتبر هذا هو الإبسلون
+
+210
+00:20:55,790 --> 00:21:01,070
+في التعريف لأنها دا عدد موجة وبيعتمد على إبسلونليش
+
+211
+00:21:01,070 --> 00:21:05,970
+حطينا 2M لحاجة في نفس يعقوب، عشان في النهاية أخلي
+
+212
+00:21:05,970 --> 00:21:10,610
+ال absolute value هذه أصغر من إبسلون، هنشوفها هنا،
+
+213
+00:21:10,610 --> 00:21:15,530
+بقطوة هذه طيب، إذا قلنا بما إن it's unconventional
+
+214
+00:21:15,530 --> 00:21:19,230
+ل X، فلأي إبسلون فيه capital N واحد، بحيث ال
+
+215
+00:21:19,230 --> 00:21:23,910
+implication هذه تتحققكذلك بما أن yn converge ل y
+
+216
+00:21:23,910 --> 00:21:28,270
+إذا for the same إبسلون لنفس الإبسلون العدد موجب
+
+217
+00:21:28,270 --> 00:21:33,090
+اللي هو معطى مسبقا يوجد عدد طبيعي N2 يعتمد على
+
+218
+00:21:33,090 --> 00:21:37,770
+إبسلون بحيث لكل N أكبر منه ساوي capital N2 بيطلع
+
+219
+00:21:37,770 --> 00:21:42,750
+absolute yn minus y أصغر من إبسلون على 2M
+
+220
+00:21:45,110 --> 00:21:49,830
+الان خلّينا ناخد capital N الأكبر من N واحد و N
+
+221
+00:21:49,830 --> 00:21:54,610
+اتنين هذي و هذي كلاهما يعتمد على epsilon اذا
+
+222
+00:21:54,610 --> 00:22:00,490
+capital N تعتمد على epsilon و عدد طبيعي الان لكل
+
+223
+00:22:00,490 --> 00:22:07,370
+عدد طبيعي أكبر من أو ساوي capital N هذا بيطلع و
+
+224
+00:22:07,370 --> 00:22:11,570
+capital N هي من تعريفه هذا أكبر من أو ساوي capital
+
+225
+00:22:11,570 --> 00:22:20,710
+N واحدو أيضا أكبر من أوساوي capital N2أذن بيطلع
+
+226
+00:22:20,710 --> 00:22:27,350
+عندي absolute xn yn minus xy أصغر من أو يساوي M و
+
+227
+00:22:27,350 --> 00:22:32,870
+absolute yn minus y بما أن small n هذي أكبر من أو
+
+228
+00:22:32,870 --> 00:22:38,410
+يساوي capital N اتنين أذن من هنا بيطلع الفرق هذا
+
+229
+00:22:38,410 --> 00:22:43,590
+أصغر من epsilon على اتنين M أصغر من epsilon على
+
+230
+00:22:43,590 --> 00:22:49,900
+اتنين M في Mو كذلك small n هذه أكبر من أو ساوي n
+
+231
+00:22:49,900 --> 00:22:55,460
+واحد لما small n أكبر من أو ساوي n واحد بطلع بقدر
+
+232
+00:22:55,460 --> 00:23:00,880
+أشيل absolute xn minus x و أحط أصغر من epsilon على
+
+233
+00:23:00,880 --> 00:23:08,660
+اتنين M الان M بتروح مع M و M بتروح مع M بضل عندي
+
+234
+00:23:08,660 --> 00:23:11,700
+epsilon على اتنين زاد epsilon على اتنين بطلع
+
+235
+00:23:11,700 --> 00:23:17,540
+epsilon تمام؟ و هذا اللي بدنا ي ..فاكرين في بداية
+
+236
+00:23:17,540 --> 00:23:21,160
+البرنامج احنا عايزين في النهاية نثبت ان ال
+
+237
+00:23:21,160 --> 00:23:25,700
+absolute value هذه اصغر من epsilon اصغر من epsilon
+
+238
+00:23:27,020 --> 00:23:30,340
+for any given epsilon وفعلا هي for any epsilon
+
+239
+00:23:30,340 --> 00:23:34,760
+أكبر من 0 أثبتنا أن يوجد capital N أعداد طبيعية
+
+240
+00:23:34,760 --> 00:23:38,580
+تامد على epsilon بحيث لكل N أكبر من أو ساوي
+
+241
+00:23:38,580 --> 00:23:43,000
+capital N طلع ال absolute value للفرق هذا أصغر من
+
+242
+00:23:43,000 --> 00:23:48,820
+epsilon، إذن حسب التعريف بطلع limit xn في yn بساوي
+
+243
+00:23:48,820 --> 00:23:55,170
+x في y، تمام؟إن الهدف برهن برهان بكمل برهان جزء
+
+244
+00:23:55,170 --> 00:23:59,730
+بيه بالمثل ممكن تبرهن الأجزاء الأخرى فمطلوب منكم
+
+245
+00:23:59,730 --> 00:24:04,130
+اتبرهنوها اعتقد انه في الكتاب المقرر في نظرية
+
+246
+00:24:04,130 --> 00:24:11,010
+تلاتة اتنين تلاتة in the textbook فيها براهين بعض
+
+247
+00:24:11,010 --> 00:24:16,110
+الأجزاء أو ربما كلهم فحاولوا تكتبوا البرهان الأول
+
+248
+00:24:16,110 --> 00:24:19,630
+لوحدكم وإذا ماعرفتوش اخرقوا البرهان في الكتاب
+
+249
+00:24:19,630 --> 00:24:27,040
+حاولوا تفهموهإذا في أي شيء مش واضح اكتب البرهان في
+
+250
+00:24:27,040 --> 00:24:34,740
+ورقة و أعطينيها عشان أصلح لكيها okay
+
+251
+00:24:34,740 --> 00:24:42,560
+إذا هذه قوانين النهايات هذه قوانين النهايات طيب في
+
+252
+00:24:42,560 --> 00:24:47,480
+كمان قوانين أخرى أو خواس أخرى للنهايات
+
+253
+00:24:52,790 --> 00:24:57,850
+فمثلا في النظرية هذه النظرية هذه بتقول لو في عندي
+
+254
+00:24:57,850 --> 00:25:01,810
+convergence sequence xn convergence ل x و ال
+
+255
+00:25:01,810 --> 00:25:06,750
+sequence xn حدودها غير سالبة كل حدودها غير سالبة ف
+
+256
+00:25:06,750 --> 00:25:11,630
+ال limit تبعتها ايضا لازم تكون غير سالبة و البرهان
+
+257
+00:25:11,630 --> 00:25:16,590
+سهل by contradiction prove by contradiction assume
+
+258
+00:25:16,590 --> 00:25:21,170
+on the contrary ان ال x سالبة
+
+259
+00:25:24,860 --> 00:25:29,200
+أنا عايز أثبت x أكبر من أو ساوى 0 النافي تبعها x
+
+260
+00:25:29,200 --> 00:25:36,380
+أصغر من 0 صح؟الان خدي epsilon بالساوي سالب x إذا
+
+261
+00:25:36,380 --> 00:25:40,300
+ال x سالب هذا سالب x عدد موجبة، إذا هاي الآن في
+
+262
+00:25:40,300 --> 00:25:44,180
+عندي epsilon موجبة وفي عندي من الفرض x n converge
+
+263
+00:25:44,180 --> 00:25:49,440
+ل x إذا من تعريف epsilon capital N للنهايات بما أن
+
+264
+00:25:49,440 --> 00:25:52,680
+x n converge ل x إذا يوجد capital N يعتمد على ال
+
+265
+00:25:52,680 --> 00:25:56,780
+epsilon بحيث لكل n أكبر لو ساوي capital Nالمسافة
+
+266
+00:25:56,780 --> 00:26:02,540
+بين xn و x أصغر من y طيب فك ال absolute value هذه
+
+267
+00:26:02,540 --> 00:26:09,260
+شيلها بصير المتباينة هذه xn minus x أصغر من y أكبر
+
+268
+00:26:09,260 --> 00:26:10,420
+من سالب y
+
+269
+00:26:14,450 --> 00:26:18,770
+خدي هذا جزء من المتباينة و ادى x على الناحية
+
+270
+00:26:18,770 --> 00:26:24,410
+التانية فبصير هذا بيقدي ان xn أصغر من أو ساوي x
+
+271
+00:26:24,410 --> 00:26:29,290
+زاد epsilon الكلام هذا صحيح لكل n أكبر من أو ساوي
+
+272
+00:26:29,290 --> 00:26:32,410
+capital N تمام؟
+
+273
+00:26:34,150 --> 00:26:39,850
+طيب الان خدي N خدي N هنا خدي ال N الصغيرة هذه
+
+274
+00:26:39,850 --> 00:26:44,130
+بيساوي capital N فبصير X capital N أصغر من ما
+
+275
+00:26:44,130 --> 00:26:48,970
+بيساوي X زاد Epsilon صح عوض عن ال Epsilon ال
+
+276
+00:26:48,970 --> 00:26:56,310
+Epsilon بيساوي سالب X إذن عوض عن X بسالب عن
+
+277
+00:26:56,310 --> 00:27:02,260
+Epsilon سالب X فهذا بيطلع sevenإذا أنا طلع عندي x
+
+278
+00:27:02,260 --> 00:27:06,960
+رقم capital N أصغر من صفر وهذا جدّيني
+
+279
+00:27:06,960 --> 00:27:13,180
+contradiction ليه؟ لأنه أنا فارق في النظرية أن كل
+
+280
+00:27:13,180 --> 00:27:19,000
+حدود ال sequence كلهم غير سالبين كلهم يعني حدود
+
+281
+00:27:19,000 --> 00:27:24,750
+غير سالبةفكيف طلع الحد رقم N capital N سالب هذا
+
+282
+00:27:24,750 --> 00:27:30,910
+بتناقض مع الفرض هذا بكمل البرهان تمام واضح البرهان
+
+283
+00:27:30,910 --> 00:27:35,390
+طبعا على طريقة هذه إذا لو ال sequence كانت
+
+284
+00:27:35,390 --> 00:27:39,630
+convergent وكل حدودها غير سالبة فنهايتها أيضا
+
+285
+00:27:39,630 --> 00:27:40,850
+هتكون غير سالبة
+
+286
+00:27:46,250 --> 00:27:49,370
+طب لو كانت ال sequence convergent و حدودها غير
+
+287
+00:27:49,370 --> 00:27:54,610
+موجبة كل حدودها غير موجبة فنهايتها ايضا هتكون غير
+
+288
+00:27:54,610 --> 00:28:02,430
+موجبة و البرهان مشابه لبرهان نظرية السابقة النظرية
+
+289
+00:28:02,430 --> 00:28:08,730
+اللي بعدها لو
+
+290
+00:28:08,730 --> 00:28:13,390
+في عندي two sequences و التنتين convergentوالحد
+
+291
+00:28:13,390 --> 00:28:16,130
+العاملة الأولى أصغر من أو ساوي الحد العاملة
+
+292
+00:28:16,130 --> 00:28:22,110
+التانية فنهاية الأولى أصغر من أو ساوي نهاية
+
+293
+00:28:22,110 --> 00:28:31,520
+التانية والبرهان تطبيق مباشر على نظرية اللى فاتتلو
+
+294
+00:28:31,520 --> 00:28:39,000
+اخدت انا عندي احنا فرضين ان xn أصغر من أو يساوي yn
+
+295
+00:28:39,000 --> 00:28:47,380
+لكل n هذا معناه ان yn minus xn أكبر من أو يساوي
+
+296
+00:28:47,380 --> 00:28:49,700
+سفر لكل n
+
+297
+00:28:53,620 --> 00:28:59,620
+فلو أخدت zn عرفت zn على أنه الفرق بين yn و xn
+
+298
+00:28:59,620 --> 00:29:03,580
+فقلنا الفرق هذا غير سالب لكل n إذن هذه ال sequence
+
+299
+00:29:03,580 --> 00:29:09,700
+جديدة حد العام تبعها zn وهذا الحد العام غير سالب
+
+300
+00:29:09,700 --> 00:29:15,530
+لكل nوالـ sequence xn convergent وyn convergent
+
+301
+00:29:15,530 --> 00:29:20,510
+حسب قوانين
+
+302
+00:29:20,510 --> 00:29:27,790
+النهايات limit الفرق هذا موجود أو بساوي فرق
+
+303
+00:29:27,790 --> 00:29:33,430
+النهايات إذا ال sequence zn حدودها كلها غير سالبة
+
+304
+00:29:33,430 --> 00:29:36,950
+وconvergent لأنها الفرق بين two convergent
+
+305
+00:29:36,950 --> 00:29:37,530
+sequences
+
+306
+00:29:40,870 --> 00:29:45,350
+وبالتالي حسب النظرية السابقة إذا ال limit ال
+
+307
+00:29:45,350 --> 00:29:50,890
+sequence zn هذه بتطلع غير سالبة طب limit ال zn عسب
+
+308
+00:29:50,890 --> 00:29:55,830
+قوانين النهيات بيساوي limit yn سالب limit xn و ادى
+
+309
+00:29:55,830 --> 00:29:59,230
+هذه عن ناحية التانية فبيطلع limit yn أكبر من لو
+
+310
+00:29:59,230 --> 00:30:06,090
+ساوي limit xn وهذا هو المظلومة okay تمام؟ راضى؟ دى
+
+311
+00:30:06,090 --> 00:30:06,730
+اي سؤال؟
+
+312
+00:30:12,550 --> 00:30:22,370
+النتيجة هذه بتقول
+
+313
+00:30:22,370 --> 00:30:26,430
+لو كان في عندي sequence و ال sequence هذي
+
+314
+00:30:26,430 --> 00:30:35,670
+convergent و الحدود تبعتها معصورة بين a و b
+
+315
+00:30:39,050 --> 00:30:45,870
+فلازم نهايتها ايضا .. نهايتها تطلع محصورة بين
+
+316
+00:30:45,870 --> 00:30:53,930
+العددين a وb فبرهان
+
+317
+00:30:53,930 --> 00:31:03,850
+النظرية هذه بنطبق نظرية 2.9 مرتين مرة
+
+318
+00:31:04,750 --> 00:31:10,170
+على الـ sequence الثابتة a اللي الحدود تبعتها أصغر
+
+319
+00:31:10,170 --> 00:31:15,310
+من حدود ال sequence xn هذه convergence نهايتها a
+
+320
+00:31:15,310 --> 00:31:24,470
+وهذه convergence نهايتها a إذا limit limit ال
+
+321
+00:31:24,470 --> 00:31:29,710
+sequence الثابتة a بساوي a أصغر من أو ساوي limit
+
+322
+00:31:29,710 --> 00:31:32,090
+ال sequence xn
+
+323
+00:31:34,930 --> 00:31:38,930
+كذلك أنا عندي الـ sequence xn من الفرض أنا عندي
+
+324
+00:31:38,930 --> 00:31:42,770
+الـ sequence xn أصغر من أو ساوي الـ sequence اللي
+
+325
+00:31:42,770 --> 00:31:48,330
+الحد العام تبعها ثابت بيه إذا ال limit حسب النظرية
+
+326
+00:31:48,330 --> 00:31:52,570
+8 إذا ال limit xn أصغر من أو ساوي limit ال
+
+327
+00:31:52,570 --> 00:31:57,530
+sequence اللي الحد العام تبعها ثابت اللي هو بيه
+
+328
+00:31:59,560 --> 00:32:05,040
+من هنا بطلع الـ delimit xn هي أكبر من أو ساوي a
+
+329
+00:32:05,040 --> 00:32:14,280
+أصغر من أو ساوي b و هو المفروض في
+
+330
+00:32:14,280 --> 00:32:22,180
+ال squeeze theorem أو نظرية الـ sandwich في
+
+331
+00:32:22,180 --> 00:32:25,160
+ناس يسموها squeeze theorem في ناس يسموها sandwich
+
+332
+00:32:25,160 --> 00:32:31,450
+theorem هذه في التفاضل والتكاملبسموها sandwich
+
+333
+00:32:31,450 --> 00:32:39,070
+sandwich
+
+334
+00:32:39,070 --> 00:32:42,750
+firm أو
+
+335
+00:32:42,750 --> 00:32:46,990
+squeeze firm فالنظرية هذه بتقول لو في عندي three
+
+336
+00:32:46,990 --> 00:32:54,710
+sequences XN, YN و ZN و العلاقة بينهم هكذا YN
+
+337
+00:32:54,710 --> 00:33:02,050
+محصورة بين XN و ZNو لو كانت الـ sequences الحدود
+
+338
+00:33:02,050 --> 00:33:06,790
+العامة تبعتها على الأطراف convergent يعني ال
+
+339
+00:33:06,790 --> 00:33:10,530
+sequence x in convergent و ال sequence z in
+
+340
+00:33:10,530 --> 00:33:14,150
+convergent و التنتين ال limit تبعتهم متساوية
+
+341
+00:33:16,990 --> 00:33:22,390
+فلابد أو لازم أن الـ sequence المحصورة بينهم بتكون
+
+342
+00:33:22,390 --> 00:33:27,130
+أيضًا convergent ونهايتها بتساوي القيمة المشتركة
+
+343
+00:33:27,130 --> 00:33:33,070
+لنهايات للنهايات limit yn بتكون موجودة ونهايتها
+
+344
+00:33:33,070 --> 00:33:39,910
+بتساوي نهاية xn ونهاية zn البرهان
+
+345
+00:33:39,910 --> 00:33:41,090
+برضه مش صعب
+
+346
+00:33:46,210 --> 00:33:50,630
+أحنا فرضين أن limit sequence x in موجودة و limit
+
+347
+00:33:50,630 --> 00:33:57,130
+sequence z in موجودة وتنتين متساويات فدعونا نسمي
+
+348
+00:33:57,130 --> 00:34:05,230
+ال limit هذه مشتركة a عدد a الأن ناخد let epsilon
+
+349
+00:34:05,230 --> 00:34:12,210
+أكبر من صفر bبما أن ال sequence xn converge ل a،
+
+350
+00:34:12,210 --> 00:34:15,430
+إذا يوجد عدد طبيعي capital N يعتمد على إبسلون،
+
+351
+00:34:15,430 --> 00:34:20,570
+بحيث لكل N أكبر من وسوء capital N، المسافة بين xn
+
+352
+00:34:20,570 --> 00:34:27,930
+و a أصغر من إبسلونكذلك بما أن ال sequence zm
+
+353
+00:34:27,930 --> 00:34:33,630
+converge ل a إذا بقدر ألاقي لكل .. لنفس ال epsilon
+
+354
+00:34:33,630 --> 00:34:38,650
+.. لنفس ال epsilon بقدر ألاقي capital N ممكن ال
+
+355
+00:34:38,650 --> 00:34:42,970
+capital N مختلف عن ال capital N الأولى نية ففي
+
+356
+00:34:42,970 --> 00:34:46,830
+الحالة هذه باخد capital N هذا ال maximum لل N
+
+357
+00:34:46,830 --> 00:34:49,490
+الأولى و التانية زي ما شوفنا في القرآن السابق
+
+358
+00:34:50,870 --> 00:34:54,170
+وبالتالي لكل n أكبر من مسافة capital N بقدر أخلي
+
+359
+00:34:54,170 --> 00:35:01,530
+المسافة هذه أصغر من epsilon الآن
+
+360
+00:35:01,530 --> 00:35:08,190
+من الفرض و من ال implication تمانية نحصل
+
+361
+00:35:08,190 --> 00:35:11,030
+على ال implication الجديدة هذه
+
+362
+00:35:18,350 --> 00:35:25,630
+يعني من هنا انا عندي xn minus a أصغر من ي وطبعا
+
+363
+00:35:25,630 --> 00:35:33,090
+أكبر من سالب ي ومن هنا انا عندي zn minus a أصغر من
+
+364
+00:35:33,090 --> 00:35:36,770
+ي أكبر من سالب ي
+
+365
+00:35:39,750 --> 00:35:45,170
+فهي xn minus a أصغر من إبسلون لكل n أكبر من أو
+
+366
+00:35:45,170 --> 00:35:54,810
+ساوية capital N إذا هذه هذه هي من هنا أو
+
+367
+00:35:54,810 --> 00:36:00,210
+هذه عفوا أنا أخدت الجزء هذه سالب إبسلون
+
+368
+00:36:04,860 --> 00:36:12,860
+-Epsilon أصغر من Xn-A وأنا عندي Xn أصغر من يساوي
+
+369
+00:36:12,860 --> 00:36:20,000
+Yn إذا لو طرحت A من هنا وطرحت A من هنا فبيطلع عندي
+
+370
+00:36:20,000 --> 00:36:27,080
+Xn-A أصغر من يساوي Yn-Aو نفس الحاجة لو طرحت a من
+
+371
+00:36:27,080 --> 00:36:32,620
+هنا طب يطلع y n minus a أصغر من أو ساوي z n minus
+
+372
+00:36:32,620 --> 00:36:41,320
+a و من هنا هاندي z n minus a من هذا الجزء z n
+
+373
+00:36:41,320 --> 00:36:43,460
+minus a أصغر من إبسن
+
+374
+00:36:48,860 --> 00:36:54,460
+إذاً هذه الـ implication بتقول باختصار لكل N أكبر
+
+375
+00:36:54,460 --> 00:37:00,240
+من أو ساوي capital N أنا عندي طلع يطلع عندي YN
+
+376
+00:37:00,240 --> 00:37:08,620
+minus A أصغر من أبسلون أكبر من سالب أبسلون أو
+
+377
+00:37:08,620 --> 00:37:13,140
+absolute YN minus A أصغر من أبسلون لكل N أكبر من
+
+378
+00:37:13,140 --> 00:37:18,280
+أو ساوي capital Nأذن هين أثبتنا أنه for any
+
+379
+00:37:18,280 --> 00:37:22,860
+epsilon أكبر من سفر يوجد capital N يعتمد على
+
+380
+00:37:22,860 --> 00:37:27,080
+epsilon عدد طبيعي بحيث لكل N أكبر من أو ساوي
+
+381
+00:37:27,080 --> 00:37:30,720
+capital N ال absolute value هذي أصغر من epsilon
+
+382
+00:37:30,720 --> 00:37:34,200
+اذا by epsilon capital N definition of limit بطلع
+
+383
+00:37:34,200 --> 00:37:39,640
+عندي limit هذا معناه ان limit ال sequence yn as n
+
+384
+00:37:39,640 --> 00:37:45,530
+tends to infinity بتساوي العدد Aو هذا اللي بدنا
+
+385
+00:37:45,530 --> 00:37:51,790
+يعني .. هذا اللي بدنا يعني .. okay تمام؟ إذن هذا
+
+386
+00:37:51,790 --> 00:37:55,790
+هو برهان ال sandwich أو ال squeeze ال theorem
+
+387
+00:37:55,790 --> 00:38:00,710
+تمام؟
+
+388
+00:38:00,710 --> 00:38:04,810
+هاي ناخد بعض ال .. بعض الأمثلة .. بعض الأمثلة
+
+389
+00:38:20,690 --> 00:38:25,770
+عايزين نثبت انه limit ال sequence لحد العام تبعها
+
+390
+00:38:25,770 --> 00:38:31,910
+الكاسر هذا بساوة ستة فزي
+
+391
+00:38:31,910 --> 00:38:38,270
+ما كنا نعمل في تفاضل ألف أو با بنجسم بسط مقام على
+
+392
+00:38:38,270 --> 00:38:46,130
+n لأكبر أساللي هو n تربية فبصير عندي المقدار ده
+
+393
+00:38:46,130 --> 00:38:49,790
+بيساوي اتنين على n على واحد زاد .. على واحد ..
+
+394
+00:38:49,790 --> 00:38:54,490
+واحد على n تربية زاد واحد انا عندي ال sequence
+
+395
+00:38:54,490 --> 00:38:59,190
+واحد على n ال limit تبعتها سفر اذا اتنين على n ال
+
+396
+00:38:59,190 --> 00:39:06,360
+limit تبعتها سفرالمقام واحد على M تربية نهايتها
+
+397
+00:39:06,360 --> 00:39:10,580
+سفر اذا واحد زاد واحد على M تربية نهايتها واحد زاد
+
+398
+00:39:10,580 --> 00:39:19,860
+سفر بطلع واحد لا يساوي سفر تمام اذا ال limit الكسر
+
+399
+00:39:19,860 --> 00:39:27,000
+هذا بساوي limit الكسر اللي هان الان الكسر هذا
+
+400
+00:39:27,000 --> 00:39:31,490
+بتكون من bust مقام ال bustالمقام عبارة عن الـ
+
+401
+00:39:31,490 --> 00:39:35,670
+sequence 2 على M المقام عبارة عن الـ sequence الحد
+
+402
+00:39:35,670 --> 00:39:41,130
+العام تبعه 1 زاد 1 على M تربيه الآن limit ال
+
+403
+00:39:41,130 --> 00:39:44,950
+sequence في المقام شفنا بالساوي واحد لا تساوي سفر
+
+404
+00:39:44,950 --> 00:39:50,370
+عشان هيك قدرت اقول ان limit limit الكسر بساوي
+
+405
+00:39:50,370 --> 00:39:53,330
+limit البسط على limit المقام لكن لو limit المقام
+
+406
+00:39:53,330 --> 00:39:59,590
+بالساوي سفر فما بقدر استخدم القانون هذا ماشي الحل
+
+407
+00:40:02,990 --> 00:40:06,970
+limit الكثر هذا بيساوي limit الكثر اللي هان وهذا
+
+408
+00:40:06,970 --> 00:40:11,170
+طبعا لأن limit المقام بيساويش سفر فبساوي limit ال
+
+409
+00:40:11,170 --> 00:40:15,170
+bus على limit المقام حسب قوانين النهايات limit ال
+
+410
+00:40:15,170 --> 00:40:20,610
+bus سفر limit المقام واحد اذا ال limit في النهاية
+
+411
+00:40:20,610 --> 00:40:29,880
+تمتها سفر تمام المثال التاني هذابإنه نثبت ان الـ
+
+412
+00:40:29,880 --> 00:40:35,960
+sequence الحد العام تبعها xn بساوي sin n على n ال
+
+413
+00:40:35,960 --> 00:40:41,520
+limit تبعتها بساوي 0 طيب احنا عارفين من تفاض الألف
+
+414
+00:40:41,520 --> 00:40:46,060
+ان absolute sin x أصغر من أو ساوي واحد لكل x أخر
+
+415
+00:40:46,060 --> 00:40:54,660
+او وبالتالي لكل عدد طبيعي n absolute او sin n أكبر
+
+416
+00:40:54,660 --> 00:41:00,960
+من أو ساوي سالب واحدأصغر من أو يساوي واحد طيب لكل
+
+417
+00:41:00,960 --> 00:41:07,460
+عدد طبيعي ان واحد على ان عدد موجب فلو ضربنا
+
+418
+00:41:07,460 --> 00:41:12,800
+المتباينة هذه في العدد الموجب واحد على ان فبصير
+
+419
+00:41:12,800 --> 00:41:13,820
+شكلها هكذا
+
+420
+00:41:18,240 --> 00:41:21,680
+فبتصير ال sequence بعد ما نضربها أو المتباينة
+
+421
+00:41:21,680 --> 00:41:25,120
+الأخيرة بعد ما نضربها في واحد على N بصير شكلها
+
+422
+00:41:25,120 --> 00:41:30,240
+هكذا سالب واحد على N أصلا ما يساوي sign N على N
+
+423
+00:41:30,240 --> 00:41:35,120
+أصلا ما يساوي واحد على N طيب ال sequence هذه ال
+
+424
+00:41:35,120 --> 00:41:42,860
+limit تبعتها سفر و ال sequence هذه أيضا نهايتها
+
+425
+00:41:42,860 --> 00:41:45,960
+سالب واحد في سفر بطلع سفر
+
+426
+00:41:49,290 --> 00:41:52,830
+إذا by squeeze theorem أو by sandwich theorem
+
+427
+00:41:52,830 --> 00:41:56,210
+limit ال sequence اللي في الوسط اللي محصورة في
+
+428
+00:41:56,210 --> 00:42:03,950
+النصر بيطلع أيضا موجودة و بيساوي السفر okay تمام
+
+429
+00:42:03,950 --> 00:42:09,150
+طب لو كانت ال limit هذه بيساوي سفر وهذه بيساوي
+
+430
+00:42:09,150 --> 00:42:13,670
+واحد ماقدرش أطبق ال squeeze theorem لازم ال
+
+431
+00:42:13,670 --> 00:42:17,630
+sequences اللي على أطرافمش بس يكونوا convergent
+
+432
+00:42:17,630 --> 00:42:23,390
+ويلهم نفس ال limit تكون نفس القيمة okay تمام؟ اذا
+
+433
+00:42:23,390 --> 00:42:30,690
+يعني هذا .. هذا بعض الأمثلة المرة الجاية طبعا
+
+434
+00:42:30,690 --> 00:42:37,350
+هنكمل .. اذا
+
+435
+00:42:37,350 --> 00:42:42,330
+المرة الجاية هنكمل نشوف بعض نظريات النهايات و ..
+
+436
+00:42:43,750 --> 00:42:50,310
+هنخلص اللي هو section اتنين اتنين مش يعني ضايل
+
+437
+00:42:50,310 --> 00:42:58,470
+كتير و هحللكم بعض التمرين و ناخد ال homework ل
+
+438
+00:42:58,470 --> 00:43:03,070
+section تلات اتنين okay تمام فحاولوا تحضروا هذا
+
+439
+00:43:03,070 --> 00:43:12,050
+الكلام للمحاضرة الجاية و نشوف مع بعض الحاجات هذهو
+
+440
+00:43:12,050 --> 00:43:18,550
+نشرحها okay إذا انتهت المحاضرة نكمل ان شاء الله
+
+441
+00:43:18,550 --> 00:43:19,450
+المرة الجاية
+
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/r7wN576DqQ0_raw.json b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/r7wN576DqQ0_raw.json
new file mode 100644
index 0000000000000000000000000000000000000000..41ad54b0f9d4f3dfc6279ba0e8b30bc7bcef2169
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/r7wN576DqQ0_raw.json
@@ -0,0 +1 @@
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طيب احنا بما نثبت انه بما نثبت ان ال absolute value", "tokens": [2655, 15042, 2304, 16472, 23758, 2423, 28820, 10943, 20328, 5016, 1829, 5016, 5296, 28820, 2031, 8978, 257, 3714, 46456, 46538, 18871, 2423, 269, 4032, 5296, 28820, 4032, 8978, 8717, 36178, 2423, 30543, 2655, 3714, 29245, 23328, 2423, 2031, 8978, 2423, 8289, 7630, 5296, 269, 23032, 1829, 3555, 1975, 5016, 8315, 4724, 15042, 8717, 12984, 3555, 2655, 16472, 3224, 4724, 15042, 8717, 12984, 3555, 2655, 16472, 2423, 8236, 2158], "avg_logprob": -0.25747283299763996, "compression_ratio": 1.5276073619631902, "no_speech_prob": 0.0, "words": [{"start": 234.51, "end": 234.97, "word": "تمام", "probability": 0.488067626953125}, {"start": 234.97, "end": 235.15, "word": " ان", "probability": 0.194091796875}, {"start": 235.15, "end": 235.37, "word": " هذا", "probability": 0.9033203125}, {"start": 235.37, "end": 235.75, "word": " الكلام", "probability": 0.9148763020833334}, {"start": 235.75, "end": 236.87, "word": " صحيح", "probability": 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6225, 47302, 11622, 9957, 8717, 12984, 3555, 2655, 14739, 8236, 479, 295, 1783, 5551, 9381, 17082, 2288, 9154, 11296, 7251, 3794, 995, 45865, 376], "avg_logprob": -0.4187911199895959, "compression_ratio": 1.0084745762711864, "no_speech_prob": 0.0, "words": [{"start": 259.64000000000004, "end": 260.6, "word": "أو", "probability": 0.25360107421875}, {"start": 260.6, "end": 261.56, "word": " جدنا", "probability": 0.75927734375}, {"start": 261.56, "end": 261.9, "word": " delta", "probability": 0.42041015625}, {"start": 261.9, "end": 262.38, "word": " neighborhood", "probability": 0.295166015625}, {"start": 262.38, "end": 263.38, "word": " لـC", "probability": 0.4337565104166667}, {"start": 263.38, "end": 275.24, "word": " نحن", "probability": 0.4750162760416667}, {"start": 275.24, "end": 276.36, "word": " عايزين", "probability": 0.8616943359375}, {"start": 276.36, "end": 276.7, "word": " نثبت", "probability": 0.970947265625}, {"start": 276.7, "end": 276.84, "word": " أن", "probability": 0.5390625}, {"start": 276.84, "end": 277.3, "word": " absolute", "probability": 0.76513671875}, {"start": 277.3, "end": 277.86, "word": " F", "probability": 0.332275390625}, {"start": 277.86, "end": 278.14, "word": " of", "probability": 0.39501953125}, {"start": 278.14, "end": 278.56, "word": " X", "probability": 0.96142578125}, {"start": 278.56, "end": 279.66, "word": " أصغر", "probability": 0.9659423828125}, {"start": 279.66, "end": 280.06, "word": " منها", "probability": 0.929443359375}, {"start": 280.06, "end": 280.6, "word": " يساوي", "probability": 0.77911376953125}, {"start": 280.6, "end": 280.94, "word": " M", "probability": 0.85986328125}], "temperature": 1.0}, {"id": 11, "seek": 30430, "start": 282.52, "end": 304.3, "text": "أو there exists M أكبر من السفر بحيث أن absolute F of X أصغر من أو ساوي M لكل X هنتمي إلى A تقاطع D Delta of C مش هيك تعريف اللي هو المتبين هذا طيب we have two cases", "tokens": [10721, 2407, 456, 8198, 376, 5551, 4117, 26890, 9154, 21136, 5172, 2288, 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{"start": 290.8, "end": 291.24, "word": " X", "probability": 0.9814453125}, {"start": 291.24, "end": 292.76, "word": " هنتمي", "probability": 0.831298828125}, {"start": 292.76, "end": 293.2, "word": " إلى", "probability": 0.91259765625}, {"start": 293.2, "end": 294.18, "word": " A", "probability": 0.6513671875}, {"start": 294.18, "end": 295.22, "word": " تقاطع", "probability": 0.8973388671875}, {"start": 295.22, "end": 295.54, "word": " D", "probability": 0.80859375}, {"start": 295.54, "end": 296.1, "word": " Delta", "probability": 0.6025390625}, {"start": 296.1, "end": 297.3, "word": " of", "probability": 0.689453125}, {"start": 297.3, "end": 297.76, "word": " C", "probability": 0.82763671875}, {"start": 297.76, "end": 298.6, "word": " مش", "probability": 0.45703125}, {"start": 298.6, "end": 298.82, "word": " هيك", "probability": 0.98193359375}, {"start": 298.82, "end": 299.3, "word": " تعريف", "probability": 0.9694010416666666}, {"start": 299.3, "end": 299.62, "word": " اللي", "probability": 0.81787109375}, {"start": 299.62, "end": 299.74, "word": " هو", "probability": 0.8759765625}, {"start": 299.74, "end": 300.28, "word": " المتبين", "probability": 0.917236328125}, {"start": 300.28, "end": 300.66, "word": " هذا", "probability": 0.26171875}, {"start": 300.66, "end": 302.48, "word": " طيب", "probability": 0.9583333333333334}, {"start": 302.48, "end": 303.24, "word": " we", "probability": 0.41943359375}, {"start": 303.24, "end": 303.56, "word": " have", "probability": 0.97509765625}, {"start": 303.56, "end": 303.8, "word": " two", "probability": 0.9287109375}, {"start": 303.8, "end": 304.3, "word": " cases", "probability": 0.943359375}], "temperature": 1.0}, {"id": 12, "seek": 33920, "start": 311.24, "end": 339.2, "text": "كاس واحد إذا كانت الـ C تنتمي إلى A ال cluster point C هي ال cluster point ال C is a cluster point لست A فممكن ال C تنتمي ل A و ممكن ما تنتميش إذا في أنتي حالتين لو كانت ال C تنتمي ل A in this case in this case", "tokens": [4117, 32277, 36764, 24401, 11933, 15730, 25961, 2655, 2423, 39184, 383, 6055, 29399, 2304, 1829, 30731, 316, 2423, 13630, 935, 383, 39896, 2423, 13630, 935, 2423, 383, 307, 257, 13630, 935, 5296, 14851, 316, 6156, 2304, 43020, 2423, 383, 6055, 29399, 2304, 1829, 5296, 316, 4032, 3714, 43020, 19446, 6055, 29399, 2304, 1829, 8592, 11933, 15730, 8978, 14739, 31371, 11331, 6027, 2655, 9957, 45164, 25961, 2655, 2423, 383, 6055, 29399, 2304, 1829, 5296, 316, 294, 341, 1389, 294, 341, 1389], "avg_logprob": -0.2656249941131215, "compression_ratio": 1.8580246913580247, "no_speech_prob": 5.960464477539063e-08, "words": [{"start": 311.24, "end": 311.8, "word": "كاس", "probability": 0.12451171875}, {"start": 311.8, "end": 312.88, "word": " واحد", "probability": 0.78369140625}, {"start": 312.88, "end": 313.78, "word": " إذا", "probability": 0.6531982421875}, {"start": 313.78, "end": 314.24, "word": " كانت", "probability": 0.945556640625}, {"start": 314.24, "end": 314.48, "word": " الـ", "probability": 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" ال", "probability": 0.74365234375}, {"start": 319.98, "end": 320.42, "word": " C", "probability": 0.63623046875}, {"start": 320.42, "end": 321.36, "word": " is", "probability": 0.609375}, {"start": 321.36, "end": 321.7, "word": " a", "probability": 0.9462890625}, {"start": 321.7, "end": 322.2, "word": " cluster", "probability": 0.94384765625}, {"start": 322.2, "end": 322.9, "word": " point", "probability": 0.98876953125}, {"start": 322.9, "end": 324.4, "word": " لست", "probability": 0.57470703125}, {"start": 324.4, "end": 324.82, "word": " A", "probability": 0.88037109375}, {"start": 324.82, "end": 327.12, "word": " فممكن", "probability": 0.9401041666666666}, {"start": 327.12, "end": 327.36, "word": " ال", "probability": 0.9384765625}, {"start": 327.36, "end": 327.54, "word": " C", "probability": 0.93896484375}, {"start": 327.54, "end": 328.1, "word": " تنتمي", "probability": 0.98828125}, {"start": 328.1, "end": 328.2, "word": " ل", "probability": 0.525390625}, {"start": 328.2, "end": 328.4, "word": " A", "probability": 0.76123046875}, {"start": 328.4, "end": 328.54, "word": " و", "probability": 0.9208984375}, {"start": 328.54, "end": 328.82, "word": " ممكن", "probability": 0.79052734375}, {"start": 328.82, "end": 328.96, "word": " ما", "probability": 0.58203125}, {"start": 328.96, "end": 329.68, "word": " تنتميش", "probability": 0.9599609375}, {"start": 329.68, "end": 331.48, "word": " إذا", "probability": 0.827392578125}, {"start": 331.48, "end": 331.64, "word": " في", "probability": 0.79541015625}, {"start": 331.64, "end": 331.92, "word": " أنتي", "probability": 0.506591796875}, {"start": 331.92, "end": 332.4, "word": " حالتين", "probability": 0.9749755859375}, {"start": 332.4, "end": 332.94, "word": " لو", "probability": 0.4912109375}, {"start": 332.94, "end": 333.34, "word": " كانت", "probability": 0.978515625}, {"start": 333.34, "end": 333.48, "word": " ال", "probability": 0.9404296875}, {"start": 333.48, "end": 333.62, "word": " C", "probability": 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"probability": 0.98486328125}, {"start": 394.12, "end": 394.26, "word": " عن", "probability": 0.98193359375}, {"start": 394.26, "end": 394.44, "word": " الـ", "probability": 0.967529296875}, {"start": 394.44, "end": 394.66, "word": " C", "probability": 0.87939453125}], "temperature": 1.0}, {"id": 15, "seek": 42696, "start": 400.34, "end": 426.96, "text": "ففي الحالة هذه بيطلع absolute F of X أصغر من واحد زاد absolute L وهذا العدد أصغر من أو يساوي M ففي كل الأحوال هذا الكلام صحيح لكل X تنتمي إلى A تقاطة U Delta of C وهو المطلوب طبعاً نشوف الحالة التانية Case 2", "tokens": [5172, 41185, 21542, 6027, 3660, 29538, 4724, 1829, 9566, 1211, 3615, 8236, 479, 295, 1783, 5551, 9381, 17082, 2288, 9154, 36764, 24401, 30767, 18513, 8236, 441, 37037, 15730, 18863, 3215, 3215, 5551, 9381, 17082, 2288, 9154, 34051, 7251, 3794, 995, 45865, 376, 6156, 41185, 28242, 16247, 5016, 2407, 6027, 23758, 2423, 28820, 10943, 20328, 5016, 1829, 5016, 5296, 28820, 1783, 6055, 29399, 2304, 1829, 30731, 316, 6055, 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"probability": 0.45654296875}, {"start": 418.2, "end": 418.58, "word": " Delta", "probability": 0.5322265625}, {"start": 418.58, "end": 419.26, "word": " of", "probability": 0.7294921875}, {"start": 419.26, "end": 419.84, "word": " C", "probability": 0.72998046875}, {"start": 419.84, "end": 422.78, "word": " وهو", "probability": 0.862060546875}, {"start": 422.78, "end": 423.28, "word": " المطلوب", "probability": 0.9874267578125}, {"start": 423.28, "end": 424.32, "word": " طبعاً", "probability": 0.8560546875}, {"start": 424.32, "end": 424.68, "word": " نشوف", "probability": 0.8748372395833334}, {"start": 424.68, "end": 425.18, "word": " الحالة", "probability": 0.9718424479166666}, {"start": 425.18, "end": 425.8, "word": " التانية", "probability": 0.9728190104166666}, {"start": 425.8, "end": 426.58, "word": " Case", "probability": 0.59521484375}, {"start": 426.58, "end": 426.96, "word": " 2", "probability": 0.6259765625}], "temperature": 1.0}, {"id": 16, "seek": 45731, "start": 428.17, "end": 457.31, "text": "إن الـ X لا تنتمي إلى A in this case we have absolute F in this case we take M بساوي واحد زاد absolute L", "tokens": [28814, 1863, 2423, 39184, 1783, 20193, 6055, 29399, 2304, 1829, 30731, 316, 294, 341, 1389, 321, 362, 8236, 479, 294, 341, 1389, 321, 747, 376, 4724, 3794, 995, 45865, 36764, 24401, 30767, 18513, 8236, 441], "avg_logprob": -0.3196614442600144, "compression_ratio": 1.1491228070175439, "no_speech_prob": 0.0, "words": [{"start": 428.17, "end": 428.49, "word": "إن", "probability": 0.6943359375}, {"start": 428.49, "end": 428.71, "word": " الـ", "probability": 0.57275390625}, {"start": 428.71, "end": 429.01, "word": " X", "probability": 0.50341796875}, {"start": 429.01, "end": 429.31, "word": " لا", "probability": 0.82568359375}, {"start": 429.31, "end": 430.05, "word": " تنتمي", "probability": 0.955078125}, {"start": 430.05, "end": 430.29, "word": " إلى", "probability": 0.63623046875}, {"start": 430.29, "end": 430.63, "word": " A", "probability": 0.3994140625}, {"start": 430.63, "end": 433.77, "word": " in", "probability": 0.2117919921875}, {"start": 433.77, "end": 434.09, "word": " this", "probability": 0.9658203125}, {"start": 434.09, "end": 434.57, "word": " case", "probability": 0.91845703125}, {"start": 434.57, "end": 440.81, "word": " we", "probability": 0.37548828125}, {"start": 440.81, "end": 441.43, "word": " have", "probability": 0.9208984375}, {"start": 441.43, "end": 444.23, "word": " absolute", "probability": 0.68408203125}, {"start": 444.23, "end": 444.97, "word": " F", "probability": 0.5810546875}, {"start": 444.97, "end": 446.33, "word": " in", "probability": 0.572265625}, {"start": 446.33, "end": 446.67, "word": " this", "probability": 0.9619140625}, {"start": 446.67, "end": 447.19, "word": " case", "probability": 0.9365234375}, {"start": 447.19, "end": 448.09, "word": " we", "probability": 0.66455078125}, {"start": 448.09, "end": 448.75, "word": " take", "probability": 0.8798828125}, {"start": 448.75, "end": 452.39, "word": " M", "probability": 0.92626953125}, {"start": 452.39, "end": 453.33, "word": " بساوي", "probability": 0.8226318359375}, {"start": 453.33, "end": 455.55, "word": " واحد", "probability": 0.917724609375}, {"start": 455.55, "end": 456.07, "word": " زاد", "probability": 0.6510009765625}, {"start": 456.07, "end": 456.65, "word": " absolute", "probability": 0.869140625}, {"start": 456.65, "end": 457.31, "word": " L", "probability": 0.78369140625}], "temperature": 1.0}, {"id": 17, "seek": 47403, "start": 460.85, "end": 474.03, "text": "بالحالة هاريب ناخد M بساوية واحد زاد absolute L باستخدام المتباينة هذه الأخيرة اللي هي سميها star", "tokens": [3555, 6027, 5016, 6027, 3660, 8032, 9640, 1829, 3555, 8717, 47283, 3215, 376, 4724, 3794, 995, 2407, 10632, 36764, 24401, 30767, 18513, 8236, 441, 4724, 995, 14851, 9778, 3215, 10943, 9673, 2655, 3555, 995, 9957, 3660, 29538, 16247, 9778, 48923, 13672, 1829, 39896, 8608, 2304, 1829, 11296, 3543], "avg_logprob": -0.5985331827280472, "compression_ratio": 1.3015873015873016, "no_speech_prob": 0.0, "words": [{"start": 460.85, "end": 461.67, "word": "بالحالة", "probability": 0.5357177734375}, {"start": 461.67, "end": 461.97, "word": " هاريب", "probability": 0.533294677734375}, {"start": 461.97, "end": 462.37, "word": " ناخد", "probability": 0.6884765625}, {"start": 462.37, "end": 462.65, "word": " M", "probability": 0.1966552734375}, {"start": 462.65, "end": 463.07, "word": " بساوية", "probability": 0.4537841796875}, {"start": 463.07, "end": 463.59, "word": " واحد", "probability": 0.93896484375}, {"start": 463.59, "end": 464.45, "word": " زاد", "probability": 0.5745849609375}, {"start": 464.45, "end": 464.87, "word": " absolute", "probability": 0.331787109375}, {"start": 464.87, "end": 465.55, "word": " L", "probability": 0.09912109375}, {"start": 465.55, "end": 469.67, "word": " باستخدام", "probability": 0.7953287760416666}, {"start": 469.67, "end": 470.45, "word": " المتباينة", "probability": 0.7416788736979166}, {"start": 470.45, "end": 470.67, "word": " هذه", "probability": 0.55126953125}, {"start": 470.67, "end": 471.25, "word": " الأخيرة", "probability": 0.84375}, {"start": 471.25, "end": 472.67, "word": " اللي", "probability": 0.702392578125}, {"start": 472.67, "end": 472.89, "word": " هي", "probability": 0.64306640625}, {"start": 472.89, "end": 473.53, "word": " سميها", "probability": 0.71868896484375}, {"start": 473.53, "end": 474.03, "word": " star", "probability": 0.4853515625}], "temperature": 1.0}, {"id": 18, "seek": 51267, "start": 483.26, "end": 512.68, "text": "بطلع عندي absolute f of x إذا كانت الـ c لا تنتمي إلى a فبصير هذه a و إطرح منها c هي a صح؟ وبالتالي الكلام هذا بصير أصغر من 1 زاد absolute L اللي هو أصغر من أو ساوي M وهذا صحيح لكل x ينتمي إلى a تقاطع بي دلتا", "tokens": [3555, 9566, 1211, 3615, 18871, 16254, 8236, 283, 295, 2031, 11933, 15730, 25961, 2655, 2423, 39184, 269, 20193, 6055, 29399, 2304, 1829, 30731, 257, 6156, 3555, 9381, 13546, 29538, 257, 4032, 11933, 9566, 2288, 5016, 9154, 11296, 269, 39896, 257, 20328, 5016, 22807, 46599, 6027, 2655, 6027, 1829, 2423, 28820, 10943, 23758, 4724, 9381, 13546, 5551, 9381, 17082, 2288, 9154, 502, 30767, 18513, 8236, 441, 13672, 1829, 31439, 5551, 9381, 17082, 2288, 9154, 34051, 8608, 995, 45865, 376, 37037, 15730, 20328, 5016, 1829, 5016, 5296, 28820, 2031, 7251, 29399, 2304, 1829, 30731, 257, 6055, 4587, 41193, 3615, 4724, 1829, 11778, 1211, 2655, 995], "avg_logprob": -0.28245193654528034, "compression_ratio": 1.6470588235294117, "no_speech_prob": 0.0, "words": [{"start": 483.26, "end": 483.78, "word": "بطلع", "probability": 0.59942626953125}, {"start": 483.78, "end": 484.04, "word": " عندي", "probability": 0.55352783203125}, {"start": 484.04, "end": 484.54, "word": " absolute", "probability": 0.6669921875}, {"start": 484.54, "end": 485.3, "word": " f", "probability": 0.361572265625}, {"start": 485.3, "end": 485.58, "word": " of", "probability": 0.50537109375}, {"start": 485.58, "end": 486.06, "word": " x", "probability": 0.845703125}, {"start": 486.06, "end": 487.2, "word": " إذا", "probability": 0.6707763671875}, {"start": 487.2, "end": 487.56, "word": " كانت", "probability": 0.943603515625}, {"start": 487.56, "end": 490.66, "word": " الـ", "probability": 0.493896484375}, {"start": 490.66, "end": 490.66, "word": " c", "probability": 0.23095703125}, {"start": 490.66, "end": 492.0, "word": " لا", "probability": 0.578125}, {"start": 492.0, "end": 492.66, "word": " تنتمي", "probability": 0.9310302734375}, {"start": 492.66, "end": 492.8, "word": " إلى", "probability": 0.38330078125}, {"start": 492.8, "end": 493.04, "word": " a", "probability": 0.90771484375}, {"start": 493.04, "end": 495.48, "word": " فبصير", "probability": 0.844970703125}, {"start": 495.48, "end": 495.84, "word": " هذه", "probability": 0.74072265625}, {"start": 495.84, "end": 496.2, "word": " a", "probability": 0.8291015625}, {"start": 496.2, "end": 496.4, "word": " و", "probability": 0.84375}, {"start": 496.4, "end": 496.7, "word": " إطرح", "probability": 0.708740234375}, {"start": 496.7, "end": 497.0, "word": " منها", "probability": 0.934814453125}, {"start": 497.0, "end": 497.28, "word": " c", "probability": 0.56591796875}, {"start": 497.28, "end": 497.56, "word": " هي", "probability": 0.9404296875}, {"start": 497.56, "end": 497.86, "word": " a", "probability": 0.9482421875}, {"start": 497.86, "end": 498.96, "word": " صح؟", "probability": 0.6555989583333334}, {"start": 498.96, "end": 499.94, "word": " وبالتالي", "probability": 0.9287109375}, {"start": 499.94, "end": 500.36, "word": " الكلام", "probability": 0.8839518229166666}, {"start": 500.36, "end": 500.84, "word": " هذا", "probability": 0.92578125}, {"start": 500.84, "end": 501.38, "word": " بصير", "probability": 0.8831380208333334}, {"start": 501.38, "end": 502.02, "word": " أصغر", "probability": 0.9906005859375}, {"start": 502.02, "end": 502.36, "word": " من", "probability": 0.994140625}, {"start": 502.36, "end": 503.22, "word": " 1", "probability": 0.462646484375}, {"start": 503.22, "end": 503.76, "word": " زاد", "probability": 0.480224609375}, {"start": 503.76, "end": 504.3, "word": " absolute", "probability": 0.90234375}, {"start": 504.3, "end": 504.94, "word": " L", "probability": 0.346923828125}, {"start": 504.94, "end": 505.66, "word": " اللي", "probability": 0.87353515625}, {"start": 505.66, "end": 505.86, "word": " هو", "probability": 0.9853515625}, {"start": 505.86, "end": 506.36, "word": " أصغر", "probability": 0.9921875}, {"start": 506.36, "end": 506.62, "word": " من", "probability": 0.9951171875}, {"start": 506.62, "end": 506.8, "word": " أو", "probability": 0.4384765625}, {"start": 506.8, "end": 507.4, "word": " ساوي", "probability": 0.8678385416666666}, {"start": 507.4, "end": 507.68, "word": " M", "probability": 0.744140625}, {"start": 507.68, "end": 508.06, "word": " وهذا", "probability": 0.7166748046875}, {"start": 508.06, "end": 508.72, "word": " صحيح", "probability": 0.9852294921875}, {"start": 508.72, "end": 509.7, "word": " لكل", "probability": 0.948486328125}, {"start": 509.7, "end": 510.0, "word": " x", "probability": 0.84619140625}, {"start": 510.0, "end": 510.58, "word": " ينتمي", "probability": 0.890869140625}, {"start": 510.58, "end": 510.72, "word": " إلى", "probability": 0.9541015625}, {"start": 510.72, "end": 511.0, "word": " a", "probability": 0.86328125}, {"start": 511.0, "end": 511.78, "word": " تقاطع", "probability": 0.845947265625}, {"start": 511.78, "end": 512.08, "word": " بي", "probability": 0.3602294921875}, {"start": 512.08, "end": 512.68, "word": " دلتا", "probability": 0.75750732421875}], "temperature": 1.0}, {"id": 19, "seek": 53324, "start": 513.84, "end": 533.24, "text": "وبسيط لأن الـ C مش موجودة في إيه فاطرحها أو ماطرحاش مابتفرجش لأن هنا استخدمنا star و هنا برضه استخدمنا star then by star ال absolute value هنا طلعت", "tokens": [37746, 3794, 1829, 9566, 5296, 33456, 2423, 39184, 383, 37893, 3714, 29245, 23328, 3660, 8978, 11933, 1829, 3224, 6156, 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" show", "probability": 0.95068359375}], "temperature": 1.0}, {"id": 26, "seek": 72459, "start": 696.35, "end": 724.59, "text": "إن الـ function تبعتنا الـ function f هذه اللي عرضناها that الـ function f is not is not bounded on", "tokens": [28814, 1863, 2423, 39184, 2445, 6055, 3555, 34268, 8315, 2423, 39184, 2445, 283, 29538, 13672, 1829, 6225, 43042, 8315, 11296, 300, 2423, 39184, 2445, 283, 307, 406, 307, 406, 37498, 322], "avg_logprob": -0.27832030691206455, "compression_ratio": 1.368421052631579, "no_speech_prob": 0.0, "words": [{"start": 696.35, "end": 696.69, "word": "إن", "probability": 0.5552978515625}, {"start": 696.69, "end": 697.01, "word": " الـ", "probability": 0.5869140625}, {"start": 697.01, "end": 697.41, "word": " function", "probability": 0.87939453125}, {"start": 697.41, "end": 698.47, "word": " تبعتنا", "probability": 0.7061767578125}, {"start": 698.47, "end": 704.61, "word": " الـ", "probability": 0.81591796875}, {"start": 704.61, "end": 709.75, "word": " function", "probability": 0.943359375}, {"start": 709.75, "end": 710.33, "word": " f", "probability": 0.59033203125}, {"start": 710.33, "end": 710.81, "word": " هذه", "probability": 0.7138671875}, {"start": 710.81, "end": 711.01, "word": " اللي", "probability": 0.906005859375}, {"start": 711.01, "end": 712.89, "word": " عرضناها", "probability": 0.9698486328125}, {"start": 712.89, "end": 713.45, "word": " that", "probability": 0.43505859375}, {"start": 713.45, "end": 715.91, "word": " الـ", "probability": 0.777587890625}, {"start": 715.91, "end": 716.21, "word": " function", "probability": 0.9609375}, {"start": 716.21, "end": 716.71, "word": " f", "probability": 0.92724609375}, {"start": 716.71, "end": 717.23, "word": " is", "probability": 0.94384765625}, {"start": 717.23, "end": 717.69, "word": " not", "probability": 0.96875}, {"start": 717.69, "end": 719.13, "word": " is", "probability": 0.642578125}, {"start": 719.13, "end": 719.53, "word": " not", "probability": 0.96923828125}, 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"probability": 0.453125}], "temperature": 1.0}, {"id": 28, "seek": 78184, "start": 759.0, "end": 781.84, "text": "عشان أثبت أن الدالة ليست bounded ليس .. ماليهاش limit عند السفر حسب النظرية يكفي أثبات أن الدالة هذه is not bounded على أي جوار للسفر لأن لو كانت bounded على جوار واحد للسفر", "tokens": [3615, 8592, 7649, 5551, 12984, 3555, 2655, 14739, 32748, 6027, 3660, 32239, 14851, 37498, 32239, 3794, 4386, 19446, 20292, 3224, 33599, 4948, 43242, 21136, 5172, 2288, 11331, 35457, 28239, 19913, 2288, 10632, 7251, 4117, 41185, 5551, 12984, 3555, 9307, 14739, 32748, 6027, 3660, 29538, 307, 406, 37498, 15844, 36632, 10874, 2407, 9640, 24976, 3794, 5172, 2288, 5296, 33456, 45164, 25961, 2655, 37498, 15844, 10874, 2407, 9640, 36764, 24401, 24976, 3794, 5172, 2288], "avg_logprob": -0.17765410795603712, "compression_ratio": 1.6666666666666667, "no_speech_prob": 0.0, "words": [{"start": 759.0, "end": 759.4, "word": "عشان", "probability": 0.7828776041666666}, {"start": 759.4, "end": 759.72, 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{"start": 1132.65, "end": 1133.19, "word": " لبرهان", "probability": 0.9735107421875}, {"start": 1133.19, "end": 1133.73, "word": " النظرية", "probability": 0.85235595703125}, {"start": 1133.73, "end": 1134.29, "word": " هذه", "probability": 0.84423828125}, {"start": 1134.29, "end": 1135.55, "word": " خوف", "probability": 0.57110595703125}, {"start": 1135.55, "end": 1135.95, "word": " one", "probability": 0.61328125}], "temperature": 1.0}, {"id": 42, "seek": 117273, "start": 1143.81, "end": 1172.73, "text": "يوزن epsilon delta definition زي اللي ما شوفنا في حالة النهايات ال limits لل sequences هنا استخدمنا epsilon capital N definition هنا هنستخدم epsilon delta definition فمثلا يعني لو بالبداية to show", "tokens": [1829, 2407, 11622, 1863, 17889, 8289, 7123, 30767, 1829, 13672, 1829, 19446, 13412, 38688, 8315, 8978, 11331, 6027, 3660, 28239, 11296, 1829, 9307, 2423, 10406, 24976, 22978, 34105, 44713, 9778, 40448, 8315, 17889, 4238, 426, 7123, 34105, 8032, 1863, 14851, 9778, 40448, 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"end": 1197.0, "text": "لو بدأت بالجزء الأول مثلا بي show limit ل f plus g of x as x tends to c بساوي n plus m let epsilon أكبر من السفر be given since", "tokens": [1211, 2407, 47525, 10721, 2655, 20666, 7435, 11622, 38207, 16247, 12610, 50113, 15040, 4724, 1829, 855, 4948, 5296, 283, 1804, 290, 295, 2031, 382, 2031, 12258, 281, 269, 4724, 3794, 995, 45865, 297, 1804, 275, 718, 17889, 5551, 4117, 26890, 9154, 21136, 5172, 2288, 312, 2212, 1670], "avg_logprob": -0.36555990514655906, "compression_ratio": 1.2014388489208634, "no_speech_prob": 0.0, "words": [{"start": 1174.8, "end": 1175.08, "word": "لو", "probability": 0.5472412109375}, {"start": 1175.08, "end": 1175.38, "word": " بدأت", "probability": 0.605712890625}, {"start": 1175.38, "end": 1175.86, "word": " بالجزء", "probability": 0.80535888671875}, {"start": 1175.86, "end": 1176.26, "word": " الأول", "probability": 0.9345703125}, {"start": 1176.26, "end": 1176.7, "word": " مثلا", "probability": 0.867431640625}, {"start": 1176.7, 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to", "probability": 0.98291015625}, {"start": 1184.74, "end": 1185.02, "word": " c", "probability": 0.88525390625}, {"start": 1185.02, "end": 1185.8, "word": " بساوي", "probability": 0.838623046875}, {"start": 1185.8, "end": 1186.24, "word": " n", "probability": 0.497802734375}, {"start": 1186.24, "end": 1186.72, "word": " plus", "probability": 0.892578125}, {"start": 1186.72, "end": 1187.22, "word": " m", "probability": 0.450927734375}, {"start": 1187.22, "end": 1189.62, "word": " let", "probability": 0.69873046875}, {"start": 1189.62, "end": 1191.28, "word": " epsilon", "probability": 0.5390625}, {"start": 1191.28, "end": 1191.84, "word": " أكبر", "probability": 0.9404296875}, {"start": 1191.84, "end": 1192.1, "word": " من", "probability": 0.9853515625}, {"start": 1192.1, "end": 1192.66, "word": " السفر", "probability": 0.7188313802083334}, {"start": 1192.66, "end": 1192.86, "word": " be", "probability": 0.79296875}, {"start": 1192.86, "end": 1193.3, "word": " given", "probability": 0.8623046875}, {"start": 1193.3, "end": 1197.0, "word": " since", "probability": 0.85986328125}], "temperature": 1.0}, {"id": 44, "seek": 121620, "start": 1201.26, "end": 1216.2, "text": "لما يكون f of x as x tends to c بساوي L هنا هناك delta one تعتمد على أبسلان عدد موزة لكل x ينتمي إلى a", "tokens": [1211, 15042, 7251, 30544, 283, 295, 2031, 382, 2031, 3928, 82, 281, 269, 4724, 3794, 995, 45865, 441, 34105, 34105, 4117, 8289, 472, 6055, 34268, 2304, 3215, 15844, 5551, 3555, 3794, 1211, 7649, 6225, 3215, 3215, 3714, 2407, 11622, 3660, 5296, 28820, 2031, 7251, 29399, 2304, 1829, 30731, 257], "avg_logprob": -0.6837499952316284, "compression_ratio": 1.204724409448819, "no_speech_prob": 2.384185791015625e-07, "words": [{"start": 1201.2599999999998, "end": 1202.1599999999999, "word": "لما", "probability": 0.132110595703125}, {"start": 1202.1599999999999, "end": 1203.06, "word": " يكون", "probability": 0.3477783203125}, {"start": 1203.06, "end": 1203.22, "word": " f", "probability": 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1243.75, "text": "و absolute x minus c أصغر من الـ delta و 1 أكبر من 0 هذا بتضمن أن absolute f of x minus l أصغر من إبسلن أتنين هذا من تعريف epsilon delta ال element also أيضا since", "tokens": [2407, 8236, 2031, 3175, 269, 5551, 9381, 17082, 2288, 9154, 2423, 39184, 8289, 4032, 502, 5551, 4117, 26890, 9154, 1958, 23758, 39894, 11242, 27842, 14739, 8236, 283, 295, 2031, 3175, 287, 5551, 9381, 17082, 2288, 9154, 11933, 3555, 3794, 1211, 1863, 5551, 2655, 1863, 9957, 23758, 9154, 37279, 16572, 5172, 17889, 8289, 2423, 4478, 611, 36632, 11242, 995, 1670], "avg_logprob": -0.4541666716337204, "compression_ratio": 1.414012738853503, "no_speech_prob": 0.0, "words": [{"start": 1216.87, "end": 1217.23, "word": "و", "probability": 0.5185546875}, {"start": 1217.23, "end": 1217.77, "word": " absolute", "probability": 0.282958984375}, {"start": 1217.77, "end": 1218.31, "word": " x", "probability": 0.65771484375}, {"start": 1218.31, "end": 1218.77, "word": " minus", "probability": 0.420654296875}, {"start": 1218.77, "end": 1219.07, "word": " c", "probability": 0.52734375}, {"start": 1219.07, "end": 1219.55, "word": " أصغر", "probability": 0.943359375}, {"start": 1219.55, "end": 1219.75, "word": " من", "probability": 0.98779296875}, {"start": 1219.75, "end": 1219.87, "word": " الـ", "probability": 0.30389404296875}, {"start": 1219.87, "end": 1220.13, "word": " delta", "probability": 0.5673828125}, {"start": 1220.13, "end": 1220.33, "word": " و", "probability": 0.52392578125}, {"start": 1220.33, "end": 1220.53, "word": " 1", "probability": 0.2481689453125}, {"start": 1220.53, "end": 1221.03, "word": " أكبر", "probability": 0.9384765625}, {"start": 1221.03, "end": 1221.27, "word": " من", "probability": 0.98388671875}, {"start": 1221.27, "end": 1221.67, "word": " 0", "probability": 0.4755859375}, {"start": 1221.67, "end": 1222.95, "word": " هذا", "probability": 0.369140625}, {"start": 1222.95, "end": 1223.67, "word": " بتضمن", "probability": 0.83740234375}, {"start": 1223.67, "end": 1224.51, "word": " أن", "probability": 0.7548828125}, {"start": 1224.51, "end": 1225.13, "word": " absolute", "probability": 0.7353515625}, {"start": 1225.13, "end": 1225.75, "word": " f", "probability": 0.7626953125}, {"start": 1225.75, "end": 1226.05, "word": " of", "probability": 0.67919921875}, {"start": 1226.05, "end": 1226.45, "word": " x", "probability": 0.9736328125}, {"start": 1226.45, "end": 1227.09, "word": " minus", "probability": 0.93798828125}, {"start": 1227.09, "end": 1227.61, "word": " l", "probability": 0.5322265625}, {"start": 1227.61, "end": 1228.93, "word": " أصغر", "probability": 0.9864501953125}, {"start": 1228.93, "end": 1229.15, "word": " من", "probability": 0.9921875}, {"start": 1229.15, "end": 1229.75, "word": " إبسلن", "probability": 0.541552734375}, {"start": 1229.75, "end": 1230.41, "word": " أتنين", "probability": 0.5361328125}, {"start": 1230.41, "end": 1235.43, "word": " هذا", "probability": 0.22509765625}, {"start": 1235.43, "end": 1235.81, "word": " من", "probability": 0.9208984375}, {"start": 1235.81, "end": 1236.45, "word": " تعريف", "probability": 0.9884440104166666}, {"start": 1236.45, "end": 1237.85, "word": " epsilon", "probability": 0.285400390625}, {"start": 1237.85, "end": 1238.21, "word": " delta", "probability": 0.57958984375}, {"start": 1238.21, "end": 1238.45, "word": " ال", "probability": 0.38623046875}, {"start": 1238.45, "end": 1238.81, "word": " element", "probability": 0.441650390625}, {"start": 1238.81, "end": 1240.47, "word": " also", "probability": 0.493896484375}, {"start": 1240.47, "end": 1241.09, "word": " أيضا", "probability": 0.8914388020833334}, {"start": 1241.09, "end": 1243.75, "word": " since", "probability": 0.70361328125}], "temperature": 1.0}, {"id": 46, "seek": 126715, "start": 1246.39, "end": 1267.15, "text": "أنا عندي فرد من limit لل function g of x as x tends to c exist و بيسوي عدد حقيقي M فليه for the same epsilon for the same given epsilon من تعريف ال limit يوجد Delta 2", "tokens": [10721, 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30767, 16373, 3215, 8236, 290, 295, 2031, 3175, 376, 3714, 19913, 3555, 2407, 9566, 3660, 6024, 48506, 4724, 995, 14851, 9778, 3215, 10943, 2423, 39184, 3543], "avg_logprob": -0.3301504530288555, "compression_ratio": 1.3194444444444444, "no_speech_prob": 0.0, "words": [{"start": 1397.94, "end": 1398.78, "word": "وباستخدام", "probability": 0.8567708333333334}, {"start": 1398.78, "end": 1398.98, "word": " الـ", "probability": 0.340087890625}, {"start": 1398.98, "end": 1399.4, "word": " triangle", "probability": 0.32958984375}, {"start": 1399.4, "end": 1400.18, "word": " inequality", "probability": 0.97021484375}, {"start": 1400.18, "end": 1400.56, "word": " هذا", "probability": 0.6767578125}, {"start": 1400.56, "end": 1401.06, "word": " أصغر", "probability": 0.9498291015625}, {"start": 1401.06, "end": 1401.22, "word": " من", "probability": 0.93994140625}, {"start": 1401.22, "end": 1401.52, "word": " أوي", "probability": 0.468994140625}, {"start": 1401.52, "end": 1402.06, "word": " ساوي", 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"end": 1446.68, "word": " value", "probability": 0.9931640625}, {"start": 1446.68, "end": 1446.8, "word": " ل", "probability": 0.86865234375}, {"start": 1446.8, "end": 1446.96, "word": " X", "probability": 0.51220703125}, {"start": 1446.96, "end": 1447.3, "word": " minus", "probability": 0.6318359375}, {"start": 1447.3, "end": 1447.64, "word": " C", "probability": 0.84228515625}, {"start": 1447.64, "end": 1448.24, "word": " أصغر", "probability": 0.9942626953125}, {"start": 1448.24, "end": 1448.44, "word": " من", "probability": 0.99560546875}, {"start": 1448.44, "end": 1448.82, "word": " delta", "probability": 0.85009765625}, {"start": 1448.82, "end": 1449.58, "word": " واحد", "probability": 0.94140625}, {"start": 1449.58, "end": 1450.26, "word": " أصغر", "probability": 0.936767578125}, {"start": 1450.26, "end": 1450.42, "word": " من", "probability": 0.99755859375}, {"start": 1450.42, "end": 1450.7, "word": " delta", "probability": 0.87744140625}, {"start": 1450.7, "end": 1451.24, 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"end": 1510.5, "word": " هي", "probability": 0.2059326171875}, {"start": 1510.5, "end": 1511.42, "word": " ايه", "probability": 0.8080240885416666}, {"start": 1511.42, "end": 1512.5, "word": " وabsolute", "probability": 0.72125244140625}, {"start": 1512.5, "end": 1512.78, "word": " x", "probability": 0.93701171875}, {"start": 1512.78, "end": 1513.18, "word": " minus", "probability": 0.80419921875}, {"start": 1513.18, "end": 1513.46, "word": " c", "probability": 0.67529296875}, {"start": 1513.46, "end": 1513.84, "word": " اكبر", "probability": 0.8907877604166666}, {"start": 1513.84, "end": 1514.0, "word": " من", "probability": 0.9951171875}, {"start": 1514.0, "end": 1514.32, "word": " سفر", "probability": 0.9568684895833334}, {"start": 1514.32, "end": 1514.76, "word": " اصغر", "probability": 0.897705078125}, {"start": 1514.76, "end": 1514.92, "word": " من", "probability": 0.99365234375}, {"start": 1514.92, "end": 1515.46, "word": " الدلتا", "probability": 0.80877685546875}, {"start": 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minus", "probability": 0.85986328125}, {"start": 1525.53, "end": 1525.93, "word": " ال", "probability": 0.82958984375}, {"start": 1525.93, "end": 1526.19, "word": " limit", "probability": 0.9580078125}, {"start": 1526.19, "end": 1527.11, "word": " المقترحها", "probability": 0.840625}, {"start": 1527.11, "end": 1528.27, "word": " أصغر", "probability": 0.9315185546875}, {"start": 1528.27, "end": 1528.53, "word": " من", "probability": 0.99462890625}, {"start": 1528.53, "end": 1528.97, "word": " epsilon", "probability": 0.51953125}, {"start": 1528.97, "end": 1532.15, "word": " طبعا؟", "probability": 0.75908203125}, {"start": 1532.15, "end": 1532.93, "word": " اذا", "probability": 0.6273193359375}, {"start": 1532.93, "end": 1533.39, "word": " حسب", "probability": 0.9853515625}, {"start": 1533.39, "end": 1535.13, "word": " التعريف", "probability": 0.98681640625}, {"start": 1535.13, "end": 1536.15, "word": " بما", "probability": 0.842529296875}, {"start": 1536.15, "end": 1536.29, "word": " 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1884.03, "end": 1884.47, "word": " n", "probability": 0.5986328125}, {"start": 1884.47, "end": 1884.73, "word": " tends", "probability": 0.41552734375}, {"start": 1884.73, "end": 1884.93, "word": " to", "probability": 0.90771484375}, {"start": 1884.93, "end": 1885.41, "word": " infinity", "probability": 0.93359375}, {"start": 1885.41, "end": 1886.77, "word": " بساوي", "probability": 0.7618408203125}, {"start": 1886.77, "end": 1889.81, "word": " ال", "probability": 0.6025390625}, {"start": 1889.81, "end": 1890.19, "word": " limit", "probability": 0.9814453125}], "temperature": 1.0}, {"id": 70, "seek": 191140, "start": 1891.7, "end": 1911.4, "text": "حاصل ضرب two functions عند أي x أو xn هذا عبارة عن f of xn ضرب g of xn لأن هذا من تعريف حاصل ضرب two functions الآن هذه عبارة عن sequence وهذه عبارة عن sequence", "tokens": [5016, 33546, 1211, 48812, 25513, 732, 6828, 43242, 36632, 2031, 34051, 2031, 77, 23758, 6225, 3555, 9640, 3660, 18871, 283, 295, 2031, 77, 48812, 25513, 290, 295, 2031, 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"probability": 0.92138671875}, {"start": 1902.92, "end": 1903.18, "word": " من", "probability": 0.96533203125}, {"start": 1903.18, "end": 1903.88, "word": " تعريف", "probability": 0.9275716145833334}, {"start": 1903.88, "end": 1904.72, "word": " حاصل", "probability": 0.93212890625}, {"start": 1904.72, "end": 1904.96, "word": " ضرب", "probability": 0.996337890625}, {"start": 1904.96, "end": 1905.16, "word": " two", "probability": 0.94677734375}, {"start": 1905.16, "end": 1905.74, "word": " functions", "probability": 0.9091796875}, {"start": 1905.74, "end": 1907.76, "word": " الآن", "probability": 0.5892333984375}, {"start": 1907.76, "end": 1908.14, "word": " هذه", "probability": 0.422607421875}, {"start": 1908.14, "end": 1908.58, "word": " عبارة", "probability": 0.9969482421875}, {"start": 1908.58, "end": 1908.82, "word": " عن", "probability": 0.99755859375}, {"start": 1908.82, "end": 1909.6, "word": " sequence", "probability": 0.95263671875}, {"start": 1909.6, "end": 1910.12, "word": " وهذه", "probability": 0.79833984375}, {"start": 1910.12, "end": 1910.52, "word": " عبارة", "probability": 0.9951171875}, {"start": 1910.52, "end": 1910.74, "word": " عن", "probability": 0.98779296875}, {"start": 1910.74, "end": 1911.4, "word": " sequence", "probability": 0.93798828125}], "temperature": 1.0}, {"id": 71, "seek": 193130, "start": 1912.76, "end": 1931.3, "text": "و limit الـ sequence الأولى exist و limit الـ sequence التانية exist إذا by limit theorems لsequences إن ال limit حاصل الضرب بساوي حاصل ضرب ال limits فهذا بساوي limit f of xn ضرب", "tokens": [2407, 4948, 2423, 39184, 8310, 16247, 12610, 7578, 2514, 4032, 4948, 2423, 39184, 8310, 16712, 7649, 10632, 2514, 11933, 15730, 538, 4948, 10299, 2592, 5296, 11834, 2667, 36145, 2423, 4948, 11331, 33546, 1211, 6024, 114, 25513, 4724, 3794, 995, 45865, 11331, 33546, 1211, 48812, 25513, 2423, 10406, 6156, 3224, 15730, 4724, 3794, 995, 45865, 4948, 283, 295, 2031, 77, 48812, 25513], "avg_logprob": -0.35483869717967126, 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ال limit الأخيرة هدى بالساوي L by sequential criterion اذا كانت limit f of x اما x او لا c موجودة وبالساوي L واندي فيه sequence نهايتها C فنهاية صورة ال sequence under F بالساوي L وكذلك", "tokens": [9566, 1829, 3555, 19446, 8032, 3215, 7578, 2423, 4948, 16247, 9778, 48923, 8032, 3215, 7578, 20666, 3794, 995, 45865, 441, 538, 42881, 46691, 1975, 15730, 25961, 2655, 4948, 283, 295, 2031, 1975, 15042, 2031, 1975, 2407, 20193, 269, 3714, 29245, 23328, 3660, 46599, 6027, 3794, 995, 45865, 441, 4032, 7649, 16254, 8978, 3224, 8310, 8717, 11296, 36081, 11296, 383, 6156, 1863, 11296, 10632, 20328, 13063, 3660, 2423, 8310, 833, 479, 20666, 3794, 995, 45865, 441, 4032, 4117, 8848, 23275], "avg_logprob": -0.3126953214406967, "compression_ratio": 1.5675675675675675, "no_speech_prob": 0.0, "words": [{"start": 2032.71, "end": 2033.07, "word": "طيب", "probability": 0.6697591145833334}, {"start": 2033.07, "end": 2033.27, "word": " ما", "probability": 0.5927734375}, {"start": 2033.27, "end": 2033.77, 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"end": 2043.93, "word": " limit", "probability": 0.9697265625}, {"start": 2043.93, "end": 2044.23, "word": " f", "probability": 0.6123046875}, {"start": 2044.23, "end": 2044.43, "word": " of", "probability": 0.6015625}, {"start": 2044.43, "end": 2044.69, "word": " x", "probability": 0.76806640625}, {"start": 2044.69, "end": 2044.85, "word": " اما", "probability": 0.417724609375}, {"start": 2044.85, "end": 2045.05, "word": " x", "probability": 0.60595703125}, {"start": 2045.05, "end": 2045.25, "word": " او", "probability": 0.524139404296875}, {"start": 2045.25, "end": 2045.45, "word": " لا", "probability": 0.14111328125}, {"start": 2045.45, "end": 2045.69, "word": " c", "probability": 0.5771484375}, {"start": 2045.69, "end": 2046.35, "word": " موجودة", "probability": 0.9658203125}, {"start": 2046.35, "end": 2046.89, "word": " وبالساوي", "probability": 0.830224609375}, {"start": 2046.89, "end": 2047.17, "word": " L", "probability": 0.94287109375}, {"start": 2047.17, "end": 2048.19, "word": " واندي", "probability": 0.7556966145833334}, {"start": 2048.19, "end": 2048.43, "word": " فيه", "probability": 0.719970703125}, {"start": 2048.43, "end": 2048.89, "word": " sequence", "probability": 0.98046875}, {"start": 2048.89, "end": 2049.47, "word": " نهايتها", "probability": 0.878662109375}, {"start": 2049.47, "end": 2049.79, "word": " C", "probability": 0.63037109375}, {"start": 2049.79, "end": 2051.15, "word": " فنهاية", "probability": 0.9263916015625}, {"start": 2051.15, "end": 2051.69, "word": " صورة", "probability": 0.9612630208333334}, {"start": 2051.69, "end": 2051.85, "word": " ال", "probability": 0.95947265625}, {"start": 2051.85, "end": 2052.27, "word": " sequence", "probability": 0.99169921875}, {"start": 2052.27, "end": 2052.67, "word": " under", "probability": 0.84375}, {"start": 2052.67, "end": 2053.09, "word": " F", "probability": 0.51416015625}, {"start": 2053.09, "end": 2053.85, "word": " بالساوي", "probability": 0.94140625}, {"start": 2053.85, "end": 2054.15, "word": " L", "probability": 0.9912109375}, {"start": 2054.15, "end": 2055.97, "word": " وكذلك", "probability": 0.9114990234375}], "temperature": 1.0}, {"id": 77, "seek": 208365, "start": 2056.93, "end": 2083.65, "text": "نفس الحاجة بما ان limit g and c موجودة و بيساوي m و xn نهايتها c اذا by sequential criterion limit g سورة ال xn under g بيساوي m اذا هذا by sequential criterion وهذا اللي احنا عايزين نثبته", "tokens": [1863, 36178, 21542, 26108, 3660, 4724, 15042, 16472, 4948, 290, 293, 269, 3714, 29245, 23328, 3660, 4032, 4724, 1829, 3794, 995, 45865, 275, 4032, 2031, 77, 8717, 11296, 36081, 11296, 269, 1975, 15730, 538, 42881, 46691, 4948, 290, 8608, 13063, 3660, 2423, 2031, 77, 833, 290, 4724, 1829, 3794, 995, 45865, 275, 1975, 15730, 23758, 538, 42881, 46691, 37037, 15730, 13672, 1829, 1975, 5016, 8315, 6225, 47302, 11622, 9957, 8717, 12984, 3555, 2655, 3224], "avg_logprob": -0.3045833190282186, "compression_ratio": 1.5798816568047338, "no_speech_prob": 0.0, "words": 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{"start": 2061.33, "end": 2061.49, "word": " و", "probability": 0.8076171875}, {"start": 2061.49, "end": 2062.13, "word": " xn", "probability": 0.667724609375}, {"start": 2062.13, "end": 2063.55, "word": " نهايتها", "probability": 0.85009765625}, {"start": 2063.55, "end": 2063.89, "word": " c", "probability": 0.759765625}, {"start": 2063.89, "end": 2064.47, "word": " اذا", "probability": 0.55706787109375}, {"start": 2064.47, "end": 2064.73, "word": " by", "probability": 0.744140625}, {"start": 2064.73, "end": 2065.25, "word": " sequential", "probability": 0.93603515625}, {"start": 2065.25, "end": 2065.97, "word": " criterion", "probability": 0.92236328125}, {"start": 2065.97, "end": 2066.89, "word": " limit", "probability": 0.83203125}, {"start": 2066.89, "end": 2067.37, "word": " g", "probability": 0.8828125}, {"start": 2067.37, "end": 2068.01, "word": " سورة", "probability": 0.7222086588541666}, {"start": 2068.01, "end": 2068.25, "word": " ال", "probability": 0.436279296875}, {"start": 2068.25, "end": 2068.85, "word": " xn", "probability": 0.84375}, {"start": 2068.85, "end": 2069.33, "word": " under", "probability": 0.79052734375}, {"start": 2069.33, "end": 2069.83, "word": " g", "probability": 0.943359375}, {"start": 2069.83, "end": 2071.17, "word": " بيساوي", "probability": 0.975}, {"start": 2071.17, "end": 2071.63, "word": " m", "probability": 0.83642578125}, {"start": 2071.63, "end": 2072.55, "word": " اذا", "probability": 0.755126953125}, {"start": 2072.55, "end": 2072.93, "word": " هذا", "probability": 0.7373046875}, {"start": 2072.93, "end": 2073.43, "word": " by", "probability": 0.8974609375}, {"start": 2073.43, "end": 2074.55, "word": " sequential", "probability": 0.9755859375}, {"start": 2074.55, "end": 2076.51, "word": " criterion", "probability": 0.93701171875}, {"start": 2076.51, "end": 2082.41, "word": " وهذا", "probability": 0.8193359375}, {"start": 2082.41, "end": 2082.55, "word": " اللي", "probability": 0.941650390625}, {"start": 2082.55, "end": 2082.69, "word": " احنا", "probability": 0.9480794270833334}, {"start": 2082.69, "end": 2083.03, "word": " عايزين", "probability": 0.935791015625}, {"start": 2083.03, "end": 2083.65, "word": " نثبته", "probability": 0.71761474609375}], "temperature": 1.0}, {"id": 78, "seek": 210911, "start": 2088.11, "end": 2109.11, "text": "عايزين نثبت ان limit f ضارب g او ال image لسيكوينس x in under ال function f ضارب g بالساوية ال M هاي بدينا ب limit ال image لسيكوينس x in under f g وطلعت هي بالساوية ال M اذا by sequential", "tokens": [3615, 47302, 11622, 9957, 8717, 12984, 3555, 2655, 16472, 4948, 283, 48812, 9640, 3555, 290, 1975, 2407, 2423, 3256, 5296, 3794, 1829, 4117, 2407, 9957, 3794, 2031, 294, 833, 2423, 2445, 283, 48812, 9640, 3555, 290, 20666, 3794, 995, 2407, 10632, 2423, 376, 8032, 47302, 4724, 16254, 8315, 4724, 4948, 2423, 3256, 5296, 3794, 1829, 4117, 2407, 9957, 3794, 2031, 294, 833, 283, 290, 4032, 9566, 1211, 34268, 39896, 20666, 3794, 995, 2407, 10632, 2423, 376, 1975, 15730, 538, 42881], "avg_logprob": -0.36651234567901236, "compression_ratio": 1.6728395061728396, "no_speech_prob": 0.0, "words": [{"start": 2088.1100000000006, "end": 2088.8300000000004, "word": "عايزين", "probability": 0.56134033203125}, {"start": 2088.8300000000004, "end": 2089.55, "word": " نثبت", "probability": 0.9332275390625}, {"start": 2089.55, "end": 2090.09, "word": " ان", "probability": 0.490478515625}, {"start": 2090.09, "end": 2090.55, "word": " limit", "probability": 0.27099609375}, {"start": 2090.55, "end": 2091.17, "word": " f", "probability": 0.498779296875}, {"start": 2091.17, "end": 2091.71, "word": " ضارب", "probability": 0.6751302083333334}, {"start": 2091.71, "end": 2091.95, "word": " g", "probability": 0.78955078125}, {"start": 2091.95, "end": 2092.65, "word": " او", "probability": 0.671142578125}, {"start": 2092.65, "end": 2092.75, "word": " ال", "probability": 0.63037109375}, {"start": 2092.75, "end": 2092.99, "word": " image", "probability": 0.7109375}, 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2097.55, "end": 2097.89, "word": " هاي", "probability": 0.57513427734375}, {"start": 2097.89, "end": 2098.27, "word": " بدينا", "probability": 0.7545572916666666}, {"start": 2098.27, "end": 2098.51, "word": " ب", "probability": 0.8671875}, {"start": 2098.51, "end": 2099.03, "word": " limit", "probability": 0.93212890625}, {"start": 2099.03, "end": 2099.77, "word": " ال", "probability": 0.890625}, {"start": 2099.77, "end": 2100.01, "word": " image", "probability": 0.9306640625}, {"start": 2100.01, "end": 2100.67, "word": " لسيكوينس", "probability": 0.9591238839285714}, {"start": 2100.67, "end": 2100.87, "word": " x", "probability": 0.91552734375}, {"start": 2100.87, "end": 2101.13, "word": " in", "probability": 0.892578125}, {"start": 2101.13, "end": 2101.45, "word": " under", "probability": 0.8544921875}, {"start": 2101.45, "end": 2101.75, "word": " f", "probability": 0.73486328125}, {"start": 2101.75, "end": 2102.05, "word": " g", "probability": 0.78369140625}, {"start": 2102.05, "end": 2102.89, "word": " وطلعت", "probability": 0.73211669921875}, {"start": 2102.89, "end": 2103.09, "word": " هي", "probability": 0.474853515625}, {"start": 2103.09, "end": 2103.61, "word": " بالساوية", "probability": 0.890625}, {"start": 2103.61, "end": 2103.85, "word": " ال", "probability": 0.814453125}, {"start": 2103.85, "end": 2104.63, "word": " M", "probability": 0.98828125}, {"start": 2104.63, "end": 2107.11, "word": " اذا", "probability": 0.58380126953125}, {"start": 2107.11, "end": 2108.33, "word": " by", "probability": 0.72314453125}, {"start": 2108.33, "end": 2109.11, "word": " sequential", "probability": 0.91796875}], "temperature": 1.0}, {"id": 79, "seek": 213751, "start": 2126.03, "end": 2137.51, "text": "البرهين الأجزاء الأخرى مماثلة ال proof of the other parts", "tokens": [6027, 26890, 3224, 9957, 16247, 7435, 11622, 16606, 16247, 34740, 7578, 3714, 2304, 5718, 104, 37977, 2423, 8177, 295, 264, 661, 3166], "avg_logprob": -0.4551630512527797, "compression_ratio": 1.0897435897435896, "no_speech_prob": 0.0, "words": [{"start": 2126.0299999999997, "end": 2127.35, "word": "البرهين", "probability": 0.58258056640625}, {"start": 2127.35, "end": 2127.85, "word": " الأجزاء", "probability": 0.715484619140625}, {"start": 2127.85, "end": 2128.31, "word": " الأخرى", "probability": 0.9580078125}, {"start": 2128.31, "end": 2129.05, "word": " مماثلة", "probability": 0.78173828125}, {"start": 2129.05, "end": 2130.49, "word": " ال", "probability": 0.49072265625}, {"start": 2130.49, "end": 2131.07, "word": " proof", "probability": 0.118408203125}, {"start": 2131.07, "end": 2133.65, "word": " of", "probability": 0.88671875}, {"start": 2133.65, "end": 2134.05, "word": " the", "probability": 0.7255859375}, {"start": 2134.05, "end": 2134.53, "word": " other", "probability": 0.8349609375}, {"start": 2134.53, "end": 2137.51, "word": " parts", "probability": 0.85791015625}], "temperature": 1.0}, {"id": 80, "seek": 216334, "start": 2140.0, "end": 2163.34, "text": "is similar مشابه exercise it اتمرن عليهم اتمرن علي كتابة البرهيم اكتبوهم okay تمام إذا مافي عندي نهائي من القرآن لlimits", "tokens": [271, 2531, 37893, 16758, 3224, 5380, 309, 1975, 2655, 29973, 1863, 25894, 16095, 1975, 2655, 29973, 1863, 25894, 9122, 2655, 16758, 3660, 2423, 26890, 3224, 32640, 1975, 4117, 2655, 3555, 2407, 16095, 1392, 46811, 10943, 11933, 15730, 19446, 41185, 18871, 16254, 8717, 3224, 16373, 1829, 9154, 25062, 2288, 148, 48506, 5296, 4197, 1208], "avg_logprob": -0.2693865801449175, "compression_ratio": 1.3642857142857143, "no_speech_prob": 0.0, "words": [{"start": 2140.0, "end": 2140.7, "word": "is", "probability": 0.93017578125}, {"start": 2140.7, "end": 2142.64, "word": " similar", "probability": 0.9638671875}, {"start": 2142.64, "end": 2144.8, "word": " مشابه", "probability": 0.86083984375}, {"start": 2144.8, "end": 2147.22, "word": " exercise", "probability": 0.5078125}, {"start": 2147.22, "end": 2150.56, "word": " it", "probability": 0.92626953125}, {"start": 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{"start": 2360.06, "end": 2360.44, "word": " on", "probability": 0.87646484375}, {"start": 2360.44, "end": 2360.76, "word": " M", "probability": 0.38671875}, {"start": 2360.76, "end": 2367.94, "word": " إذاً", "probability": 0.4449462890625}, {"start": 2367.94, "end": 2368.38, "word": " to", "probability": 0.3857421875}, {"start": 2368.38, "end": 2368.98, "word": " prove", "probability": 0.935546875}, {"start": 2368.98, "end": 2371.38, "word": " one", "probability": 0.6337890625}, {"start": 2371.38, "end": 2371.94, "word": " and", "probability": 0.970703125}, {"start": 2371.94, "end": 2372.82, "word": " two", "probability": 0.95556640625}, {"start": 2372.82, "end": 2374.48, "word": " use", "probability": 0.60888671875}, {"start": 2374.48, "end": 2375.18, "word": " above", "probability": 0.958984375}, {"start": 2375.18, "end": 2376.4, "word": " theorem", "probability": 0.8388671875}, {"start": 2376.4, "end": 2378.58, "word": " and", "probability": 0.9111328125}, {"start": 2378.58, "end": 2379.28, "word": " induction", "probability": 0.958984375}], "temperature": 1.0}, {"id": 89, "seek": 239684, "start": 2382.28, "end": 2396.84, "text": "invection on edge هنا في اثبات الجزء التالت to prove تلاتة", "tokens": [259, 303, 882, 322, 4691, 34105, 8978, 1975, 12984, 3555, 9307, 25724, 11622, 38207, 16712, 6027, 2655, 281, 7081, 6055, 1211, 9307, 3660], "avg_logprob": -0.523763008415699, "compression_ratio": 1.05, "no_speech_prob": 0.0, "words": [{"start": 2382.28, "end": 2383.2, "word": "invection", "probability": 0.3661397298177083}, {"start": 2383.2, "end": 2383.58, "word": " on", "probability": 0.6103515625}, {"start": 2383.58, "end": 2384.7, "word": " edge", "probability": 0.367919921875}, {"start": 2384.7, "end": 2389.14, "word": " هنا", "probability": 0.088623046875}, {"start": 2389.14, "end": 2389.32, "word": " في", "probability": 0.8857421875}, {"start": 2389.32, "end": 2389.72, "word": " اثبات", "probability": 0.84930419921875}, {"start": 2389.72, "end": 2390.1, "word": " الجزء", "probability": 0.8953450520833334}, {"start": 2390.1, "end": 2390.68, "word": " التالت", "probability": 0.9554036458333334}, {"start": 2390.68, "end": 2392.4, "word": " to", "probability": 0.2327880859375}, {"start": 2392.4, "end": 2393.0, "word": " prove", "probability": 0.9501953125}, {"start": 2393.0, "end": 2396.84, "word": " تلاتة", "probability": 0.8795166015625}], "temperature": 1.0}, {"id": 90, "seek": 242342, "start": 2400.32, "end": 2423.42, "text": "أخذ F1 بساوي F2 بساوي Fn بساوي F in part اتنين to get the result", "tokens": [10721, 9778, 8848, 479, 16, 4724, 3794, 995, 45865, 479, 17, 4724, 3794, 995, 45865, 479, 77, 4724, 3794, 995, 45865, 479, 294, 644, 1975, 2655, 1863, 9957, 281, 483, 264, 1874], "avg_logprob": -0.37642045454545453, "compression_ratio": 1.16, "no_speech_prob": 0.0, "words": [{"start": 2400.32, "end": 2400.9, "word": "أخذ", "probability": 0.3780517578125}, {"start": 2400.9, "end": 2403.04, "word": " F1", "probability": 0.5418701171875}, {"start": 2403.04, "end": 2404.08, "word": " بساوي", "probability": 0.635467529296875}, {"start": 2404.08, "end": 2405.04, "word": " F2", "probability": 0.89794921875}, {"start": 2405.04, "end": 2405.9, "word": " بساوي", "probability": 0.9620361328125}, {"start": 2405.9, "end": 2407.84, "word": " Fn", "probability": 0.733154296875}, {"start": 2407.84, "end": 2408.76, "word": " بساوي", "probability": 0.954345703125}, {"start": 2408.76, "end": 2409.26, "word": " F", "probability": 0.78125}, {"start": 2409.26, "end": 2410.56, "word": " in", "probability": 0.416259765625}, {"start": 2410.56, "end": 2411.32, "word": " part", "probability": 0.94384765625}, {"start": 2411.32, "end": 2413.96, "word": " اتنين", "probability": 0.733642578125}, {"start": 2413.96, "end": 2417.96, "word": " to", "probability": 0.194091796875}, {"start": 2417.96, "end": 2418.4, "word": " get", "probability": 0.953125}, {"start": 2418.4, "end": 2422.96, "word": " the", "probability": 0.76708984375}, {"start": 2422.96, "end": 2423.42, "word": " result", "probability": 0.9716796875}], "temperature": 1.0}, {"id": 91, "seek": 245931, "start": 2430.71, "end": 2459.31, "text": "يعني لدرهان الكلام هذا إذا كانت limit F عن C بساوي L فعوضي او خدي F واحد هي F و F اتنين F و كلهم بساوي F فكأن إذا هنا limit F to the power N and X هيطلع بساوي L مضروبة في نفس M المرات اللي هو إذا هذا الجزء التالت corollary على الجزء التاني", "tokens": [40228, 22653, 5296, 3215, 2288, 3224, 7649, 2423, 28820, 10943, 23758, 11933, 15730, 25961, 2655, 4948, 479, 18871, 383, 4724, 3794, 995, 45865, 441, 6156, 3615, 2407, 11242, 1829, 1975, 2407, 16490, 16254, 479, 36764, 24401, 39896, 479, 4032, 479, 1975, 2655, 1863, 9957, 479, 4032, 28242, 16095, 4724, 3794, 995, 45865, 479, 6156, 4117, 33456, 11933, 15730, 34105, 4948, 479, 281, 264, 1347, 426, 293, 1783, 39896, 9566, 1211, 3615, 4724, 3794, 995, 45865, 441, 3714, 11242, 32887, 49401, 8978, 8717, 36178, 376, 9673, 2288, 9307, 13672, 1829, 31439, 11933, 15730, 23758, 25724, 11622, 38207, 16712, 6027, 2655, 1181, 1833, 822, 15844, 25724, 11622, 38207, 16712, 7649, 1829], "avg_logprob": -0.19502841478044336, "compression_ratio": 1.6506550218340612, "no_speech_prob": 5.960464477539063e-08, "words": [{"start": 2430.71, "end": 2431.19, "word": "يعني", "probability": 0.8564453125}, {"start": 2431.19, "end": 2432.01, "word": " لدرهان", "probability": 0.697119140625}, {"start": 2432.01, "end": 2432.51, "word": " الكلام", "probability": 0.9195963541666666}, {"start": 2432.51, "end": 2432.81, "word": " هذا", "probability": 0.92919921875}, {"start": 2432.81, "end": 2433.03, "word": " إذا", "probability": 0.785888671875}, {"start": 2433.03, "end": 2433.39, "word": " كانت", "probability": 0.95263671875}, {"start": 2433.39, "end": 2433.63, "word": " limit", "probability": 0.8876953125}, {"start": 2433.63, "end": 2433.99, "word": " F", "probability": 0.23828125}, {"start": 2433.99, "end": 2434.27, "word": " عن", "probability": 0.54638671875}, {"start": 2434.27, "end": 2434.97, "word": " C", "probability": 0.79833984375}, {"start": 2434.97, "end": 2435.51, "word": " بساوي", "probability": 0.68341064453125}, {"start": 2435.51, "end": 2435.81, "word": " L", "probability": 0.9208984375}, {"start": 2435.81, "end": 2438.51, "word": " فعوضي", "probability": 0.9125}, {"start": 2438.51, "end": 2438.69, "word": " او", "probability": 0.78125}, {"start": 2438.69, "end": 2438.97, "word": " خدي", "probability": 0.64306640625}, {"start": 2438.97, "end": 2439.19, "word": " F", "probability": 0.92578125}, {"start": 2439.19, "end": 2439.61, "word": " واحد", "probability": 0.8662109375}, {"start": 2439.61, "end": 2439.81, "word": " هي", "probability": 0.8349609375}, {"start": 2439.81, "end": 2440.13, "word": " F", "probability": 0.96044921875}, {"start": 2440.13, "end": 2440.25, "word": " و", "probability": 0.92431640625}, {"start": 2440.25, "end": 2440.41, "word": " F", "probability": 0.83349609375}, {"start": 2440.41, "end": 2440.79, "word": " اتنين", "probability": 0.865234375}, {"start": 2440.79, "end": 2441.13, "word": " F", "probability": 0.712890625}, {"start": 2441.13, "end": 2441.85, "word": " و", "probability": 0.9140625}, {"start": 2441.85, "end": 2442.33, "word": " كلهم", "probability": 0.82568359375}, {"start": 2442.33, "end": 2443.39, "word": " بساوي", "probability": 0.8773193359375}, {"start": 2443.39, "end": 2443.75, "word": " F", "probability": 0.98876953125}, {"start": 2443.75, "end": 2445.93, "word": " فكأن", "probability": 0.79931640625}, {"start": 2445.93, "end": 2446.23, "word": " إذا", "probability": 0.657470703125}, {"start": 2446.23, "end": 2446.51, "word": " هنا", "probability": 0.93701171875}, {"start": 2446.51, "end": 2447.49, "word": " limit", "probability": 0.943359375}, {"start": 2447.49, "end": 2448.05, "word": " F", "probability": 0.95654296875}, {"start": 2448.05, "end": 2448.79, "word": " to", "probability": 0.92724609375}, {"start": 2448.79, "end": 2448.93, "word": " the", "probability": 0.9208984375}, {"start": 2448.93, "end": 2449.27, "word": " power", "probability": 0.87451171875}, {"start": 2449.27, "end": 2449.61, "word": " N", "probability": 0.748046875}, {"start": 2449.61, "end": 2450.17, "word": " and", "probability": 0.89892578125}, {"start": 2450.17, "end": 2450.57, "word": " X", "probability": 0.98095703125}, {"start": 2450.57, "end": 2451.89, "word": " هيطلع", "probability": 0.89892578125}, {"start": 2451.89, "end": 2452.41, "word": " بساوي", "probability": 0.9447021484375}, {"start": 2452.41, "end": 2452.71, "word": " L", "probability": 0.98486328125}, {"start": 2452.71, "end": 2453.19, "word": " مضروبة", "probability": 0.847900390625}, {"start": 2453.19, "end": 2453.35, "word": " في", "probability": 0.9462890625}, {"start": 2453.35, "end": 2453.59, "word": " نفس", "probability": 0.99462890625}, {"start": 2453.59, "end": 2453.95, "word": " M", "probability": 0.59130859375}, {"start": 2453.95, "end": 2454.57, "word": " المرات", "probability": 0.8313802083333334}, {"start": 2454.57, "end": 2454.73, "word": " اللي", "probability": 0.6759033203125}, {"start": 2454.73, "end": 2454.89, "word": " هو", "probability": 0.99267578125}, {"start": 2454.89, "end": 2455.63, "word": " إذا", "probability": 0.7059326171875}, {"start": 2455.63, "end": 2456.05, "word": " هذا", "probability": 0.6259765625}, {"start": 2456.05, "end": 2456.49, "word": " الجزء", "probability": 0.9576822916666666}, {"start": 2456.49, "end": 2456.95, "word": " التالت", "probability": 0.90185546875}, {"start": 2456.95, "end": 2457.57, "word": " corollary", "probability": 0.8504231770833334}, {"start": 2457.57, "end": 2458.39, "word": " على", "probability": 0.7421875}, {"start": 2458.39, "end": 2458.79, "word": " الجزء", "probability": 0.9910481770833334}, {"start": 2458.79, "end": 2459.31, "word": " التاني", "probability": 0.7857259114583334}], "temperature": 1.0}, {"id": 92, "seek": 248360, "start": 2460.54, "end": 2483.6, "text": "Okay تمام؟ Okay إذا بنوقف هنا وإن شاء الله المرة الجاية هنأكل أمثلة على المظريات هذه ونحاول ناخد قصارى أخرى لل limits of functions إذا نكتفي بهذا القدر ونشوفكم إن شاء الله يوم الأثنين", "tokens": [8297, 46811, 10943, 22807, 1033, 11933, 15730, 44945, 30543, 5172, 34105, 4032, 28814, 1863, 13412, 16606, 21984, 9673, 25720, 25724, 995, 10632, 8032, 1863, 10721, 28820, 5551, 2304, 12984, 37977, 15844, 9673, 19913, 16572, 9307, 29538, 4032, 1863, 5016, 995, 12610, 8717, 47283, 3215, 12174, 9381, 9640, 7578, 5551, 34740, 7578, 24976, 10406, 295, 6828, 11933, 15730, 8717, 4117, 2655, 41185, 39627, 15730, 25062, 3215, 2288, 4032, 1863, 8592, 38688, 24793, 36145, 13412, 16606, 21984, 7251, 20498, 16247, 12984, 1863, 9957], "avg_logprob": -0.19950457753204717, "compression_ratio": 1.5477386934673367, "no_speech_prob": 1.1920928955078125e-07, "words": [{"start": 2460.54, "end": 2460.96, "word": "Okay", "probability": 0.7470703125}, {"start": 2460.96, "end": 2464.04, "word": " تمام؟", "probability": 0.6458333333333334}, {"start": 2464.04, "end": 2464.3, "word": " Okay", "probability": 0.259521484375}, {"start": 2464.3, "end": 2465.26, "word": " إذا", "probability": 0.5799560546875}, {"start": 2465.26, "end": 2466.16, "word": " بنوقف", "probability": 0.88134765625}, {"start": 2466.16, "end": 2466.48, "word": " هنا", "probability": 0.994140625}, {"start": 2466.48, "end": 2466.86, "word": " وإن", "probability": 0.7505696614583334}, {"start": 2466.86, "end": 2466.96, "word": " شاء", "probability": 0.98486328125}, {"start": 2466.96, "end": 2467.06, "word": " الله", "probability": 0.95263671875}, {"start": 2467.06, "end": 2467.3, "word": " المرة", "probability": 0.9228515625}, {"start": 2467.3, "end": 2467.6, "word": " الجاية", "probability": 0.7760416666666666}, {"start": 2467.6, "end": 2468.22, "word": " هنأكل", "probability": 0.7352294921875}, {"start": 2468.22, "end": 2469.62, "word": " أمثلة", "probability": 0.98193359375}, {"start": 2469.62, "end": 2469.88, "word": " على", "probability": 0.86767578125}, {"start": 2469.88, "end": 2470.6, "word": " المظريات", "probability": 0.8006591796875}, {"start": 2470.6, "end": 2471.02, "word": " هذه", "probability": 0.890625}, {"start": 2471.02, "end": 2474.34, "word": " ونحاول", "probability": 0.96533203125}, {"start": 2474.34, "end": 2474.76, "word": " ناخد", "probability": 0.8671875}, {"start": 2474.76, "end": 2475.24, "word": " قصارى", "probability": 0.77587890625}, {"start": 2475.24, "end": 2476.1, "word": " أخرى", "probability": 0.9860026041666666}, {"start": 2476.1, "end": 2476.78, "word": " لل", "probability": 0.36669921875}, {"start": 2476.78, "end": 2477.16, "word": " limits", "probability": 0.96337890625}, {"start": 2477.16, "end": 2477.4, "word": " of", "probability": 0.9404296875}, {"start": 2477.4, "end": 2477.98, "word": " functions", "probability": 0.84375}, {"start": 2477.98, "end": 2478.78, "word": " إذا", "probability": 0.845703125}, {"start": 2478.78, "end": 2479.26, "word": " نكتفي", "probability": 0.79473876953125}, {"start": 2479.26, "end": 2479.54, "word": " بهذا", "probability": 0.961181640625}, {"start": 2479.54, "end": 2480.06, "word": " القدر", "probability": 0.9562174479166666}, {"start": 2480.06, "end": 2481.44, "word": " ونشوفكم", "probability": 0.92666015625}, {"start": 2481.44, "end": 2481.6, "word": " إن", "probability": 0.82470703125}, {"start": 2481.6, "end": 2481.74, "word": " شاء", "probability": 0.990478515625}, {"start": 2481.74, "end": 2481.94, "word": " الله", "probability": 0.96728515625}, {"start": 2481.94, "end": 2482.98, "word": " يوم", "probability": 0.971923828125}, {"start": 2482.98, "end": 2483.6, "word": " الأثنين", "probability": 0.8533935546875}], "temperature": 1.0}], "language": "ar", "language_probability": 1.0, "duration": 2484.5815, "duration_after_vad": 2241.769374999989}
\ No newline at end of file
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/t35SQVeADNY_postprocess.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/t35SQVeADNY_postprocess.srt
new file mode 100644
index 0000000000000000000000000000000000000000..a76a12d40fcdf22b14469b0182d80c8db1afc9cf
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/t35SQVeADNY_postprocess.srt
@@ -0,0 +1,1560 @@
+1
+00:00:21,400 --> 00:00:26,440
+المرة اللى فاتت احنا يعني اخدنا ال Archimedean
+
+2
+00:00:26,440 --> 00:00:33,820
+property وشوفنا فيه صور متعددة لها او صور مختلفة
+
+3
+00:00:33,820 --> 00:00:39,140
+اللى هي الصور التلاتة هذه فكل
+
+4
+00:00:39,140 --> 00:00:43,160
+واحدة من الخواصة دى تعتبر Archimedean property او
+
+5
+00:00:43,160 --> 00:00:48,670
+مكافئة لخاصية Archimedes او Archimedesو أهم واحدة
+
+6
+00:00:48,670 --> 00:00:55,930
+في الصور التلاتة اللي هي الخاصية دي D اللي بتقول
+
+7
+00:00:55,930 --> 00:01:01,490
+أو بتنص على أنه لأي عدد حقيقي موج ابواي بقدر ألاقي
+
+8
+00:01:01,490 --> 00:01:06,650
+عدد طبيعي مقلوبه أصغر من العدد الموج ابواي
+
+9
+00:01:10,990 --> 00:01:15,490
+وشوفت المرة اللي فتحت البرهين تبعت الأجزاء التلاتة
+
+10
+00:01:15,490 --> 00:01:21,530
+هذه و هذا مثال قولنا هذا مثال على set و ال set هذه
+
+11
+00:01:21,530 --> 00:01:26,950
+bounded و أثبتنا .. شوفنا كيف نثبت أن ال infimum
+
+12
+00:01:26,950 --> 00:01:34,090
+تبعها بساوى 7 إذن هذا كان مجرد مثال على كيف نثبت
+
+13
+00:01:34,090 --> 00:01:36,170
+أو كيف نوجد ال infimum لل set
+
+14
+00:01:40,130 --> 00:01:53,370
+الان في أنا نظرية كتير مهمة احنا أخدنا في ال
+
+15
+00:01:53,370 --> 00:01:59,570
+citation السابقة أثبتنا كان في عندي example اعتقد
+
+16
+00:01:59,570 --> 00:02:05,850
+او fact انه
+
+17
+00:02:05,850 --> 00:02:18,320
+ال ..لا يوجد .. لا يوجد T ينتبه لل Q بحيث انه T
+
+18
+00:02:18,320 --> 00:02:28,060
+تربية بساوي اتنين مافيش عدد نسبي مربع بساوي اتنين
+
+19
+00:02:28,060 --> 00:02:36,220
+او بمعنى اخر .. او بمعنى اخر جذر الاتنين ليس عدد
+
+20
+00:02:36,220 --> 00:02:43,960
+نسبيوهذا أثبتناه في مثال سابق أثبوت؟ في هذا المثال
+
+21
+00:02:43,960 --> 00:02:52,580
+افترضنا ضمنيا أن جذر 2 هذا هو عدد حقيقي لأن هذا
+
+22
+00:02:52,580 --> 00:02:58,260
+عبارة عن عدد حقيقي اليوم
+
+23
+00:02:58,260 --> 00:03:06,400
+هنرجع ونثبت في نظرية 1.17 أن جذر 2 هذا فعلا هو عدد
+
+24
+00:03:06,400 --> 00:03:13,130
+حقيقيبنثبت انه يوجد عدد حقيقي اللي هو اسمه جدر
+
+25
+00:03:13,130 --> 00:03:21,050
+اتنين هذا العدد موجود ممكن يعني اثبات انه موجود
+
+26
+00:03:21,050 --> 00:03:30,290
+اذا النظرية هذه بتقول يوجد عدد حقيقي موجب مربع
+
+27
+00:03:30,290 --> 00:03:35,510
+بساوة اتنين بمعنى اخر جدر اتنين هو عدد حقيقي
+
+28
+00:03:38,190 --> 00:03:43,250
+فالبرهان ذلك ده ما .. هذا ما نسميه يعني هنا
+
+29
+00:03:43,250 --> 00:03:48,910
+البرهان constructive proof أو يعني برهان بناء أو
+
+30
+00:03:48,910 --> 00:03:54,670
+بالبناء لأنه بنا نثبت ال existence تبع شغلة فلازم
+
+31
+00:03:54,670 --> 00:03:57,790
+هذا الشغلة اللي بنا نثبت وجودها بنا نبنيها بنا
+
+32
+00:03:57,790 --> 00:04:05,160
+نكونها ففي عملية بناء هناففي البراهين اللي زي هذه
+
+33
+00:04:05,160 --> 00:04:09,740
+بنعرف set وبنعرف شغلات بنثبت أنها هي الحاجات
+
+34
+00:04:09,740 --> 00:04:16,360
+المطلوبة ف let S نعرف مجموعة S على أنها كل العناصر
+
+35
+00:04:16,360 --> 00:04:23,500
+S في R كل الأعداد الحقيقية S اللي بتكون غير سالبة
+
+36
+00:04:23,500 --> 00:04:29,860
+ومربعها أصغر من الإتنين الآن
+
+37
+00:04:29,860 --> 00:04:36,210
+المجموعة هذه غير خاليةليه؟ لأن الواحد .. الواحد
+
+38
+00:04:36,210 --> 00:04:41,650
+ينتمي إلها تعالى نتأكد ليه الواحد ينتمي للست هذه؟
+
+39
+00:04:41,650 --> 00:04:46,890
+لأن الواحد بيحقق الشرقين تبعات الست أول شيء واحد
+
+40
+00:04:46,890 --> 00:04:51,870
+أكبر من أو ساوي سفر هذا صحيح واحد تربية بيساوي
+
+41
+00:04:51,870 --> 00:04:55,390
+واحد أصغر من الأتنين لأن هذه المجموعة فيها على
+
+42
+00:04:55,390 --> 00:05:01,490
+الأقل الواحد وبالتالي هذه مجموعة غير خالية non
+
+43
+00:05:01,490 --> 00:05:02,070
+-empty
+
+44
+00:05:05,790 --> 00:05:16,350
+كذلك المجموعة هذه bounded above by two لو
+
+45
+00:05:16,350 --> 00:05:26,350
+كان بنثبت أن T أصغر من أو يساوي اتنين لكل T في S
+
+46
+00:05:34,900 --> 00:05:38,820
+T أصغر من او سوى اتنين لكل T في S يعني اتنين upper
+
+47
+00:05:38,820 --> 00:05:43,480
+bound لست S فالبرهان
+
+48
+00:05:43,480 --> 00:05:47,700
+ذلك بالتناقض برهنها
+
+49
+00:05:47,700 --> 00:05:53,160
+بالتناقض assume أنه
+
+50
+00:05:53,160 --> 00:06:01,430
+يوجد T0 كيف بتدن في الجملة هذه؟بدل لكل T في S يوجد
+
+51
+00:06:01,430 --> 00:06:08,330
+T0 في S such that T0 بدل أصغر من أو يساوي بيصير
+
+52
+00:06:08,330 --> 00:06:15,030
+أكبر من الأتنين هذا بيقدي أن T0 تربية أكبر من
+
+53
+00:06:15,030 --> 00:06:25,110
+أربعة صح؟ و أربعة أكبر من الأتنين طب T0 هذا ينتمي
+
+54
+00:06:25,110 --> 00:06:35,900
+ل Sمعناته T0 تربيع هذا أصغر من الأتنين صح؟ أظبط؟
+
+55
+00:06:35,900 --> 00:06:39,660
+إذا بيطلع عندى اتنين أكبر من الأتنين اتنين أكبر من
+
+56
+00:06:39,660 --> 00:06:44,720
+الأتنين contradiction تناخد تمام؟ إذا هذا التناخد
+
+57
+00:06:44,720 --> 00:06:45,920
+بيثبت لك ليه؟
+
+58
+00:06:51,390 --> 00:06:55,630
+إذا انا هقدر اقلين بيقوللي ان ال 6S is bounded
+
+59
+00:06:55,630 --> 00:07:05,110
+above by 2 ال 6S محدودة من أعلى by 2 طيب
+
+60
+00:07:05,110 --> 00:07:11,830
+لإن أنا فيه اندي 6S non-empty و bounded aboveإذا
+
+61
+00:07:11,830 --> 00:07:17,190
+by completeness property ال 6S هذه لازم يكون فيه
+
+62
+00:07:17,190 --> 00:07:23,290
+لها supremum it has a supremum in R say دعنا نسميه
+
+63
+00:07:23,290 --> 00:07:28,870
+X إذا ال X اللي هو ال supremum ل S هذا exist in R
+
+64
+00:07:28,870 --> 00:07:34,390
+by completeness property هذا ال X اللي هو ال
+
+65
+00:07:34,390 --> 00:07:42,060
+supremum ل S برضه بطلع أكبر من واحد ليه؟لأن خمسة
+
+66
+00:07:42,060 --> 00:07:47,780
+على أربعة ينتمي ل S عنصر في S لهذا صح تعالى نتأكد
+
+67
+00:07:47,780 --> 00:07:53,080
+الخمسة على أربعة أكبر من أو ساوى سفر وخمسة على
+
+68
+00:07:53,080 --> 00:07:57,040
+أربعة تربيع اللي هو خمسة عشرين على ستاشر أصغر من
+
+69
+00:07:57,040 --> 00:08:03,030
+الإتنينإذن هذا العدد ينتمي ل S وال X هذا supremum
+
+70
+00:08:03,030 --> 00:08:08,870
+ل S وبالتالي X upper bound ل S ال X upper bound ل
+
+71
+00:08:08,870 --> 00:08:14,410
+6S إذن أكبر من أو ساوي كل عناصرها إذن ال X أكبر من
+
+72
+00:08:14,410 --> 00:08:19,070
+أو ساوي خمسة على أربعة وخمسة على أربعة أكبر من
+
+73
+00:08:19,070 --> 00:08:25,470
+واحد إذن X أكبر من واحد تمام؟ إذن هين أثبتنا إن ال
+
+74
+00:08:25,470 --> 00:08:31,610
+6 يوجدعدد حقيقي x اللي هو الـ suprem ل S وهذا الـ
+
+75
+00:08:31,610 --> 00:08:39,910
+x أكبر من واحد الآن هذا الـ x هو .. بنثبت هو الجذر
+
+76
+00:08:39,910 --> 00:08:42,690
+التربيعي للإتنين
+
+77
+00:08:49,400 --> 00:08:53,760
+الان هذا ال X اللي هو طلع موجود by completeness of
+
+78
+00:08:53,760 --> 00:08:58,600
+property وعدد اكبر من واحد بنثبت انه مربع و بساوي
+
+79
+00:08:58,600 --> 00:09:05,180
+اتنين يعني هو العدد المطلوب ان ال X هو جدر اتنين
+
+80
+00:09:11,980 --> 00:09:15,800
+فإذا أثبتنا ال claim هذا بتكون كملنا برهانة نظرية
+
+81
+00:09:15,800 --> 00:09:19,760
+طيب هذا ال claim برضه متبرهن بالتناقض assume ال
+
+82
+00:09:19,760 --> 00:09:25,060
+contrary إن x تربية بسويش اتنين if by tricotomy
+
+83
+00:09:25,060 --> 00:09:30,020
+property إما x تربية أصغر من اتنين أو أكبر من
+
+84
+00:09:30,020 --> 00:09:34,160
+اتنين إذا أنا في عندي حالتين هنا لازم أخدهم في
+
+85
+00:09:34,160 --> 00:09:40,610
+الاعتبارعشان أكمل برهان ال claim الحالة الأولى
+
+86
+00:09:40,610 --> 00:09:47,190
+فيها extra بيها أصغر من الإتنين و بتدصل في الحالة
+
+87
+00:09:47,190 --> 00:09:51,790
+هذه لتناقض عشان أثبت ال claim لإن ال claim بدينا
+
+88
+00:09:51,790 --> 00:09:56,810
+برهانه بالتناقض assume the contrary فرضنا النقيد
+
+89
+00:09:58,620 --> 00:10:02,360
+تعالى نشوف كيف نصل لتناقض في الحالة هذه انا عندي n
+
+90
+00:10:02,360 --> 00:10:07,640
+لكل n عدد طبيعى n تربيع اكبر من او ساوى n و
+
+91
+00:10:07,640 --> 00:10:10,700
+بالتاني مقلوب n تربيع دايما اصغر من او ساوى مقلوب
+
+92
+00:10:10,700 --> 00:10:18,500
+ال n هذا صحيح الان x زائد واحد على n لكل تربيع ربع
+
+93
+00:10:18,500 --> 00:10:23,820
+بطلع x تربيع زاد اتنين زاد ضعف الاول في التانى زاد
+
+94
+00:10:23,820 --> 00:10:31,370
+مربع التانىالان بدي أشيل 1 على n تربية هدي أشيلها
+
+95
+00:10:31,370 --> 00:10:37,250
+و أستبدلها بحاجة أكبر من .. أكبر من او ساوية اللي
+
+96
+00:10:37,250 --> 00:10:42,030
+هو 1 على n لأن هنا لما أستبدل هذه ب1 على n بصير
+
+97
+00:10:42,030 --> 00:10:47,250
+هذا أصغر من او ساوية x تربية زاد 2x على n زاد 1
+
+98
+00:10:47,250 --> 00:10:51,410
+على n و لما أخد من الحدين هدول 1 على n عامل مشترك
+
+99
+00:10:51,410 --> 00:10:53,610
+بيضل 2x زاد 1
+
+100
+00:11:02,350 --> 00:11:07,870
+تمام؟ اذا انا عندي أصبح ال X زاد واحد على M لكل
+
+101
+00:11:07,870 --> 00:11:15,110
+تربية أصغر من أو ساوي المقدار هذا طيب احنا من ال
+
+102
+00:11:15,110 --> 00:11:20,150
+.. احنا فرضين X تربية أصغر من 2 اذا 2 سالب X تربية
+
+103
+00:11:20,150 --> 00:11:25,410
+بيطلع عدد موجبو ال X عدد اكبر من واحد، اذا اتنين X
+
+104
+00:11:25,410 --> 00:11:30,050
+زاد واحد عدد موجب، اذا كسر بسطه موجب و مقامه موجب،
+
+105
+00:11:30,050 --> 00:11:34,330
+بيطلع عدد موجب، اذا حسب ال property اللي انا
+
+106
+00:11:34,330 --> 00:11:38,670
+علمتها بال star لأي
+
+107
+00:11:38,670 --> 00:11:43,530
+عدد موجب، قولنا يوجد عدد طبيعي N0 مقلوب و أصغر من
+
+108
+00:11:43,530 --> 00:11:49,810
+العدد الموجبأدرج الطرفين في العدد الموجود 2x زاد 1
+
+109
+00:11:49,810 --> 00:11:56,990
+بصير المتباينة هذه شكلها هكذا الان
+
+110
+00:11:56,990 --> 00:12:01,710
+من المتباينة تلاتة هاي المتباينة تلاتة هاي عندي x
+
+111
+00:12:01,710 --> 00:12:05,590
+انا
+
+112
+00:12:05,590 --> 00:12:11,000
+عندي يوجد in zero ينتمي ل inطبعا هذا الكلام
+
+113
+00:12:11,000 --> 00:12:15,160
+المتباينة تلاتة هذه صحيحة لكل العداد الطبيعي N
+
+114
+00:12:15,160 --> 00:12:21,240
+وبالتالي صحيحة للعدد الطبيعي N0 إذا نتعالوا نقرأ
+
+115
+00:12:21,240 --> 00:12:26,380
+المتباينة تلاتة ونبدل N ب N0 فبصير X زاد واحد على
+
+116
+00:12:26,380 --> 00:12:29,840
+N0 ترديد أصغر من
+
+117
+00:12:34,400 --> 00:12:39,760
+مفروض أصغر منها يساوي X تربية زياد واحد على N Zero
+
+118
+00:12:39,760 --> 00:12:47,860
+في اتنين X زياد واحد وواحد
+
+119
+00:12:47,860 --> 00:12:53,640
+على N Zero في اتنين X زياد واحد أصغر من اتنين سالب
+
+120
+00:12:53,640 --> 00:13:00,200
+X تربية إذن هذا هيصير أصغر من اتنين سالب X تربية Y
+
+121
+00:13:00,200 --> 00:13:02,880
+X تربية اللي هو بيطلع اتنين
+
+122
+00:13:08,840 --> 00:13:14,640
+إذا أنا في عندي الآن عدد العدد اللي هو X زياد واحد
+
+123
+00:13:14,640 --> 00:13:21,600
+على N Zero مربعه أصغر من الإتنين إذا هذا العدد
+
+124
+00:13:21,600 --> 00:13:25,320
+اللي هو X زياد واحد على N Zero هو عدد غير سالب
+
+125
+00:13:25,320 --> 00:13:29,320
+ومربعه أصغر من الإتنين إذا هذا ينتمي للمجموع S
+
+126
+00:13:29,320 --> 00:13:33,880
+اللي عرفته في بداية البرهان إذا هذا العدد ينتمي
+
+127
+00:13:33,880 --> 00:13:34,920
+للمجموع S
+
+128
+00:13:37,690 --> 00:13:44,950
+و هذا بي .. بيقدّي انه ال .. العدد هذا مش ممكن
+
+129
+00:13:44,950 --> 00:13:53,790
+يكون upper bound للست S هذا
+
+130
+00:13:53,790 --> 00:13:58,650
+مش ممكن يكون upper bound للست S لأن ال X هو upper
+
+131
+00:13:58,650 --> 00:14:04,390
+bound هو ال supreme و
+
+132
+00:14:04,390 --> 00:14:05,850
+بالتالي
+
+133
+00:14:14,060 --> 00:14:18,100
+إحنا أثبتنا إن X هي ال upper bound لكن لجينا X زاد
+
+134
+00:14:18,100 --> 00:14:22,100
+واحد على N not هي ثانوية ل S فبالتالي في عنصر أكبر
+
+135
+00:14:22,100 --> 00:14:31,140
+من X فهذا ال contradiction مظبوط لأن هذا
+
+136
+00:14:31,140 --> 00:14:38,730
+عنصر في S وبالتالي هذا بيقدّي هذا بيقدّيإن X
+
+137
+00:14:38,730 --> 00:14:47,890
+ماهواش upper bound لأنه ال X أصغر من واحد من X
+
+138
+00:14:47,890 --> 00:14:54,650
+زائد واحد على N زيرو وهذا
+
+139
+00:14:54,650 --> 00:14:55,590
+ينتمي ل S
+
+140
+00:15:01,160 --> 00:15:06,760
+فهذا عنصر في S و أكبر من X هذا معناه ان ال X هذا
+
+141
+00:15:06,760 --> 00:15:12,420
+مش ممكن يكون upper bound ل S وهذا بتناقض مع كون X
+
+142
+00:15:12,420 --> 00:15:18,160
+هو upper bound لأنه supreme ل S احنا قلنا ان ال X
+
+143
+00:15:18,160 --> 00:15:22,840
+هذا اللي في ال claim ايه ال X هذا هو supreme ل S
+
+144
+00:15:22,840 --> 00:15:27,180
+وبالتالي upper bound ل Sو هنا وصلنا لأن ال X مش
+
+145
+00:15:27,180 --> 00:15:31,160
+ممكن يكون upper bound ل S لأنه أصغر من عنصر ما
+
+146
+00:15:31,160 --> 00:15:34,720
+ينتمي ل S إذا فيه تناقض لأن هنا وصلنا إلى
+
+147
+00:15:34,720 --> 00:15:41,480
+contradiction إذا فرضنا X بيه أصغر من 2 وصلتنا ل
+
+148
+00:15:41,480 --> 00:15:46,760
+contradiction بالمثل بالمثل
+
+149
+00:15:46,760 --> 00:15:51,480
+في الحالة التانية في
+
+150
+00:15:51,480 --> 00:15:56,980
+الحالة التانيةلو فرضنا ان x ثربيه المرة دي اكبر من
+
+151
+00:15:56,980 --> 00:16:03,100
+الاتنين فبرضه ممكن نصل بنفس الطريقة و بنفس الاسلوب
+
+152
+00:16:03,100 --> 00:16:09,220
+في النهاية الى contradiction نصل الى contradiction
+
+153
+00:16:09,220 --> 00:16:15,100
+وبالتالي اذا احنا فرضنا انه احنا كنا كان ال claim
+
+154
+00:16:15,100 --> 00:16:20,520
+تبعناالذكروا في البرنامج احنا كنا عايزين نثبت ان x
+
+155
+00:16:20,520 --> 00:16:26,060
+تربية بساوة اتنين ففرضنا ان x تربية لات ساوة اتنين
+
+156
+00:16:26,060 --> 00:16:33,520
+وبالتالي هذا بيقدر ان x تربية اصغر من اتنين او x
+
+157
+00:16:33,520 --> 00:16:38,200
+تربية اكبر من اتنين وفي كل حالة من الحالات هذه
+
+158
+00:16:38,200 --> 00:16:43,520
+وصلنا الى تناقضإذا التناقض هذا سببه انه اذا احنا
+
+159
+00:16:43,520 --> 00:16:49,620
+فرضنا هو فرضنا ان extra b لايه ساو اتنين هذا بثبت
+
+160
+00:16:49,620 --> 00:16:54,980
+ال claim اللي هو extra b بساو اتنين و بكمل البرهان
+
+161
+00:16:54,980 --> 00:17:03,440
+okay تمام إذا هذا هو برهان ال .. النظرية هذه إذا
+
+162
+00:17:03,440 --> 00:17:09,440
+الأن أصبح عندي في نظريتين واحدة بتقول انهجذر اتنين
+
+163
+00:17:09,440 --> 00:17:14,660
+هو عدد حقيقي ونظرية
+
+164
+00:17:14,660 --> 00:17:19,300
+تانية او حقيقة تانية ان جذر اتنين ليس عدد نصي
+
+165
+00:17:36,150 --> 00:17:44,930
+طيب في كمان نتيجة مهمة على النتائج هذه او نظرية
+
+166
+00:17:44,930 --> 00:17:49,490
+مهمة اللي هي density theorem نظرية الكثافة ال
+
+167
+00:17:49,490 --> 00:17:55,530
+density theorem ال density theorem بتقول بين اي
+
+168
+00:17:55,530 --> 00:18:01,230
+عددين حقيقيين يوجد عدد مثل يعني لو انا في عندى
+
+169
+00:18:02,110 --> 00:18:11,650
+عددين حقيقيين X أصغر من Y هاي X وهاي Y فيوجد عدد
+
+170
+00:18:11,650 --> 00:18:20,390
+نسبي R ينتمي لـ Q بحيث أن R أكبر من X أصغر من Y أو
+
+171
+00:18:20,390 --> 00:18:27,320
+بمعنى آخر، لو في عندي فترة مفتوحة X وYفلازم هذه
+
+172
+00:18:27,320 --> 00:18:34,100
+الفترة وفي هذه الفترة غير خالية فلازم نلاقي R
+
+173
+00:18:34,100 --> 00:18:41,700
+ينتمي للفترة المفتوحة و R هذا عدد نسمة اذا ال
+
+174
+00:18:41,700 --> 00:18:45,520
+density theorem بكل بساطة بتقول between any two
+
+175
+00:18:45,520 --> 00:18:50,990
+real numbers there exists a rational number اوAny
+
+176
+00:18:50,990 --> 00:18:55,030
+non-empty open interval contains a rational number
+
+177
+00:18:55,030 --> 00:19:02,170
+نفس الصيرة أو نفس ال statement البرهان مش صعب
+
+178
+00:19:02,170 --> 00:19:05,210
+تعالى
+
+179
+00:19:05,210 --> 00:19:10,410
+نشوف البرهان أنا عندي فرض ان X أصغر من Y فرض X
+
+180
+00:19:10,410 --> 00:19:16,790
+أصغر من Yو بالتالي y سالب x عدد موجب اذا by the
+
+181
+00:19:16,790 --> 00:19:20,330
+Archimedean property بقدر الاقي عدد هنا المفروض
+
+182
+00:19:20,330 --> 00:19:28,050
+يكون جزء b فby
+
+183
+00:19:28,050 --> 00:19:31,590
+Archimedean property واحد ستاشر بيه اللي اعطيناها
+
+184
+00:19:31,590 --> 00:19:37,930
+إشارة نجمة لأي عدد موجب زي هذا يوجد عدد طبيعي n
+
+185
+00:19:37,930 --> 00:19:45,930
+مقلوب وأصغر من عدد الموجبالان بدي اعرف ان اكبر عدد
+
+186
+00:19:45,930 --> 00:19:50,190
+صحيح اصغر من او يساوي n x هذه دالة اكبر عدد صحيح
+
+187
+00:19:50,190 --> 00:19:55,290
+واضيف عليها واحد طبعا هذا عدد صحيح زائد واحد
+
+188
+00:19:55,290 --> 00:19:59,930
+الواحد عدد صحيح فمجموعهم بطلع عدد صحيح اذا انا
+
+189
+00:19:59,930 --> 00:20:04,250
+عرفت عدد صحيح m ينتمي الى z على انه the greatest
+
+190
+00:20:04,250 --> 00:20:09,110
+integer less than or equal to n x plus oneالان
+
+191
+00:20:09,110 --> 00:20:16,650
+بتثبت انه في عندى claim بتثبت ان هذا العدد m بيحقق
+
+192
+00:20:16,650 --> 00:20:22,110
+المتباينة هذه اذا
+
+193
+00:20:22,110 --> 00:20:31,330
+انا عايز اثبت انه ال claim تبعى one انه m negative
+
+194
+00:20:31,330 --> 00:20:35,850
+one less than or equal n x less than m
+
+195
+00:20:39,470 --> 00:20:46,730
+طيب من تعريف دالة أكبر عدد صحيح by definition of
+
+196
+00:20:46,730 --> 00:20:52,970
+the greatest integer function هذا العدد دايما أكبر
+
+197
+00:20:52,970 --> 00:20:58,150
+من نفسه الأدد اللي جوا سالب واحد وأصغر من أو ساوي
+
+198
+00:20:58,150 --> 00:21:05,870
+نفسه صح؟هذا العدد أكبر عدد صحيح أصغر من أو ساوي NX
+
+199
+00:21:05,870 --> 00:21:11,970
+يعني نفسر نفسه إذا هذا العدد أصغر من أو ساوي NX و
+
+200
+00:21:11,970 --> 00:21:18,250
+أكبر من NX للمطرح منها واحد طيب ننجل الواحد عن
+
+201
+00:21:18,250 --> 00:21:22,970
+ناحية التانية فبصرا اندي NX أصغر من أكبر عدد صحيح
+
+202
+00:21:22,970 --> 00:21:28,690
+أصغر من أو ساوي NX زاد واحدوهذا من هنا أصغر من أو
+
+203
+00:21:28,690 --> 00:21:39,010
+يساوي NX زايد واحد الان هذا سمينا احنا هذا M فبطلع
+
+204
+00:21:39,010 --> 00:21:45,930
+عندي M أكبر من NX أصغر من أو يساوي NX زايد واحد
+
+205
+00:21:45,930 --> 00:21:54,340
+نطرح واحد من كل الأطراف فبطلع عندي M سالب واحدم
+
+206
+00:21:54,340 --> 00:22:02,180
+سالب واحد اصغر من او ساوي نكس و نكس من هنا نكس
+
+207
+00:22:02,180 --> 00:22:09,160
+اصغر من م وهذا اللي بدنا يقنعه لان هنا برهن ال
+
+208
+00:22:09,160 --> 00:22:13,200
+claim لان
+
+209
+00:22:13,200 --> 00:22:18,960
+من هذا ال claim من
+
+210
+00:22:18,960 --> 00:22:21,800
+هذا ال claim نستنتج
+
+211
+00:22:24,560 --> 00:22:33,840
+هذه الكلام اللي برهناه لو أنا ضربت ..
+
+212
+00:22:33,840 --> 00:22:37,100
+لو
+
+213
+00:22:37,100 --> 00:22:43,940
+ضفت واحد لكل الأطراف فبصير عندي M أصغر من أو يساوي
+
+214
+00:22:43,940 --> 00:22:50,160
+NX زايد واحد أصغر من M زايد واحد
+
+215
+00:22:56,000 --> 00:23:05,300
+و لو جسمت على N أو ضربت في N نضرب في 1 على N فهذا
+
+216
+00:23:05,300 --> 00:23:15,400
+بيقدر ان M على N أصغر من أو يساوي X زائد 1 على N
+
+217
+00:23:26,950 --> 00:23:31,950
+طيب انا عندي .. انا
+
+218
+00:23:31,950 --> 00:23:34,110
+عندي من نفس المتباينة هذه
+
+219
+00:24:07,700 --> 00:24:15,700
+أنا عندي 1 على n أصغر من y minus x أنا
+
+220
+00:24:15,700 --> 00:24:23,580
+عندي 1 على n 1 على n أصغر من y سالب x هذا بيقدي
+
+221
+00:24:23,580 --> 00:24:33,260
+أنه x زاد 1 على n أصغر من y لأن هذا أصغر من y تمام
+
+222
+00:24:39,380 --> 00:24:45,180
+هو أكبر من X نعم؟ هو أكبر من X لأن X هيت واحدة
+
+223
+00:24:45,180 --> 00:24:55,720
+عنها اه و بدي هذا يكون أكبر من X اه
+
+224
+00:24:55,720 --> 00:25:02,780
+M على N أكبر اه من هنا من
+
+225
+00:25:02,780 --> 00:25:10,740
+هنا اجسم على M لما أضرب في واحد على Nفبصير عندي M
+
+226
+00:25:10,740 --> 00:25:17,860
+على N أكبر من X نظمت هيك صح؟ إذا أصبح عندي أنا
+
+227
+00:25:17,860 --> 00:25:25,220
+بحصل على X أصغر من M على N أصغر من Y الآن هذا
+
+228
+00:25:25,220 --> 00:25:33,060
+عبارة عن عدد نسبي هذا هو هذا العدد هو ال R هذا عدد
+
+229
+00:25:33,060 --> 00:25:38,000
+نسبي لأن عدد النسبي عبارة عن عدد صحيح على عدد صحيح
+
+230
+00:25:38,000 --> 00:25:44,470
+لا يساوي سفرإذن هذه أثبتت إن يوجد عدد نسبي أكبر من
+
+231
+00:25:44,470 --> 00:25:49,790
+X وأصغر من Y أو يوجد عدد نسبي بين X وY إذن هذا
+
+232
+00:25:49,790 --> 00:26:02,470
+بيكمل البرهان تبع ال density theorem في
+
+233
+00:26:02,470 --> 00:26:07,870
+كمان نتيجة تانية برضه بنسميها density theorem أو
+
+234
+00:26:07,870 --> 00:26:11,800
+نتيجة على ال density theoremو في بعض الناس برضه
+
+235
+00:26:11,800 --> 00:26:15,620
+نفسها بيسموها density theorem ف ال density theorem
+
+236
+00:26:15,620 --> 00:26:19,100
+هذه التانية أو النتيجة عليها بتقول ان بين اي
+
+237
+00:26:19,100 --> 00:26:24,440
+between any two real numbers x and y there exists
+
+238
+00:26:24,440 --> 00:26:29,860
+an irrational number يوجد عدد z بين x و y وهذا z
+
+239
+00:26:29,860 --> 00:26:36,100
+irrational number اذا
+
+240
+00:26:36,100 --> 00:26:47,170
+كمان مرةلو في عندى اي عددين حقيقيين x و y هو
+
+241
+00:26:47,170 --> 00:26:56,830
+x أصغر من y فبنقدر نلاقي r أو p وهذا ينتمي ل r
+
+242
+00:26:56,830 --> 00:27:06,170
+minus q عدد غير نسبي بين أكبر من x و أصغر من y
+
+243
+00:27:07,390 --> 00:27:12,950
+البرهان تبع النتيجة هذه اول
+
+244
+00:27:12,950 --> 00:27:18,050
+شي احنا لسه مثبتين قبل شوية ان جدر اتنين is a real
+
+245
+00:27:18,050 --> 00:27:22,730
+number جدر اتنين كان هو supremum ل 6 وبالتالي
+
+246
+00:27:22,730 --> 00:27:27,690
+exist in R واثبتنا ان جدر اتنين هذا اكبر من صفر
+
+247
+00:27:27,690 --> 00:27:31,910
+حتى اثبتنا انه اكبر من واحد كان ال X اكبر من واحد
+
+248
+00:27:31,910 --> 00:27:37,900
+وبالتالي موجةإن جدر اتنين عدد موجب وبالتالي مقلوبه
+
+249
+00:27:37,900 --> 00:27:46,480
+عدر موجب لإن أنا لما يكون عندي x أصغر من y و أضرب
+
+250
+00:27:46,480 --> 00:27:50,440
+المتباينة هذه في واحد على جدر اتنين اللي هو عدد
+
+251
+00:27:50,440 --> 00:27:55,040
+موجب فتصير المتباينة هذه x على جدر اتنين أصغر من y
+
+252
+00:27:55,040 --> 00:28:01,030
+على جدر اتنينتمام الان اذا هذا real number وهذا
+
+253
+00:28:01,030 --> 00:28:05,630
+real number by density theorem between any two
+
+254
+00:28:05,630 --> 00:28:09,270
+real numbers there exist irrational number عدد
+
+255
+00:28:09,270 --> 00:28:13,750
+نسبي سمي R طبعا هذا العدد النسبي لا يساوي سفر لان
+
+256
+00:28:13,750 --> 00:28:20,170
+دي عداد موجة باتنين اذا انا في عندي عدد نسبي R بين
+
+257
+00:28:20,170 --> 00:28:25,310
+العددين الحقيقين حسب ال density theorem الان لو
+
+258
+00:28:25,310 --> 00:28:32,530
+اخدت لو عرفتلو عرفت العدد z على انه r ضرب جذر
+
+259
+00:28:32,530 --> 00:28:38,110
+اتنين فهذا بيطلع عدد غير نسبي تعالوا نبرهم ذلك
+
+260
+00:28:38,110 --> 00:28:42,950
+لإثبات ان العدد z اللي هو r في جذر اتنين عدد غير
+
+261
+00:28:42,950 --> 00:28:49,210
+نسبي افرض otherwise يعني خلاف ذلك افرض خلاف ذلك
+
+262
+00:28:49,210 --> 00:28:56,250
+يعني افرض ان العدد هذا نسبي افرض
+
+263
+00:28:56,250 --> 00:29:02,030
+ان هذا نسبيإذا لما جسمه هذا هو R في جدر اتنين عدد
+
+264
+00:29:02,030 --> 00:29:08,130
+نسبي و R احنا فرضيه و R طبعا هذا عدد نسبي إذا عدد
+
+265
+00:29:08,130 --> 00:29:11,770
+نسبي على عدد نسبي بيطلع عدد نسبي إذا هذا هيطلع عدد
+
+266
+00:29:11,770 --> 00:29:15,810
+نسبي اللي هو بيساوي جدر اتنين وهذا تناقض مع
+
+267
+00:29:15,810 --> 00:29:21,490
+الحقيقة اللي أثبتناها سابقا إذا .. إذا هذا العدد
+
+268
+00:29:21,490 --> 00:29:28,090
+لازم .. مش ممكن يكون نسبي لازم يكون غير نسبي طيب
+
+269
+00:29:31,920 --> 00:29:37,240
+أضرب المتباينة هذه في
+
+270
+00:29:37,240 --> 00:29:42,500
+جدر اتنين لو ضربنا المتباينة هذه في جدر اتنين
+
+271
+00:29:42,500 --> 00:29:45,620
+فبصير
+
+272
+00:29:45,620 --> 00:29:52,440
+عندي X أصغر من R في جدر اتنين اللي سمناها Z أصغر
+
+273
+00:29:52,440 --> 00:29:53,340
+من Y
+
+274
+00:29:55,970 --> 00:30:00,250
+وبالتالي هنا أثبتنا أن يوجد عدد غير نسبي وهذا
+
+275
+00:30:00,250 --> 00:30:05,710
+العدد الغير نسبي between X and Y وهو المطلوب هذا
+
+276
+00:30:05,710 --> 00:30:10,870
+هو المطلوب تمام؟
+
+277
+00:30:10,870 --> 00:30:14,250
+في
+
+278
+00:30:14,250 --> 00:30:22,590
+أي سؤال أو استفسار؟ إن ذلك يكون احنا خلصنا أربع
+
+279
+00:30:22,590 --> 00:30:29,150
+سكاش م الشتر واحدو هاي في تمارين على ال section
+
+280
+00:30:29,150 --> 00:30:40,510
+هذا و نبدأ section جديد ال
+
+281
+00:30:40,510 --> 00:30:43,950
+section الجديد هذا بتحدث عن ال intervals الفترات
+
+282
+00:30:43,950 --> 00:30:50,710
+طبعا كلكم درستوا تفاضل و تكامل و مرت عليكم أنواع
+
+283
+00:30:50,710 --> 00:30:59,210
+الفترات في course تفاضل و تكامللو أخذت اي عددين
+
+284
+00:30:59,210 --> 00:31:08,710
+حقيقياً a و b و a أصغر من أو ساول b فممكن نعرف ال
+
+285
+00:31:08,710 --> 00:31:13,730
+open interval على أن كل العداد الحقيقية x اللي بين
+
+286
+00:31:13,730 --> 00:31:20,010
+a و b فهذا ال open interval from a to b و فيه ممكن
+
+287
+00:31:20,010 --> 00:31:24,580
+نعرف closed interval from a to bعلى انها كل ال
+
+288
+00:31:24,580 --> 00:31:27,820
+real numbers اللي أكبر من أو ساوي a أصغر من أو
+
+289
+00:31:27,820 --> 00:31:34,880
+ساوي b ال half open فيه برضه نوع من الفترات بنسميه
+
+290
+00:31:34,880 --> 00:31:42,700
+half open أو half closed على الصورة هذه .. هذه
+
+291
+00:31:42,700 --> 00:31:47,890
+بنسميها half openأو half closed يعني مفتوحة من جهة
+
+292
+00:31:47,890 --> 00:31:53,010
+ومغلقة من جهة أخرى فهذه طبعا واضح تعريفة كل ال X
+
+293
+00:31:53,010 --> 00:31:59,250
+اللي أكبر من A أثر من أو ساوى B وهذه زيها في كمان
+
+294
+00:31:59,250 --> 00:32:05,650
+نوع أخر من ال intervals زي ال interval هذه بسميها
+
+295
+00:32:05,650 --> 00:32:14,430
+open left ray شعاع أيصر مفتوح يعني هايخط الأعداد
+
+296
+00:32:14,430 --> 00:32:18,970
+ليه بيسموها left ray لأن فعلا قاملة زي الشعار هاي
+
+297
+00:32:18,970 --> 00:32:26,830
+هذا خط الأعداد وهي بي عدد حقيقي فالفترة المفتوحة
+
+298
+00:32:26,830 --> 00:32:32,370
+من سالب من النهاية إلى بي هي الفترة هذه هي مصدر
+
+299
+00:32:32,370 --> 00:32:36,350
+الصورة أن هنا في مصدر ضوء وهذا الضوء بيشع إلى
+
+300
+00:32:36,350 --> 00:32:43,140
+اليسارفهذه الفترة هي من سالب مقالة نهاية إلى B
+
+301
+00:32:43,140 --> 00:32:48,580
+open interval صح؟
+
+302
+00:32:48,580 --> 00:32:53,400
+كل الأعداد الحقيقية اللي أصغر من B إذن هذه تعريفها
+
+303
+00:32:53,400 --> 00:32:57,460
+كل الأعداد الحقيقية اللي أصغر من B بالمثل ممكن أن
+
+304
+00:32:57,460 --> 00:33:06,040
+أعرف right ray right ray زي ذلك يعني D ومقالة
+
+305
+00:33:06,040 --> 00:33:11,230
+نهايةأو A وما إلى نهاية كل الإعداد الحقيقي اللي
+
+306
+00:33:11,230 --> 00:33:13,190
+أكبر أو على يمين ال A
+
+307
+00:33:16,510 --> 00:33:21,690
+طبعا ممكن نعرف closed left tray لأن لو أخلت النقطة
+
+308
+00:33:21,690 --> 00:33:28,090
+هذه موجودة ضمن الفترة يعني أغلقت الفترة زيك فهذه
+
+309
+00:33:28,090 --> 00:33:32,890
+بسميها closed left tray و كذلك لو أخدت ال beam
+
+310
+00:33:32,890 --> 00:33:37,730
+أضفتها على الفترة فهذه بسميها closed right tray
+
+311
+00:33:37,730 --> 00:33:45,660
+وفي طبعا هذه ال ..الـ two-sided ray إي ده صح
+
+312
+00:33:45,660 --> 00:33:51,000
+التعبير اللي هو خط الأعداد كله كل الأعداد بحثوا في
+
+313
+00:33:51,000 --> 00:33:54,060
+ملاحظة
+
+314
+00:33:54,060 --> 00:34:01,860
+هنا أن الأول تلت أنواع من ال intervals هدول نسميهم
+
+315
+00:34:01,860 --> 00:34:05,580
+bounded هم
+
+316
+00:34:05,580 --> 00:34:11,460
+فعلا bounded الفترات هذه محصورة محدودة ممكن نحصرها
+
+317
+00:34:13,670 --> 00:34:19,010
+أما الأنواع .. الأنواع ال .. ال .. الأخرى الخامسة
+
+318
+00:34:19,010 --> 00:34:25,350
+الأخرى هذه unbounded .. unbounded
+
+319
+00:34:25,350 --> 00:34:29,030
+غير
+
+320
+00:34:29,030 --> 00:34:33,230
+محصورة و غير محدودة الفترات هذه غير محدودة هذا
+
+321
+00:34:33,230 --> 00:34:37,810
+شعاع أيصر، لاحظ الشعاع هذا مقدرش ايه أcipher عليه
+
+322
+00:34:39,020 --> 00:34:44,220
+يعني بيمتد إلى سالب من النهاية أو بيمتد إلى من
+
+323
+00:34:44,220 --> 00:34:49,580
+النهاية، إذا الفترات هذه unbounded، النوح هذا من
+
+324
+00:34:49,580 --> 00:34:53,580
+الفترات bounded وبالتالي الفترات متجسمة إلى two
+
+325
+00:34:53,580 --> 00:34:58,260
+types bounded intervals وunbounded intervals
+
+326
+00:35:13,980 --> 00:35:26,320
+طيب في ملاحظة هنا على الفترات لو
+
+327
+00:35:26,320 --> 00:35:29,740
+أخدت أي interval بغض النظر هل هي bounded ولا
+
+328
+00:35:29,740 --> 00:35:35,200
+unbounded open, closed, half-open, half-closed بغض
+
+329
+00:35:35,200 --> 00:35:39,280
+النظر أي واحدة من الأنواع التمانية اللي شفناهم لو
+
+330
+00:35:39,280 --> 00:35:48,350
+I كانت أي intervalف ال interval I بتحقق الخصية هذه
+
+331
+00:35:48,350 --> 00:35:54,130
+اي interval بتحقق الخصية هذه ان انا لو اخدت اي
+
+332
+00:35:54,130 --> 00:36:00,070
+انصرين في الفترة و واحد اصغر من التاني فالعناصر
+
+333
+00:36:00,070 --> 00:36:05,770
+اللي بينهم بتكون موجودة في الفترة يعني هي ان عندي
+
+334
+00:36:05,770 --> 00:36:11,730
+فترة مفتوحة مغلقة او
+
+335
+00:36:13,570 --> 00:36:20,230
+left ray مثلا و
+
+336
+00:36:20,230 --> 00:36:29,330
+ال symbol فترة هذه مثلا a,b او مثلا high b فلو
+
+337
+00:36:29,330 --> 00:36:34,790
+أخدت سواء هذه الفترة او هذه او اي واحدة من انواع
+
+338
+00:36:34,790 --> 00:36:44,960
+التمنية و جيت أخدت عنصرين x و yو X أصغر من Y
+
+339
+00:36:44,960 --> 00:36:50,660
+موجودين في الفترة هذه فالفترة هذه بتتمتع بالخاصية
+
+340
+00:36:50,660 --> 00:36:58,480
+أن أي عنصر في الفترة هذه المغلقة من X و Y لو أخدت
+
+341
+00:36:58,480 --> 00:37:05,200
+أي تي يعني كل عناصر الفترة المغلقة هذه كلها بتطلع
+
+342
+00:37:05,200 --> 00:37:11,840
+موجودة في الفترة الكبيرة نفس الحاجة هنا هاي أخدي X
+
+343
+00:37:13,020 --> 00:37:24,020
+هو Y فكل العناصر T في الفترة المغلقة موجودة
+
+344
+00:37:24,020 --> 00:37:33,590
+في lift-ray اذا اي فترة Iلو أخدت داخلها أي عنصرين
+
+345
+00:37:33,590 --> 00:37:37,910
+X أزرع من Y فكل العناصر في الفترة المغلقة من X ل Y
+
+346
+00:37:37,910 --> 00:37:42,050
+أو الفترة المغلقة من X ل Y تطلع مجموعة جزئية من
+
+347
+00:37:42,050 --> 00:37:48,710
+الفترة اللي أنا ماخدها okay؟ الآن نظرية أشر أخر من
+
+348
+00:37:48,710 --> 00:37:57,000
+عشرين بتقوللي أنه العكس صحيح العكس صحيح يعنيلو أنا
+
+349
+00:37:57,000 --> 00:38:02,320
+فيه عندي مجموعة جزئية S مجموعة جزئية من R ال
+
+350
+00:38:02,320 --> 00:38:07,140
+cardinal number تبع S أكبر من أوي ساوى اتنين ال
+
+351
+00:38:07,140 --> 00:38:10,780
+cardinal number هنا يعني معناته عدد عناصر ال set
+
+352
+00:38:10,780 --> 00:38:15,740
+تجربة
+
+353
+00:38:15,740 --> 00:38:21,720
+يعني roughly speaking يعني معناه انه يعني اذا كانت
+
+354
+00:38:21,720 --> 00:38:25,460
+ال set finite فهذا عدد عناصرها is infinite طبعا
+
+355
+00:38:26,560 --> 00:38:37,180
+بيكون يعني له تعريف عام من عدد العناصر فده كانت S
+
+356
+00:38:37,180 --> 00:38:41,200
+مجموعة جزئية و cardinalها أكبر من أو ساو اتنين
+
+357
+00:38:41,200 --> 00:38:48,380
+يعني المجموعة فيها على الأقل عنصرين أو أكتر وكانت
+
+358
+00:38:48,380 --> 00:38:52,980
+المجموعة S هذه بتحقق الخاصية أربعة
+
+359
+00:38:55,110 --> 00:39:01,230
+فلازم ال 6 هذه تطلع interval كمان مرة احنا قلنا في
+
+360
+00:39:01,230 --> 00:39:08,050
+الملاحظة هذه اي interval بتحقق خاصية 4 اللي
+
+361
+00:39:08,050 --> 00:39:13,310
+شرحناها هنا الان العكس لو في عندي مجموعة جزئية S
+
+362
+00:39:13,310 --> 00:39:17,530
+من R انا مش عارف انها فترة ولا لا مجرد مجموعة
+
+363
+00:39:17,530 --> 00:39:22,510
+جزئية من R وفيها على الأقل عنصرين تحتوي على الأقل
+
+364
+00:39:22,510 --> 00:39:23,650
+عنصرين او اكتر
+
+365
+00:39:26,310 --> 00:39:31,210
+وإذا كانت المجموعة الجزئية S هذه بتحقق خاصية أربعة
+
+366
+00:39:31,210 --> 00:39:38,390
+فلازم هذه الست S تطلع فترة واحدة من الفترات
+
+367
+00:39:38,390 --> 00:39:43,970
+التمانية وبرهان
+
+368
+00:39:43,970 --> 00:39:54,130
+النظرية هذه برضه برهان طويل شوية لكن بعتمد
+
+369
+00:39:54,130 --> 00:40:01,730
+على أربعةأربع حالات فمن
+
+370
+00:40:01,730 --> 00:40:08,250
+جسم البرهان إلى أربع حالات الحالة الأولى ان ال set
+
+371
+00:40:08,250 --> 00:40:14,070
+S بتكون bounded محصورة ال set بتكون bounded إذا
+
+372
+00:40:14,070 --> 00:40:19,030
+كانت bounded above and bounded below أو S is
+
+373
+00:40:19,030 --> 00:40:24,090
+bounded above but not below يعني محدودا من أعلى
+
+374
+00:40:24,090 --> 00:40:29,760
+وليست محدودا من أسفلوهذا وارد أو S is bounded
+
+375
+00:40:29,760 --> 00:40:33,280
+below but not above يعني محدودا من أسفل وليست
+
+376
+00:40:33,280 --> 00:40:37,520
+محدودا من الأعلى الحالة الرابعة أن S تكون neither
+
+377
+00:40:37,520 --> 00:40:41,180
+bounded above nor below يعني لا محدودا من أسفل ولا
+
+378
+00:40:41,180 --> 00:40:45,200
+محدودا من الأعلى زي R زي مجموعة الأداء من حفظين
+
+379
+00:40:45,200 --> 00:40:52,670
+تمام؟ وفي كل حالة بنا نثبت أن S تطلع فترةفي كل
+
+380
+00:40:52,670 --> 00:40:58,750
+حالة بنا نثبت المقاس تطلع فترة فيعني هاي انا
+
+381
+00:40:58,750 --> 00:41:04,130
+مثبتلكم الحالة الأولى و التانية و الحالتين التالتة
+
+382
+00:41:04,130 --> 00:41:11,330
+و الرابعة ممكن يعني برهانهم بالمثل فحدؤوكم تقرؤوا
+
+383
+00:41:11,330 --> 00:41:16,450
+البرهان لأن هذا البرهان مجرد تفاصيلات حدؤوكم انكم
+
+384
+00:41:16,450 --> 00:41:18,870
+تقرؤوا البرهان تفهموه لأنه مش هشلحهم
+
+385
+00:41:21,750 --> 00:41:25,570
+فحاولوا تقرؤوه تفهموه و انا متأكد انكم هتفهموه
+
+386
+00:41:25,570 --> 00:41:28,910
+فنتوقف
+
+387
+00:41:28,910 --> 00:41:33,130
+عند هذا .. نكتفي بهذا القدر و ان شاء الله المرة
+
+388
+00:41:33,130 --> 00:41:36,850
+الجاية هنتحدث
+
+389
+00:41:36,850 --> 00:41:43,510
+عن موضوع nested intervals okay؟ في حد عنده اي سؤال
+
+390
+00:41:43,510 --> 00:41:44,330
+او صف سؤال؟
+
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/t35SQVeADNY_raw.json b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/t35SQVeADNY_raw.json
new file mode 100644
index 0000000000000000000000000000000000000000..19de4ccc2f59e805dd1aa9e1192f45664743d408
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/t35SQVeADNY_raw.json
@@ -0,0 +1 @@
+{"segments": [{"id": 1, "seek": 4698, "start": 21.4, "end": 46.98, "text": "المرة اللى فاتت احنا يعني اخدنا ال Archimedean property وشوفنا فيه صور متعددة لها او صور مختلفة اللى هي الصور التلاتة هذه فكل واحدة من الخواصة دى تعتبر Archimedean property او مكافئة لخاصية Archimedes او Archimedes", "tokens": [45340, 25720, 13672, 7578, 6156, 9307, 2655, 1975, 5016, 8315, 37495, 22653, 1975, 9778, 3215, 8315, 2423, 10984, 332, 4858, 282, 4707, 4032, 8592, 38688, 8315, 8978, 3224, 20328, 13063, 44650, 22488, 41891, 5296, 11296, 1975, 2407, 20328, 13063, 3714, 46456, 46538, 3660, 13672, 7578, 39896, 31767, 13063, 16712, 1211, 9307, 3660, 29538, 6156, 28820, 36764, 24401, 3660, 9154, 33962, 2407, 33546, 3660, 11778, 7578, 6055, 34268, 26890, 10984, 332, 4858, 282, 4707, 1975, 2407, 3714, 4117, 31845, 19986, 3660, 5296, 9778, 33546, 10632, 10984, 332, 48490, 1975, 2407, 10984, 332, 48490], "avg_logprob": -0.12584004863615958, "compression_ratio": 1.7268041237113403, "no_speech_prob": 0.0, 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أهم واحدة في الصور التلاتة اللي هي الخاصية دي D اللي بتقول أو بتنص على أنه لأي عدد حقيقي موج ابواي بقدر ألاقي عدد طبيعي مقلوبه أصغر من العدد الموج ابواي", "tokens": [2407, 5551, 16095, 36764, 24401, 3660, 8978, 31767, 13063, 16712, 1211, 9307, 3660, 13672, 1829, 39896, 33962, 33546, 10632, 11778, 1829, 413, 13672, 1829, 39894, 39648, 34051, 39894, 1863, 9381, 15844, 14739, 3224, 5296, 10721, 1829, 6225, 3215, 3215, 11331, 38436, 38436, 3714, 29245, 48127, 14407, 1829, 4724, 28543, 2288, 5551, 15040, 38436, 6225, 3215, 3215, 23032, 21292, 3615, 1829, 3714, 4587, 1211, 37746, 3224, 5551, 9381, 17082, 2288, 9154, 18863, 3215, 3215, 9673, 29245, 48127, 14407, 1829], "avg_logprob": -0.15812895569620253, "compression_ratio": 1.7579617834394905, "no_speech_prob": 0.0, "words": [{"start": 47.77, "end": 48.01, "word": "و", "probability": 0.9189453125}, {"start": 48.01, "end": 48.35, "word": " أهم", "probability": 0.7452392578125}, {"start": 48.35, "end": 48.67, "word": " واحدة", "probability": 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تربية أكبر من أربعة صح؟ و أربعة أكبر من الأتنين طب T0 هذا ينتمي ل S", "tokens": [3555, 3215, 1211, 5296, 28820, 314, 8978, 318, 7251, 29245, 3215, 314, 15, 8978, 318, 1270, 300, 314, 15, 47525, 1211, 5551, 9381, 17082, 2288, 9154, 34051, 7251, 3794, 995, 45865, 4724, 1829, 9381, 13546, 5551, 4117, 26890, 9154, 16247, 2655, 1863, 9957, 23758, 4724, 1829, 4587, 16254, 14739, 314, 15, 6055, 25513, 10632, 5551, 4117, 26890, 9154, 5551, 25513, 27884, 20328, 5016, 22807, 4032, 5551, 25513, 27884, 5551, 4117, 26890, 9154, 16247, 2655, 1863, 9957, 23032, 3555, 314, 15, 23758, 7251, 29399, 2304, 1829, 5296, 318], "avg_logprob": -0.21093749898401173, "compression_ratio": 1.670731707317073, "no_speech_prob": 0.0, "words": [{"start": 359.47, "end": 359.99, "word": "بدل", "probability": 0.6387532552083334}, {"start": 359.99, "end": 360.37, "word": " لكل", "probability": 0.6741943359375}, {"start": 360.37, "end": 360.59, "word": " T", "probability": 0.5703125}, {"start": 360.59, "end": 360.85, 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between any two real numbers there exists a rational number او", "tokens": [5172, 1211, 31377, 2304, 29538, 27188, 2655, 25720, 4032, 41185, 29538, 27188, 2655, 25720, 32771, 13546, 16490, 6027, 10632, 6156, 1211, 31377, 2304, 8717, 15040, 38436, 497, 7251, 29399, 2304, 1829, 24976, 5172, 2655, 25720, 9673, 5172, 2655, 2407, 5016, 3660, 4032, 497, 23758, 6225, 3215, 3215, 8717, 38251, 3660, 1975, 15730, 2423, 10305, 20904, 4724, 28820, 4724, 3794, 41193, 3660, 39894, 39648, 1296, 604, 732, 957, 3547, 456, 8198, 257, 15090, 1230, 1975, 2407], "avg_logprob": -0.19500410968535825, "compression_ratio": 1.5183246073298429, "no_speech_prob": 0.0, "words": [{"start": 1106.1, "end": 1107.02, "word": "فلازم", "probability": 0.779541015625}, {"start": 1107.02, "end": 1107.32, "word": " هذه", "probability": 0.87646484375}, {"start": 1107.32, "end": 1107.82, "word": " الفترة", "probability": 0.98583984375}, {"start": 1107.82, "end": 1108.5, "word": " وفي", "probability": 0.4307861328125}, {"start": 1108.5, "end": 1108.72, "word": " هذه", "probability": 0.93603515625}, {"start": 1108.72, "end": 1109.34, "word": " الفترة", "probability": 0.95654296875}, {"start": 1109.34, "end": 1109.64, "word": " غير", "probability": 0.935546875}, {"start": 1109.64, "end": 1110.2, "word": " خالية", "probability": 0.9866536458333334}, {"start": 1110.2, "end": 1112.3, "word": " فلازم", "probability": 0.8602294921875}, {"start": 1112.3, "end": 1112.98, "word": " نلاقي", "probability": 0.9269205729166666}, {"start": 1112.98, "end": 1114.1, "word": " R", "probability": 0.5947265625}, {"start": 1114.1, "end": 1115.88, "word": " ينتمي", "probability": 0.78717041015625}, {"start": 1115.88, "end": 1116.46, "word": " للفترة", "probability": 0.9232177734375}, {"start": 1116.46, "end": 1117.1, "word": " المفتوحة", "probability": 0.9742024739583334}, {"start": 1117.1, "end": 1117.24, "word": " و", "probability": 0.57470703125}, {"start": 1117.24, "end": 1117.66, "word": " R", "probability": 0.416748046875}, {"start": 1117.66, "end": 1118.12, "word": " هذا", "probability": 0.736328125}, {"start": 1118.12, "end": 1119.28, "word": " عدد", "probability": 0.9593098958333334}, {"start": 1119.28, "end": 1120.02, "word": " نسمة", "probability": 0.6700846354166666}, {"start": 1120.02, "end": 1120.98, "word": " اذا", "probability": 0.58984375}, {"start": 1120.98, "end": 1121.7, "word": " ال", "probability": 0.95751953125}, {"start": 1121.7, "end": 1122.1, "word": " density", "probability": 0.91357421875}, {"start": 1122.1, "end": 1122.52, "word": " theorem", "probability": 0.87255859375}, {"start": 1122.52, "end": 1122.9, "word": " بكل", "probability": 0.91064453125}, {"start": 1122.9, "end": 1123.38, "word": " بساطة", "probability": 0.9876708984375}, {"start": 1123.38, "end": 1123.86, "word": " بتقول", "probability": 0.95654296875}, {"start": 1123.86, "end": 1124.84, "word": " between", "probability": 0.88818359375}, {"start": 1124.84, "end": 1125.24, "word": " any", "probability": 0.9140625}, {"start": 1125.24, "end": 1125.52, "word": " two", "probability": 0.8974609375}, {"start": 1125.52, "end": 1125.8, "word": " real", "probability": 0.9658203125}, {"start": 1125.8, "end": 1126.44, "word": " numbers", "probability": 0.8955078125}, {"start": 1126.44, "end": 1127.18, "word": " there", "probability": 0.8720703125}, {"start": 1127.18, "end": 1127.64, "word": " exists", "probability": 0.4853515625}, {"start": 1127.64, "end": 1127.84, "word": " a", "probability": 0.685546875}, {"start": 1127.84, "end": 1128.16, "word": " rational", "probability": 0.919921875}, {"start": 1128.16, "end": 1128.68, "word": " number", "probability": 0.951171875}, {"start": 1128.68, "end": 1129.1, "word": " او", "probability": 0.818603515625}], "temperature": 1.0}, {"id": 43, "seek": 115139, "start": 1130.49, "end": 1151.39, "text": "Any non-empty open interval contains a rational number نفس الصيرة أو نفس ال statement البرهان مش صعب تعالى نشوف البرهان أنا عندي فرض ان X أصغر من Y فرض X أصغر من Y", "tokens": [29647, 2107, 12, 4543, 88, 1269, 15035, 8306, 257, 15090, 1230, 8717, 36178, 31767, 48923, 34051, 8717, 36178, 2423, 5629, 2423, 26890, 3224, 7649, 37893, 20328, 3615, 3555, 37279, 6027, 7578, 8717, 8592, 38688, 2423, 26890, 3224, 7649, 41850, 18871, 16254, 6156, 43042, 16472, 1783, 5551, 9381, 17082, 2288, 9154, 398, 6156, 43042, 1783, 5551, 9381, 17082, 2288, 9154, 398], "avg_logprob": -0.20350922326572607, "compression_ratio": 1.4181818181818182, "no_speech_prob": 0.0, "words": [{"start": 1130.49, "end": 1130.99, "word": "Any", "probability": 0.334716796875}, {"start": 1130.99, "end": 1131.47, "word": " non", "probability": 0.8525390625}, {"start": 1131.47, "end": 1131.97, "word": "-empty", "probability": 0.888671875}, {"start": 1131.97, "end": 1132.29, "word": " open", "probability": 0.85888671875}, {"start": 1132.29, "end": 1132.89, "word": " interval", "probability": 0.982421875}, {"start": 1132.89, "end": 1133.81, "word": " contains", "probability": 0.814453125}, {"start": 1133.81, "end": 1134.03, "word": " a", "probability": 0.82177734375}, {"start": 1134.03, "end": 1134.39, "word": " rational", "probability": 0.93212890625}, {"start": 1134.39, "end": 1135.03, "word": " number", "probability": 0.9560546875}, {"start": 1135.03, "end": 1135.47, "word": " نفس", "probability": 0.6771240234375}, {"start": 1135.47, "end": 1136.03, "word": " الصيرة", "probability": 0.5677490234375}, {"start": 1136.03, "end": 1137.05, "word": " أو", "probability": 0.79736328125}, {"start": 1137.05, "end": 1137.51, "word": " نفس", "probability": 0.9638671875}, {"start": 1137.51, "end": 1137.87, "word": " ال", "probability": 0.73046875}, {"start": 1137.87, "end": 1138.47, "word": " statement", "probability": 0.72021484375}, {"start": 1138.47, "end": 1141.35, "word": " البرهان", "probability": 0.83050537109375}, {"start": 1141.35, "end": 1141.63, "word": " مش", "probability": 0.890625}, {"start": 1141.63, "end": 1142.17, "word": " صعب", "probability": 0.9490559895833334}, {"start": 1142.17, "end": 1145.21, "word": " تعالى", "probability": 0.783203125}, {"start": 1145.21, "end": 1145.49, "word": " نشوف", "probability": 0.9933268229166666}, {"start": 1145.49, "end": 1146.27, "word": " البرهان", "probability": 0.9635009765625}, {"start": 1146.27, "end": 1146.95, "word": " أنا", "probability": 0.33544921875}, {"start": 1146.95, "end": 1147.33, "word": " عندي", "probability": 0.83642578125}, {"start": 1147.33, "end": 1147.73, "word": " فرض", "probability": 0.81298828125}, {"start": 1147.73, "end": 1147.93, "word": " ان", "probability": 0.599609375}, {"start": 1147.93, "end": 1148.33, "word": " X", "probability": 0.48681640625}, {"start": 1148.33, "end": 1148.83, "word": " أصغر", "probability": 0.9298095703125}, {"start": 1148.83, "end": 1149.01, "word": " من", "probability": 0.986328125}, {"start": 1149.01, "end": 1149.33, "word": " Y", "probability": 0.9912109375}, {"start": 1149.33, "end": 1150.05, "word": " فرض", "probability": 0.82373046875}, {"start": 1150.05, "end": 1150.41, "word": " X", "probability": 0.95654296875}, {"start": 1150.41, "end": 1150.89, "word": " أصغر", "probability": 0.9876708984375}, {"start": 1150.89, "end": 1151.09, "word": " من", "probability": 0.99072265625}, {"start": 1151.09, "end": 1151.39, "word": " Y", "probability": 0.9951171875}], "temperature": 1.0}, {"id": 44, "seek": 117971, "start": 1152.97, "end": 1179.71, "text": "و بالتالي y سالب x عدد موجب اذا by the Archimedean property بقدر الاقي عدد هنا المفروض يكون جزء b فby Archimedean property واحد ستاشر بيه اللي اعطيناها إشارة نجمة لأي عدد موجب زي هذا يوجد عدد طبيعي n مقلوب وأصغر من عدد الموجب", "tokens": [2407, 20666, 2655, 6027, 1829, 288, 8608, 6027, 3555, 2031, 6225, 3215, 3215, 3714, 29245, 3555, 1975, 15730, 538, 264, 10984, 332, 4858, 282, 4707, 4724, 28543, 2288, 1975, 15040, 38436, 6225, 3215, 3215, 34105, 9673, 5172, 32887, 11242, 7251, 30544, 10874, 11622, 38207, 272, 6156, 2322, 10984, 332, 4858, 282, 4707, 36764, 24401, 8608, 2655, 33599, 2288, 4724, 1829, 3224, 13672, 1829, 1975, 3615, 9566, 1829, 8315, 11296, 11933, 8592, 9640, 3660, 8717, 7435, 46007, 5296, 10721, 1829, 6225, 3215, 3215, 3714, 29245, 3555, 30767, 1829, 23758, 7251, 29245, 3215, 6225, 3215, 3215, 23032, 21292, 3615, 1829, 297, 3714, 4587, 1211, 37746, 36725, 9381, 17082, 2288, 9154, 6225, 3215, 3215, 9673, 29245, 3555], "avg_logprob": -0.2067934723004051, "compression_ratio": 1.6422018348623852, "no_speech_prob": 0.0, "words": [{"start": 1152.97, "end": 1153.19, "word": "و", "probability": 0.44091796875}, {"start": 1153.19, "end": 1153.83, "word": " بالتالي", "probability": 0.82403564453125}, {"start": 1153.83, "end": 1154.25, "word": " y", "probability": 0.367919921875}, {"start": 1154.25, "end": 1154.91, "word": " سالب", "probability": 0.6988932291666666}, {"start": 1154.91, "end": 1155.17, "word": " x", "probability": 0.8916015625}, {"start": 1155.17, "end": 1155.53, "word": " عدد", "probability": 0.9612630208333334}, {"start": 1155.53, "end": 1156.01, "word": " موجب", "probability": 0.9912109375}, {"start": 1156.01, "end": 1156.35, "word": " اذا", "probability": 0.58746337890625}, {"start": 1156.35, "end": 1156.59, "word": " by", "probability": 0.76123046875}, {"start": 1156.59, "end": 1156.79, "word": " the", "probability": 0.7763671875}, {"start": 1156.79, "end": 1157.55, "word": " Archimedean", "probability": 0.78228759765625}, {"start": 1157.55, "end": 1158.29, "word": " property", "probability": 0.88623046875}, {"start": 1158.29, "end": 1158.89, "word": " بقدر", "probability": 0.7880859375}, {"start": 1158.89, "end": 1159.31, "word": " الاقي", "probability": 0.641845703125}, {"start": 1159.31, "end": 1159.79, "word": " عدد", "probability": 0.9881184895833334}, {"start": 1159.79, "end": 1159.93, "word": " هنا", "probability": 0.09722900390625}, {"start": 1159.93, "end": 1160.33, "word": " المفروض", "probability": 0.921142578125}, {"start": 1160.33, "end": 1160.87, "word": " يكون", "probability": 0.94677734375}, {"start": 1160.87, "end": 1163.23, "word": " جزء", "probability": 0.921875}, {"start": 1163.23, "end": 1163.71, "word": " b", "probability": 0.319580078125}, {"start": 1163.71, "end": 1168.05, "word": " فby", "probability": 0.544921875}, {"start": 1168.05, "end": 1168.89, "word": " Archimedean", "probability": 0.9468994140625}, {"start": 1168.89, "end": 1169.47, "word": " property", "probability": 0.95263671875}, {"start": 1169.47, "end": 1169.77, "word": " واحد", "probability": 0.969482421875}, {"start": 1169.77, "end": 1170.37, "word": " ستاشر", "probability": 0.8624267578125}, {"start": 1170.37, "end": 1170.63, "word": " بيه", "probability": 0.598876953125}, {"start": 1170.63, "end": 1170.77, "word": " اللي", "probability": 0.96142578125}, {"start": 1170.77, "end": 1171.59, "word": " اعطيناها", "probability": 0.7474772135416666}, {"start": 1171.59, "end": 1172.05, "word": " إشارة", "probability": 0.84637451171875}, {"start": 1172.05, "end": 1172.49, "word": " نجمة", "probability": 0.9786783854166666}, {"start": 1172.49, "end": 1174.27, "word": " لأي", "probability": 0.91552734375}, {"start": 1174.27, "end": 1174.65, "word": " عدد", "probability": 0.9959309895833334}, {"start": 1174.65, "end": 1175.11, "word": " موجب", "probability": 0.9949544270833334}, {"start": 1175.11, "end": 1175.39, "word": " زي", "probability": 0.965087890625}, {"start": 1175.39, "end": 1175.73, "word": " هذا", "probability": 0.94580078125}, {"start": 1175.73, "end": 1176.65, "word": " يوجد", "probability": 0.9703776041666666}, {"start": 1176.65, "end": 1177.05, "word": " عدد", "probability": 0.9943033854166666}, {"start": 1177.05, "end": 1177.67, "word": " طبيعي", "probability": 0.973388671875}, {"start": 1177.67, "end": 1177.93, "word": " n", "probability": 0.5400390625}, {"start": 1177.93, "end": 1178.45, "word": " مقلوب", "probability": 0.9498291015625}, {"start": 1178.45, "end": 1178.83, "word": " وأصغر", "probability": 0.8970947265625}, {"start": 1178.83, "end": 1178.95, "word": " من", "probability": 0.9755859375}, {"start": 1178.95, "end": 1179.21, "word": " عدد", "probability": 0.9026692708333334}, {"start": 1179.21, "end": 1179.71, "word": " الموجب", "probability": 0.9903971354166666}], "temperature": 1.0}, {"id": 45, "seek": 120741, "start": 1181.05, "end": 1207.41, "text": "الان بدي اعرف ان اكبر عدد صحيح اصغر من او يساوي n x هذه دالة اكبر عدد صحيح واضيف عليها واحد طبعا هذا عدد صحيح زائد واحد الواحد عدد صحيح فمجموعهم بطلع عدد صحيح اذا انا عرفت عدد صحيح m ينتمي الى z على انه the greatest integer less than or equal to n x plus one", "tokens": [6027, 7649, 4724, 16254, 1975, 3615, 28480, 16472, 1975, 4117, 26890, 6225, 3215, 3215, 20328, 5016, 1829, 5016, 1975, 9381, 17082, 2288, 9154, 1975, 2407, 7251, 3794, 995, 45865, 297, 2031, 29538, 11778, 6027, 3660, 1975, 4117, 26890, 6225, 3215, 3215, 20328, 5016, 1829, 5016, 4032, 46958, 33911, 25894, 11296, 36764, 24401, 23032, 3555, 3615, 995, 23758, 6225, 3215, 3215, 20328, 5016, 1829, 5016, 30767, 16373, 3215, 36764, 24401, 2423, 14407, 24401, 6225, 3215, 3215, 20328, 5016, 1829, 5016, 6156, 2304, 7435, 2304, 45367, 16095, 4724, 9566, 1211, 3615, 6225, 3215, 3215, 20328, 5016, 1829, 5016, 1975, 15730, 1975, 8315, 6225, 28480, 2655, 6225, 3215, 3215, 20328, 5016, 1829, 5016, 275, 7251, 29399, 2304, 1829, 2423, 7578, 710, 15844, 16472, 3224, 264, 6636, 24922, 1570, 813, 420, 2681, 281, 297, 2031, 1804, 472], "avg_logprob": -0.1436567156394916, "compression_ratio": 1.8311111111111111, "no_speech_prob": 0.0, "words": [{"start": 1181.05, "end": 1181.49, "word": "الان", "probability": 0.61767578125}, {"start": 1181.49, "end": 1181.81, "word": " بدي", "probability": 0.734130859375}, {"start": 1181.81, "end": 1182.45, "word": " اعرف", "probability": 0.82373046875}, {"start": 1182.45, "end": 1183.43, "word": " ان", "probability": 0.43701171875}, {"start": 1183.43, "end": 1185.59, "word": " اكبر", "probability": 0.9386393229166666}, {"start": 1185.59, 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1568.71, "word": " ال", "probability": 0.86376953125}, {"start": 1568.71, "end": 1569.03, "word": " density", "probability": 0.95654296875}, {"start": 1569.03, "end": 1569.49, "word": " theorem", "probability": 0.91552734375}], "temperature": 1.0}, {"id": 58, "seek": 159740, "start": 1570.34, "end": 1597.4, "text": "و في بعض الناس برضه نفسها بيسموها density theorem ف ال density theorem هذه التانية أو النتيجة عليها بتقول ان بين اي between any two real numbers x and y there exists an irrational number يوجد عدد z بين x و y وهذا z irrational number اذا كمان مرة", "tokens": [2407, 8978, 45030, 11242, 2423, 8315, 3794, 4724, 43042, 3224, 8717, 36178, 11296, 4724, 1829, 38251, 2407, 11296, 10305, 20904, 6156, 2423, 10305, 20904, 29538, 16712, 7649, 10632, 34051, 28239, 31371, 7435, 3660, 25894, 11296, 39894, 39648, 16472, 49374, 1975, 1829, 1296, 604, 732, 957, 3547, 2031, 293, 288, 456, 8198, 364, 39914, 1230, 7251, 29245, 3215, 6225, 3215, 3215, 710, 49374, 2031, 4032, 288, 37037, 15730, 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{"start": 1579.1, "end": 1580.6, "word": " between", "probability": 0.62841796875}, {"start": 1580.6, "end": 1581.1, "word": " any", "probability": 0.892578125}, {"start": 1581.1, "end": 1581.36, "word": " two", "probability": 0.89794921875}, {"start": 1581.36, "end": 1581.68, "word": " real", "probability": 0.95849609375}, {"start": 1581.68, "end": 1582.22, "word": " numbers", "probability": 0.78173828125}, {"start": 1582.22, "end": 1582.5, "word": " x", "probability": 0.421630859375}, {"start": 1582.5, "end": 1582.78, "word": " and", "probability": 0.63427734375}, {"start": 1582.78, "end": 1583.18, "word": " y", "probability": 0.98291015625}, {"start": 1583.18, "end": 1583.92, "word": " there", "probability": 0.82958984375}, {"start": 1583.92, "end": 1584.44, "word": " exists", "probability": 0.5859375}, {"start": 1584.44, "end": 1584.96, "word": " an", "probability": 0.83935546875}, {"start": 1584.96, "end": 1585.58, "word": " irrational", "probability": 0.90234375}, {"start": 1585.58, "end": 1586.24, "word": " number", "probability": 0.974609375}, {"start": 1586.24, "end": 1586.82, "word": " يوجد", "probability": 0.7916666666666666}, {"start": 1586.82, "end": 1587.92, "word": " عدد", "probability": 0.9832356770833334}, {"start": 1587.92, "end": 1588.16, "word": " z", "probability": 0.89404296875}, {"start": 1588.16, "end": 1588.44, "word": " بين", "probability": 0.97802734375}, {"start": 1588.44, "end": 1588.74, "word": " x", "probability": 0.8828125}, {"start": 1588.74, "end": 1588.9, "word": " و", "probability": 0.471923828125}, {"start": 1588.9, "end": 1589.12, "word": " y", "probability": 0.57958984375}, {"start": 1589.12, "end": 1589.52, "word": " وهذا", "probability": 0.84521484375}, {"start": 1589.52, "end": 1589.86, "word": " z", "probability": 0.76611328125}, {"start": 1589.86, "end": 1590.38, "word": " irrational", "probability": 0.453125}, {"start": 1590.38, "end": 1591.28, "word": " number", "probability": 0.9716796875}, {"start": 1591.28, "end": 1596.1, "word": " اذا", "probability": 0.6239013671875}, {"start": 1596.1, "end": 1597.02, "word": " كمان", "probability": 0.9423828125}, {"start": 1597.02, "end": 1597.4, "word": " مرة", "probability": 0.98876953125}], "temperature": 1.0}, {"id": 59, "seek": 162617, "start": 1599.29, "end": 1626.17, "text": "لو في عندى اي عددين حقيقيين x و y هو x أصغر من y فبنقدر نلاقي r أو p وهذا ينتمي ل r minus q عدد غير نسبي بين أكبر من x و أصغر من y", "tokens": [1211, 2407, 8978, 43242, 7578, 1975, 1829, 6225, 3215, 3215, 9957, 11331, 38436, 38436, 9957, 2031, 4032, 288, 31439, 2031, 5551, 9381, 17082, 2288, 9154, 288, 6156, 3555, 1863, 28543, 2288, 8717, 15040, 38436, 367, 34051, 280, 37037, 15730, 7251, 29399, 2304, 1829, 5296, 367, 3175, 9505, 6225, 3215, 3215, 32771, 13546, 8717, 35457, 1829, 49374, 5551, 4117, 26890, 9154, 2031, 4032, 5551, 9381, 17082, 2288, 9154, 288], "avg_logprob": -0.25271738266599353, "compression_ratio": 1.4685314685314685, "no_speech_prob": 0.0, "words": [{"start": 1599.29, "end": 1599.63, "word": "لو", "probability": 0.7998046875}, {"start": 1599.63, "end": 1599.87, "word": " في", "probability": 0.25}, {"start": 1599.87, "end": 1600.29, "word": " عندى", "probability": 0.700439453125}, {"start": 1600.29, "end": 1600.53, "word": " اي", "probability": 0.719970703125}, {"start": 1600.53, "end": 1601.01, "word": " عددين", "probability": 0.9500732421875}, {"start": 1601.01, "end": 1601.91, "word": " حقيقيين", "probability": 0.887939453125}, {"start": 1601.91, "end": 1603.23, "word": " x", "probability": 0.379150390625}, {"start": 1603.23, "end": 1603.49, "word": " و", "probability": 0.76611328125}, {"start": 1603.49, "end": 1603.87, "word": " y", "probability": 0.857421875}, {"start": 1603.87, "end": 1607.17, "word": " هو", "probability": 0.33544921875}, {"start": 1607.17, "end": 1607.53, "word": " x", "probability": 0.93701171875}, {"start": 1607.53, "end": 1608.07, "word": " أصغر", "probability": 0.8642578125}, {"start": 1608.07, "end": 1608.29, "word": " من", "probability": 0.9912109375}, {"start": 1608.29, "end": 1608.75, "word": " y", "probability": 0.9814453125}, {"start": 1608.75, "end": 1609.57, "word": " فبنقدر", "probability": 0.79658203125}, {"start": 1609.57, "end": 1610.15, "word": " نلاقي", "probability": 0.8792317708333334}, {"start": 1610.15, "end": 1610.71, "word": " r", "probability": 0.3193359375}, {"start": 1610.71, "end": 1612.23, "word": " أو", "probability": 0.60498046875}, {"start": 1612.23, "end": 1613.47, "word": " p", "probability": 0.432861328125}, {"start": 1613.47, "end": 1615.29, "word": " وهذا", "probability": 0.756103515625}, {"start": 1615.29, "end": 1616.01, "word": " ينتمي", "probability": 0.900146484375}, {"start": 1616.01, "end": 1616.25, "word": " ل", "probability": 0.83447265625}, {"start": 1616.25, "end": 1616.83, "word": " r", "probability": 0.1873779296875}, {"start": 1616.83, "end": 1617.93, "word": " minus", "probability": 0.6875}, {"start": 1617.93, "end": 1618.45, "word": " q", "probability": 0.73046875}, {"start": 1618.45, "end": 1620.47, "word": " عدد", "probability": 0.9822591145833334}, {"start": 1620.47, "end": 1621.03, "word": " غير", "probability": 0.994873046875}, {"start": 1621.03, "end": 1621.87, "word": " نسبي", "probability": 0.7879231770833334}, {"start": 1621.87, "end": 1623.39, "word": " بين", "probability": 0.354736328125}, {"start": 1623.39, "end": 1623.83, "word": " أكبر", "probability": 0.9488932291666666}, {"start": 1623.83, "end": 1624.03, "word": " من", "probability": 0.9951171875}, {"start": 1624.03, "end": 1624.41, "word": " x", "probability": 0.9697265625}, {"start": 1624.41, "end": 1625.15, "word": " و", "probability": 0.52734375}, {"start": 1625.15, "end": 1625.67, "word": " أصغر", "probability": 0.9482421875}, {"start": 1625.67, "end": 1625.89, "word": " من", "probability": 0.99658203125}, {"start": 1625.89, "end": 1626.17, "word": " y", "probability": 0.98291015625}], "temperature": 1.0}, {"id": 60, "seek": 165335, "start": 1627.39, "end": 1653.35, "text": "البرهان تبع النتيجة هذه اول شي احنا لسه مثبتين قبل شوية ان جدر اتنين is a real number جدر اتنين كان هو supremum ل 6 وبالتالي exist in R واثبتنا ان جدر اتنين هذا اكبر من صفر حتى اثبتنا انه اكبر من واحد كان ال X اكبر من واحد وبالتالي موجة", "tokens": [6027, 26890, 3224, 7649, 6055, 3555, 3615, 28239, 31371, 7435, 3660, 29538, 1975, 12610, 44049, 1975, 5016, 8315, 5296, 3794, 3224, 3714, 12984, 3555, 2655, 9957, 12174, 36150, 13412, 2407, 10632, 16472, 10874, 3215, 2288, 1975, 2655, 1863, 9957, 307, 257, 957, 1230, 10874, 3215, 2288, 1975, 2655, 1863, 9957, 25961, 31439, 23710, 449, 5296, 1386, 46599, 6027, 2655, 6027, 1829, 2514, 294, 497, 4032, 5718, 104, 3555, 2655, 8315, 16472, 10874, 3215, 2288, 1975, 2655, 1863, 9957, 23758, 1975, 4117, 26890, 9154, 20328, 5172, 2288, 11331, 49975, 1975, 12984, 3555, 2655, 8315, 16472, 3224, 1975, 4117, 26890, 9154, 36764, 24401, 25961, 2423, 1783, 1975, 4117, 26890, 9154, 36764, 24401, 46599, 6027, 2655, 6027, 1829, 3714, 29245, 3660], "avg_logprob": -0.1646533678559696, "compression_ratio": 1.8888888888888888, "no_speech_prob": 0.0, "words": [{"start": 1627.3899999999999, "end": 1628.07, "word": "البرهان", "probability": 0.8717041015625}, {"start": 1628.07, "end": 1628.33, "word": " تبع", "probability": 0.6950276692708334}, {"start": 1628.33, "end": 1628.97, "word": " النتيجة", "probability": 0.9439697265625}, {"start": 1628.97, "end": 1629.43, "word": " هذه", "probability": 0.6259765625}, {"start": 1629.43, "end": 1632.95, "word": " اول", "probability": 0.6727294921875}, {"start": 1632.95, "end": 1633.13, "word": " شي", "probability": 0.630859375}, {"start": 1633.13, "end": 1633.37, "word": " احنا", "probability": 0.9358723958333334}, {"start": 1633.37, "end": 1633.67, "word": " لسه", "probability": 0.91552734375}, {"start": 1633.67, "end": 1634.41, "word": " مثبتين", "probability": 0.92685546875}, {"start": 1634.41, "end": 1635.29, "word": " قبل", "probability": 0.836181640625}, {"start": 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"end": 1641.59, "word": " supremum", "probability": 0.6845703125}, {"start": 1641.59, "end": 1641.75, "word": " ل", "probability": 0.9189453125}, {"start": 1641.75, "end": 1642.11, "word": " 6", "probability": 0.44775390625}, {"start": 1642.11, "end": 1642.73, "word": " وبالتالي", "probability": 0.8755859375}, {"start": 1642.73, "end": 1643.13, "word": " exist", "probability": 0.82177734375}, {"start": 1643.13, "end": 1643.39, "word": " in", "probability": 0.884765625}, {"start": 1643.39, "end": 1643.91, "word": " R", "probability": 0.94189453125}, {"start": 1643.91, "end": 1645.89, "word": " واثبتنا", "probability": 0.8498942057291666}, {"start": 1645.89, "end": 1646.05, "word": " ان", "probability": 0.85400390625}, {"start": 1646.05, "end": 1646.37, "word": " جدر", "probability": 0.9879557291666666}, {"start": 1646.37, "end": 1646.67, "word": " اتنين", "probability": 0.985595703125}, {"start": 1646.67, "end": 1646.87, "word": " هذا", "probability": 0.3408203125}, {"start": 1646.87, "end": 1647.19, "word": " اكبر", "probability": 0.9231770833333334}, {"start": 1647.19, "end": 1647.35, "word": " من", "probability": 0.99609375}, {"start": 1647.35, "end": 1647.69, "word": " صفر", "probability": 0.7832845052083334}, {"start": 1647.69, "end": 1648.03, "word": " حتى", "probability": 0.6419677734375}, {"start": 1648.03, "end": 1648.73, "word": " اثبتنا", "probability": 0.97783203125}, {"start": 1648.73, "end": 1648.93, "word": " انه", "probability": 0.80859375}, {"start": 1648.93, "end": 1649.21, "word": " اكبر", "probability": 0.9560546875}, {"start": 1649.21, "end": 1649.37, "word": " من", "probability": 0.99560546875}, {"start": 1649.37, "end": 1649.71, "word": " واحد", "probability": 0.95751953125}, {"start": 1649.71, "end": 1649.99, "word": " كان", "probability": 0.8876953125}, {"start": 1649.99, "end": 1650.17, "word": " ال", "probability": 0.931640625}, {"start": 1650.17, "end": 1650.43, "word": " X", "probability": 0.78271484375}, {"start": 1650.43, "end": 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diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/tKghhKfPOf0_raw.json b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/tKghhKfPOf0_raw.json
new file mode 100644
index 0000000000000000000000000000000000000000..f4e4255ab12cf9a178ce885f728293d188d9a517
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/tKghhKfPOf0_raw.json
@@ -0,0 +1 @@
+{"segments": [{"id": 1, "seek": 4610, "start": 21.68, "end": 46.1, "text": "بسم الله الرحمن الرحيم في اللقاء هذا هحنحاول نكمل ما بدأناه في section خمسة اتنين اخر نظرية أخدناها في المحاضرة السابقة كانت ال boundedness theorem بتقول لو كان في اندي اف function continuous على ال interval I", "tokens": [3555, 38251, 21984, 34892, 5016, 27842, 34892, 5016, 32640, 8978, 13672, 4587, 16606, 23758, 8032, 5016, 1863, 5016, 995, 12610, 8717, 24793, 1211, 19446, 47525, 10721, 8315, 3224, 8978, 3541, 16490, 2304, 3794, 3660, 1975, 2655, 1863, 9957, 1975, 34740, 8717, 19913, 2288, 10632, 5551, 9778, 3215, 8315, 11296, 8978, 9673, 5016, 46958, 25720, 21136, 16758, 28671, 25961, 2655, 2423, 37498, 1287, 20904, 39894, 39648, 45164, 25961, 8978, 16472, 16254, 1975, 5172, 2445, 10957, 15844, 2423, 15035, 286], "avg_logprob": -0.14388845389402366, "compression_ratio": 1.5348837209302326, "no_speech_prob": 0.0, "words": [{"start": 21.68, "end": 22.02, "word": "بسم", "probability": 0.886962890625}, 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ناخد ال identity function f و ناخد ال I هي الفترة المغلقة من صفر إلى ملنهاي واضح ان الفترة I هذه ماهياش bounded", "tokens": [5172, 6027, 1863, 19913, 2288, 10632, 39894, 5172, 8592, 1211, 4032, 8032, 47302, 50113, 6027, 7251, 2407, 11242, 5016, 29910, 23275, 8717, 47283, 3215, 2423, 6575, 2445, 283, 4032, 8717, 47283, 3215, 2423, 286, 39896, 27188, 2655, 25720, 9673, 17082, 1211, 28671, 9154, 20328, 5172, 2288, 30731, 3714, 1211, 1863, 11296, 1829, 4032, 46958, 5016, 16472, 27188, 2655, 25720, 286, 29538, 19446, 3224, 1829, 33599, 37498], "avg_logprob": -0.28148319739014355, "compression_ratio": 1.4451219512195121, "no_speech_prob": 0.0, "words": [{"start": 98.58, "end": 99.58, "word": "فالنظرية", "probability": 0.8167317708333334}, {"start": 99.58, "end": 100.22, "word": " بتفشل", "probability": 0.8359375}, {"start": 100.22, "end": 100.7, "word": " و", "probability": 0.1844482421875}, {"start": 100.7, "end": 100.88, "word": " هاي", "probability": 0.3360595703125}, {"start": 100.88, 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مغلقة فع سبيل المثال consider الـ function واحد على X حيث X أكبر من سفر و أصغر من أو ساول واحد فواضح إن الفترة I هذه not closed ليست مغلقة و الدالة G of X متصلة عليها لأنها لا تحتوى السفر لأن هذه عندي function G", "tokens": [28814, 15730, 25961, 2655, 2423, 39184, 15035, 19446, 3224, 1829, 33599, 3714, 17082, 1211, 28671, 6156, 3615, 8608, 21292, 1211, 9673, 12984, 6027, 1949, 2423, 39184, 2445, 36764, 24401, 15844, 1783, 11331, 1829, 12984, 1783, 5551, 4117, 26890, 9154, 8608, 5172, 2288, 4032, 5551, 9381, 17082, 2288, 9154, 34051, 8608, 995, 12610, 36764, 24401, 6156, 2407, 46958, 5016, 36145, 27188, 2655, 25720, 286, 29538, 406, 5395, 32239, 14851, 3714, 17082, 1211, 28671, 4032, 32748, 6027, 3660, 460, 295, 1783, 44650, 36520, 3660, 25894, 11296, 5296, 33456, 11296, 20193, 6055, 33753, 2407, 7578, 21136, 5172, 2288, 5296, 33456, 29538, 18871, 16254, 2445, 460], "avg_logprob": -0.24347694521968805, "compression_ratio": 1.5809128630705394, "no_speech_prob": 0.0, "words": [{"start": 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{"start": 228.65, "end": 229.23, "word": " function", "probability": 0.947265625}, {"start": 229.23, "end": 229.65, "word": " G", "probability": 0.927734375}], "temperature": 1.0}, {"id": 9, "seek": 25739, "start": 230.62, "end": 257.4, "text": "continuous على الفترة I الفترة I ماهياش closed وبالتالي وفي نفس الوجهة ال function هذه ليست bounded على الفترة I احنا شفنا ان ال function 1 على X في بداية اللقاء اللي فات ليست bounded على الفترة من صفر إلى ملا نهاية", "tokens": [9000, 259, 12549, 15844, 27188, 2655, 25720, 286, 27188, 2655, 25720, 286, 19446, 3224, 1829, 33599, 5395, 46599, 6027, 2655, 6027, 1829, 4032, 41185, 8717, 36178, 2423, 29245, 3224, 3660, 2423, 2445, 29538, 32239, 14851, 37498, 15844, 27188, 2655, 25720, 286, 1975, 5016, 8315, 13412, 5172, 8315, 16472, 2423, 2445, 502, 15844, 1783, 8978, 4724, 28259, 10632, 13672, 4587, 16606, 13672, 1829, 6156, 9307, 32239, 14851, 37498, 15844, 27188, 2655, 25720, 9154, 20328, 5172, 2288, 30731, 3714, 15040, 8717, 11296, 10632], 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"word": " على", "probability": 0.9228515625}, {"start": 253.88, "end": 254.72, "word": " الفترة", "probability": 0.9889322916666666}, {"start": 254.72, "end": 255.94, "word": " من", "probability": 0.97705078125}, {"start": 255.94, "end": 256.42, "word": " صفر", "probability": 0.7579752604166666}, {"start": 256.42, "end": 256.6, "word": " إلى", "probability": 0.83984375}, {"start": 256.6, "end": 256.86, "word": " ملا", "probability": 0.798095703125}, {"start": 256.86, "end": 257.4, "word": " نهاية", "probability": 0.89453125}], "temperature": 1.0}, {"id": 10, "seek": 28781, "start": 259.03, "end": 287.81, "text": "و بالمثل ليست bounded على الفترة من صفر إلى واحد ممكن اعطاء برهان مشابه، إذا هذا مثال على function متصرة على فترة ولكنها ليست bounded على نفس الفترة لأن الفترة هذه السبب أن الفترة هذه ليست closed، تمام؟ إذا الفترة هذه لو ماكانتش closed، النظرية بتفشل", "tokens": [2407, 20666, 2304, 12984, 1211, 32239, 14851, 37498, 15844, 27188, 2655, 25720, 9154, 20328, 5172, 2288, 30731, 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"probability": 0.94140625}, {"start": 285.57, "end": 285.91, "word": " لو", "probability": 0.83740234375}, {"start": 285.91, "end": 286.41, "word": " ماكانتش", "probability": 0.835693359375}, {"start": 286.41, "end": 286.87, "word": " closed،", "probability": 0.7998046875}, {"start": 286.87, "end": 287.19, "word": " النظرية", "probability": 0.8641357421875}, {"start": 287.19, "end": 287.81, "word": " بتفشل", "probability": 0.9278564453125}], "temperature": 1.0}, {"id": 11, "seek": 31264, "start": 288.44, "end": 312.64, "text": "طب لو كانت closed و لا bounded، برضه بتفشل، لأن هذا الشرط ضروري و هذا الشرط ضروري كذلك ال function لازم تكون continuous على الفترة هذه، يعني لو كانت الفترة هذه closed و bounded، فال function مش continuous، فالنظرية تفشل، لأن هنا the boundedness theorem", "tokens": [9566, 3555, 45164, 25961, 2655, 5395, 4032, 20193, 37498, 12399, 4724, 43042, 3224, 39894, 5172, 8592, 1211, 12399, 5296, 33456, 23758, 25124, 2288, 9566, 48812, 2288, 13063, 1829, 4032, 23758, 25124, 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"end": 473.97, "word": " السفر", "probability": 0.9723307291666666}], "temperature": 1.0}, {"id": 17, "seek": 50352, "start": 474.68, "end": 503.52, "text": "فى نفس الواجهة Note that الفترة I بساوي سفر و واحد is closed and bounded يعني أنا مافيش عند التصال عند السفر ال function هذه H طبعا زى ما ال function هذيك", "tokens": [5172, 7578, 8717, 36178, 2423, 2407, 26108, 3224, 3660, 11633, 300, 27188, 2655, 25720, 286, 4724, 3794, 995, 45865, 8608, 5172, 2288, 4032, 36764, 24401, 307, 5395, 293, 37498, 37495, 22653, 41850, 19446, 41185, 8592, 43242, 16712, 9381, 6027, 43242, 21136, 5172, 2288, 2423, 2445, 29538, 389, 23032, 3555, 3615, 995, 30767, 7578, 19446, 2423, 2445, 8032, 8848, 1829, 4117], "avg_logprob": -0.3140368715661471, "compression_ratio": 1.4036144578313252, "no_speech_prob": 0.0, "words": [{"start": 474.68, "end": 476.1, "word": "فى", "probability": 0.5806884765625}, {"start": 476.1, "end": 476.34, "word": " نفس", "probability": 0.988037109375}, {"start": 476.34, "end": 477.68, "word": " الواجهة", "probability": 0.78955078125}, {"start": 477.68, "end": 480.08, "word": " Note", "probability": 0.43701171875}, {"start": 480.08, "end": 480.68, "word": " that", "probability": 0.80224609375}, {"start": 480.68, "end": 484.36, "word": " الفترة", "probability": 0.939453125}, {"start": 484.36, "end": 484.84, "word": " I", "probability": 0.78369140625}, {"start": 484.84, "end": 486.68, "word": " بساوي", "probability": 0.734130859375}, {"start": 486.68, "end": 487.48, "word": " سفر", "probability": 0.6990559895833334}, {"start": 487.48, "end": 487.66, "word": " و", "probability": 0.740234375}, {"start": 487.66, "end": 488.26, "word": " واحد", "probability": 0.926025390625}, {"start": 488.26, "end": 489.72, "word": " is", "probability": 0.57421875}, {"start": 489.72, "end": 490.48, "word": " closed", "probability": 0.8447265625}, {"start": 490.48, "end": 492.52, "word": " and", "probability": 0.94140625}, {"start": 492.52, "end": 493.14, "word": " bounded", "probability": 0.9794921875}, {"start": 493.14, "end": 495.54, "word": " يعني", "probability": 0.57293701171875}, {"start": 495.54, "end": 495.74, "word": " أنا", "probability": 0.712890625}, {"start": 495.74, "end": 496.04, "word": " مافيش", "probability": 0.8582356770833334}, {"start": 496.04, "end": 496.26, "word": " عند", "probability": 0.2056884765625}, {"start": 496.26, "end": 496.68, "word": " التصال", "probability": 0.7672526041666666}, {"start": 496.68, "end": 496.88, "word": " عند", "probability": 0.92236328125}, {"start": 496.88, "end": 497.44, "word": " السفر", "probability": 0.9763997395833334}, {"start": 497.44, "end": 500.22, "word": " ال", "probability": 0.45703125}, {"start": 500.22, "end": 500.66, "word": " function", "probability": 0.8984375}, {"start": 500.66, "end": 501.08, "word": " هذه", "probability": 0.541015625}, {"start": 501.08, "end": 501.38, "word": " H", "probability": 0.87255859375}, {"start": 501.38, "end": 502.18, "word": " طبعا", "probability": 0.9149169921875}, {"start": 502.18, "end": 502.42, "word": " زى", "probability": 0.7037353515625}, {"start": 502.42, "end": 502.54, "word": " ما", "probability": 0.890625}, {"start": 502.54, "end": 502.68, "word": " ال", "probability": 0.68701171875}, {"start": 502.68, "end": 502.98, "word": " function", "probability": 0.95166015625}, {"start": 502.98, "end": 503.52, "word": " هذيك", "probability": 0.62725830078125}], "temperature": 1.0}, {"id": 18, "seek": 53080, "start": 504.19, "end": 530.81, "text": "Note also that the function H is unbounded على الفترة من صفر لواحد لأنها unbounded على الفترة المفتوحة H بالساعة واحد على X على الفترة نصف المفتوحة هذه", "tokens": [49250, 611, 300, 264, 2445, 389, 307, 517, 18767, 292, 15844, 27188, 2655, 25720, 9154, 20328, 5172, 2288, 5296, 14407, 24401, 5296, 33456, 11296, 517, 18767, 292, 15844, 27188, 2655, 25720, 9673, 5172, 2655, 2407, 5016, 3660, 389, 20666, 3794, 995, 27884, 36764, 24401, 15844, 1783, 15844, 27188, 2655, 25720, 8717, 9381, 5172, 9673, 5172, 2655, 2407, 5016, 3660, 29538], "avg_logprob": -0.24987192427525756, "compression_ratio": 1.5902777777777777, "no_speech_prob": 0.0, "words": [{"start": 504.19, "end": 504.97, "word": "Note", "probability": 0.341552734375}, {"start": 504.97, "end": 505.57, "word": " also", "probability": 0.529296875}, {"start": 505.57, "end": 509.87, "word": " that", "probability": 0.6650390625}, {"start": 509.87, "end": 510.25, "word": " the", "probability": 0.388916015625}, {"start": 510.25, "end": 510.69, "word": " function", "probability": 0.9130859375}, {"start": 510.69, "end": 511.13, "word": " H", "probability": 0.376953125}, {"start": 511.13, "end": 511.71, "word": " is", "probability": 0.9111328125}, {"start": 511.71, "end": 515.89, "word": " unbounded", "probability": 0.9353841145833334}, {"start": 515.89, "end": 516.75, "word": " على", "probability": 0.453857421875}, {"start": 516.75, "end": 519.07, "word": " الفترة", "probability": 0.9739583333333334}, {"start": 519.07, "end": 519.39, "word": " من", "probability": 0.89501953125}, {"start": 519.39, "end": 520.03, "word": " صفر", "probability": 0.72119140625}, {"start": 520.03, "end": 520.59, "word": " لواحد", "probability": 0.81689453125}, {"start": 520.59, "end": 523.83, "word": " لأنها", "probability": 0.7989908854166666}, {"start": 523.83, "end": 524.85, "word": " unbounded", "probability": 0.93017578125}, {"start": 524.85, "end": 525.07, "word": " على", "probability": 0.87548828125}, {"start": 525.07, "end": 525.63, "word": " الفترة", "probability": 0.9889322916666666}, {"start": 525.63, "end": 526.49, "word": " المفتوحة", "probability": 0.9789225260416666}, {"start": 526.49, "end": 526.95, "word": " H", "probability": 0.249267578125}, {"start": 526.95, "end": 527.41, "word": " بالساعة", "probability": 0.5048828125}, {"start": 527.41, "end": 527.73, "word": " واحد", "probability": 0.985107421875}, {"start": 527.73, "end": 527.91, "word": " على", "probability": 0.76904296875}, {"start": 527.91, "end": 528.21, "word": " X", "probability": 0.51318359375}, {"start": 528.21, "end": 528.97, "word": " على", "probability": 0.81201171875}, {"start": 528.97, "end": 529.47, "word": " الفترة", "probability": 0.9783528645833334}, {"start": 529.47, "end": 529.83, "word": " نصف", "probability": 0.9454752604166666}, {"start": 529.83, "end": 530.43, "word": " المفتوحة", "probability": 0.9908040364583334}, {"start": 530.43, "end": 530.81, "word": " هذه", "probability": 0.6630859375}], "temperature": 1.0}, {"id": 19, "seek": 56016, "start": 531.59, "end": 560.17, "text": "و يقولنا إن الـ function 1 على x ليست bounded 1 على x ماهياش bounded على الفترة من 0 إلى 1 إذا النظرية تفشل لأن الـ function تبعت هنا ليست continuous عند السفر وبالتالي ليست continuous على كل الفترة الواحد هنا H is not continuous على كل الفترة I", "tokens": [2407, 7251, 39648, 8315, 36145, 2423, 39184, 2445, 502, 15844, 2031, 32239, 14851, 37498, 502, 15844, 2031, 19446, 3224, 1829, 33599, 37498, 15844, 27188, 2655, 25720, 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{"start": 548.77, "end": 550.15, "word": " عند", "probability": 0.377197265625}, {"start": 550.15, "end": 550.67, "word": " السفر", "probability": 0.8009440104166666}, {"start": 550.67, "end": 551.37, "word": " وبالتالي", "probability": 0.89208984375}, {"start": 551.37, "end": 551.81, "word": " ليست", "probability": 0.968505859375}, {"start": 551.81, "end": 552.41, "word": " continuous", "probability": 0.9638671875}, {"start": 552.41, "end": 552.75, "word": " على", "probability": 0.9228515625}, {"start": 552.75, "end": 553.03, "word": " كل", "probability": 0.9951171875}, {"start": 553.03, "end": 553.47, "word": " الفترة", "probability": 0.9729817708333334}, {"start": 553.47, "end": 553.81, "word": " الواحد", "probability": 0.7008870442708334}, {"start": 553.81, "end": 554.01, "word": " هنا", "probability": 0.99169921875}, {"start": 554.01, "end": 554.35, "word": " H", "probability": 0.33154296875}, {"start": 554.35, "end": 555.45, "word": " is", "probability": 0.81591796875}, {"start": 555.45, "end": 556.01, "word": " not", "probability": 0.96044921875}, {"start": 556.01, "end": 556.99, "word": " continuous", "probability": 0.93603515625}, {"start": 556.99, "end": 558.75, "word": " على", "probability": 0.9140625}, {"start": 558.75, "end": 559.15, "word": " كل", "probability": 0.9931640625}, {"start": 559.15, "end": 559.83, "word": " الفترة", "probability": 0.97900390625}, {"start": 559.83, "end": 560.17, "word": " I", "probability": 0.888671875}], "temperature": 1.0}, {"id": 20, "seek": 58760, "start": 561.02, "end": 587.6, "text": "لأنها مش continuous عند السفر اللي هي answer في آية عشان تكون ال function continuous على كل الفترة لازم تكون continuous عند كل نقطة في الفترة لو كانت مش continuous عند نقطة واحدة في الفترة فمايقدرش أقول continuous على الفترة تمام إذا ناخد", "tokens": [1211, 33456, 11296, 37893, 10957, 43242, 21136, 5172, 2288, 13672, 1829, 39896, 1867, 8978, 19753, 10632, 6225, 8592, 7649, 6055, 30544, 2423, 2445, 10957, 15844, 28242, 27188, 2655, 25720, 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"probability": 0.85400390625}, {"start": 564.2, "end": 564.58, "word": " answer", "probability": 0.70947265625}, {"start": 564.58, "end": 564.82, "word": " في", "probability": 0.92822265625}, {"start": 564.82, "end": 565.12, "word": " آية", "probability": 0.574951171875}, {"start": 565.12, "end": 565.88, "word": " عشان", "probability": 0.9640299479166666}, {"start": 565.88, "end": 566.26, "word": " تكون", "probability": 0.95263671875}, {"start": 566.26, "end": 566.52, "word": " ال", "probability": 0.96533203125}, {"start": 566.52, "end": 567.12, "word": " function", "probability": 0.8974609375}, {"start": 567.12, "end": 567.76, "word": " continuous", "probability": 0.9658203125}, {"start": 567.76, "end": 568.02, "word": " على", "probability": 0.833984375}, {"start": 568.02, "end": 568.4, "word": " كل", "probability": 0.98583984375}, {"start": 568.4, "end": 568.94, "word": " الفترة", "probability": 0.9361979166666666}, {"start": 568.94, "end": 570.0, "word": " لازم", "probability": 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{"start": 573.5, "end": 573.82, "word": " نقطة", "probability": 0.9827473958333334}, {"start": 573.82, "end": 574.1, "word": " واحدة", "probability": 0.9860026041666666}, {"start": 574.1, "end": 574.22, "word": " في", "probability": 0.81396484375}, {"start": 574.22, "end": 574.64, "word": " الفترة", "probability": 0.9943033854166666}, {"start": 574.64, "end": 576.08, "word": " فمايقدرش", "probability": 0.7292887369791666}, {"start": 576.08, "end": 576.3, "word": " أقول", "probability": 0.701171875}, {"start": 576.3, "end": 576.76, "word": " continuous", "probability": 0.97216796875}, {"start": 576.76, "end": 577.22, "word": " على", "probability": 0.84765625}, {"start": 577.22, "end": 578.08, "word": " الفترة", "probability": 0.9899088541666666}, {"start": 578.08, "end": 583.76, "word": " تمام", "probability": 0.91552734375}, {"start": 583.76, "end": 586.34, "word": " إذا", "probability": 0.5076904296875}, {"start": 586.34, "end": 587.6, "word": " ناخد", "probability": 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"start": 1049.72, "end": 1074.74, "text": "الـ set is non-empty لأن الفترة I هنا طبعا ليست فترة خالية، فيها على الأقل أناصر اشمال الفترة هذه is bounded", "tokens": [6027, 39184, 992, 307, 2107, 12, 4543, 88, 5296, 33456, 27188, 2655, 25720, 286, 34105, 23032, 3555, 3615, 995, 32239, 14851, 6156, 2655, 25720, 16490, 6027, 10632, 12399, 8978, 11296, 15844, 16247, 4587, 1211, 14739, 33546, 2288, 1975, 8592, 2304, 6027, 27188, 2655, 25720, 29538, 307, 37498], "avg_logprob": -0.32454428138832253, "compression_ratio": 1.2463768115942029, "no_speech_prob": 0.0, "words": [{"start": 1049.72, "end": 1050.2, "word": "الـ", "probability": 0.335357666015625}, {"start": 1050.2, "end": 1050.8, "word": " set", "probability": 0.5380859375}, {"start": 1050.8, "end": 1051.0, "word": " is", "probability": 0.822265625}, {"start": 1051.0, "end": 1051.24, "word": " non", "probability": 0.413330078125}, {"start": 1051.24, "end": 1051.74, "word": "-empty", "probability": 0.8250325520833334}, {"start": 1051.74, 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"end": 1068.62, "word": " اشمال", "probability": 0.8033447265625}, {"start": 1068.62, "end": 1069.16, "word": " الفترة", "probability": 0.8444010416666666}, {"start": 1069.16, "end": 1069.54, "word": " هذه", "probability": 0.56298828125}, {"start": 1069.54, "end": 1071.16, "word": " is", "probability": 0.8955078125}, {"start": 1071.16, "end": 1074.74, "word": " bounded", "probability": 0.69091796875}], "temperature": 1.0}, {"id": 38, "seek": 110276, "start": 1078.46, "end": 1102.76, "text": "by boundedness theorem احنا أخدنا نظرية اللي جابي لهذه كانت في اللقاء الأول بتقول لو كانت if function from I to R وكانت الدالة هذه continuous و I closed bounded interval", "tokens": [2322, 37498, 1287, 20904, 1975, 5016, 8315, 5551, 9778, 3215, 8315, 8717, 19913, 2288, 10632, 13672, 1829, 10874, 16758, 1829, 46740, 24192, 25961, 2655, 8978, 13672, 4587, 16606, 16247, 12610, 39894, 39648, 45164, 25961, 2655, 498, 2445, 490, 286, 281, 497, 4032, 41361, 2655, 32748, 6027, 3660, 29538, 10957, 4032, 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من ال boundedness ال theorem طيب وبالتالي", "tokens": [5172, 6027, 39184, 2445, 283, 39894, 9566, 1211, 3615, 37498, 15844, 27188, 2655, 25720, 286, 2423, 39184, 2445, 37498, 15844, 27188, 2655, 25720, 286, 9673, 3615, 1863, 10632, 31439, 2423, 39184, 3613, 6055, 3555, 3615, 2423, 39184, 2445, 13672, 1829, 39896, 2423, 992, 29538, 307, 257, 37498, 992, 1392, 46811, 10943, 23758, 44945, 5016, 36520, 47356, 9154, 2423, 37498, 1287, 2423, 20904, 23032, 1829, 3555, 46599, 6027, 2655, 6027, 1829], "avg_logprob": -0.29775528001113677, "compression_ratio": 1.7262569832402235, "no_speech_prob": 0.0, "words": [{"start": 1103.39, "end": 1103.79, "word": "فالـ", "probability": 0.6134440104166666}, {"start": 1103.79, "end": 1104.11, "word": " function", "probability": 0.814453125}, {"start": 1104.11, "end": 1104.39, "word": " f", "probability": 0.58056640625}, {"start": 1104.39, "end": 1104.91, "word": " بتطلع", "probability": 0.80084228515625}, {"start": 1104.91, "end": 1105.43, "word": " 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1229.67, "end": 1230.11, "word": " X", "probability": 0.67529296875}, {"start": 1230.11, "end": 1231.29, "word": " Superstar", "probability": 0.6536865234375}, {"start": 1231.29, "end": 1233.11, "word": " بساوي", "probability": 0.9093017578125}, {"start": 1233.11, "end": 1233.51, "word": " S", "probability": 0.98193359375}, {"start": 1233.51, "end": 1234.49, "word": " Superstar", "probability": 0.944091796875}], "temperature": 1.0}, {"id": 44, "seek": 126475, "start": 1237.01, "end": 1264.75, "text": "and الجزء التاني ان يوجد x lower star عنصر ما في الفترة i بحيث ان f ل x lower star بساوي s lower star فلو أثبتنا الجزء الأول معناته ال x star هذه نقطة الدالة بتاخد قيمتها العظم المطلقة عندها", "tokens": [474, 25724, 11622, 38207, 16712, 7649, 1829, 16472, 7251, 29245, 3215, 2031, 3126, 3543, 18871, 9381, 2288, 19446, 8978, 27188, 2655, 25720, 741, 4724, 5016, 1829, 12984, 16472, 283, 5296, 2031, 3126, 3543, 4724, 3794, 995, 45865, 262, 3126, 3543, 6156, 1211, 2407, 5551, 12984, 3555, 2655, 8315, 25724, 11622, 38207, 16247, 12610, 20449, 8315, 47395, 2423, 2031, 3543, 29538, 8717, 47432, 3660, 32748, 6027, 3660, 39894, 47283, 3215, 12174, 32640, 2655, 11296, 18863, 19913, 2304, 9673, 9566, 1211, 28671, 43242, 11296], "avg_logprob": -0.1795933788799378, "compression_ratio": 1.6358695652173914, "no_speech_prob": 0.0, "words": [{"start": 1237.0100000000002, "end": 1238.0700000000002, "word": "and", "probability": 0.2978515625}, {"start": 1238.0700000000002, "end": 1239.13, "word": " الجزء", "probability": 0.8857421875}, {"start": 1239.13, "end": 1239.79, "word": " التاني", "probability": 0.8946940104166666}, {"start": 1239.79, "end": 1240.73, "word": " ان", "probability": 0.295166015625}, {"start": 1240.73, "end": 1241.45, "word": " يوجد", "probability": 0.9397786458333334}, {"start": 1241.45, "end": 1241.87, "word": " x", "probability": 0.6982421875}, {"start": 1241.87, "end": 1242.37, "word": " lower", "probability": 0.5390625}, {"start": 1242.37, "end": 1242.97, "word": " star", "probability": 0.73095703125}, {"start": 1242.97, "end": 1243.71, "word": " عنصر", "probability": 0.9498697916666666}, {"start": 1243.71, "end": 1243.95, "word": " ما", "probability": 0.56787109375}, {"start": 1243.95, "end": 1244.93, "word": " في", "probability": 0.78662109375}, {"start": 1244.93, "end": 1245.43, "word": " الفترة", "probability": 0.9892578125}, {"start": 1245.43, "end": 1245.81, "word": " i", "probability": 0.3193359375}, {"start": 1245.81, "end": 1246.33, "word": " بحيث", "probability": 0.9691162109375}, {"start": 1246.33, "end": 1246.51, "word": " ان", "probability": 0.83544921875}, {"start": 1246.51, "end": 1246.93, "word": " f", "probability": 0.5078125}, {"start": 1246.93, "end": 1247.93, "word": " ل", "probability": 0.56005859375}, {"start": 1247.93, "end": 1248.25, "word": " x", "probability": 0.5166015625}, {"start": 1248.25, "end": 1248.69, "word": " lower", "probability": 0.8134765625}, {"start": 1248.69, "end": 1249.27, "word": " star", "probability": 0.91064453125}, {"start": 1249.27, "end": 1250.07, "word": " بساوي", "probability": 0.8037109375}, {"start": 1250.07, "end": 1250.47, "word": " s", "probability": 0.7490234375}, {"start": 1250.47, "end": 1250.89, "word": " lower", "probability": 0.8349609375}, {"start": 1250.89, "end": 1251.41, "word": " star", "probability": 0.93505859375}, {"start": 1251.41, "end": 1255.11, "word": " فلو", "probability": 0.95654296875}, {"start": 1255.11, "end": 1255.65, "word": " أثبتنا", "probability": 0.97978515625}, {"start": 1255.65, "end": 1256.11, "word": " الجزء", "probability": 0.9851888020833334}, {"start": 1256.11, "end": 1256.81, "word": " الأول", "probability": 0.96533203125}, {"start": 1256.81, "end": 1258.67, "word": " معناته", "probability": 0.9384765625}, {"start": 1258.67, "end": 1259.81, "word": " ال", "probability": 0.73046875}, {"start": 1259.81, "end": 1260.11, "word": " x", "probability": 0.6640625}, {"start": 1260.11, "end": 1260.65, "word": " star", "probability": 0.61572265625}, {"start": 1260.65, "end": 1261.07, "word": " هذه", "probability": 0.6748046875}, {"start": 1261.07, "end": 1261.71, "word": " نقطة", "probability": 0.9246419270833334}, {"start": 1261.71, "end": 1262.35, "word": " الدالة", "probability": 0.8107096354166666}, {"start": 1262.35, "end": 1262.85, "word": " بتاخد", "probability": 0.9300130208333334}, {"start": 1262.85, "end": 1263.35, "word": " قيمتها", "probability": 0.9910888671875}, {"start": 1263.35, "end": 1263.71, "word": " العظم", "probability": 0.6433512369791666}, {"start": 1263.71, "end": 1264.27, "word": " المطلقة", "probability": 0.9375}, {"start": 1264.27, "end": 1264.75, "word": " عندها", "probability": 0.993896484375}], "temperature": 1.0}, {"id": 45, "seek": 127833, "start": 1265.67, "end": 1278.33, "text": "وهذا معناه ان x lower star هي نقطة قيمة صغرى المطلقة لل function f لأن هذه قيمة الصغرى المطلقة للfunction f على i", "tokens": [2407, 3224, 15730, 20449, 8315, 3224, 16472, 2031, 3126, 3543, 39896, 8717, 47432, 3660, 12174, 32640, 3660, 20328, 17082, 2288, 7578, 9673, 9566, 1211, 28671, 24976, 2445, 283, 5296, 33456, 29538, 12174, 32640, 3660, 31767, 17082, 2288, 7578, 9673, 9566, 1211, 28671, 24976, 22845, 283, 15844, 741], "avg_logprob": -0.38964842570324737, "compression_ratio": 1.4227642276422765, "no_speech_prob": 0.0, "words": [{"start": 1265.67, "end": 1266.11, "word": "وهذا", "probability": 0.6798502604166666}, {"start": 1266.11, "end": 1266.53, "word": " معناه", "probability": 0.7740885416666666}, {"start": 1266.53, "end": 1266.73, "word": " ان", "probability": 0.45849609375}, {"start": 1266.73, "end": 1267.13, "word": " x", "probability": 0.53955078125}, {"start": 1267.13, "end": 1267.43, "word": " lower", "probability": 0.1258544921875}, {"start": 1267.43, "end": 1267.87, "word": " star", "probability": 0.275390625}, {"start": 1267.87, "end": 1268.19, "word": " هي", "probability": 0.87353515625}, {"start": 1268.19, "end": 1269.25, "word": " نقطة", "probability": 0.95166015625}, {"start": 1269.25, "end": 1269.99, "word": " قيمة", "probability": 0.9449869791666666}, {"start": 1269.99, "end": 1270.55, "word": " صغرى", "probability": 0.771728515625}, {"start": 1270.55, "end": 1271.51, "word": " المطلقة", "probability": 0.81219482421875}, {"start": 1271.51, "end": 1271.79, "word": " لل", "probability": 0.71533203125}, {"start": 1271.79, "end": 1272.13, "word": " function", "probability": 0.5771484375}, {"start": 1272.13, "end": 1272.49, "word": " f", "probability": 0.7470703125}, {"start": 1272.49, "end": 1272.83, "word": " لأن", "probability": 0.704345703125}, {"start": 1272.83, "end": 1273.23, "word": " هذه", "probability": 0.5595703125}, {"start": 1273.23, "end": 1275.63, "word": " قيمة", "probability": 0.7484537760416666}, {"start": 1275.63, "end": 1275.81, "word": " الصغرى", "probability": 0.83685302734375}, {"start": 1275.81, "end": 1276.59, "word": " المطلقة", "probability": 0.9256591796875}, {"start": 1276.59, "end": 1277.71, "word": " للfunction", "probability": 0.5174560546875}, {"start": 1277.71, "end": 1277.83, "word": " f", "probability": 0.8701171875}, {"start": 1277.83, "end": 1278.03, "word": " على", "probability": 0.442138671875}, {"start": 1278.03, "end": 1278.33, "word": " i", "probability": 0.472412109375}], "temperature": 1.0}, {"id": 46, "seek": 132030, "start": 1292.02, "end": 1320.3, "text": "فلو أثبتنا واحد واتنين يكون أثبتنا النظرية حنثبت واحد وبرهان التاني بالمثل مشابه فحاسبكم أنتوا تكتبوا هنا we prove one and leave the proof of two for you", "tokens": [5172, 1211, 2407, 5551, 12984, 3555, 2655, 8315, 36764, 24401, 4032, 9307, 1863, 9957, 7251, 30544, 5551, 12984, 3555, 2655, 8315, 28239, 19913, 2288, 10632, 11331, 1863, 12984, 3555, 2655, 36764, 24401, 4032, 26890, 3224, 7649, 16712, 7649, 1829, 20666, 2304, 12984, 1211, 37893, 16758, 3224, 6156, 5016, 32277, 3555, 24793, 14739, 2655, 14407, 6055, 4117, 2655, 3555, 14407, 34105, 321, 7081, 472, 293, 1856, 264, 8177, 295, 732, 337, 291], "avg_logprob": -0.182617188948724, "compression_ratio": 1.4938271604938271, "no_speech_prob": 0.0, "words": [{"start": 1292.02, "end": 1292.4, "word": "فلو", "probability": 0.8077799479166666}, {"start": 1292.4, "end": 1292.86, "word": " أثبتنا", "probability": 0.97763671875}, {"start": 1292.86, "end": 1293.12, "word": " واحد", "probability": 0.863037109375}, {"start": 1293.12, "end": 1293.5, "word": " واتنين", "probability": 0.8111572265625}, {"start": 1293.5, "end": 1293.76, "word": " يكون", "probability": 0.773681640625}, {"start": 1293.76, "end": 1294.36, "word": " أثبتنا", "probability": 0.98876953125}, {"start": 1294.36, "end": 1295.34, "word": " النظرية", "probability": 0.9681396484375}, {"start": 1295.34, "end": 1296.92, "word": " حنثبت", "probability": 0.809814453125}, {"start": 1296.92, "end": 1297.34, "word": " واحد", "probability": 0.98779296875}, {"start": 1297.34, "end": 1297.86, "word": " وبرهان", "probability": 0.95361328125}, {"start": 1297.86, "end": 1298.3, "word": " التاني", "probability": 0.8951822916666666}, {"start": 1298.3, "end": 1298.94, "word": " بالمثل", "probability": 0.9754638671875}, {"start": 1298.94, "end": 1299.8, "word": " مشابه", "probability": 0.824462890625}, {"start": 1299.8, "end": 1300.9, "word": " فحاسبكم", "probability": 0.74970703125}, {"start": 1300.9, "end": 1301.26, "word": " أنتوا", "probability": 0.56640625}, {"start": 1301.26, "end": 1302.82, "word": " تكتبوا", "probability": 0.95419921875}, {"start": 1302.82, "end": 1303.54, "word": " هنا", "probability": 0.11346435546875}, {"start": 1303.54, "end": 1303.84, "word": " we", "probability": 0.609375}, {"start": 1303.84, "end": 1304.52, "word": " prove", "probability": 0.93310546875}, {"start": 1304.52, "end": 1307.74, "word": " one", "probability": 0.86279296875}, {"start": 1307.74, "end": 1309.68, "word": " and", "probability": 0.93603515625}, {"start": 1309.68, "end": 1312.28, "word": " leave", "probability": 0.94287109375}, {"start": 1312.28, "end": 1312.56, "word": " the", "probability": 0.85498046875}, {"start": 1312.56, "end": 1313.12, "word": " proof", "probability": 0.9794921875}, {"start": 1313.12, "end": 1315.9, "word": " of", "probability": 0.96240234375}, {"start": 1315.9, "end": 1316.52, "word": " two", "probability": 0.9501953125}, {"start": 1316.52, "end": 1319.16, "word": " for", "probability": 0.962890625}, {"start": 1319.16, "end": 1320.3, "word": " you", "probability": 0.96240234375}], "temperature": 1.0}, {"id": 47, "seek": 134970, "start": 1321.58, "end": 1349.7, "text": "زي ما قلنا البرهان هذا تبع الجزء التاني مشابه للأول طيب since S star بساوي ال supremum ل F of I then for each N عدد طبيعي", "tokens": [11622, 1829, 19446, 12174, 1211, 8315, 2423, 26890, 3224, 7649, 23758, 6055, 3555, 3615, 25724, 11622, 38207, 16712, 7649, 1829, 37893, 16758, 3224, 24976, 10721, 12610, 23032, 1829, 3555, 1670, 318, 3543, 4724, 3794, 995, 45865, 2423, 23710, 449, 5296, 479, 295, 286, 550, 337, 1184, 426, 6225, 3215, 3215, 23032, 21292, 3615, 1829], "avg_logprob": -0.1953125010837208, "compression_ratio": 1.221476510067114, "no_speech_prob": 0.0, "words": [{"start": 1321.58, "end": 1321.92, "word": "زي", "probability": 0.8857421875}, {"start": 1321.92, "end": 1322.0, "word": " ما", "probability": 0.857421875}, {"start": 1322.0, "end": 1322.28, "word": " قلنا", "probability": 0.8968098958333334}, {"start": 1322.28, "end": 1322.74, "word": " البرهان", "probability": 0.8258056640625}, {"start": 1322.74, "end": 1323.06, "word": " هذا", "probability": 0.52978515625}, {"start": 1323.06, "end": 1323.62, "word": " تبع", "probability": 0.7360026041666666}, {"start": 1323.62, "end": 1323.94, "word": " الجزء", "probability": 0.8043619791666666}, {"start": 1323.94, "end": 1324.54, "word": " التاني", "probability": 0.9296875}, {"start": 1324.54, "end": 1325.9, "word": " مشابه", "probability": 0.97216796875}, {"start": 1325.9, "end": 1327.22, "word": " للأول", "probability": 0.88232421875}, {"start": 1327.22, "end": 1330.5, "word": " طيب", "probability": 0.9518229166666666}, {"start": 1330.5, "end": 1331.12, "word": " since", "probability": 0.76611328125}, {"start": 1331.12, "end": 1334.04, "word": " S", "probability": 0.359375}, {"start": 1334.04, "end": 1334.76, "word": " star", "probability": 0.1077880859375}, {"start": 1334.76, "end": 1336.28, "word": " بساوي", "probability": 0.9066162109375}, {"start": 1336.28, "end": 1337.44, "word": " ال", "probability": 0.4248046875}, {"start": 1337.44, "end": 1338.46, "word": " supremum", "probability": 0.818359375}, {"start": 1338.46, "end": 1338.88, "word": " ل", "probability": 0.84521484375}, {"start": 1338.88, "end": 1339.44, "word": " F", "probability": 0.69580078125}, {"start": 1339.44, "end": 1340.08, "word": " of", "probability": 0.88525390625}, {"start": 1340.08, "end": 1340.84, "word": " I", "probability": 0.94970703125}, {"start": 1340.84, "end": 1345.16, "word": " then", "probability": 0.77685546875}, {"start": 1345.16, "end": 1345.94, "word": " for", "probability": 0.962890625}, {"start": 1345.94, "end": 1346.74, "word": " each", "probability": 0.91748046875}, {"start": 1346.74, "end": 1348.46, "word": " N", "probability": 0.411376953125}, {"start": 1348.46, "end": 1348.92, "word": " عدد", "probability": 0.9736328125}, {"start": 1348.92, "end": 1349.7, "word": " طبيعي", "probability": 0.92578125}], "temperature": 1.0}, {"id": 48, "seek": 137424, "start": 1352.66, "end": 1374.24, "text": "S star minus one upon n is not upper bound of set F of I لأن هذا عبارة عن least upper bound", "tokens": [50, 3543, 3175, 472, 3564, 297, 307, 406, 6597, 5472, 295, 992, 479, 295, 286, 5296, 33456, 23758, 6225, 3555, 9640, 3660, 18871, 1935, 6597, 5472], "avg_logprob": -0.4887152910232544, "compression_ratio": 1.0833333333333333, "no_speech_prob": 0.0, "words": [{"start": 1352.66, "end": 1353.14, "word": "S", "probability": 0.122314453125}, {"start": 1353.14, "end": 1353.86, "word": " star", "probability": 0.047027587890625}, {"start": 1353.86, "end": 1354.5, "word": " minus", "probability": 0.36083984375}, {"start": 1354.5, "end": 1355.32, "word": " one", "probability": 0.666015625}, {"start": 1355.32, "end": 1355.68, "word": " upon", "probability": 0.313232421875}, {"start": 1355.68, "end": 1356.08, "word": " n", "probability": 0.5595703125}, {"start": 1356.08, "end": 1357.42, "word": " is", "probability": 0.74267578125}, {"start": 1357.42, "end": 1357.98, "word": " not", "probability": 0.939453125}, {"start": 1357.98, "end": 1360.44, "word": " upper", "probability": 0.5537109375}, {"start": 1360.44, "end": 1361.16, "word": " bound", "probability": 0.900390625}, {"start": 1361.16, "end": 1365.42, "word": " of", "probability": 0.84228515625}, {"start": 1365.42, "end": 1366.14, "word": " set", "probability": 0.7744140625}, {"start": 1366.14, "end": 1366.56, "word": " F", "probability": 0.580078125}, {"start": 1366.56, "end": 1366.84, "word": " of", "probability": 0.81103515625}, {"start": 1366.84, "end": 1368.46, "word": " I", "probability": 0.92578125}, {"start": 1368.46, "end": 1372.28, "word": " لأن", "probability": 0.534912109375}, {"start": 1372.28, "end": 1372.6, "word": " هذا", "probability": 0.55810546875}, {"start": 1372.6, "end": 1373.02, "word": " عبارة", "probability": 0.876220703125}, {"start": 1373.02, "end": 1373.2, "word": " عن", "probability": 0.99658203125}, {"start": 1373.2, "end": 1373.56, "word": " least", "probability": 0.74462890625}, {"start": 1373.56, "end": 1373.9, "word": " upper", "probability": 0.9208984375}, {"start": 1373.9, "end": 1374.24, "word": " bound", "probability": 0.908203125}], "temperature": 1.0}, {"id": 49, "seek": 139887, "start": 1380.43, "end": 1398.87, "text": "So وبالتالي there exists xn ينتمي للفترة I بحيث أنه F of xn ده أكبر من S star minus واحد على M", "tokens": [6455, 46599, 6027, 2655, 6027, 1829, 456, 8198, 2031, 77, 7251, 29399, 2304, 1829, 24976, 5172, 2655, 25720, 286, 4724, 5016, 1829, 12984, 14739, 3224, 479, 295, 2031, 77, 11778, 3224, 5551, 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0.456787109375}, {"start": 1388.39, "end": 1390.03, "word": " F", "probability": 0.63818359375}, {"start": 1390.03, "end": 1392.29, "word": " of", "probability": 0.666015625}, {"start": 1392.29, "end": 1393.15, "word": " xn", "probability": 0.813720703125}, {"start": 1393.15, "end": 1393.61, "word": " ده", "probability": 0.5780029296875}, {"start": 1393.61, "end": 1394.23, "word": " أكبر", "probability": 0.9563802083333334}, {"start": 1394.23, "end": 1394.65, "word": " من", "probability": 0.99462890625}, {"start": 1394.65, "end": 1395.79, "word": " S", "probability": 0.71337890625}, {"start": 1395.79, "end": 1396.47, "word": " star", "probability": 0.0802001953125}, {"start": 1396.47, "end": 1397.47, "word": " minus", "probability": 0.8056640625}, {"start": 1397.47, "end": 1398.19, "word": " واحد", "probability": 0.91357421875}, {"start": 1398.19, "end": 1398.49, "word": " على", "probability": 0.7744140625}, {"start": 1398.49, "end": 1398.87, "word": " M", "probability": 0.6279296875}], "temperature": 1.0}, {"id": 50, "seek": 141385, "start": 1400.81, "end": 1413.85, "text": "هذا العدد ليس upper bound لل set هذه طب ما إذا يوجد عنصر في ال set هذه اللي هو صورة لعنصر في I وهذا العنصر أكبر من SSR ماس واحدة", "tokens": [3224, 15730, 18863, 3215, 3215, 32239, 3794, 6597, 5472, 24976, 992, 29538, 23032, 3555, 19446, 11933, 15730, 7251, 29245, 3215, 18871, 9381, 2288, 8978, 2423, 992, 29538, 13672, 1829, 31439, 20328, 13063, 3660, 5296, 3615, 1863, 9381, 2288, 8978, 286, 37037, 15730, 18863, 1863, 9381, 2288, 5551, 4117, 26890, 9154, 12238, 49, 19446, 3794, 36764, 24401, 3660], "avg_logprob": -0.24191810319135928, "compression_ratio": 1.4577464788732395, "no_speech_prob": 0.0, "words": [{"start": 1400.81, "end": 1401.29, "word": "هذا", "probability": 0.5797119140625}, {"start": 1401.29, "end": 1401.89, "word": " العدد", "probability": 0.88525390625}, {"start": 1401.89, "end": 1402.27, "word": " ليس", "probability": 0.92578125}, {"start": 1402.27, "end": 1402.55, "word": " upper", "probability": 0.80810546875}, {"start": 1402.55, "end": 1403.03, "word": " bound", "probability": 0.92919921875}, {"start": 1403.03, "end": 1403.93, "word": " لل", "probability": 0.74462890625}, {"start": 1403.93, "end": 1404.29, "word": " set", "probability": 0.57080078125}, {"start": 1404.29, "end": 1404.75, "word": " هذه", "probability": 0.642578125}, {"start": 1404.75, "end": 1405.17, "word": " طب", "probability": 0.748046875}, {"start": 1405.17, "end": 1405.33, "word": " ما", "probability": 0.61474609375}, {"start": 1405.33, "end": 1405.57, "word": " إذا", "probability": 0.74169921875}, {"start": 1405.57, "end": 1406.05, "word": " يوجد", "probability": 0.96875}, {"start": 1406.05, "end": 1406.49, "word": " عنصر", "probability": 0.96923828125}, {"start": 1406.49, "end": 1406.63, "word": " في", "probability": 0.95654296875}, {"start": 1406.63, "end": 1406.77, "word": " ال", "probability": 0.71044921875}, {"start": 1406.77, "end": 1407.07, "word": " set", "probability": 0.98095703125}, {"start": 1407.07, "end": 1407.49, "word": " هذه", "probability": 0.861328125}, {"start": 1407.49, "end": 1408.25, "word": " اللي", "probability": 0.90625}, {"start": 1408.25, "end": 1408.49, "word": " هو", "probability": 0.98388671875}, {"start": 1408.49, "end": 1408.97, "word": " صورة", "probability": 0.9583333333333334}, {"start": 1408.97, "end": 1409.49, "word": " لعنصر", "probability": 0.89375}, {"start": 1409.49, "end": 1409.69, "word": " في", "probability": 0.869140625}, {"start": 1409.69, "end": 1410.03, "word": " I", "probability": 0.77099609375}, {"start": 1410.03, "end": 1411.43, "word": " وهذا", "probability": 0.676025390625}, {"start": 1411.43, "end": 1411.93, "word": " العنصر", "probability": 0.9852294921875}, {"start": 1411.93, "end": 1412.37, "word": " أكبر", "probability": 0.9518229166666666}, {"start": 1412.37, "end": 1412.55, "word": " من", "probability": 0.97705078125}, {"start": 1412.55, "end": 1413.11, "word": " SSR", "probability": 0.5418701171875}, {"start": 1413.11, "end": 1413.49, "word": " ماس", "probability": 0.2733154296875}, {"start": 1413.49, "end": 1413.85, "word": " واحدة", "probability": 0.9773763020833334}], "temperature": 1.0}, {"id": 51, "seek": 144242, "start": 1414.76, "end": 1442.42, "text": "الان S*) هذا عبارة عن upper bound للset F of I وهذا العنصر ينتمي ل F of I، إذن هذا أصغر من أو ساوي ال upper bound لكل المجموعة اللي بتحتوي العناصر اللي زي هذا، إذن هذا أصغر من أو ساوي S*)، okay؟ وهذا طبعا صحيح لكل N في N، كل N في N", "tokens": [6027, 7649, 318, 34634, 23758, 6225, 3555, 9640, 3660, 18871, 6597, 5472, 24976, 3854, 479, 295, 286, 37037, 15730, 18863, 1863, 9381, 2288, 7251, 29399, 2304, 1829, 5296, 479, 295, 286, 12399, 11933, 8848, 1863, 23758, 5551, 9381, 17082, 2288, 9154, 34051, 8608, 995, 45865, 2423, 6597, 5472, 5296, 28820, 9673, 7435, 2304, 2407, 27884, 13672, 1829, 39894, 33753, 45865, 18863, 1863, 33546, 2288, 13672, 1829, 30767, 1829, 23758, 12399, 11933, 8848, 1863, 23758, 5551, 9381, 17082, 2288, 9154, 34051, 8608, 995, 45865, 318, 34634, 12399, 1392, 22807, 37037, 15730, 23032, 3555, 3615, 995, 20328, 5016, 1829, 5016, 5296, 28820, 426, 8978, 426, 12399, 28242, 426, 8978, 426], "avg_logprob": -0.18176605559270317, "compression_ratio": 1.793103448275862, "no_speech_prob": 0.0, "words": [{"start": 1414.76, "end": 1415.42, "word": "الان", "probability": 0.77685546875}, {"start": 1415.42, "end": 1416.6, "word": " S*)", "probability": 0.289215087890625}, {"start": 1416.6, "end": 1416.98, "word": " هذا", "probability": 0.5107421875}, {"start": 1416.98, "end": 1417.4, "word": " عبارة", "probability": 0.9910888671875}, {"start": 1417.4, "end": 1417.64, "word": " عن", "probability": 0.9951171875}, {"start": 1417.64, "end": 1417.98, "word": " upper", "probability": 0.83935546875}, {"start": 1417.98, "end": 1418.44, "word": " bound", "probability": 0.88037109375}, {"start": 1418.44, "end": 1419.22, "word": " للset", "probability": 0.55859375}, {"start": 1419.22, "end": 1419.54, "word": " F", "probability": 0.65283203125}, {"start": 1419.54, "end": 1419.78, "word": " of", "probability": 0.66748046875}, {"start": 1419.78, "end": 1420.1, "word": " I", "probability": 0.98486328125}, {"start": 1420.1, "end": 1420.94, "word": " وهذا", "probability": 0.784912109375}, {"start": 1420.94, "end": 1421.42, "word": " العنصر", "probability": 0.9881591796875}, {"start": 1421.42, "end": 1421.88, "word": " ينتمي", "probability": 0.982666015625}, {"start": 1421.88, "end": 1422.0, "word": " ل", "probability": 0.8583984375}, {"start": 1422.0, "end": 1422.22, "word": " F", "probability": 0.64990234375}, {"start": 1422.22, "end": 1422.42, "word": " of", "probability": 0.91064453125}, {"start": 1422.42, "end": 1423.04, "word": " I،", "probability": 0.65478515625}, {"start": 1423.04, "end": 1423.46, "word": " إذن", "probability": 0.6792805989583334}, {"start": 1423.46, "end": 1423.72, "word": " هذا", "probability": 0.9169921875}, {"start": 1423.72, "end": 1424.16, "word": " 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"probability": 0.970458984375}, {"start": 1428.7, "end": 1428.92, "word": " زي", "probability": 0.974609375}, {"start": 1428.92, "end": 1429.84, "word": " هذا،", "probability": 0.8466796875}, {"start": 1429.84, "end": 1430.18, "word": " إذن", "probability": 0.8349609375}, {"start": 1430.18, "end": 1430.4, "word": " هذا", "probability": 0.96337890625}, {"start": 1430.4, "end": 1430.86, "word": " أصغر", "probability": 0.98876953125}, {"start": 1430.86, "end": 1431.02, "word": " من", "probability": 0.99169921875}, {"start": 1431.02, "end": 1431.2, "word": " أو", "probability": 0.97802734375}, {"start": 1431.2, "end": 1431.8, "word": " ساوي", "probability": 0.9615885416666666}, {"start": 1431.8, "end": 1433.36, "word": " S*)،", "probability": 0.4764811197916667}, {"start": 1433.36, "end": 1435.34, "word": " okay؟", "probability": 0.60400390625}, {"start": 1435.34, "end": 1435.66, "word": " وهذا", "probability": 0.802978515625}, {"start": 1435.66, "end": 1436.18, "word": " طبعا", "probability": 0.9774169921875}, {"start": 1436.18, "end": 1437.5, "word": " صحيح", "probability": 0.9603271484375}, {"start": 1437.5, "end": 1438.2, "word": " لكل", "probability": 0.978515625}, {"start": 1438.2, "end": 1438.64, "word": " N", "probability": 0.5693359375}, {"start": 1438.64, "end": 1439.24, "word": " في", "probability": 0.9814453125}, {"start": 1439.24, "end": 1440.6, "word": " N،", "probability": 0.705322265625}, {"start": 1440.6, "end": 1441.34, "word": " كل", "probability": 0.27197265625}, {"start": 1441.34, "end": 1441.9, "word": " N", "probability": 0.93505859375}, {"start": 1441.9, "end": 1442.16, "word": " في", "probability": 0.99072265625}, {"start": 1442.16, "end": 1442.42, "word": " N", "probability": 0.994140625}], "temperature": 1.0}, {"id": 52, "seek": 147071, "start": 1444.63, "end": 1470.71, "text": "تمام اذا ان انا في ياندي المتباينة هذه بما انه ال .. ال set since الفترة I is closed and bounded is bounded", "tokens": [39237, 10943, 1975, 15730, 16472, 1975, 8315, 8978, 7251, 7649, 16254, 9673, 2655, 3555, 995, 9957, 3660, 29538, 4724, 15042, 16472, 3224, 2423, 4386, 2423, 992, 1670, 27188, 2655, 25720, 286, 307, 5395, 293, 37498, 307, 37498], "avg_logprob": -0.2728207197628523, "compression_ratio": 1.3275862068965518, "no_speech_prob": 0.0, "words": [{"start": 1444.63, "end": 1445.15, "word": "تمام", "probability": 0.685791015625}, {"start": 1445.15, "end": 1446.45, "word": " اذا", "probability": 0.540069580078125}, {"start": 1446.45, "end": 1446.65, "word": " ان", "probability": 0.73681640625}, {"start": 1446.65, "end": 1446.95, "word": " انا", "probability": 0.932373046875}, {"start": 1446.95, "end": 1447.11, "word": " في", "probability": 0.94970703125}, {"start": 1447.11, "end": 1447.53, "word": " ياندي", "probability": 0.5479329427083334}, {"start": 1447.53, "end": 1449.09, "word": " المتباينة", "probability": 0.93359375}, {"start": 1449.09, "end": 1449.55, "word": " هذه", "probability": 0.6796875}, {"start": 1449.55, "end": 1453.01, "word": " بما", "probability": 0.93115234375}, {"start": 1453.01, "end": 1454.37, "word": " انه", "probability": 0.720458984375}, {"start": 1454.37, "end": 1455.25, "word": " ال", "probability": 0.8984375}, {"start": 1455.25, "end": 1455.57, "word": " ..", "probability": 0.50927734375}, {"start": 1455.57, "end": 1455.95, "word": " ال", "probability": 0.83349609375}, {"start": 1455.95, "end": 1456.35, "word": " set", "probability": 0.324951171875}, {"start": 1456.35, "end": 1465.01, "word": " since", "probability": 0.7919921875}, {"start": 1465.01, "end": 1465.69, "word": " الفترة", "probability": 0.9698893229166666}, {"start": 1465.69, "end": 1466.11, "word": " I", "probability": 0.89453125}, {"start": 1466.11, "end": 1466.51, "word": " is", "probability": 0.958984375}, {"start": 1466.51, "end": 1466.99, "word": " closed", "probability": 0.9189453125}, {"start": 1466.99, "end": 1467.33, "word": " and", "probability": 0.97412109375}, {"start": 1467.33, "end": 1467.91, "word": " bounded", "probability": 0.95166015625}, {"start": 1467.91, "end": 1470.21, "word": " is", "probability": 0.91259765625}, {"start": 1470.21, "end": 1470.71, "word": " bounded", "probability": 0.9501953125}], "temperature": 1.0}, {"id": 53, "seek": 150053, "start": 1474.57, "end": 1500.53, "text": "و X in .. لاحظوا X in ال sequence هذه contained in I then ال sequence X in is bounded so by Bolzano Weierstrass theorem", "tokens": [2407, 1783, 294, 4386, 20193, 5016, 19913, 14407, 1783, 294, 2423, 8310, 29538, 16212, 294, 286, 550, 2423, 8310, 1783, 294, 307, 37498, 370, 538, 14331, 89, 3730, 492, 811, 9733, 640, 20904], "avg_logprob": -0.4126838349244174, "compression_ratio": 1.146551724137931, "no_speech_prob": 0.0, "words": [{"start": 1474.57, "end": 1475.15, "word": "و", "probability": 0.8193359375}, {"start": 1475.15, "end": 1475.57, "word": " X", "probability": 0.5693359375}, {"start": 1475.57, "end": 1476.09, "word": " in", "probability": 0.2479248046875}, {"start": 1476.09, "end": 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1689.24, "word": " convergence", "probability": 0.4521484375}], "temperature": 1.0}, {"id": 61, "seek": 172152, "start": 1693.34, "end": 1721.52, "text": "و ال .. و متحقق طبعا .. نعم من هنا .. من هنا نسمي هذه قصة", "tokens": [2407, 2423, 4386, 4032, 44650, 5016, 4587, 4587, 23032, 3555, 3615, 995, 4386, 8717, 25957, 9154, 34105, 4386, 9154, 34105, 8717, 38251, 1829, 29538, 12174, 9381, 3660], "avg_logprob": -0.20856584289244243, "compression_ratio": 1.2236842105263157, "no_speech_prob": 0.0, "words": [{"start": 1693.34, "end": 1693.74, "word": "و", "probability": 0.947265625}, {"start": 1693.74, "end": 1694.4, "word": " ال", "probability": 0.60205078125}, {"start": 1694.4, "end": 1704.48, "word": " ..", "probability": 0.88720703125}, {"start": 1704.48, "end": 1704.68, "word": " و", "probability": 0.60595703125}, {"start": 1704.68, "end": 1705.48, "word": " متحقق", "probability": 0.830078125}, {"start": 1705.48, "end": 1706.14, "word": " طبعا", "probability": 0.997314453125}, {"start": 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واحد على nk", "tokens": [2322, 3543, 321, 362, 283, 47525, 1211, 2031, 294, 4724, 2031, 294, 350, 6156, 3224, 15730, 5551, 9381, 17082, 2288, 9154, 34051, 8608, 995, 45865, 262, 3543, 4032, 5551, 4117, 26890, 9154, 34051, 8608, 995, 45865, 262, 1687, 3543, 3175, 36764, 24401, 15844, 297, 74], "avg_logprob": -0.3019701171180476, "compression_ratio": 1.2881355932203389, "no_speech_prob": 0.0, "words": [{"start": 1726.3, "end": 1727.5, "word": "by", "probability": 0.256591796875}, {"start": 1727.5, "end": 1728.7, "word": " star", "probability": 0.445068359375}, {"start": 1728.7, "end": 1731.66, "word": " we", "probability": 0.333984375}, {"start": 1731.66, "end": 1732.2, "word": " have", "probability": 0.96630859375}, {"start": 1732.2, "end": 1734.06, "word": " f", "probability": 0.456298828125}, {"start": 1734.06, "end": 1735.18, "word": " بدل", "probability": 0.8447265625}, {"start": 1735.18, "end": 1735.56, "word": " x", "probability": 0.91259765625}, {"start": 1735.56, "end": 1736.06, 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2314.62, "word": "اعتبري", "probability": 0.6246795654296875}, {"start": 2314.62, "end": 2314.98, "word": " الـ", "probability": 0.6915283203125}, {"start": 2314.98, "end": 2315.48, "word": " function", "probability": 0.80810546875}, {"start": 2315.48, "end": 2317.24, "word": " f", "probability": 0.4716796875}, {"start": 2317.24, "end": 2317.54, "word": " of", "probability": 0.306884765625}, {"start": 2317.54, "end": 2318.02, "word": " x", "probability": 0.85400390625}, {"start": 2318.02, "end": 2319.98, "word": " بساوي", "probability": 0.65484619140625}, {"start": 2319.98, "end": 2320.34, "word": " x", "probability": 0.861328125}, {"start": 2320.34, "end": 2320.92, "word": " تربية", "probability": 0.8268229166666666}, {"start": 2320.92, "end": 2323.22, "word": " وطبعا", "probability": 0.857421875}, {"start": 2323.22, "end": 2323.48, "word": " هنا", "probability": 0.87939453125}, {"start": 2323.48, "end": 2323.9, "word": " x", "probability": 0.9091796875}, {"start": 2323.9, "end": 2324.58, "word": " ينتمي", "probability": 0.94677734375}, {"start": 2324.58, "end": 2324.8, "word": " إلى", "probability": 0.66796875}, {"start": 2324.8, "end": 2325.14, "word": " R", "probability": 0.55859375}, {"start": 2325.14, "end": 2326.48, "word": " ده", "probability": 0.779541015625}, {"start": 2326.48, "end": 2327.08, "word": " للتربعية", "probability": 0.69775390625}, {"start": 2327.08, "end": 2327.46, "word": " المجال", "probability": 0.81103515625}, {"start": 2327.46, "end": 2327.76, "word": " تبع", "probability": 0.79248046875}, {"start": 2327.76, "end": 2328.06, "word": " كل", "probability": 0.6064453125}, {"start": 2328.06, "end": 2328.48, "word": " العداد", "probability": 0.777587890625}, {"start": 2328.48, "end": 2329.22, "word": " الحقيقية", "probability": 0.8302001953125}, {"start": 2329.22, "end": 2330.42, "word": " خلينا", "probability": 0.779541015625}, {"start": 2330.42, "end": 2331.02, "word": " ناخد", "probability": 0.9830729166666666}, {"start": 2331.02, 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"end": 2342.42, "word": " تمام؟", "probability": 0.8966471354166666}], "temperature": 1.0}, {"id": 84, "seek": 236826, "start": 2343.7, "end": 2368.26, "text": "فال function f is continuous on I and بنلاحظ أن f عن سلب واحد بساوي f عن واحد بساوي واحد is an absolute maximum", "tokens": [5172, 6027, 2445, 283, 307, 10957, 322, 286, 293, 44945, 15040, 5016, 19913, 14739, 283, 18871, 8608, 46152, 36764, 24401, 4724, 3794, 995, 45865, 283, 18871, 36764, 24401, 4724, 3794, 995, 45865, 36764, 24401, 307, 364, 8236, 6674], "avg_logprob": -0.30128206350864506, "compression_ratio": 1.2583333333333333, "no_speech_prob": 0.0, "words": [{"start": 2343.7, "end": 2344.44, "word": "فال", "probability": 0.794921875}, {"start": 2344.44, "end": 2344.84, "word": " function", "probability": 0.29443359375}, {"start": 2344.84, "end": 2345.36, "word": " f", "probability": 0.57373046875}, {"start": 2345.36, "end": 2348.38, "word": " is", "probability": 0.192626953125}, {"start": 2348.38, "end": 2350.38, "word": " continuous", "probability": 0.88818359375}, {"start": 2350.38, "end": 2351.3, "word": " on", "probability": 0.87939453125}, {"start": 2351.3, "end": 2351.86, "word": " I", "probability": 0.7998046875}, {"start": 2351.86, "end": 2353.26, "word": " and", "probability": 0.609375}, {"start": 2353.26, "end": 2356.48, "word": " بنلاحظ", "probability": 0.8311767578125}, {"start": 2356.48, "end": 2356.66, "word": " أن", "probability": 0.619140625}, {"start": 2356.66, "end": 2357.12, "word": " f", "probability": 0.29296875}, {"start": 2357.12, "end": 2357.44, "word": " عن", "probability": 0.76806640625}, {"start": 2357.44, "end": 2357.94, "word": " سلب", "probability": 0.775146484375}, {"start": 2357.94, "end": 2358.4, "word": " واحد", "probability": 0.82470703125}, {"start": 2358.4, "end": 2359.08, "word": " بساوي", "probability": 0.77435302734375}, {"start": 2359.08, "end": 2359.42, "word": " f", "probability": 0.28125}, {"start": 2359.42, "end": 2359.66, "word": " عن", "probability": 0.95849609375}, {"start": 2359.66, "end": 2360.2, "word": " واحد", "probability": 0.9609375}, {"start": 2360.2, "end": 2361.38, "word": " بساوي", "probability": 0.9473876953125}, {"start": 2361.38, "end": 2362.0, "word": " واحد", "probability": 0.990966796875}, {"start": 2362.0, "end": 2364.0, "word": " is", "probability": 0.9130859375}, {"start": 2364.0, "end": 2364.74, "word": " an", "probability": 0.86962890625}, {"start": 2364.74, "end": 2366.36, "word": " absolute", "probability": 0.9111328125}, {"start": 2366.36, "end": 2368.26, "word": " maximum", "probability": 0.9599609375}], "temperature": 1.0}, {"id": 85, "seek": 239612, "start": 2373.02, "end": 2396.12, "text": "at x بساوي سالب واحد واحد إذا سالب واحد واحد عبارة عن absolute maximum points مظبوط هاي الدالة التربية أنا بس ماخد المجال تبعها اللي هو الفترة المغلقة والمحدودة من سالب واحد إلى واحد فطبعا رسمتها زي هيك", "tokens": [267, 2031, 4724, 3794, 995, 45865, 8608, 6027, 3555, 36764, 24401, 36764, 24401, 11933, 15730, 8608, 6027, 3555, 36764, 24401, 36764, 24401, 6225, 3555, 9640, 3660, 18871, 8236, 6674, 2793, 3714, 19913, 3555, 2407, 9566, 8032, 47302, 32748, 6027, 3660, 16712, 25513, 10632, 41850, 4724, 3794, 3714, 47283, 3215, 9673, 7435, 6027, 6055, 3555, 3615, 11296, 13672, 1829, 31439, 27188, 2655, 25720, 9673, 17082, 1211, 28671, 16070, 2304, 24401, 23328, 3660, 9154, 8608, 6027, 3555, 36764, 24401, 30731, 36764, 24401, 6156, 9566, 3555, 3615, 995, 12602, 38251, 2655, 11296, 30767, 1829, 39896, 4117], "avg_logprob": -0.1737034539593027, "compression_ratio": 1.758974358974359, "no_speech_prob": 0.0, "words": [{"start": 2373.0200000000004, "end": 2374.1400000000003, "word": "at", "probability": 0.095703125}, {"start": 2374.1400000000003, "end": 2375.26, "word": " x", "probability": 0.82568359375}, {"start": 2375.26, "end": 2375.92, "word": " بساوي", "probability": 0.595733642578125}, {"start": 2375.92, "end": 2376.5, "word": " سالب", "probability": 0.7095133463541666}, {"start": 2376.5, "end": 2376.96, "word": " واحد", "probability": 0.974365234375}, {"start": 2376.96, "end": 2377.44, "word": " واحد", "probability": 0.927490234375}, {"start": 2377.44, "end": 2379.64, "word": " إذا", "probability": 0.521820068359375}, {"start": 2379.64, "end": 2380.12, "word": " سالب", "probability": 0.91845703125}, {"start": 2380.12, "end": 2380.44, "word": " واحد", "probability": 0.99462890625}, {"start": 2380.44, "end": 2381.0, "word": " واحد", "probability": 0.98486328125}, {"start": 2381.0, "end": 2382.0, "word": " عبارة", "probability": 0.8896484375}, {"start": 2382.0, "end": 2382.22, "word": " عن", "probability": 0.9931640625}, {"start": 2382.22, "end": 2382.66, "word": " absolute", "probability": 0.78076171875}, {"start": 2382.66, "end": 2383.28, "word": " maximum", "probability": 0.94873046875}, {"start": 2383.28, "end": 2383.94, "word": " points", "probability": 0.78466796875}, {"start": 2383.94, "end": 2384.5, "word": " مظبوط", "probability": 0.829833984375}, {"start": 2384.5, "end": 2384.7, "word": " هاي", "probability": 0.569091796875}, {"start": 2384.7, "end": 2385.26, "word": " الدالة", "probability": 0.9366861979166666}, {"start": 2385.26, "end": 2386.36, "word": " التربية", "probability": 0.8396809895833334}, {"start": 2386.36, "end": 2386.62, "word": " أنا", "probability": 0.6943359375}, {"start": 2386.62, "end": 2386.9, "word": " بس", "probability": 0.9560546875}, {"start": 2386.9, "end": 2387.46, "word": " ماخد", "probability": 0.888671875}, {"start": 2387.46, "end": 2388.64, "word": " المجال", "probability": 0.98046875}, {"start": 2388.64, "end": 2389.34, "word": " تبعها", "probability": 0.9658203125}, {"start": 2389.34, "end": 2390.2, "word": " اللي", "probability": 0.908203125}, {"start": 2390.2, "end": 2390.38, "word": " هو", "probability": 0.9853515625}, {"start": 2390.38, "end": 2390.84, "word": " الفترة", "probability": 0.9886067708333334}, {"start": 2390.84, "end": 2391.44, "word": " المغلقة", "probability": 0.957275390625}, {"start": 2391.44, "end": 2392.3, "word": " والمحدودة", "probability": 0.8927734375}, {"start": 2392.3, "end": 2392.44, "word": " من", "probability": 0.9736328125}, {"start": 2392.44, "end": 2392.78, "word": " سالب", "probability": 0.9505208333333334}, {"start": 2392.78, "end": 2393.1, "word": " واحد", "probability": 0.99267578125}, {"start": 2393.1, "end": 2393.3, "word": " إلى", "probability": 0.89697265625}, {"start": 2393.3, "end": 2393.78, "word": " واحد", "probability": 0.990234375}, {"start": 2393.78, "end": 2394.9, "word": " فطبعا", "probability": 0.95517578125}, {"start": 2394.9, "end": 2395.48, "word": " رسمتها", "probability": 0.98388671875}, {"start": 2395.48, "end": 2395.74, "word": " زي", "probability": 0.9873046875}, {"start": 2395.74, "end": 2396.12, "word": " هيك", "probability": 0.5859375}], "temperature": 1.0}, {"id": 86, "seek": 242765, "start": 2400.87, "end": 2427.65, "text": "هذه الدالة التربعية ان المجال تبعها الفترة المغلقة هذه فواضح انه هيقلها absolute maximum اللي هي الواحد وال absolute maximum هذا حصلنا عليه عن مقطه تالى سالب واحد واحد نعم مظبوط صحيح وطبعا هنا فيقلها absolute minimum واحدة اللي هي السفر", "tokens": [3224, 24192, 32748, 6027, 3660, 16712, 25513, 3615, 10632, 16472, 9673, 7435, 6027, 6055, 3555, 3615, 11296, 27188, 2655, 25720, 9673, 17082, 1211, 28671, 29538, 6156, 2407, 46958, 5016, 16472, 3224, 39896, 4587, 1211, 11296, 8236, 6674, 13672, 1829, 39896, 2423, 14407, 24401, 16070, 8236, 6674, 23758, 11331, 36520, 8315, 47356, 18871, 3714, 47432, 3224, 6055, 6027, 7578, 8608, 6027, 3555, 36764, 24401, 36764, 24401, 8717, 25957, 3714, 19913, 3555, 2407, 9566, 20328, 5016, 1829, 5016, 4032, 9566, 3555, 3615, 995, 34105, 8978, 4587, 1211, 11296, 8236, 7285, 36764, 24401, 3660, 13672, 1829, 39896, 21136, 5172, 2288], "avg_logprob": -0.22305485378114545, "compression_ratio": 1.8878048780487804, "no_speech_prob": 5.960464477539063e-08, "words": [{"start": 2400.87, "end": 2401.25, "word": "هذه", "probability": 0.5335693359375}, {"start": 2401.25, "end": 2401.75, "word": " الدالة", "probability": 0.8587239583333334}, {"start": 2401.75, "end": 2402.63, "word": " التربعية", "probability": 0.78369140625}, {"start": 2402.63, "end": 2402.83, "word": " ان", "probability": 0.200927734375}, {"start": 2402.83, "end": 2403.25, "word": " المجال", "probability": 0.9296875}, {"start": 2403.25, "end": 2403.85, "word": " تبعها", "probability": 0.878662109375}, {"start": 2403.85, "end": 2404.69, "word": " الفترة", "probability": 0.9519856770833334}, {"start": 2404.69, "end": 2405.17, "word": " المغلقة", "probability": 0.94091796875}, {"start": 2405.17, "end": 2405.61, "word": " هذه", "probability": 0.7783203125}, {"start": 2405.61, "end": 2406.67, "word": " فواضح", "probability": 0.92919921875}, {"start": 2406.67, "end": 2408.01, "word": " انه", "probability": 0.6826171875}, {"start": 2408.01, "end": 2408.59, "word": " هيقلها", "probability": 0.632232666015625}, {"start": 2408.59, "end": 2409.03, "word": " absolute", "probability": 0.900390625}, {"start": 2409.03, "end": 2409.71, "word": " maximum", "probability": 0.97509765625}, {"start": 2409.71, "end": 2410.03, "word": " اللي", "probability": 0.81396484375}, {"start": 2410.03, "end": 2410.23, "word": " هي", "probability": 0.927734375}, {"start": 2410.23, "end": 2410.91, "word": " الواحد", "probability": 0.9475911458333334}, {"start": 2410.91, "end": 2412.17, "word": " وال", "probability": 0.486083984375}, {"start": 2412.17, "end": 2412.67, "word": " absolute", "probability": 0.79638671875}, {"start": 2412.67, "end": 2413.25, "word": " maximum", "probability": 0.9833984375}, {"start": 2413.25, "end": 2413.73, "word": " هذا", "probability": 0.86181640625}, {"start": 2413.73, "end": 2414.79, "word": " حصلنا", "probability": 0.8108723958333334}, {"start": 2414.79, "end": 2415.03, "word": " عليه", "probability": 0.93603515625}, {"start": 2415.03, "end": 2415.25, "word": " عن", "probability": 0.650390625}, {"start": 2415.25, "end": 2415.63, "word": " مقطه", "probability": 0.6087239583333334}, {"start": 2415.63, "end": 2415.83, "word": " تالى", "probability": 0.5651041666666666}, {"start": 2415.83, "end": 2416.25, "word": " سالب", "probability": 0.8287760416666666}, {"start": 2416.25, "end": 2416.65, "word": " واحد", "probability": 0.991455078125}, {"start": 2416.65, "end": 2417.13, "word": " واحد", "probability": 0.920166015625}, {"start": 2417.13, "end": 2419.89, "word": " نعم", "probability": 0.743408203125}, {"start": 2419.89, "end": 2422.89, "word": " مظبوط", "probability": 0.68310546875}, {"start": 2422.89, "end": 2423.61, "word": " صحيح", "probability": 0.9906005859375}, {"start": 2423.61, "end": 2424.67, "word": " وطبعا", "probability": 0.9193359375}, {"start": 2424.67, "end": 2424.87, "word": " هنا", "probability": 0.974609375}, {"start": 2424.87, "end": 2425.35, "word": " فيقلها", "probability": 0.8463134765625}, {"start": 2425.35, "end": 2425.81, "word": " absolute", "probability": 0.97265625}, {"start": 2425.81, "end": 2426.27, "word": " minimum", "probability": 0.986328125}, {"start": 2426.27, "end": 2426.81, "word": " واحدة", "probability": 0.9754231770833334}, {"start": 2426.81, "end": 2426.99, "word": " اللي", "probability": 0.96826171875}, {"start": 2426.99, "end": 2427.15, "word": " هي", "probability": 0.955078125}, {"start": 2427.15, "end": 2427.65, "word": " السفر", "probability": 0.8478190104166666}], "temperature": 1.0}, {"id": 87, "seek": 244438, "start": 2429.08, "end": 2444.38, "text": "واضح هنا أن دالة هنا المجال تبعها فترة مغلقة ومحدودة اذا by maximum minimum theorem اكيد فيه اللي هي absolute maximum وفيه اللي هي absolute minimum ال absolute minimum point هي السفر", "tokens": [2407, 46958, 5016, 34105, 14739, 11778, 6027, 3660, 34105, 9673, 7435, 6027, 6055, 3555, 3615, 11296, 6156, 2655, 25720, 3714, 17082, 1211, 28671, 4032, 2304, 24401, 23328, 3660, 1975, 15730, 538, 6674, 7285, 20904, 1975, 4117, 25708, 8978, 3224, 13672, 1829, 39896, 8236, 6674, 4032, 41185, 3224, 13672, 1829, 39896, 8236, 7285, 2423, 8236, 7285, 935, 39896, 21136, 5172, 2288], "avg_logprob": -0.2686987744003046, "compression_ratio": 1.6049382716049383, "no_speech_prob": 5.960464477539063e-08, "words": [{"start": 2429.08, "end": 2429.98, "word": "واضح", "probability": 0.8650716145833334}, {"start": 2429.98, "end": 2430.26, "word": " هنا", "probability": 0.5517578125}, {"start": 2430.26, "end": 2431.24, "word": " أن", "probability": 0.428466796875}, {"start": 2431.24, "end": 2431.76, "word": " دالة", "probability": 0.5887451171875}, {"start": 2431.76, "end": 2432.0, "word": " هنا", "probability": 0.2152099609375}, {"start": 2432.0, "end": 2432.62, "word": " المجال", "probability": 0.90185546875}, {"start": 2432.62, "end": 2433.14, "word": " تبعها", "probability": 0.843017578125}, {"start": 2433.14, "end": 2433.58, "word": " فترة", "probability": 0.9894205729166666}, {"start": 2433.58, "end": 2434.04, "word": " مغلقة", "probability": 0.9398193359375}, {"start": 2434.04, "end": 2434.94, "word": " ومحدودة", "probability": 0.95966796875}, {"start": 2434.94, "end": 2435.5, "word": " اذا", "probability": 0.57806396484375}, {"start": 2435.5, "end": 2435.78, "word": " by", "probability": 0.763671875}, {"start": 2435.78, "end": 2436.2, "word": " maximum", "probability": 0.93603515625}, {"start": 2436.2, "end": 2436.54, "word": " minimum", "probability": 0.662109375}, {"start": 2436.54, "end": 2436.98, "word": " theorem", "probability": 0.79443359375}, {"start": 2436.98, "end": 2437.74, "word": " اكيد", "probability": 0.88232421875}, {"start": 2437.74, "end": 2438.64, "word": " فيه", "probability": 0.61669921875}, {"start": 2438.64, "end": 2438.8, "word": " اللي", "probability": 0.6295166015625}, {"start": 2438.8, "end": 2438.9, "word": " هي", "probability": 0.2396240234375}, {"start": 2438.9, "end": 2439.3, "word": " absolute", "probability": 0.94873046875}, {"start": 2439.3, "end": 2439.84, "word": " maximum", "probability": 0.9833984375}, {"start": 2439.84, "end": 2440.22, "word": " وفيه", "probability": 0.8221028645833334}, {"start": 2440.22, "end": 2440.32, "word": " اللي", "probability": 0.985595703125}, {"start": 2440.32, "end": 2440.38, "word": " هي", "probability": 0.96728515625}, {"start": 2440.38, "end": 2440.76, "word": " absolute", "probability": 0.9775390625}, {"start": 2440.76, "end": 2441.28, "word": " minimum", "probability": 0.990234375}, {"start": 2441.28, "end": 2441.98, "word": " ال", "probability": 0.7353515625}, {"start": 2441.98, "end": 2442.38, "word": " absolute", "probability": 0.81201171875}, {"start": 2442.38, "end": 2442.86, "word": " minimum", "probability": 0.974609375}, {"start": 2442.86, "end": 2443.5, "word": " point", "probability": 0.98779296875}, {"start": 2443.5, "end": 2443.82, "word": " هي", "probability": 0.923828125}, {"start": 2443.82, "end": 2444.38, "word": " السفر", "probability": 0.795654296875}], "temperature": 1.0}, {"id": 88, "seek": 247182, "start": 2444.9, "end": 2471.82, "text": "الـ absolute maximum point هنا في نقطتين النظرية مجرد يعني مجرد يوجد على الأقل نقطة واحدة في end absolute maximum فلو كانوا تنتين لا ضرر لا بأس تمام okay إذن يعني هذه بعض الملاحظات على ال maximum theorem ال ..", "tokens": [6027, 39184, 8236, 6674, 935, 34105, 8978, 8717, 47432, 2655, 9957, 28239, 19913, 2288, 10632, 3714, 7435, 2288, 3215, 37495, 22653, 3714, 7435, 2288, 3215, 7251, 29245, 3215, 15844, 16247, 4587, 1211, 8717, 47432, 3660, 36764, 24401, 3660, 8978, 917, 8236, 6674, 6156, 1211, 2407, 25961, 14407, 6055, 29399, 9957, 20193, 48812, 2288, 2288, 20193, 4724, 10721, 3794, 46811, 10943, 1392, 11933, 8848, 1863, 37495, 22653, 29538, 45030, 11242, 9673, 15040, 5016, 19913, 9307, 15844, 2423, 6674, 20904, 2423, 4386], "avg_logprob": -0.21064814226126966, "compression_ratio": 1.605, "no_speech_prob": 0.0, "words": [{"start": 2444.9, "end": 2445.28, "word": "الـ", "probability": 0.3187255859375}, {"start": 2445.28, "end": 2445.64, "word": " absolute", "probability": 0.40478515625}, {"start": 2445.64, "end": 2446.36, "word": " maximum", "probability": 0.9619140625}, {"start": 2446.36, "end": 2446.84, "word": " point", "probability": 0.97705078125}, {"start": 2446.84, "end": 2447.04, "word": " هنا", "probability": 0.8671875}, {"start": 2447.04, "end": 2447.18, "word": " في", "probability": 0.76171875}, {"start": 2447.18, "end": 2447.78, "word": " نقطتين", "probability": 0.8543701171875}, {"start": 2447.78, "end": 2448.52, "word": " النظرية", "probability": 0.9144287109375}, {"start": 2448.52, "end": 2448.9, "word": " مجرد", "probability": 0.8175048828125}, {"start": 2448.9, "end": 2449.18, "word": " يعني", "probability": 0.69384765625}, {"start": 2449.18, "end": 2450.38, "word": " مجرد", "probability": 0.8155517578125}, {"start": 2450.38, "end": 2450.88, "word": " يوجد", "probability": 0.9313151041666666}, {"start": 2450.88, "end": 2451.22, "word": " على", "probability": 0.96826171875}, {"start": 2451.22, "end": 2451.72, "word": " الأقل", "probability": 0.9685872395833334}, {"start": 2451.72, "end": 2452.14, "word": " نقطة", "probability": 0.9881184895833334}, {"start": 2452.14, "end": 2452.54, "word": " واحدة", "probability": 0.9679361979166666}, {"start": 2452.54, "end": 2452.68, "word": " في", "probability": 0.9599609375}, {"start": 2452.68, "end": 2452.88, "word": " end", "probability": 0.78125}, {"start": 2452.88, "end": 2453.28, "word": " absolute", "probability": 0.87353515625}, {"start": 2453.28, "end": 2453.82, "word": " maximum", "probability": 0.97900390625}, {"start": 2453.82, "end": 2454.14, "word": " فلو", "probability": 0.92626953125}, {"start": 2454.14, "end": 2454.44, "word": " كانوا", "probability": 0.66455078125}, {"start": 2454.44, "end": 2454.94, "word": " تنتين", "probability": 0.8790690104166666}, {"start": 2454.94, "end": 2455.72, "word": " لا", "probability": 0.8056640625}, {"start": 2455.72, "end": 2456.66, "word": " ضرر", "probability": 0.8365885416666666}, {"start": 2456.66, "end": 2458.1, "word": " لا", "probability": 0.489501953125}, {"start": 2458.1, "end": 2458.68, "word": " بأس", "probability": 0.9674479166666666}, {"start": 2458.68, "end": 2460.14, "word": " تمام", "probability": 0.6671142578125}, {"start": 2460.14, "end": 2460.7, "word": " okay", "probability": 0.5947265625}, {"start": 2460.7, "end": 2463.16, "word": " إذن", "probability": 0.618896484375}, {"start": 2463.16, "end": 2463.5, "word": " يعني", "probability": 0.96533203125}, {"start": 2463.5, "end": 2463.76, "word": " هذه", "probability": 0.88720703125}, {"start": 2463.76, "end": 2464.16, "word": " بعض", "probability": 0.973876953125}, {"start": 2464.16, "end": 2465.02, "word": " الملاحظات", "probability": 0.97890625}, {"start": 2465.02, "end": 2465.42, "word": " على", "probability": 0.88671875}, {"start": 2465.42, "end": 2465.8, "word": " ال", "probability": 0.68408203125}, {"start": 2465.8, "end": 2466.3, "word": " maximum", "probability": 0.98486328125}, {"start": 2466.3, "end": 2468.08, "word": " theorem", "probability": 0.90478515625}, {"start": 2468.08, "end": 2471.44, "word": " ال", "probability": 0.86376953125}, {"start": 2471.44, "end": 2471.82, "word": " ..", "probability": 0.86572265625}], "temperature": 1.0}, {"id": 89, "seek": 250439, "start": 2482.53, "end": 2504.39, "text": "أعتقد أن احنا هنكتفي بهذا القدر و ان شاء الله بنكمل ال section هذا في المحاضرة القادمة فاشكركم لحسن استماعكم و ان شاء الله نلتقي يوم الأتنين", "tokens": [10721, 34268, 28543, 14739, 1975, 5016, 8315, 8032, 1863, 4117, 2655, 41185, 39627, 15730, 25062, 3215, 2288, 4032, 16472, 13412, 16606, 21984, 44945, 24793, 1211, 2423, 3541, 23758, 8978, 9673, 5016, 46958, 25720, 25062, 18513, 46007, 6156, 33599, 4117, 31747, 2304, 5296, 5016, 3794, 1863, 44713, 15042, 3615, 24793, 4032, 16472, 13412, 16606, 21984, 8717, 1211, 2655, 38436, 7251, 20498, 16247, 2655, 1863, 9957], "avg_logprob": -0.11123798076923076, "compression_ratio": 1.603896103896104, "no_speech_prob": 1.7881393432617188e-07, "words": [{"start": 2482.5299999999997, "end": 2483.89, "word": "أعتقد", "probability": 0.8553059895833334}, {"start": 2483.89, "end": 2484.05, "word": " أن", "probability": 0.74560546875}, {"start": 2484.05, "end": 2484.37, "word": " احنا", "probability": 0.8642578125}, {"start": 2484.37, "end": 2485.21, "word": " هنكتفي", "probability": 0.950390625}, {"start": 2485.21, "end": 2486.65, "word": " بهذا", "probability": 0.974853515625}, {"start": 2486.65, "end": 2487.49, "word": " القدر", "probability": 0.9964192708333334}, {"start": 2487.49, "end": 2489.35, "word": " و", "probability": 0.953125}, {"start": 2489.35, "end": 2489.49, "word": " ان", "probability": 0.55419921875}, {"start": 2489.49, "end": 2489.71, "word": " شاء", "probability": 0.981689453125}, {"start": 2489.71, "end": 2489.81, "word": " الله", "probability": 0.96142578125}, {"start": 2489.81, "end": 2490.63, "word": " بنكمل", "probability": 0.9837239583333334}, {"start": 2490.63, "end": 2491.73, "word": " ال", "probability": 0.9736328125}, {"start": 2491.73, "end": 2492.21, "word": " section", "probability": 0.93701171875}, {"start": 2492.21, "end": 2492.59, "word": " هذا", "probability": 0.418701171875}, {"start": 2492.59, "end": 2492.83, "word": " في", "probability": 0.88427734375}, {"start": 2492.83, "end": 2493.51, "word": " المحاضرة", "probability": 0.98046875}, {"start": 2493.51, "end": 2494.31, "word": " القادمة", "probability": 0.9959309895833334}, {"start": 2494.31, "end": 2496.85, "word": " فاشكركم", "probability": 0.785986328125}, {"start": 2496.85, "end": 2499.23, "word": " لحسن", "probability": 0.9320068359375}, {"start": 2499.23, "end": 2499.99, "word": " استماعكم", "probability": 0.955322265625}, {"start": 2499.99, "end": 2500.35, "word": " و", "probability": 0.9951171875}, {"start": 2500.35, "end": 2500.55, "word": " ان", "probability": 0.93408203125}, {"start": 2500.55, "end": 2500.75, "word": " شاء", "probability": 0.99462890625}, {"start": 2500.75, "end": 2501.01, "word": " الله", "probability": 0.9638671875}, {"start": 2501.01, "end": 2503.25, "word": " نلتقي", "probability": 0.966552734375}, {"start": 2503.25, "end": 2503.95, "word": " يوم", "probability": 0.759033203125}, {"start": 2503.95, "end": 2504.39, "word": " الأتنين", "probability": 0.8399658203125}], "temperature": 1.0}], "language": "ar", "language_probability": 1.0, "duration": 2505.34025, "duration_after_vad": 2290.6299999999906}
\ No newline at end of file
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/u5C7kmY6Ltw.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/u5C7kmY6Ltw.srt
new file mode 100644
index 0000000000000000000000000000000000000000..1ef435018d1564281e309d81d277065165130adc
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/u5C7kmY6Ltw.srt
@@ -0,0 +1,1515 @@
+1
+00:00:20,100 --> 00:00:29,020
+بسم الله الرحمن الرحيم ناخذ اليوم الجزء المتعلق بال
+
+2
+00:00:29,020 --> 00:00:33,400
+composition of continuous functions لو كان في عندي 
+
+3
+00:00:33,400 --> 00:00:38,960
+اقترانين متصلين فهنشوف أن ال composition تبعهم
+
+4
+00:00:38,960 --> 00:00:47,360
+بيطلع أيضًا متصل على مجال الدالة الأولى وبعد هيك
+
+5
+00:00:47,360 --> 00:00:54,360
+بعد ما نقلص هنبدأ ب section خمسة تلاتة التي تتحدث
+
+6
+00:00:54,360 --> 00:00:58,100
+عن ال continuous functions على ال intervals أو على
+
+7
+00:00:58,100 --> 00:01:06,420
+الفترات أولًا بناخد النظرية التي خاصة بال
+
+8
+00:01:06,420 --> 00:01:18,240
+composition of continuous functions النظرية
+
+9
+00:01:18,240 --> 00:01:19,360
+هذه تقول
+
+10
+00:01:22,140 --> 00:01:34,600
+Let a و b be subsets of R و if f from a to b is
+
+11
+00:01:34,600 --> 00:01:38,960
+continuous at 
+
+12
+00:01:38,960 --> 00:01:47,700
+c تنتمي إلى a and g function
+
+13
+00:01:51,030 --> 00:01:59,250
+from a to R هنا و g function from b to R إذا مجال
+
+14
+00:01:59,250 --> 00:02:05,010
+ال function g هو ال set b مجال ال function f هو ال
+
+15
+00:02:05,010 --> 00:02:10,370
+set a be
+
+16
+00:02:10,370 --> 00:02:14,070
+continuous at
+
+17
+00:02:17,610 --> 00:02:27,310
+B بساوي F of C تنتمي إلى 
+
+18
+00:02:27,310 --> 00:02:32,530
+الـ B where
+
+19
+00:02:32,530 --> 00:02:41,410
+طبعًا ال range لل function F is contained in B عشان
+
+20
+00:02:41,410 --> 00:02:45,990
+يكون ال composition لل two functions f و g يكون
+
+21
+00:02:45,990 --> 00:02:50,590
+معرف لازم range ال f يكون subset من domain ال g
+
+22
+00:02:50,590 --> 00:03:00,410
+then في الحالة هذه the composite function the
+
+23
+00:03:00,410 --> 00:03:10,680
+composite function f circle g أو g circle f G
+
+24
+00:03:10,680 --> 00:03:15,660
+Circle F from
+
+25
+00:03:15,660 --> 00:03:26,360
+A to R is continuous تطلع continuous at C
+
+26
+00:03:39,140 --> 00:03:45,580
+أنا في عندي function يعني المجال تبعها A و
+
+27
+00:03:45,580 --> 00:03:57,140
+F function من A ل R و كانتيل يوسع عند النقطة C و في
+
+28
+00:03:57,140 --> 00:04:05,180
+عندي function ثانية يعني عندي ال 6 هذه F of A
+
+29
+00:04:09,920 --> 00:04:17,400
+F of A subset من الـ B المجموعة هذه نسميها B وهي F
+
+30
+00:04:17,400 --> 00:04:24,140
+of A subset فالآن وفي function G continuous
+
+31
+00:04:24,140 --> 00:04:28,840
+function G
+
+32
+00:04:28,840 --> 00:04:31,140
+is continuous عند النقطة
+
+33
+00:04:36,550 --> 00:04:43,070
+النقطة f of c بيبساوي f of c إذا ال function g
+
+34
+00:04:43,070 --> 00:04:48,450
+فاندي f continuous عند c تنتمي ل a و g continuous
+
+35
+00:04:48,450 --> 00:04:57,110
+عند b التي هي صورة ال f and c إذا في المحصلة تطلع
+
+36
+00:04:57,110 --> 00:05:04,420
+عندي ال function طبعًا هي المفروض تطلع g of b ففي 
+
+37
+00:05:04,420 --> 00:05:07,220
+المحصلة بيطلع عند الـ function G circle F
+
+38
+00:05:07,220 --> 00:05:13,620
+continuous تطلع continuous عند النقطة C طبعًا عشان
+
+39
+00:05:13,620 --> 00:05:17,400
+ال composition هذا يكون معرف لازم range ال F التي
+
+40
+00:05:17,400 --> 00:05:22,020
+هو ده هو F of A يكون subset من ال B التي هو domain
+
+41
+00:05:22,020 --> 00:05:29,160
+ال function G وهذا طبعًا احنا فرضيناه لبرهان ذلك
+
+42
+00:05:29,160 --> 00:05:33,380
+هنستخدم الـ neighborhood definition of continuity
+
+43
+00:05:33,380 --> 00:05:48,380
+at a point ف let W be an epsilon neighborhood of G
+
+44
+00:05:48,380 --> 00:06:02,050
+circle F ل C التي هي G ل F of C التي هي جي او بي
+
+45
+00:06:02,050 --> 00:06:10,710
+صح؟ ونثبت أنه يوجد Delta neighborhood للـC التي
+
+46
+00:06:10,710 --> 00:06:15,210
+هي تنتمي لـA بحيث أن صورة ال neighborhood هذا
+
+47
+00:06:15,210 --> 00:06:19,110
+subset من الـW neighborhood هنا تعريف ال
+
+48
+00:06:19,110 --> 00:06:22,730
+neighborhood definition للاتصال عن النقطة طيب
+
+49
+00:06:22,730 --> 00:06:25,650
+since
+
+50
+00:06:28,450 --> 00:06:40,530
+الفنشن الـ g is continuous at b التي هي بالساوي f
+
+51
+00:06:40,530 --> 00:06:47,430
+of c تنتمي ل b وهي في عندي w neighborhood w is
+
+52
+00:06:47,430 --> 00:06:52,810
+epsilon neighborhood ل g of b إذا من تعريف الاتصال
+
+53
+00:06:52,810 --> 00:07:04,400
+عن النقطة يوجد delta neighborhood نسميه V of ال
+
+54
+00:07:04,400 --> 00:07:12,120
+B بحيث أن لكل
+
+55
+00:07:12,120 --> 00:07:24,000
+Y ينتمي إلى B تقاطع ال V لازم يطلع عندي صورة ال YG
+
+56
+00:07:24,000 --> 00:07:35,680
+of Y تنتمي لـ W الـ
+
+57
+00:07:35,680 --> 00:07:39,460
+function
+
+58
+00:07:39,460 --> 00:07:45,700
+G continuous عند B في مجالها وبالتالي حسب الاتصال
+
+59
+00:07:45,700 --> 00:07:53,520
+لأي epsilon neighborhood لصورة B يوجد delta
+
+60
+00:07:53,520 --> 00:07:59,820
+neighborhood V للـ B بحيث أن لكل Y في ال
+
+61
+00:07:59,820 --> 00:08:04,520
+neighborhood و في مجال الدالة صورته تطلع موجودة في
+
+62
+00:08:04,520 --> 00:08:08,720
+ال epsilon neighborhood نسمي هذه ال implication
+
+63
+00:08:08,720 --> 00:08:19,400
+star طيب since أنا
+
+64
+00:08:19,400 --> 00:08:20,620
+عندي ال function
+
+65
+00:08:25,730 --> 00:08:34,370
+Since الـ function F is continuous at
+
+66
+00:08:34,370 --> 00:08:43,550
+C و هي NDV عبارة عن delta neighborhood للـ B التي
+
+67
+00:08:43,550 --> 00:08:47,810
+هي F of C إذا 
+
+68
+00:08:47,810 --> 00:08:57,710
+يوجد Gamma neighborhood نسميه U of
+
+69
+00:08:57,710 --> 00:09:01,570
+...
+
+70
+00:09:01,570 --> 00:09:14,250
+of C بحيث أن لكل X ينتمي إلى A تقاطع ال U يطلع
+
+71
+00:09:14,250 --> 00:09:20,410
+عندي صورة ال X تنتمي ل ال U
+
+72
+00:09:24,690 --> 00:09:43,130
+نسمي هذه double star إذا
+
+73
+00:09:43,130 --> 00:09:59,970
+now from star and double star أثبتنا أنه يوجد Gamma
+
+74
+00:09:59,970 --> 00:10:08,110
+neighborhood التي هو U of C تنتمي إلى A بحيث أن
+
+75
+00:10:08,110 --> 00:10:20,830
+كل X ينتمي إلى A تقاطع U هذا يؤدي أن F
+
+76
+00:10:20,830 --> 00:10:21,510
+of X
+
+77
+00:10:26,740 --> 00:10:42,500
+تنتمي إلى لأ
+
+78
+00:10:42,500 --> 00:10:47,120
+هذا المفروض يطلع تنتمي
+
+79
+00:10:47,120 --> 00:10:52,100
+يعني
+
+80
+00:10:52,100 --> 00:10:53,980
+المفروض تنتمي إلى V
+
+81
+00:10:58,230 --> 00:11:05,750
+الـ F continuous and الـ C و V neighborhood ل B 
+
+82
+00:11:05,750 --> 00:11:07,110
+التي هي صورة الـ C
+
+83
+00:11:10,240 --> 00:11:14,920
+Gamma neighbourhood U للـC بحيث لكل X مجال الدالة
+
+84
+00:11:14,920 --> 00:11:19,740
+وفي الـGamma neighbourhood صورة الـX هتطلع موجودة
+
+85
+00:11:19,740 --> 00:11:24,560
+في الـV neighbourhood الـneighbourhood لF of C V
+
+86
+00:11:24,560 --> 00:11:29,140
+التي هو neighbourhood لمين of B التي هي بالساوي F
+
+87
+00:11:29,140 --> 00:11:32,380
+of C إذن هذا الكلام صحيح
+
+88
+00:11:39,160 --> 00:11:49,360
+الآن من star هذا يؤدي أن f of x ينتمي إلى f of a
+
+89
+00:11:49,360 --> 00:11:57,340
+تقاطع ... تقاطع
+
+90
+00:11:57,340 --> 00:11:58,860
+f of u
+
+91
+00:12:06,740 --> 00:12:13,500
+وعندي هذا ال F of A من الفرض subset من B و ال F of
+
+92
+00:12:13,500 --> 00:12:17,120
+U أو
+
+93
+00:12:17,120 --> 00:12:23,120
+لكل X ينتمي ل U F of X ينتمي ل V هذا موجود في V
+
+94
+00:12:23,120 --> 00:12:25,640
+وبالتالي التقاطع هذا
+
+95
+00:12:32,220 --> 00:12:41,340
+إذا هذا يؤدي أن f of x تنتمي إلى b تقاطع ال v هذا
+
+96
+00:12:41,340 --> 00:12:51,520
+باستخدام double star أن
+
+97
+00:12:51,520 --> 00:12:59,560
+لكل x في a تقاطع u f of x تنتمي لf of u التي هي
+
+98
+00:12:59,560 --> 00:13:01,120
+subset من b
+
+99
+00:13:10,220 --> 00:13:17,340
+أو ممكن حتى كمان هنا بلاش هذه ممكن نحط هنا V لأن
+
+100
+00:13:17,340 --> 00:13:23,840
+هي لكل X في A تقاطع U F of X ينتمي ل B و لكل X
+
+101
+00:13:23,840 --> 00:13:29,620
+ينتمي إلى A تقاطع U ما ده subset من A فبيطلع F of X
+
+102
+00:13:29,620 --> 00:13:37,660
+ينتمي ل F of A إذا F of X ينتمي ل F of A و ينتمي ل
+
+103
+00:13:37,660 --> 00:13:46,030
+ال B طيب الآن من star لما
+
+104
+00:13:46,030 --> 00:13:53,010
+أكون عندي لو سمي هذه y لما أكون عندي y تنتمي ل b
+
+105
+00:13:53,010 --> 00:13:59,830
+تقاطع v فبيطلع عندي g of y التي هي g ل f of x
+
+106
+00:13:59,830 --> 00:14:07,730
+تنتمي ل w طب ما هذه عبارة عن g circle f ل x
+
+107
+00:14:12,350 --> 00:14:24,070
+الآن since w was an arbitrary epsilon neighborhood
+
+108
+00:14:24,070 --> 00:14:29,110
+of g
+
+109
+00:14:29,110 --> 00:14:37,270
+circle f of c we
+
+110
+00:14:37,270 --> 00:14:49,290
+get from the neighborhood definition of
+
+111
+00:14:49,290 --> 00:14:57,770
+continuity that G
+
+112
+00:14:57,770 --> 00:15:03,650
+circle F is continuous at
+
+113
+00:15:03,650 --> 00:15:12,870
+C وهو المطلوب كمان مرة ما معنى أن الـ function هذه
+
+114
+00:15:12,870 --> 00:15:16,590
+ما معنى أن الـ composite function التي هي G circle
+
+115
+00:15:16,590 --> 00:15:21,530
+if it's continuous in C هذا حسب التعريف التعريف
+
+116
+00:15:21,530 --> 00:15:27,370
+الجوار للاتصال عند نقطة هذا معناه لأي W neighborhood
+
+117
+00:15:27,370 --> 00:15:30,810
+لصورة الـ C يوجد
+
+118
+00:15:32,310 --> 00:15:40,450
+Gamma neighborhood U لـ C بحيث لكل X في مجال A
+
+119
+00:15:40,450 --> 00:15:48,430
+تقاطع Gamma neighborhood تطلع صورة X هذه تنتمي لـ
+
+120
+00:15:48,430 --> 00:15:52,830
+Y neighborhood التي بدأنا فيه وبالتالي حسب التعريف
+
+121
+00:15:52,830 --> 00:15:57,550
+أن الـ function هي continuous and آسف okay إذا هذا
+
+122
+00:15:57,550 --> 00:16:04,270
+برهنت النظرية تمامًا في أي سؤال في أي صفارة على
+
+123
+00:16:04,270 --> 00:16:08,350
+البرهان ناخذ
+
+124
+00:16:08,350 --> 00:16:14,830
+أمثلة أو تطبيقات على نظرية هذه نظرية هذه كثير قوية
+
+125
+00:16:14,830 --> 00:16:19,210
+لأن هنشوف الجهة التي بتبرهن أن نظريات سابقة برهنها
+
+126
+00:16:19,210 --> 00:16:28,330
+سابقا فمثلاً
+
+127
+00:16:28,330 --> 00:16:30,070
+عندي ال function
+
+128
+00:16:38,720 --> 00:16:47,700
+بالمناسبة النظرية دي ممكن طبعًا أن نُعممها وبدل
+
+129
+00:16:47,700 --> 00:16:52,720
+ما تكون صحيحة على الاتصال عند نقطة تصبح صحيحة على
+
+130
+00:16:52,720 --> 00:16:57,420
+الاتصال على مجموعة يعني لو كانت if continuous على
+
+131
+00:16:57,420 --> 00:17:02,680
+كل مجموعة a on
+
+132
+00:17:02,680 --> 00:17:04,700
+a and g is continuous
+
+133
+00:17:06,910 --> 00:17:13,690
+على كل المجال تبعها on b then the composite
+
+134
+00:17:13,690 --> 00:17:21,690
+function is continuous على كل ال a وهو نتيجة عن
+
+135
+00:17:21,690 --> 00:17:27,890
+النظرية السابقة الآن
+
+136
+00:17:27,890 --> 00:17:37,550
+لو أخذنا let جي وان او اكس بساوي اكس ساين اكس و اكس
+
+137
+00:17:37,550 --> 00:17:42,950
+ينتمي الار claim
+
+138
+00:17:42,950 --> 00:17:46,910
+جي
+
+139
+00:17:46,910 --> 00:17:54,750
+is continuous on R let
+
+140
+00:17:54,750 --> 00:18:00,710
+ابسلون اكبر من الصفر be given choose
+
+141
+00:18:06,010 --> 00:18:11,910
+دلتا بساوي إبسلون، عدد موجة بيعتمد على إبسلون،
+
+142
+00:18:11,910 --> 00:18:20,410
+then لو كان X ينتمي إلى مجال الدالة اللي هو R و
+
+143
+00:18:20,410 --> 00:18:24,930
+absolute X minus C
+
+144
+00:18:36,770 --> 00:18:42,650
+خلّيني أقول هنا fix C
+
+145
+00:18:42,650 --> 00:18:52,290
+ينتمي إلى R and let epsilon 
+
+146
+00:18:52,290 --> 00:18:56,770
+أكبر من صفر be given أنا بدأ أثبت أن الـ
+
+147
+00:18:56,770 --> 00:19:01,990
+function G continuous على الـ R فبأثبت الـ C ينتمي
+
+148
+00:19:01,990 --> 00:19:05,450
+إلى R و أثبت أن الـ function continuous عند النقطة 
+
+149
+00:19:05,450 --> 00:19:10,720
+C الأول بعدين بقول بما أنه C was arbitrary إذا ال
+
+150
+00:19:10,720 --> 00:19:15,600
+function continuous على كل ال R فلأي إبسلون أكبر
+
+151
+00:19:15,600 --> 00:19:20,580
+من صفر ت choose دلتا بساوي إبسلون then لكل X في
+
+152
+00:19:20,580 --> 00:19:25,580
+R و absolute X minus C أصغر من دلتا هذا بيقدر أنه
+
+153
+00:19:25,580 --> 00:19:34,900
+absolute G1 of Xg1 of x minus g1 of c اللي هو
+
+154
+00:19:34,900 --> 00:19:41,700
+بيساوي absolute absolute x minus absolute c طبعا
+
+155
+00:19:41,700 --> 00:19:44,800
+هذا by triangle inequality أصغر من أو يساوي
+
+156
+00:19:44,800 --> 00:19:54,440
+absolute x minus c ومن الفرض ومن
+
+157
+00:19:54,440 --> 00:19:58,960
+الفرض أنا ماخد ال x لحياتي اللي مسافة بينها وبين
+
+158
+00:19:58,960 --> 00:20:02,660
+ال c أصغر من دلتاعشان هيك انا اختارت الـ delta
+
+159
+00:20:02,660 --> 00:20:07,080
+بالساوية epsilon عشان يطلع absolute g1 ل x minus
+
+160
+00:20:07,080 --> 00:20:14,520
+g1 ل c أصغر من epsilon و بما انه since epsilon
+
+161
+00:20:14,520 --> 00:20:23,160
+أكبر من صفر was arbitrary اذا g1 is continuous
+
+162
+00:20:27,330 --> 00:20:29,730
+بما أن الـ C ينتمي إلى الـ R كان عبارة عن الـ
+
+163
+00:20:29,730 --> 00:20:50,630
+arbitrary فالفنشن G1 مستمر على كل الـ R تمام فهو
+
+164
+00:20:50,630 --> 00:21:03,110
+فنشن من A إلى R is continuous is continuous on
+
+165
+00:21:03,110 --> 00:21:18,650
+a then by composition theorem بيطلع
+
+166
+00:21:18,650 --> 00:21:31,550
+عندي g و g onecircle F absolute F is continuous on
+
+167
+00:21:31,550 --> 00:21:37,770
+A هاي ال composition theorem او النتيجة تبعتها انا 
+
+168
+00:21:37,770 --> 00:21:46,190
+لسه اثبات ان ال G1 function from R to R is
+
+169
+00:21:46,190 --> 00:21:50,730
+continuous وفي
+
+170
+00:21:50,730 --> 00:21:51,830
+عندي ال
+
+171
+00:21:54,980 --> 00:22:06,640
+عندي f function from a to r f
+
+172
+00:22:06,640 --> 00:22:10,500
+function from a to r continuous انا فرض ان f is
+
+173
+00:22:10,500 --> 00:22:20,000
+continuous على a و ال function التاني g1 g1
+
+174
+00:22:20,000 --> 00:22:24,680
+function from r to rأيضا أثبتنا أنها continuous
+
+175
+00:22:24,680 --> 00:22:28,980
+فلو كانت هذه continuous من A ل B فال composition
+
+176
+00:22:28,980 --> 00:22:33,960
+تبعهم G1 circle F اللي بيطلع absolute F تطلع
+
+177
+00:22:33,960 --> 00:22:37,080
+continuous على A وهذه نظرية أثبتناها المرة اللي
+
+178
+00:22:37,080 --> 00:22:45,400
+فاتت فهي برهان أخر لهذه النظرية تمام مثال
+
+179
+00:22:45,400 --> 00:22:50,560
+تاني let G2
+
+180
+00:22:57,490 --> 00:23:04,270
+let let مثال تاني let g2 of x function بيساوي
+
+181
+00:23:04,270 --> 00:23:08,810
+square root of x و x أكبر من أو يساوي صفر طبعا ده
+
+182
+00:23:08,810 --> 00:23:11,910
+اللي تجد التربية بس معرفة للأعداد الحقيقية غير
+
+183
+00:23:11,910 --> 00:23:19,510
+السالبة claim g2 أثبتنا احنا ملاش claim by
+
+184
+00:23:19,510 --> 00:23:24,800
+previous exercise أجيبنالكم في الامتحان حتى مثال في
+
+185
+00:23:24,800 --> 00:23:37,120
+الامتحان by a previous exercise g2
+
+186
+00:23:37,120 --> 00:23:43,020
+is continuous تطلع continuous على كل فترة من صفر
+
+187
+00:23:43,020 --> 00:23:47,140
+إلى ما لا نهاية هذا السؤال في الامتحان المناسب أخر
+
+188
+00:23:47,140 --> 00:23:51,300
+سؤال في الامتحان now
+
+189
+00:23:53,630 --> 00:24:02,570
+إذا .. إذا من a إلى r هو
+
+190
+00:24:02,570 --> 00:24:08,890
+مثلًا إن f من
+
+191
+00:24:08,890 --> 00:24:19,050
+x أكبر من أو يساوي صفر لكل x تنتمي إلى a و f
+
+192
+00:24:19,050 --> 00:24:20,130
+مستمر
+
+193
+00:24:23,230 --> 00:24:29,110
+on A then
+
+194
+00:24:29,110 --> 00:24:33,810
+by composition theorem
+
+195
+00:24:33,810 --> 00:24:40,130
+بيطلع
+
+196
+00:24:40,130 --> 00:24:48,210
+G to circle F اللي هي بساوي جذر ال F is continuous
+
+197
+00:24:48,210 --> 00:24:57,810
+على المجموعة Aوهذا بيعطي برهان آخر لنظرية أخدناها
+
+198
+00:24:57,810 --> 00:25:02,570
+في آخر محاضرة بتقول لو كانت F دالة غير سالب على
+
+199
+00:25:02,570 --> 00:25:07,630
+المجال تبعها ومتصلة على مجالها ف ال square root لل
+
+200
+00:25:07,630 --> 00:25:12,010
+function بتطلع أيضا متصلة على مجالها صح؟ وهذا
+
+201
+00:25:12,010 --> 00:25:17,170
+بيعتبر برهان آخر غير عن البرهان اللي أخدناه في
+
+202
+00:25:17,170 --> 00:25:19,210
+المحاضرة السابقة okay تمام؟
+
+203
+00:25:29,990 --> 00:25:35,230
+بمثال ثلاثة we
+
+204
+00:25:35,230 --> 00:25:41,310
+know ان ال function f
+
+205
+00:25:41,310 --> 00:25:49,690
+of x بساوي sin x is continuous اثبتنا ان الدالي
+
+206
+00:25:49,690 --> 00:25:53,490
+هاد continuous on R صح؟
+
+207
+00:25:58,290 --> 00:26:04,870
+خلّيني أسميها g of x وفي
+
+208
+00:26:04,870 --> 00:26:18,670
+عندي f of x بساوي واحد على x و
+
+209
+00:26:18,670 --> 00:26:23,050
+x لا يساوي صفر is continuous
+
+210
+00:26:29,880 --> 00:26:34,600
+for all x لا يساوي صفر ده لما اتصل معدى عند
+
+211
+00:26:34,600 --> 00:26:41,940
+الصفر صح is continuous on a اللي هي بساوي r معدى
+
+212
+00:26:41,940 --> 00:26:46,520
+الصفر
+
+213
+00:26:46,520 --> 00:26:55,960
+hence by composition theorem
+
+214
+00:26:55,960 --> 00:27:04,070
+اللي هو النتيجة هذه بيطلع عندي g circle f of x اللي
+
+215
+00:27:04,070 --> 00:27:14,350
+هو g ل f of x اللي هي بتطلع بالساوي sin واحد على x
+
+216
+00:27:14,350 --> 00:27:21,390
+is continuous على
+
+217
+00:27:21,390 --> 00:27:29,730
+كل الأعداد الحقيقية معدا الصفر Okay فالـ function
+
+218
+00:27:29,730 --> 00:27:37,410
+هذه متصلة معدا عند الصفر وهي البرهان تبعها ان هذه
+
+219
+00:27:37,410 --> 00:27:43,230
+كلها تطبيقات على ال composition theorem فهذا
+
+220
+00:27:43,230 --> 00:27:49,390
+بورجيكم قوة ال composition theorem تمام؟ هيك ممكن
+
+221
+00:27:49,390 --> 00:27:54,610
+خلصنا section اتنين خمسة اتنين خلينا نبدأ في
+
+222
+00:27:54,610 --> 00:27:56,350
+section خمسة تلاتة
+
+223
+00:28:15,560 --> 00:28:30,180
+continuous functions on
+
+224
+00:28:30,180 --> 00:28:40,780
+intervals الفترات
+
+225
+00:28:40,780 --> 00:28:47,270
+المتصلة على الفترات قبل ان نبدأ في هذا الموضوع
+
+226
+00:28:47,270 --> 00:28:58,450
+نحتاج التعريف التالي a function f from A to R is
+
+227
+00:28:58,450 --> 00:29:04,630
+bounded on
+
+228
+00:29:04,630 --> 00:29:14,650
+المجموعة A إذا تحقّق الشرط التالي يوجد M عدد موجب
+
+229
+00:29:14,650 --> 00:29:22,090
+بحيث أنه absolute F of X أصغر بيكون أصغر من أو
+
+230
+00:29:22,090 --> 00:29:31,670
+يساوي M لكل X ينتمي إلى A that
+
+231
+00:29:31,670 --> 00:29:39,030
+is هذا معناه أنه the range
+
+232
+00:29:42,830 --> 00:29:52,470
+The range of F which is the set F of A which is
+
+233
+00:29:52,470 --> 00:29:57,950
+the whole F of X حيث X ينتمي إلى A
+
+234
+00:30:12,380 --> 00:30:18,740
+is a bounded subset
+
+235
+00:30:18,740 --> 00:30:23,320
+of
+
+236
+00:30:23,320 --> 00:30:28,240
+R الـ
+
+237
+00:30:28,240 --> 00:30:31,400
+function بتكون bounded function إذا كان ال range
+
+238
+00:30:31,400 --> 00:30:35,520
+تبعها كمجموعة بتطلع bounded set إيه يعني bounded
+
+239
+00:30:35,520 --> 00:30:40,080
+set يعني يوجد عدد موجب بحيث القيمة المطلقة لكل
+
+240
+00:30:40,080 --> 00:30:55,220
+عناصر ال set هذه أصغر من أو يساوي الـ M طيب
+
+241
+00:30:55,220 --> 00:30:59,660
+لو بدي أقول ما معنى أن تكون الدالة unbounded على
+
+242
+00:30:59,660 --> 00:31:09,660
+المجموعة A لأن هنا F from A to R is unbounded
+
+243
+00:31:13,140 --> 00:31:18,900
+is unbounded on
+
+244
+00:31:18,900 --> 00:31:26,580
+a if الشرط
+
+245
+00:31:26,580 --> 00:31:36,020
+التالي يتحقق خلينا
+
+246
+00:31:36,020 --> 00:31:37,220
+ننفي الشرط هذا
+
+247
+00:31:41,360 --> 00:31:47,660
+ننفي الشرط بقى الـ bounded فيوجد M عدد موجب لكل
+
+248
+00:31:47,660 --> 00:31:57,700
+بيصير لكل M عدد موجب يوجد X يوجد X يعتمد على M في
+
+249
+00:31:57,700 --> 00:32:03,600
+مجال الدالة بحيث انه نفي المتباينة هذا يتحقق اللي هو
+
+250
+00:32:03,600 --> 00:32:17,160
+absolute F of X بيطلع أكبر من أو يساوي أكبر
+
+251
+00:32:17,160 --> 00:32:22,500
+من أو يساوي ال M أكبر من أو يساوي ال M هذا معناه أن
+
+252
+00:32:22,500 --> 00:32:27,700
+الدالة ما تكونش bounded على المجال تبعها معناه أنه
+
+253
+00:32:27,700 --> 00:32:31,920
+لأي عدد موجب بقدر ألاقي عنصر يعتمد على عدد الموجب
+
+254
+00:32:31,920 --> 00:32:35,520
+هذا في مجال الدالة والقيمة المطلقة
+
+255
+00:32:38,330 --> 00:32:44,270
+قيمة الدالة عند العنصر أكبر من أو يساوي ال M فمثال
+
+256
+00:32:44,270 --> 00:32:51,010
+على ذلك هاي
+
+257
+00:32:51,010 --> 00:33:00,450
+مثال example لو
+
+258
+00:33:00,450 --> 00:33:14,520
+أخدت ال function f of x show that f of x بيساوي
+
+259
+00:33:14,520 --> 00:33:22,240
+واحد على x و x أكبر من صفر is unbounded
+
+260
+00:33:22,240 --> 00:33:26,500
+على
+
+261
+00:33:26,500 --> 00:33:31,680
+المجموعة a اللي هي الفترة المفتوحة من صفر إلى ما لا نهائي
+
+262
+00:33:31,680 --> 00:33:32,320
+نهائي
+
+263
+00:33:49,080 --> 00:33:57,820
+proof حسب تعريف ال unboundedness بدي اخد ابدأ let
+
+264
+00:33:57,820 --> 00:34:06,960
+m be an arbitrarily positive number be given بدي
+
+265
+00:34:06,960 --> 00:34:14,440
+أرد عليه باكس يعتمد xm بحيث ان المتباينة هذه تتحقق
+
+266
+00:34:14,440 --> 00:34:24,160
+ف choose إذاً لهذا العدد الموجب هختار XM X يعتمد
+
+267
+00:34:24,160 --> 00:34:31,920
+على M على إنه واحد على M زائد واحد فهذا بالتأكيد
+
+268
+00:34:31,920 --> 00:34:39,440
+عدد موجب لأن M عدد موجب صح؟ فبالتالي هذا ينتمي
+
+269
+00:34:39,440 --> 00:34:44,180
+للفترة المفتوحة من صفر إلى ما لا نهاية يعني اللي هي ال A
+
+270
+00:34:47,860 --> 00:34:53,380
+يعني هاي لأي عدد موجب يعني وجدت there exist xm
+
+271
+00:34:53,380 --> 00:34:59,800
+ينتمي للمجال تبع الدالة وهذا يثبت أنه يحقق
+
+272
+00:34:59,800 --> 00:35:04,340
+المتباينة هذا then
+
+273
+00:35:04,340 --> 00:35:15,540
+absolute f of xm بساوي absolute مقلوب
+
+274
+00:35:17,150 --> 00:35:26,330
+الـ Xm هو absolute واحد على Xm وهذا بيساوي M زائد
+
+275
+00:35:26,330 --> 00:35:32,290
+واحد هو مفروض اكتب absolute M زائد واحد اللي هو M
+
+276
+00:35:32,290 --> 00:35:39,450
+زائد واحد لأن M عدد موجب وهذا أكبر من M لأن هذه
+
+277
+00:35:39,450 --> 00:35:47,910
+المتباينة اتحققت أكبر من M لأن نفي أصغر من أو يساوي
+
+278
+00:35:47,910 --> 00:35:53,330
+مفروض يكون أكبر وهي أثبتت لأي عدد موجب
+
+279
+00:35:53,330 --> 00:35:58,570
+بيوجد XM في مجال الدالة و Absolute صورة ال XM أكبر
+
+280
+00:35:58,570 --> 00:36:05,470
+من M okay تمام حسب التعريف أو ال remark بنطلع ال 
+
+281
+00:36:05,470 --> 00:36:09,830
+function تبعتي unbounded على الفترة المفتوحة من 
+
+282
+00:36:09,830 --> 00:36:12,630
+صفر إلى ما لا نهاية تمام
+
+283
+00:36:15,080 --> 00:36:28,840
+الآن في عندي boundedness theorem في
+
+284
+00:36:28,840 --> 00:36:41,220
+نظرية مهمة اسمها boundedness theorem boundedness
+
+285
+00:36:47,400 --> 00:36:54,560
+bound this theorem let
+
+286
+00:36:54,560 --> 00:37:04,580
+خلينا ناخد I let I be closed and bounded interval
+
+287
+00:37:04,580 --> 00:37:10,740
+be closed and 
+
+288
+00:37:10,740 --> 00:37:16,500
+bounded interval
+
+289
+00:37:18,720 --> 00:37:27,740
+لو كانت F F from I to R is continuous on
+
+290
+00:37:27,740 --> 00:37:41,980
+I then F is bounded on 
+
+291
+00:37:41,980 --> 00:37:49,110
+I لأن لو كان المجال ده closed bounded interval وكانت
+
+292
+00:37:49,110 --> 00:37:53,950
+الـ function continuous على I فلابد أنَّها تطلع
+
+293
+00:37:53,950 --> 00:38:05,750
+bounded على الفترة I وبرهان ذلك البرهان سهل proof
+
+294
+00:38:05,750 --> 00:38:13,590
+برهان بالتناقض assume on
+
+295
+00:38:13,590 --> 00:38:14,330
+contrary
+
+296
+00:38:18,360 --> 00:38:23,540
+إذا كان الـ F
+
+297
+00:38:23,540 --> 00:38:29,060
+غير مجموعة على 
+
+298
+00:38:29,060 --> 00:38:34,620
+I فبعد 
+
+299
+00:38:34,620 --> 00:38:35,780
+ذلك بمعالجة أعلى
+
+300
+00:38:43,910 --> 00:38:46,770
+by above remark اللي هي الـ remark اللي مسحناها
+
+301
+00:38:46,770 --> 00:38:55,790
+اللي هي تعطيني تعريف الـ unboundedness لكل n for
+
+302
+00:38:55,790 --> 00:39:10,890
+every n عدد طبيعي يوجد x يعتمد على n في الفترة I 
+
+303
+00:39:10,890 --> 00:39:12,250
+في مجال الدالة
+
+304
+00:39:15,350 --> 00:39:24,050
+بحيث أنه بحيث
+
+305
+00:39:24,050 --> 00:39:35,750
+أنه صورة الـ x<sub>n</sub> أكبر من ... أوي أكبر من n
+
+306
+00:39:35,750 --> 00:39:45,140
+أثبت الـ function F is unbounded on I هذا معناه لكل 
+
+307
+00:39:45,140 --> 00:39:50,140
+n عدد موجب وبالتالي لكل n عدد طبيعي لأن العداد 
+
+308
+00:39:50,140 --> 00:39:56,180
+الطبيعي كلها موجبة يوجد x يعتمد على n في مجال الدالة
+
+309
+00:39:56,180 --> 00:40:02,240
+بحيث f of x<sub>n</sub> يكون أكبر من n
+
+310
+00:40:11,940 --> 00:40:17,960
+طيب ما هذا بتديني طبعا
+
+311
+00:40:17,960 --> 00:40:22,340
+هنا في absolute value احنا نسينا الـ absolute value
+
+312
+00:40:22,340 --> 00:40:29,060
+إذا يوجد هيك بنكون كوّننا so there exist a
+
+313
+00:40:29,060 --> 00:40:38,180
+sequence x<sub>n</sub> contained in I such
+
+314
+00:40:38,180 --> 00:40:38,740
+that
+
+315
+00:40:42,110 --> 00:40:48,930
+absolute f of x<sub>n</sub> أكبر من n for all n belong to N
+
+316
+00:40:48,930 --> 00:40:55,690
+خليني أسميها (*) طيب
+
+317
+00:40:55,690 --> 00:41:03,770
+since الـ sequence x<sub>n</sub> contained
+
+318
+00:41:03,770 --> 00:41:07,990
+in
+
+319
+00:41:07,990 --> 00:41:08,530
+I
+
+320
+00:41:11,610 --> 00:41:18,170
+والـ I عبارة عن الفترة المغلقة من A لـ B إذا x<sub>n</sub>
+
+321
+00:41:18,170 --> 00:41:26,690
+is bounded لأن كل حدود الـ sequence محصورة x<sub>n</sub>
+
+322
+00:41:26,690 --> 00:41:32,070
+أكبر من أو يساوي A أصغر من أو يساوي B لكل n هذه الـ
+
+323
+00:41:32,070 --> 00:41:35,350
+sequence محدودة من أسفل ومحدودة من أعلى كل حدودها
+
+324
+00:41:35,350 --> 00:41:39,630
+محصورة بين A وB إذا الـ sequence هذه bounded صح؟
+
+325
+00:41:40,270 --> 00:41:44,810
+وبالتالي So
+
+326
+00:41:44,810 --> 00:41:52,470
+by Bolzano-Weierstrass theorem نظرية Bolzano
+
+327
+00:41:52,470 --> 00:41:55,410
+-Weierstrass بتقول كل bounded sequence has a 
+
+328
+00:41:55,410 --> 00:41:59,790
+convergent subsequence لأن هنا there exists a
+
+329
+00:41:59,790 --> 00:42:08,950
+subsequence سمِّيها x<sub>n<sub>k</sub></sub> of الـ sequence x<sub>n</sub>
+
+330
+00:42:12,270 --> 00:42:19,590
+which converges وهذه الـ subsequence converges say
+
+331
+00:42:19,590 --> 00:42:27,550
+خلِّينا نفترض أنَّه limit لـ subsequence x<sub>n<sub>k</sub></sub> as k
+
+332
+00:42:27,550 --> 00:42:36,790
+tends to infinity يساوي x تمام؟ since
+
+333
+00:42:38,330 --> 00:42:45,570
+الـ x<sub>n<sub>k</sub></sub> أكبر من أو يساوي A أصغر من أو يساوي B لكل
+
+334
+00:42:45,570 --> 00:42:51,190
+k في N الـ sequence x<sub>n</sub> contained in I وهذه
+
+335
+00:42:51,190 --> 00:42:55,090
+subsequence منها لأن عناصرها موجودة في I بالتالي
+
+336
+00:42:55,090 --> 00:43:01,610
+محصورة بين A وB والـ sequence هذه converge لـ x إذا
+
+337
+00:43:01,610 --> 00:43:06,010
+حسب نظرية سابقة لما تكون الـ sequence محصورة بين a
+
+338
+00:43:06,010 --> 00:43:13,670
+و b وconvergent فنهايتها أيضًا we get from previous
+
+339
+00:43:13,670 --> 00:43:22,850
+theorem أنَّ x تنتمي إلى I أو 
+
+340
+00:43:22,850 --> 00:43:28,310
+تنتمي للفترة المغلقة من a إلى b اللي هي I تمام
+
+341
+00:43:32,130 --> 00:43:39,430
+طيب بما أنَّ F continuous على I hence F is
+
+342
+00:43:39,430 --> 00:43:46,710
+continuous at x لأن F continuous على كل الفترة I و
+
+343
+00:43:46,710 --> 00:43:55,890
+x تنتمي لـ I لأن F continuous at x طيب so by
+
+344
+00:43:55,890 --> 00:43:57,910
+sequential criterion
+
+345
+00:43:59,960 --> 00:44:04,080
+بسيطة لشركات مستمرين هيفي عند سيكوينس كونتينيوس ل
+
+346
+00:44:04,080 --> 00:44:12,200
+x<sub>n</sub> و F continuous عند الـ x إذا بسيطة لشركات مستمرين
+
+347
+00:44:12,200 --> 00:44:19,640
+نحصل على أنَّه limit الـ image لـ subsequence x<sub>n<sub>k</sub></sub>
+
+348
+00:44:19,640 --> 00:44:25,380
+as k tends to infinity يساوي F of x
+
+349
+00:44:30,470 --> 00:44:36,850
+طيب ما هذا معناه أنَّ الـ sequence هذه أو الـ image لـ
+
+350
+00:44:36,850 --> 00:44:40,330
+subsequence اللي هي sequence أخرى convergent
+
+351
+00:44:40,330 --> 00:44:53,750
+وبالتالي فهي bounded okay تمام إذا it follows thus
+
+352
+00:44:53,750 --> 00:44:57,210
+the sequence
+
+353
+00:45:00,200 --> 00:45:08,660
+اللي هي f of x<sub>n<sub>k</sub></sub> من k يساوي واحد إلى infinity
+
+354
+00:45:08,660 --> 00:45:15,980
+is bounded لأن أنا convergence كل convergence
+
+355
+00:45:15,980 --> 00:45:22,380
+sequence is bounded حسب نظرية السابقة وبالتالي so
+
+356
+00:45:22,380 --> 00:45:31,870
+there exist So there exist m عدد موجب بحيث أنَّه
+
+357
+00:45:31,870 --> 00:45:44,450
+absolute
+
+358
+00:45:44,450 --> 00:45:52,730
+f of x<sub>n<sub>k</sub></sub> تطلع أصغر من أو يساوي m لكل 
+
+359
+00:46:07,550 --> 00:46:19,170
+by our chameleon by our
+
+360
+00:46:19,170 --> 00:46:30,390
+chameleon property يوجد k ينتمي إلى N أو خلِّيني
+
+361
+00:46:30,390 --> 00:46:41,390
+أسمِّيه k<sub>0</sub> ينتمي إلى N بحيث أنَّه k<sub>0</sub> أكبر من m صح؟
+
+362
+00:46:41,390 --> 00:46:45,690
+لأي عدد حقيقي سواء موجب أو سالب فيه دائمًا عدد
+
+363
+00:46:45,690 --> 00:46:58,760
+طبيعي أكبر منه now by (*) and (**) we have
+
+364
+00:46:58,760 --> 00:47:10,740
+لدينا أنَّه الـ m أكبر من أو يساوي absolute F of x<sub>n<sub>k<sub>0</sub></sub></sub>
+
+365
+00:47:10,740 --> 00:47:18,320
+k<sub>0</sub> صح؟ هاي من هنا خذي k يساوي k<sub>0</sub> فهذا الكلام صحيح
+
+366
+00:47:18,320 --> 00:47:21,500
+هذا من (**)
+
+367
+00:47:24,010 --> 00:47:32,970
+ومن (*) هذا أكبر من ... إذا من (*) f of x<sub>n<sub>k<sub>0</sub></sub></sub>
+
+368
+00:47:32,970 --> 00:47:41,450
+أكبر من n<sub>k<sub>0</sub></sub> هذا صحيح لكل عناصر الـ sequence x<sub>n</sub>
+
+369
+00:47:41,450 --> 00:47:50,330
+وبالتالي صحيح لكل عناصر الـ subsequence x<sub>n<sub>k</sub></sub> و n<sub>k</sub>
+
+370
+00:47:50,330 --> 00:47:54,410
+k<sub>0</sub> من تعريف الـ subsequence دائمًا أكبر من أو يساوي
+
+371
+00:47:54,410 --> 00:48:01,570
+k<sub>0</sub> ومن هنا k<sub>0</sub> أكبر من m فنحن
+
+372
+00:48:01,570 --> 00:48:10,210
+نحصل على m أكبر من m Contradiction فالتناقض هذا
+
+373
+00:48:10,210 --> 00:48:14,610
+يقول لي أنَّ الـ assumption تبعي الـ assumption تبعنا
+
+374
+00:48:16,810 --> 00:48:22,790
+إنَّ الـ function F is unbounded on I كان assumption
+
+375
+00:48:22,790 --> 00:48:26,870
+خاطئ، إذا ليس صح أنَّ الـ function تطلع bounded على I
+
+376
+00:48:26,870 --> 00:48:32,850
+وهذا بيكمل برهان النظرية، okay تمام؟ واضح؟
+
+377
+00:48:47,090 --> 00:48:53,650
+طيب ناخد احنا شوية break نكتفي
+
+378
+00:48:53,650 --> 00:49:00,710
+بهذا القدر وهناخد إن شاء الله break ونواصل في
+
+379
+00:49:00,710 --> 00:49:01,650
+اللقاء القادم
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/uTGQ5_eGie4.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/uTGQ5_eGie4.srt
new file mode 100644
index 0000000000000000000000000000000000000000..7becefd48a748790a67696a61813af372c3fbad3
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/uTGQ5_eGie4.srt
@@ -0,0 +1,1655 @@
+1
+00:00:20,650 --> 00:00:27,710
+السلام عليكم اليوم إن شاء الله هنكمل ما ابتدأناه
+
+2
+00:00:27,710 --> 00:00:35,620
+سابقا بخصوص موضوع الكوشي sequences  آخر نظرية
+
+3
+00:00:35,620 --> 00:00:41,060
+هناخدها بالنسبة لهذا الموضوع هتكون النظرية التالية
+
+4
+00:00:41,060 --> 00:00:45,580
+لكن في الأول خلينا نراجع بس تعريف الـ Cauchy 
+
+5
+00:00:45,580 --> 00:00:48,600
+sequence فطبعا تعريف الـ Cauchy sequence زي ما
+
+6
+00:00:48,600 --> 00:00:52,560
+أنتم شايفين sequence of real numbers is Cauchy if
+
+7
+00:00:52,560 --> 00:00:55,500
+and only if for every epsilon فيه capital N
+
+8
+00:00:55,500 --> 00:01:00,200
+natural number بحيث for every N و M bigger than or
+
+9
+00:01:00,200 --> 00:01:06,260
+equal capital N المقارنة بين xn و xm أقل من إبسلون
+
+10
+00:01:06,260 --> 00:01:12,980
+Remark في ملاحظة هنا it can be easily shown يعني من
+
+11
+00:01:12,980 --> 00:01:17,540
+السهل إثبات أنه sequence of real numbers is Cauchy
+
+12
+00:01:17,540 --> 00:01:23,420
+if and only if limit the distance between xn و xm
+
+13
+00:01:23,420 --> 00:01:29,700
+بالساوي صفر whenever n و m tends to infinity يعني
+
+14
+00:01:29,700 --> 00:01:34,060
+المعنى آخر لما N و M تكون large، و N و M large
+
+15
+00:01:34,060 --> 00:01:39,660
+enough، ال distance between XN و XM بتكون very 
+
+16
+00:01:39,660 --> 00:01:44,860
+small، تقول لصفر، فهو نفس المعنى تقريبا، you can
+
+17
+00:01:44,860 --> 00:01:50,260
+write a proof for this remark و it is easy زي ما
+
+18
+00:01:50,260 --> 00:01:54,310
+هو مذكور في الـ definition a sequence of real
+
+19
+00:01:54,310 --> 00:02:02,570
+numbers بنسميها contractive إذا وجد ثابت c عدد بين
+
+20
+00:02:02,570 --> 00:02:08,630
+صفر و واحد عدد موجب بحيث أنه المسافة بين xn plus
+
+21
+00:02:08,630 --> 00:02:14,830
+two و xn plus one أصغر من أو يساوي ثابت c في
+
+22
+00:02:14,830 --> 00:02:20,140
+المسافة بين xn plus one minus xn وهذا الكلام طبعا
+
+23
+00:02:20,140 --> 00:02:24,920
+بيكون متحقق for every natural number in الثابت C
+
+24
+00:02:24,920 --> 00:02:33,760
+هذا بيسمى the constant of the contractive sequence
+
+25
+00:02:33,760 --> 00:02:39,140
+الآن النظرية اللي حكيت عنها في المقدمة هي النظرية
+
+26
+00:02:39,140 --> 00:02:46,000
+التالية theorem رقم
+
+27
+00:02:46,000 --> 00:02:47,860
+خمسة وعشرين
+
+28
+00:02:51,670 --> 00:02:59,830
+النظرية هذه بتقول every contractive sequence is Cauchy كل contractive
+
+29
+00:02:59,830 --> 00:03:08,050
+sequence بتكون Cauchy البرهان مش صعب proof for
+
+30
+00:03:08,050 --> 00:03:15,230
+every natural number in apply
+
+31
+00:03:15,230 --> 00:03:20,170
+the defining... apply
+
+32
+00:03:22,820 --> 00:03:28,900
+the defining condition ال
+
+33
+00:03:28,900 --> 00:03:34,980
+defining condition اللي هو هذا الشرط تبع الـ
+
+34
+00:03:34,980 --> 00:03:39,120
+contractive of contractive sequence
+
+35
+00:03:39,120 --> 00:03:42,720
+of contractive sequence لأن هنطبق التعريف هذا to
+
+36
+00:03:46,850 --> 00:03:52,790
+of contractive sequence  لأن هنطبق التعريف هذا to 
+
+37
+00:03:52,790 --> 00:03:56,830
+get لنحصل
+
+38
+00:03:56,830 --> 00:03:57,450
+على mainly
+
+39
+00:04:01,950 --> 00:04:08,910
+هي عندي absolute xn من التعريف هي absolute xn plus
+
+40
+00:04:08,910 --> 00:04:16,110
+two minus xn plus one less than or equal to c في
+
+41
+00:04:16,110 --> 00:04:25,890
+absolute xn plus one negative xn الآن
+
+42
+00:04:25,890 --> 00:04:35,390
+من نفس التعريف هذا بدل n هناك ب N سالب واحد فبصير
+
+43
+00:04:35,390 --> 00:04:40,010
+الطرف الشمال هكذا وهذا بيطلع أصغر من الـ absolute
+
+44
+00:04:40,010 --> 00:04:45,070
+value هذي أصغر من أو يساوي C وفي عندي C تانية فبصير
+
+45
+00:04:45,070 --> 00:04:51,470
+C تربيع في absolute XN minus
+
+46
+00:04:51,470 --> 00:04:58,190
+XN minus واحد تمام؟
+
+47
+00:04:59,310 --> 00:05:05,730
+والآن ممكن من الـ defining condition هذا أطبخه على
+
+48
+00:05:05,730 --> 00:05:11,670
+الـ absolute value هذه يعني أبدل n هناك ب n سالب
+
+49
+00:05:11,670 --> 00:05:17,410
+اتنين فبصير n بصير الطرف الشمال absolute xn minus
+
+50
+00:05:17,410 --> 00:05:22,540
+xn minus واحدهذا هيطلع أصغر من أو يساوي C في
+
+51
+00:05:22,540 --> 00:05:27,860
+absolute value تانية وفي عندي C تربيع فهيصير عندي
+
+52
+00:05:27,860 --> 00:05:34,960
+C تكعيب في absolute xn minus one minus xn minus
+
+53
+00:05:34,960 --> 00:05:42,000
+two وطبعا لو استمرينا على هذا النمط هنصل في الآخر
+
+54
+00:05:42,000 --> 00:05:47,620
+خالص إلى أصغر من أو يساوي c أُس n في absolute x2
+
+55
+00:05:47,620 --> 00:05:52,980
+negative x1 وبعد هيك بنوقف خلاص لأن هنكرر تطبيق
+
+56
+00:05:52,980 --> 00:05:57,460
+الـ defining condition هذا n من المرات وبعد هيك
+
+57
+00:05:57,460 --> 00:06:02,240
+هنوقف لأنه خلاص هنصل لـ absolute x2 minus x1 خلصت
+
+58
+00:06:02,240 --> 00:06:04,600
+الحدود أظبط؟
+
+59
+00:06:06,830 --> 00:06:12,490
+تمام إذا أنا عندي طلع absolute xn plus two minus
+
+60
+00:06:12,490 --> 00:06:17,730
+xn plus one less than or equal c to n في absolute
+
+61
+00:06:17,730 --> 00:06:23,150
+x two negative x one الكلام هذا صحيح for every n
+
+62
+00:06:23,150 --> 00:06:31,250
+كل الأعداد الطبيعية n الآن من الـ inequality هذه
+
+63
+00:06:34,380 --> 00:06:39,140
+this inequality and the triangle inequality
+
+64
+00:06:39,140 --> 00:06:50,600
+متباينة المثلث بيؤدوا and geometric progression
+
+65
+00:07:04,750 --> 00:07:11,290
+imply انه... بيقود أنه for... لو أخدت M أكبر من N
+
+66
+00:07:11,290 --> 00:07:14,870
+فبيطلع
+
+67
+00:07:14,870 --> 00:07:23,930
+عندي absolute XM minus XN هذا هيكون باستخدام
+
+68
+00:07:23,930 --> 00:07:36,720
+متباينة المثلث هأرح من XM لو طرحت من XM XM-1 ورجعتها
+
+69
+00:07:36,720 --> 00:07:50,860
+ورجعتها وبعدين هأطرح XM-2 ورجعها إلى
+
+70
+00:07:50,860 --> 00:08:02,450
+أن أصل إلى xn زائد واحد minus xm شو اللي عملته هنا
+
+71
+00:08:02,450 --> 00:08:09,790
+أنا طرحت من xm xm minus one ورجعتها بعدين طرحت xm
+
+72
+00:08:09,790 --> 00:08:14,790
+سالب اتنين ورجعتها وهكذا طرحت xn زائد واحد و
+
+73
+00:08:14,790 --> 00:08:19,550
+رجعتها وطبعا هاي سالب xn هو بوقف لأنه بعد كده
+
+74
+00:08:19,550 --> 00:08:30,360
+خلاص الآن من ال... يسمى المتباينة بال star فمن
+
+75
+00:08:30,360 --> 00:08:36,460
+الـ star this inequality اللي هي الـ star لو
+
+76
+00:08:36,460 --> 00:08:46,240
+أخذت هنا M بدلت N زائد 2 بدلتها ب M معناته N بساوي
+
+77
+00:08:46,240 --> 00:08:56,130
+M سالب 2 صح؟ وبالتالي هذا بيصير أصغر
+
+78
+00:08:56,130 --> 00:09:03,190
+من أو يساوي C أُس M اللي هي M سالب اتنين في
+
+79
+00:09:03,190 --> 00:09:09,750
+absolute X2 نيجاتيب X1 إذن هذا باستخدام المتباينة
+
+80
+00:09:09,750 --> 00:09:14,290
+الـ star حيث الـ M هذه أخدتها بساوي N زائد 2
+
+81
+00:09:14,290 --> 00:09:19,390
+وبالتالي إذا الـ N بساوي M نيجاتيف Two فالـ absolute
+
+82
+00:09:19,390 --> 00:09:23,350
+value الأولانية أصغر حسب الـ star أصغر من أو يساوي c
+
+83
+00:09:23,350 --> 00:09:27,650
+to m minus 2 في absolute x2 minus 1 Similarly
+
+84
+00:09:27,650 --> 00:09:31,730
+باستخدام الـ star المتباينة الـ star الـ absolute
+
+85
+00:09:31,730 --> 00:09:36,490
+value التاني هذه أو second term هذا بيطلع أصغر من
+
+86
+00:09:36,490 --> 00:09:43,530
+أو يساوي c to m minus 3 في absolute x2 minus x1 و
+
+87
+00:09:43,530 --> 00:09:44,130
+هكذا
+
+88
+00:09:46,560 --> 00:09:51,660
+إلى أن نصل آخر حد هيطلع عند you c to n minus واحد
+
+89
+00:09:51,660 --> 00:09:58,380
+في absolute x2 minus x1 طيب
+
+90
+00:09:58,380 --> 00:10:01,880
+نأخذ الآن عامل مشترك
+
+91
+00:10:04,360 --> 00:10:11,340
+هذا بيساوي c to n negative two زائد c to n
+
+92
+00:10:11,340 --> 00:10:17,800
+negative three زائد وهكذا to c to n negative one
+
+93
+00:10:17,800 --> 00:10:23,460
+كل هذا مضروب في العامل المشترك absolute x two
+
+94
+00:10:23,460 --> 00:10:28,540
+minus x one المجموع
+
+95
+00:10:28,540 --> 00:10:36,720
+هذا هأخذ عامل مشترك c to n minus واحد منه فهيبقى
+
+96
+00:10:36,720 --> 00:10:46,020
+لدي هنا c to m minus n minus 1 وهنا هأخذ لو قسمت
+
+97
+00:10:46,020 --> 00:10:52,420
+هذا على c to n minus 1 هيطلع c to m minus n minus
+
+98
+00:10:52,420 --> 00:10:55,800
+2
+
+99
+00:10:55,800 --> 00:11:06,070
+وهكذا إلى آخر حد هيكون 1 وطبعا كل هذا مضروب في
+
+100
+00:11:06,070 --> 00:11:17,890
+absolute x2 minus x1 y ساوي c to n negative one
+
+101
+00:11:17,890 --> 00:11:22,450
+الآن هذه عبارة عن geometric progression متوالية
+
+102
+00:11:22,450 --> 00:11:30,230
+هندسية الحد الأول فيها واحد والأساس تبعها c فمجموع
+
+103
+00:11:30,230 --> 00:11:33,930
+المتوالية الهندسية أو الـ geometric progression
+
+104
+00:11:33,930 --> 00:11:40,210
+مجموع بساوي الحد الأول واحد سالب الحد الأخير
+
+105
+00:11:40,210 --> 00:11:47,170
+مضروب في الأساس اللي هو C فبيطلع C to M negative N
+
+106
+00:11:47,170 --> 00:11:54,070
+على واحد سالب الأساس إذن هذا مجموع الـ geometric
+
+107
+00:11:54,070 --> 00:11:58,910
+progression كل هذا مضروب في absolute X2 negative
+
+108
+00:11:58,910 --> 00:12:04,610
+X1 طيب
+
+109
+00:12:04,610 --> 00:12:10,330
+أنا عندي الـ C أنا
+
+110
+00:12:10,330 --> 00:12:15,630
+عندي الـ C الـ C
+
+111
+00:12:15,630 --> 00:12:22,980
+عدد بين 0 و 1 وبالتالي c to m minus n بيبقى العدد
+
+112
+00:12:22,980 --> 00:12:32,020
+بين صفر وواحد وبالتالي إذا واحد سالب c اكيد هيطلع
+
+113
+00:12:32,020 --> 00:12:37,880
+أكبر من صفر أصغر
+
+114
+00:12:37,880 --> 00:12:38,440
+من واحد
+
+115
+00:12:41,670 --> 00:12:47,450
+إذا هاي أنا عندي هذا هستبدله بأصغر من هاي c to n
+
+116
+00:12:47,450 --> 00:12:53,570
+negative one الآن هذا الكسر الـ bus تبعه واحد سالب
+
+117
+00:12:53,570 --> 00:12:59,970
+c to n minus n أصغر من واحد فهذا الكسر أصغر من
+
+118
+00:12:59,970 --> 00:13:06,150
+واحد على واحد سالب c أصغر من واحد على واحد سالب c
+
+119
+00:13:06,150 --> 00:13:10,510
+في absolute x to negative x one
+
+120
+00:13:14,990 --> 00:13:23,230
+طيب أنا عندي برضه أخذنا قبل هيك مثال بيقول إذا كان
+
+121
+00:13:23,230 --> 00:13:29,790
+C أكبر من صفر أصغر من واحد هذا بيقود إن الـ limit لـ
+
+122
+00:13:29,790 --> 00:13:35,290
+C to N أو C to N سالب واحد لما N تقل لـ infinity
+
+123
+00:13:35,290 --> 00:13:39,410
+بيساوي صفر فبالاستخدام المثال هذا اللي أثبتناه قبل
+
+124
+00:13:39,410 --> 00:13:47,130
+هيك بنلاحظ إن c to n سالب واحد هنا هذا ثابت وهذا
+
+125
+00:13:47,130 --> 00:13:54,710
+ثابت فالـ c to n سالب واحد تقول للسفر في ثابت تقول
+
+126
+00:13:54,710 --> 00:13:59,730
+لثابت في صفر يعني صفر إذا هذا المقدار كله tends to
+
+127
+00:13:59,730 --> 00:14:03,590
+zero as n tends to infinity
+
+128
+00:14:07,000 --> 00:14:12,180
+وبالتالي إذا هيك احنا أثبتنا إن الـ limit لـ
+
+129
+00:14:12,180 --> 00:14:19,740
+absolute xm minus xn لما الـ M تقول لـ infinity
+
+130
+00:14:19,740 --> 00:14:23,680
+بتساوي
+
+131
+00:14:23,680 --> 00:14:32,280
+صفر وطبعا الـ M أنا ماخذها هنا الـ M ماخذها M أكبر
+
+132
+00:14:32,280 --> 00:14:36,860
+من N فلما الـ M تقول لـ infinity الـ M أيضا تقول
+
+133
+00:14:36,860 --> 00:14:42,000
+لإنفينيتي إذا هنا ممكن أحط and M تقول لإنفينيتي
+
+134
+00:14:42,000 --> 00:14:46,220
+إذا هنا أثبتنا إن الـ limit لـ absolute XM minus XN
+
+135
+00:14:46,220 --> 00:14:51,540
+as N and M both tends to infinity بساوي صفر
+
+136
+00:14:51,540 --> 00:14:56,800
+وبالتالي إذا by above remark حسب الـ remark اللي
+
+137
+00:14:56,800 --> 00:15:01,200
+فوق الـ sequence XM is Cauchy
+
+138
+00:15:04,210 --> 00:15:12,090
+وهذا بيكمل برهان النظرية تمام؟ واضح؟
+
+139
+00:15:12,090 --> 00:15:16,690
+في أي استفسار؟ في أي سؤال على البرهان؟ في أي قطة
+
+140
+00:15:16,690 --> 00:15:23,570
+مش واضحة؟ is there any question؟ 
+
+141
+00:15:23,570 --> 00:15:27,790
+Okay, then this ends.
+
+142
+00:15:30,870 --> 00:15:37,390
+section ثلاثة خمسة and we are going to start a new 
+
+143
+00:15:37,390 --> 00:15:44,490
+section هنبدأ section جديد وهذا ال section بتحدث 
+
+144
+00:15:44,490 --> 00:15:55,870
+عن موضوع properly divergent sequences section 
+
+145
+00:15:55,870 --> 00:16:00,450
+three point six
+
+146
+00:16:02,990 --> 00:16:11,770
+properly .. properly .. divergent .. divergent ..
+
+147
+00:16:11,770 --> 00:16:12,710
+sequences
+
+148
+00:16:17,180 --> 00:16:21,080
+أي يعني properly .. properly divergent sequence
+
+149
+00:16:21,080 --> 00:16:27,140
+احنا شفنا قبل هيك أمثلة examples about divergence
+
+150
+00:16:27,140 --> 00:16:31,200
+sequences من الـ divergence sequences هذه كان 
+
+151
+00:16:31,200 --> 00:16:36,480
+negative one to n بيقولوا أنه هذه has a divergence
+
+152
+00:16:36,480 --> 00:16:40,640
+برضه
+
+153
+00:16:40,640 --> 00:16:48,480
+sequence n Divergent means infinity وبالتالي 
+
+154
+00:16:48,480 --> 00:16:51,480
+Divergent الـ sequence negative and intense
+
+155
+00:16:51,480 --> 00:16:59,480
+وبالتالي Divergent وهكذا في كتير sequences مر
+
+156
+00:16:59,480 --> 00:17:04,520
+علينا sequences على الأقل هدول إذا ماكانش أكتر
+
+157
+00:17:04,520 --> 00:17:11,930
+وشوفنا إن كل ال sequences هذي are divergent هناك 
+
+158
+00:17:11,930 --> 00:17:17,130
+لحد الآن ما تحدثناش عن تفريق التفريق في الـ
+
+159
+00:17:17,130 --> 00:17:22,070
+divergence كنا نقول إن إذا كانت ال sequence
+
+160
+00:17:22,070 --> 00:17:26,550
+ماليهاش limit أو تقول لـ infinity أو سالب infinity 
+
+161
+00:17:26,550 --> 00:17:30,130
+فكنا نقول ال sequence is divergent أو not 
+
+162
+00:17:30,130 --> 00:17:34,210
+convergent اليوم ال divergence sequences هنجزقهم
+
+163
+00:17:34,210 --> 00:17:40,450
+إلى نوعين في sequences هنقول عنهم divergent وفي 
+
+164
+00:17:40,450 --> 00:17:44,850
+نوع معين من الـ divergence sequences هنسميهم
+
+165
+00:17:44,850 --> 00:17:49,810
+properly divergent إذن الـ sequences اللي زي هدول 
+
+166
+00:17:49,810 --> 00:17:54,890
+اللي ال limit بتاعتهم إما infinity أو negative
+
+167
+00:17:54,890 --> 00:17:59,110
+infinity طبعا هدول ال sequences are divergent لكن
+
+168
+00:17:59,110 --> 00:18:02,170
+هذا النوع من ال divergence sequences هنسميه 
+
+169
+00:18:02,170 --> 00:18:06,410
+properly divergent متباعدة تباعدا صحيحا
+
+170
+00:18:09,530 --> 00:18:13,590
+Okay إذا الـ sequences التلاتي هدول all of
+
+171
+00:18:13,590 --> 00:18:17,690
+them are divergent كلهم ممكن نقول عنهم divergent 
+
+172
+00:18:17,690 --> 00:18:23,710
+لكن التنتين هدول الأخرانيين ممكن نقول عنهم أيضا 
+
+173
+00:18:23,710 --> 00:18:27,490
+properly divergent أما هذه مقدرش أقول عنها
+
+174
+00:18:27,490 --> 00:18:32,450
+properly divergent مجرد divergent okay تمام إذا 
+
+175
+00:18:32,450 --> 00:18:38,270
+نكتب التعريفات هذه definition
+
+176
+00:18:42,170 --> 00:18:49,570
+اثنين ستة وعشرين let
+
+177
+00:18:49,570 --> 00:18:52,590
+x n be sequence of real numbers
+
+178
+00:18:56,340 --> 00:19:05,960
+نقول إن xn tends to infinity أو limit xn as n
+
+179
+00:19:05,960 --> 00:19:14,500
+tends to infinity بيساوي infinity إذا تحقق الشرط
+
+180
+00:19:14,500 --> 00:19:15,220
+التالي
+
+181
+00:19:23,170 --> 00:19:31,350
+for every real number alpha there exists capital N
+
+182
+00:19:31,350 --> 00:19:41,150
+depends on alpha natural number such that xn such
+
+183
+00:19:41,150 --> 00:19:47,270
+that لو كان n أكبر 
+
+184
+00:19:47,270 --> 00:19:51,150
+من أو يساوي capital N هذا بقدر أن xn أكبر من alpha
+
+185
+00:19:53,550 --> 00:19:56,650
+اللي بتتكلموا لو سمحتوا ما تتكلمش امنع الكلام
+
+186
+00:19:59,910 --> 00:20:04,290
+إذا ما معنى ال sequence Xn tends to infinity أو 
+
+187
+00:20:04,290 --> 00:20:09,410
+limit لها بيساوي infinity معناه لأي number ألفة
+
+188
+00:20:09,410 --> 00:20:12,670
+نقدر نجد الـ number natural نقدر نجد الـ number
+
+189
+00:20:12,670 --> 00:20:13,090
+الnatural نقدر نجد الـ number
+
+190
+00:20:13,090 --> 00:20:14,290
+الnatural نقدر نجد الـ number natural نقدر نجد الـ
+
+191
+00:20:14,290 --> 00:20:14,970
+number natural نقدر نجد الـ number natural 
+
+192
+00:20:14,970 --> 00:20:21,010
+نقدر نجد الـ number natural نقدر نجد الـ number 
+
+193
+00:20:21,010 --> 00:20:25,370
+الnatural نقدر نجد الـ number natural نقدر نجد الـ
+
+194
+00:20:25,370 --> 00:20:26,370
+number natural نقدر نجد الـ number natural نقدر
+
+195
+00:20:26,370 --> 00:20:27,990
+نجد الـ number natural نقدر نجد الـ number natural
+
+196
+00:20:27,990 --> 00:20:32,450
+نقدر نجد الـ number من capital N أو كل حدودها for 
+
+197
+00:20:32,450 --> 00:20:37,490
+large N أكبر من أي عدد Alpha أي عدد حقيقي Alpha
+
+198
+00:20:37,490 --> 00:20:41,810
+عشوائي okay إذا قدرنا نخلي حدود ال sequence أكبر 
+
+199
+00:20:41,810 --> 00:20:46,330
+من أي عدد حقيقي مهما كان فال sequence معناته
+
+200
+00:20:46,330 --> 00:20:54,110
+نهايتها infinity بالمثل ممكن نعرف نقول xn tends
+
+201
+00:20:54,110 --> 00:21:01,710
+to negative infinity أو limit xn as n tends to
+
+202
+00:21:01,710 --> 00:21:07,450
+infinity بيساوي negative infinity إذا تحقق الشرط
+
+203
+00:21:07,450 --> 00:21:17,190
+التالي for every beta real number يوجد capital N
+
+204
+00:21:17,190 --> 00:21:24,390
+يعتمد على ال beta natural number such that لكل n
+
+205
+00:21:24,390 --> 00:21:29,750
+أكبر من أو يساوي capital N بيطلع عندي xn أصغر من
+
+206
+00:21:29,750 --> 00:21:34,770
+beta إذا لو قدرت أخلي حدود ال sequence أصغر من أي
+
+207
+00:21:34,770 --> 00:21:40,640
+عدد حقيقي beta for large N فكل ال sequence هذه 
+
+208
+00:21:40,640 --> 00:21:44,780
+بيقول إنها tends to negative infinity أو ال limit
+
+209
+00:21:44,780 --> 00:21:51,020
+بتاعتها is negative infinity هاي أمثلة examples
+
+210
+00:21:51,020 --> 00:22:03,600
+show
+
+211
+00:22:03,600 --> 00:22:06,180
+that limit
+
+212
+00:22:07,910 --> 00:22:13,670
+ال sequence n بيساوي infinity احنا كلنا عارفين
+
+213
+00:22:13,670 --> 00:22:18,790
+that the sequence of natural number ال limit
+
+214
+00:22:18,790 --> 00:22:26,490
+بتاعتها infinity لكن ممكن نثبت الآن هذا باستخدام
+
+215
+00:22:26,490 --> 00:22:31,590
+التعريف فنشوف
+
+216
+00:22:31,590 --> 00:22:37,560
+هاي البرهان proof حسب التعريف بدنا نبدأ بقول let 
+
+217
+00:22:37,560 --> 00:22:45,180
+alpha belonging to R be given ناخذ 
+
+218
+00:22:45,180 --> 00:22:52,520
+alpha عدد حقيقي عشوائي by
+
+219
+00:22:52,520 --> 00:23:00,260
+Archimedean property حسب خاصية Archimedes
+
+220
+00:23:04,680 --> 00:23:13,220
+يوجد capital N عدد طبيعي بحيث إنه capital N أكبر 
+
+221
+00:23:13,220 --> 00:23:19,740
+من ال alpha  نظبط؟ هذا حسب ال Archimedean property
+
+222
+00:23:19,740 --> 00:23:22,240
+now
+
+223
+00:23:24,690 --> 00:23:28,990
+لو أخدت n bigger than or equal capital N هذا هيقدي 
+
+224
+00:23:28,990 --> 00:23:35,330
+إن xn ال sequence هذه لحد العام xn تبعها إيش
+
+225
+00:23:35,330 --> 00:23:42,730
+بيساوي بيساوي n الآن ال n هذه small n أكبر منه
+
+226
+00:23:42,730 --> 00:23:47,010
+بيساوي capital N وأنا بختار capital N by
+
+227
+00:23:47,010 --> 00:23:49,350
+Archimedean property أكبر من alpha 
+
+228
+00:23:55,340 --> 00:24:01,740
+وبالتالي إذا الـ implication هذه تتحقق
+
+229
+00:24:01,740 --> 00:24:08,220
+هنا أثبتنا for any alpha تنتمي لـ R there exists 
+
+230
+00:24:08,220 --> 00:24:12,600
+natural number يعتمد على alpha هيوجه يعتمد على 
+
+231
+00:24:12,600 --> 00:24:18,160
+alpha N مرتبطة بـ alpha بحيث لكل n أكبر من أو يساوي 
+
+232
+00:24:18,160 --> 00:24:22,740
+capital N بيطلع عندي xn أكبر من alpha therefore
+
+233
+00:24:27,120 --> 00:24:31,940
+alpha capital N definition of limit 
+
+234
+00:24:38,380 --> 00:24:43,960
+أو alpha capital N definition بيطلع عندي limit xn
+
+235
+00:24:43,960 --> 00:24:50,940
+بيساوي infinity طبعا هنا xn مقصود فيها الحد العام
+
+236
+00:24:50,940 --> 00:24:56,440
+لل sequence n يعني xn بيساوي n okay تمام واضح 
+
+237
+00:24:56,440 --> 00:25:04,360
+البرهان طب هاي مثال تاني show that إنه limit الـ 
+
+238
+00:25:04,360 --> 00:25:06,800
+sequence اللي الحد اللي عام تبعها negative
+
+239
+00:25:06,800 --> 00:25:16,080
+n تربيع equals negative infinity هنطبق
+
+240
+00:25:16,080 --> 00:25:23,660
+التعريف في الجزء الثاني ونشوف كيف ممكن نثبت إن الـ 
+
+241
+00:25:23,660 --> 00:25:28,940
+implication كيف لأي beta بقدر ألاقي capital N
+
+242
+00:25:28,940 --> 00:25:35,040
+يعتمد على beta بحيث إن ال implication هذه هي تتحقق 
+
+243
+00:25:35,040 --> 00:25:40,320
+هاي 
+
+244
+00:25:40,320 --> 00:25:45,800
+البرهان proof بالمناسبة
+
+245
+00:25:45,800 --> 00:25:51,980
+أنا ما عنديش يعني عصا سحرية عشان أعرف مسبقا لأي
+
+246
+00:25:51,980 --> 00:25:58,440
+beta كيف أختار ال n ما عنديش عصا سحرية فبنعمل 
+
+247
+00:25:58,440 --> 00:26:05,060
+analysis تحليل وبنكتشف كيف نختار الـ capital N لأي
+
+248
+00:26:05,060 --> 00:26:12,500
+given beta إذا هنا حد البرهان بـ let beta بأي real 
+
+249
+00:26:12,500 --> 00:26:16,800
+number بـ given طبعا
+
+250
+00:26:16,800 --> 00:26:21,540
+حسب التعريف عشان أثبت إنه ال sequence الحد العام
+
+251
+00:26:21,540 --> 00:26:22,760
+تبعها xn 
+
+252
+00:26:25,210 --> 00:26:29,610
+بيساوي سالب n تربيع عشان أثبت إن ال limit لل
+
+253
+00:26:29,610 --> 00:26:33,830
+sequence هذه بيساوي negative infinity بدي أثبت بدي 
+
+254
+00:26:33,830 --> 00:26:37,470
+أرد على ال given beta هذه ب capital N تعتمد عليها
+
+255
+00:26:37,470 --> 00:26:42,510
+بحيث إن ال implication هذه تتحقق فبنيجي بنعمل زي 
+
+256
+00:26:42,510 --> 00:26:47,510
+ما عملنا في تعريف epsilon capital N للنهايات أنا 
+
+257
+00:26:47,510 --> 00:26:53,340
+بقول من الآخر أنا عايز إن xn اللي هي سالب n تربيع 
+
+258
+00:26:53,340 --> 00:26:58,840
+بدي هذه في النهاية for any given beta real
+
+259
+00:26:58,840 --> 00:27:05,520
+number بدي xn اللي هي negative n squared بديها 
+
+260
+00:27:05,520 --> 00:27:12,100
+less than beta تمام؟
+
+261
+00:27:13,650 --> 00:27:19,790
+وطبعا هذا لكل n أكبر من أو يساوي capital N فما هي
+
+262
+00:27:19,790 --> 00:27:27,090
+ال n أنا بدي أجيب ال n اللي بتخلي هذا الكلام صحيح
+
+263
+00:27:27,090 --> 00:27:31,270
+أنا
+
+264
+00:27:31,270 --> 00:27:36,030
+بعرف إنه لازم ال n تكون أكبر من أو يساوي capital N
+
+265
+00:27:36,030 --> 00:27:38,630
+وبالتالي 
+
+266
+00:27:40,310 --> 00:27:47,610
+هذا بيقدي إن n تربيع أكبر من أو يساوي n صح؟ أكبر
+
+267
+00:27:47,610 --> 00:27:52,550
+من أو يساوي capital N وهذا
+
+268
+00:27:52,550 --> 00:27:59,530
+بيقدي إن سالب n تربيع أصغر من أو يساوي سالب
+
+269
+00:27:59,530 --> 00:28:00,290
+capital N
+
+270
+00:28:05,790 --> 00:28:13,770
+وأنا بدي هذا يطلع أصغر من ال beta عشان يطلع xn 
+
+271
+00:28:13,770 --> 00:28:22,870
+أصغر من beta صح؟ إذا بسأل نفسي متى هذا سالب n 
+
+272
+00:28:22,870 --> 00:28:27,670
+تربيع اللي هو xn أصغر من beta  لإنما negative
+
+273
+00:28:27,670 --> 00:28:33,880
+capital N أصغر من beta إذا for any for any beta
+
+274
+00:28:33,880 --> 00:28:40,360
+belonging to R يقول هنا for any beta belonging to
+
+275
+00:28:40,360 --> 00:28:42,640
+R يقول هنا for any beta belonging to R يقول هنا
+
+276
+00:28:42,640 --> 00:28:44,360
+for any beta belonging to R يقول هنا for any beta
+
+277
+00:28:44,360 --> 00:28:44,580
+belonging to R يقول هنا for any beta belonging to
+
+278
+00:28:44,580 --> 00:28:44,600
+for any beta belonging to R يقول هنا for any beta
+
+279
+00:28:44,600 --> 00:28:44,700
+belonging to R يقول هنا for any beta belonging to
+
+280
+00:28:44,700 --> 00:28:44,720
+R يقول هنا for any beta belonging to R يقول هنا 
+
+281
+00:28:44,720 --> 00:28:44,740
+for any beta belonging to R يقول هنا for any beta
+
+282
+00:28:44,740 --> 00:28:50,520
+R يقول هنا for any beta belonging to R يقول هنا
+
+283
+00:28:50,520 --> 00:28:54,400
+for any beta belonging to R يقول هنا for any beta
+
+284
+00:28:54,400 --> 00:29:04,050
+belonging to R يقول وهذا بقدر أختاره by Archimedean
+
+285
+00:29:04,050 --> 00:29:11,170
+property by Archimedean property لأي real number
+
+286
+00:29:11,170 --> 00:29:15,610
+beta سالب beta is real number و بقدر ألاقي capital
+
+287
+00:29:15,610 --> 00:29:20,410
+N أكبر من أي real number by Archimedean property
+
+288
+00:29:20,410 --> 00:29:23,810
+إذا capital N اللي أنا عايزها أكبر من ال given
+
+289
+00:29:23,810 --> 00:29:29,270
+بسالب ال given beta أكبر من سالب ال given beta إذا 
+
+290
+00:29:29,270 --> 00:29:36,450
+هنا باجي بقول let beta be given, it choose using
+
+291
+00:29:38,220 --> 00:29:44,220
+الـ Archimedean property capital
+
+292
+00:29:44,220 --> 00:29:50,640
+N عدد طبيعي natural number such that capital N
+
+293
+00:29:50,640 --> 00:29:56,660
+أكبر من negative beta وهذا مقدر أعمله by
+
+294
+00:29:56,660 --> 00:30:03,340
+Archimedean property الآن تعالى نشوف إذا أنا لأي
+
+295
+00:30:03,340 --> 00:30:08,660
+beta وجدت عدد طبيعي هي بيعتمد على beta هاي
+
+296
+00:30:08,660 --> 00:30:12,420
+capital N تعتمد على beta مرتبطة فيها بالمتباينة
+
+297
+00:30:12,420 --> 00:30:19,340
+هذه الآن فاضل باقي أثبت أنه لو أخدت أي small n أكبر 
+
+298
+00:30:19,340 --> 00:30:25,540
+من أو يساوي capital N بدي أثبت أن هذا بيقدي أن xn
+
+299
+00:30:25,540 --> 00:30:30,820
+أصغر من beta طيب هاي n أكبر من أو يساوي capital N
+
+300
+00:30:30,820 --> 00:30:35,400
+شوف أن هذا بيقدي أن n تربيع أكبر من أو يساوي small
+
+301
+00:30:35,400 --> 00:30:41,760
+n لأي عداد طبيعي طيب small n أكبر من أو يساوي
+
+302
+00:30:41,760 --> 00:30:42,520
+capital N
+
+303
+00:30:45,500 --> 00:30:51,940
+و capital N أكبر من negative beta حسب اختيارنا إذا
+
+304
+00:30:51,940 --> 00:30:56,480
+هذا بيقدي اضرب في سالب واحد إذا هذا بيقدي ان xn
+
+305
+00:30:56,480 --> 00:31:01,520
+اللي هي سالب أو negative n تربيع اضرب في سالب واحد
+
+306
+00:31:01,520 --> 00:31:07,760
+هذا بيصير أصغر من beta وهذه
+
+307
+00:31:07,760 --> 00:31:12,340
+هي ال implication اللي أنا عايز أحققها صح؟ إذا أنا
+
+308
+00:31:12,340 --> 00:31:20,580
+هيندي صار by definition حققت أنه for any beta يوجد
+
+309
+00:31:20,580 --> 00:31:26,100
+capital N عدد طبيعي يعتمد على ال beta بحيث لكل N
+
+310
+00:31:26,100 --> 00:31:32,640
+أكبر من أو يساوي capital N هذا بيقدي ان Xn أصغر من
+
+311
+00:31:32,640 --> 00:31:35,060
+beta therefore by definition
+
+312
+00:31:37,910 --> 00:31:44,330
+حسب التعريف الثاني بطلع عندي limit xn equals
+
+313
+00:31:44,330 --> 00:31:53,950
+negative infinity وهو المطلوب تمام هيك؟ في أي
+
+314
+00:31:53,950 --> 00:31:59,250
+سؤال؟ في أي استفسار؟
+
+315
+00:31:59,250 --> 00:32:04,730
+طيب نأخذ الآن نظرية 
+
+316
+00:32:19,760 --> 00:32:32,580
+خليني أغير الجملة theorem
+
+317
+00:32:32,580 --> 00:32:40,260
+رقمها ثمانية عشر هذه
+
+318
+00:32:40,260 --> 00:32:49,040
+تعتبر monotone convergence theorem for properly
+
+319
+00:32:55,550 --> 00:33:03,530
+divergent sequences monotone
+
+320
+00:33:03,530 --> 00:33:06,630
+convergence theorem for properly divergent
+
+321
+00:33:06,630 --> 00:33:15,410
+sequences النظرية هذه بتنص على ان a monotone .. a
+
+322
+00:33:15,410 --> 00:33:19,450
+monotone sequence
+
+323
+00:33:21,930 --> 00:33:35,070
+in R is properly is properly divergent if
+
+324
+00:33:35,070 --> 00:33:41,270
+and only if it is unbounded
+
+325
+00:33:41,270 --> 00:33:46,230
+in 
+
+326
+00:33:46,230 --> 00:33:50,950
+fact في حقيقة الأمر in fact
+
+327
+00:33:54,620 --> 00:34:00,520
+إذا Xn غير
+
+328
+00:34:00,520 --> 00:34:06,580
+مجموعة ومحدود
+
+329
+00:34:06,580 --> 00:34:09,740
+ومحدود
+
+330
+00:34:09,740 --> 00:34:13,280
+فإن
+
+331
+00:34:13,280 --> 00:34:18,400
+Xn يتنقل إلى الانفصال
+
+332
+00:34:27,110 --> 00:34:34,510
+وإذا كان Xn غير مجموعة ومتناقص
+
+333
+00:34:34,510 --> 00:34:38,090
+فإن
+
+334
+00:34:38,090 --> 00:34:43,330
+سيكوان Xn يتنقص إلى  ..... 
+
+335
+00:34:43,330 --> 00:34:45,210
+.....
+
+336
+00:34:45,210 --> 00:34:50,090
+.....
+
+337
+00:34:50,090 --> 00:34:50,850
+.....
+
+338
+00:34:50,850 --> 00:34:50,990
+.....
+
+339
+00:34:50,990 --> 00:34:51,050
+.....
+
+340
+00:35:00,740 --> 00:35:12,340
+الأول ..الأول مصدر يتبع من 
+
+341
+00:35:12,340 --> 00:35:16,960
+الـ monotone convergence theorem
+
+342
+00:35:21,340 --> 00:35:24,060
+الـ first statement اللي هو بيقول a monotone
+
+343
+00:35:24,060 --> 00:35:29,100
+sequence of real numbers is properly divergent if
+
+344
+00:35:29,100 --> 00:35:32,300
+and only if it is unbounded هذا نتيجة على الـ
+
+345
+00:35:32,300 --> 00:35:35,740
+monotone convergence theorem لأن الـ monotone
+
+346
+00:35:35,740 --> 00:35:40,340
+convergence theorem اللي أخذناها قبل هيك بتقول a
+
+347
+00:35:40,340 --> 00:35:45,600
+monotone sequence is convergent if and only if it
+
+348
+00:35:45,600 --> 00:35:51,040
+is bounded فمن نفس النظرية ومن نفس النص ممكن نقول
+
+349
+00:35:51,040 --> 00:35:54,960
+الـ monotone sequence is not convergent أو
+
+350
+00:35:54,960 --> 00:36:00,580
+divergent if and only if it is unbounded بغض النظر
+
+351
+00:36:00,580 --> 00:36:04,800
+ال divergence هنا شو نوعه okay إذا ال face
+
+352
+00:36:04,800 --> 00:36:10,000
+statement العبارة الأولى هذه لحد هنانتيجة على أو
+
+353
+00:36:10,000 --> 00:36:15,900
+corollary to the monotone convergence theorem الآن
+
+354
+00:36:15,900 --> 00:36:22,400
+بدنا نثبت الأجزاء واحد واثنين الآن خلينا نثبت
+
+355
+00:36:22,400 --> 00:36:28,260
+الجزء الأول والثاني برهانه بالمثل similar to one
+
+356
+00:36:28,260 --> 00:36:32,660
+إذا هنا نثبت الجزء الأول assume
+
+357
+00:36:35,940 --> 00:36:45,260
+إن Xin is a sequence of real numbers is unbounded
+
+358
+00:36:45,260 --> 00:36:48,540
+and
+
+359
+00:36:48,540 --> 00:36:51,560
+increasing
+
+360
+00:37:02,370 --> 00:37:09,050
+بنثبت أن الـ sequence xn properly
+
+361
+00:37:09,050 --> 00:37:10,890
+divergent to infinity
+
+362
+00:37:28,250 --> 00:37:33,650
+طيب بس نستذكر هنا في هذه المناسبة خلينا نستذكر
+
+363
+00:37:33,650 --> 00:37:39,950
+تعريف ال bounded sequence definition a sequence x
+
+364
+00:37:39,950 --> 00:37:47,230
+in contained in R is bounded is
+
+365
+00:37:47,230 --> 00:37:54,550
+bounded if and only if there exists positive real
+
+366
+00:37:54,550 --> 00:38:04,160
+number أو حتى عدد حقيقي M بحيث أنه absolute Xn
+
+367
+00:38:04,160 --> 00:38:13,300
+أصغر من أو يساوي M for every N ينتمي ل N مش هيك
+
+368
+00:38:13,300 --> 00:38:20,010
+تعريف ال bounded sequence؟ وطبعا دائما لأي sequence
+
+369
+00:38:20,010 --> 00:38:24,510
+دائما ال xn بالمناسبة أصغر من أو يساوي القيمة 
+
+370
+00:38:24,510 --> 00:38:28,130
+المطلقة تبعته أي real number is less than or equal
+
+371
+00:38:28,130 --> 00:38:34,630
+to its absolute value هذا مافيش فيها شكل الآن تعالوا
+
+372
+00:38:34,630 --> 00:38:39,390
+نعمل negation لهذا ما معنى ان هنا هقول in a sense
+
+373
+00:38:39,390 --> 00:38:47,030
+ال sequence xn is unbounded احنا فرضين أن ال
+
+374
+00:38:47,030 --> 00:38:51,770
+sequence تبعتي unbounded و increasing بما أن ال
+
+375
+00:38:51,770 --> 00:38:57,530
+sequence xn is unbounded
+
+376
+00:38:57,530 --> 00:39:03,150
+فلكل
+
+377
+00:39:03,150 --> 00:39:10,730
+n then
+
+378
+00:39:10,730 --> 00:39:24,330
+for every alpha for every alpha عدد حقيقي يوجد
+
+379
+00:39:24,330 --> 00:39:29,210
+by
+
+380
+00:39:29,210 --> 00:39:40,080
+Archimedean property لا يوجد N يعتمد على Alpha عدد
+
+381
+00:39:40,080 --> 00:39:48,060
+طبيعي بحيث ان ال X رقم capital N Alpha هذا بيطلع
+
+382
+00:39:48,060 --> 00:39:55,700
+أكبر من Alpha يعني
+
+383
+00:39:55,700 --> 00:40:00,360
+لو كان هذا ال M اسمها Alpha
+
+384
+00:40:03,420 --> 00:40:07,900
+معناه أن الـ sequence هذه تكون bounded فما معناه
+
+385
+00:40:07,900 --> 00:40:10,860
+أن الـ sequence هذه تكون unbounded معناه أننا بدنا
+
+386
+00:40:10,860 --> 00:40:15,480
+ننفي الشرط هذا عشان أنفي الشرط هذا هذا معناه أن
+
+387
+00:40:15,480 --> 00:40:20,920
+بدل يوجد alpha موجبة لكل alpha سواء موجبة أو سالبة
+
+388
+00:40:20,920 --> 00:40:25,980
+لكل alpha for any alpha يوجد
+
+389
+00:40:28,130 --> 00:40:34,230
+بدل لكل n عدد طبيعي يوجد عدد طبيعي capital N يوجد
+
+390
+00:40:34,230 --> 00:40:41,740
+واحد capital N عدد طبيعي ما في لكل يوجد بحيث أن هذا
+
+391
+00:40:41,740 --> 00:40:48,720
+الـ xn بدل
+
+392
+00:40:48,720 --> 00:40:52,880
+ما هي أصغر من أو يساوي Alpha نفي xn أصغر من أو يساوي
+
+393
+00:40:52,880 --> 00:41:00,880
+Alpha هو أكبر من Alpha تمام؟ إذن هذا هو نفي الشرط
+
+394
+00:41:00,880 --> 00:41:04,940
+هذا أو نفي boundedness إذا الـ sequence unbounded
+
+395
+00:41:04,940 --> 00:41:09,400
+معناه لأي عدد حقيقي Alpha فهي عدد طبيعي يعتمد على 
+
+396
+00:41:09,400 --> 00:41:14,680
+Alpha بحيث ان ال X المؤشر تبعه capital N بيطلع
+
+397
+00:41:14,680 --> 00:41:19,340
+أكبر من Alpha طيب الآن بما أن ال sequence
+
+398
+00:41:19,340 --> 00:41:27,540
+increasing as ال sequence Xn is increasing احنا
+
+399
+00:41:27,540 --> 00:41:35,200
+فرضين انها متزايدة we get نحصل على لو كان n أكبر من 
+
+400
+00:41:35,200 --> 00:41:41,620
+أو يساوي n of alpha فهذا بالتأكيد هيقدي أن x
+
+401
+00:41:41,620 --> 00:41:47,060
+المؤشر تبعها small n أكبر من أو يساوي x المؤشر
+
+402
+00:41:47,060 --> 00:41:52,840
+تبعها n of alpha وهذا من هنا أكبر من alpha
+
+403
+00:41:56,150 --> 00:42:02,150
+إذن خليني ألخص شو عملنا احنا أثبتنا الآن أن for
+
+404
+00:42:02,150 --> 00:42:06,570
+every alpha real number there exists a natural
+
+405
+00:42:06,570 --> 00:42:14,310
+number depends on alpha هذا هو بحيث أنه لكل N أكبر
+
+406
+00:42:14,310 --> 00:42:20,530
+من أو يساوي capital N طلع ندي xn أكبر من alpha هذا
+
+407
+00:42:20,530 --> 00:42:25,310
+إذا حسب التعريف الأولاني حسب تعريف one by
+
+408
+00:42:25,310 --> 00:42:32,350
+definition one طبعا هذا معناه ان limit xn يساوي
+
+409
+00:42:32,350 --> 00:42:39,170
+infinity وهو المطلوب okay برهان الجزء الثاني the
+
+410
+00:42:39,170 --> 00:42:46,290
+proof of part اثنين is similar
+
+411
+00:42:49,840 --> 00:42:55,180
+to one فحاسبكم انتم تكتبوا البرهان تبعه okay
+
+412
+00:42:55,180 --> 00:43:01,640
+تمام وهيك بنكون كملنا النظرية إذا هنوقف هنا و
+
+413
+00:43:01,640 --> 00:43:10,420
+بنكمل إن شاء الله الموضوع هذا المرة القادمة فشوفكم
+
+414
+00:43:10,420 --> 00:43:11,720
+إن شاء الله المرة الجاية على فيه
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/unjPK5-iKR8.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/unjPK5-iKR8.srt
new file mode 100644
index 0000000000000000000000000000000000000000..e2fd88ad1624e50f7e3470f80a4f4da93f835bc3
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/unjPK5-iKR8.srt
@@ -0,0 +1,1447 @@
+1
+00:00:00,000 --> 00:00:01,100
+موسيقى
+
+2
+00:00:19,710 --> 00:00:25,810
+السلام عليكم إن شاء الله اليوم هنكمل section أربعة
+
+3
+00:00:25,810 --> 00:00:31,470
+اللي بدأناها المرة اللي فاتت في المرة
+
+4
+00:00:31,470 --> 00:00:36,810
+السابقة أخذنا نظريات النهايات أو قوانين النهايات
+
+5
+00:00:37,440 --> 00:00:44,420
+للـ functions وكنتائج
+
+6
+00:00:44,420 --> 00:00:50,220
+على قوانين النهايات لدينا النظرية التالية theorem
+
+7
+00:00:50,220 --> 00:00:55,020
+واحد
+
+8
+00:00:56,630 --> 00:01:04,910
+إذا P of X يساوي A N في X to N زائد A N minus one
+
+9
+00:01:04,910 --> 00:01:12,290
+X to N minus one زائد وهكذا زائد A one في X زائد
+
+10
+00:01:12,290 --> 00:01:16,450
+A zero is a polynomial
+
+11
+00:01:25,480 --> 00:01:33,060
+Polynomial function of X then
+
+12
+00:01:33,060 --> 00:01:47,820
+for any C real number limit P of X as X tends to C
+
+13
+00:01:47,820 --> 00:01:53,740
+يساوي قيمة الـ polynomial عند C then if
+
+14
+00:01:56,280 --> 00:02:06,620
+R of X يساوي P
+
+15
+00:02:06,620 --> 00:02:15,740
+of X over Q of X where
+
+16
+00:02:15,740 --> 00:02:18,860
+P
+
+17
+00:02:18,860 --> 00:02:22,580
+و Q are polynomial
+
+18
+00:02:26,290 --> 00:02:41,850
+الـ X and if Q of C لا تساوي صفر then limit لـ R
+
+19
+00:02:41,850 --> 00:02:50,930
+of X as X tends to C يساوي R محسوب عن C البرهان
+
+20
+00:02:50,930 --> 00:02:59,720
+سهلة  البرهان الأول note first
+
+21
+00:02:59,720 --> 00:03:03,500
+that
+
+22
+00:03:03,500 --> 00:03:08,180
+limit
+
+23
+00:03:08,180 --> 00:03:17,480
+لـ X as X tends to C يساوي C يقدّي أن ال limit لـ
+
+24
+00:03:17,480 --> 00:03:25,240
+X to the power K as X tends to C يساوي C to K for
+
+25
+00:03:25,240 --> 00:03:30,580
+all K يساوي صفر واحد اثنين إلى ما لا نهاية
+
+26
+00:03:33,860 --> 00:03:37,040
+أثبتنا قبل ذلك أن ال limit لل identity function
+
+27
+00:03:37,040 --> 00:03:42,500
+لما X تقول لـ C يساوي C أثبتنا ذلك باستخدام تعريف
+
+28
+00:03:42,500 --> 00:03:46,480
+epsilon delta قلنا لأي given epsilon choose delta
+
+29
+00:03:46,480 --> 00:03:51,080
+يساوي epsilon الآن
+
+30
+00:03:51,080 --> 00:03:57,010
+حسب نظريات النهايات limit X to the power K يساوي
+
+31
+00:03:57,010 --> 00:04:01,030
+limit X ضرب نفسها مضروبة في نفسها كم المرات
+
+32
+00:04:01,030 --> 00:04:05,250
+أخذنا في المحاضرة الفاترة أن limit F of X الكل
+
+33
+00:04:05,250 --> 00:04:12,470
+أس M يساوي limit F of X الكل أس M فهذا يطلع
+
+34
+00:04:12,470 --> 00:04:20,490
+limit X هذا يطلع C limit تبع الـ X C هذا يطلع C
+
+35
+00:04:20,490 --> 00:04:23,610
+أس K الآن
+
+36
+00:04:28,400 --> 00:04:35,780
+by limit by limit
+
+37
+00:04:35,780 --> 00:04:47,060
+theorem نظرية النهايات يطلع عندي ال limit لـ P of X
+
+38
+00:04:47,060 --> 00:04:52,140
+as X tends to C يساوي ال limit هي عندي الـ
+
+39
+00:04:52,140 --> 00:04:57,810
+polynomial limit المجموعة أثبتنا أن limit المجموعة
+
+40
+00:04:57,810 --> 00:05:04,650
+بيستخدم مجموعة limits ف limit X to limit الحد الأول
+
+41
+00:05:04,650 --> 00:05:09,510
+زائد
+
+42
+00:05:09,510 --> 00:05:11,750
+limit الحد الثاني
+
+43
+00:05:18,500 --> 00:05:25,660
+وهكذا إذا limit A1 في X as X tends to C زائد limit
+
+44
+00:05:25,660 --> 00:05:35,370
+A0 as X tends to C والآن limit حاصل ثابت في دالة
+
+45
+00:05:35,370 --> 00:05:42,350
+يساوي ثابت A N المعاملات هذه A N و A N سالب واحد
+
+46
+00:05:42,350 --> 00:05:46,450
+الأخرى دي كلها ثوابت ف limit ثابت في دالة يساوي
+
+47
+00:05:46,450 --> 00:05:54,710
+ثابت في limit X أس N اللي هي C أس N زائد limit
+
+48
+00:05:54,710 --> 00:06:02,540
+ثابت في دالة X أس N سالب واحد يطلع C أس N سالب
+
+49
+00:06:02,540 --> 00:06:10,780
+واحد وهكذا إلى A واحد في limit X لما X تقوى لـ C
+
+50
+00:06:10,780 --> 00:06:17,180
+يساوي C limit A Zero يطلع A Zero الآن هذه هي نفس
+
+51
+00:06:17,180 --> 00:06:24,820
+الـ Polynomial المحسوب عند X يساوي C إذن هذا يثبت
+
+52
+00:06:24,820 --> 00:06:33,430
+الجزء الأول من النظرية لإثبات الجزء الثاني Assume
+
+53
+00:06:33,430 --> 00:06:46,170
+أن الـ Q عند الـ C لا يساوي صفر ثم
+
+54
+00:06:46,170 --> 00:06:55,590
+Limit لـ R of X لما X تقول إلى C يساوي Limit البسط
+
+55
+00:06:58,130 --> 00:07:06,090
+P of X as X tends to C على Limit Q of X لما X تقول
+
+56
+00:07:06,090 --> 00:07:12,810
+إلى C الآن
+
+57
+00:07:12,810 --> 00:07:16,790
+باستخدام الجزء الأول من النظرية هذه كثيرة حدود
+
+58
+00:07:16,790 --> 00:07:24,430
+وبالتالي Limit لها عن C يساوي قيمتها عن C و Q of X
+
+59
+00:07:24,430 --> 00:07:30,350
+برضه كثيرة حدود القانون اللي عند أي عدد C يساوي
+
+60
+00:07:30,350 --> 00:07:38,130
+قيمتها عن C ومع أن Q of C هنا لا يساوي صفر فبقدر
+
+61
+00:07:38,130 --> 00:07:43,650
+استخدم القانون تبع Limit الكسر يساوي أو Limit قسم
+
+62
+00:07:43,650 --> 00:07:46,530
+الدالتين يساوي خارج قسم ال Limits
+
+63
+00:07:51,050 --> 00:07:55,270
+لو Q of C يساوي صفر ما أقدرش أعمل الكلام هذا لو الـ
+
+64
+00:07:55,270 --> 00:07:59,190
+Limit هذه يساوي صفر ما أقدرش أوزع ال Limit على الـ
+
+65
+00:07:59,190 --> 00:08:03,610
+بسط و المقام طب ما هذا بالضبط هو عبارة عن الـ
+
+66
+00:08:03,610 --> 00:08:10,090
+function R محسوب عن C وهذا يكمل برهان الجزء الثاني
+
+67
+00:08:10,090 --> 00:08:12,230
+okay هذه أمثلة
+
+68
+00:08:23,910 --> 00:08:29,950
+Find limit لـ
+
+69
+00:08:29,950 --> 00:08:35,650
+X تربيع سالب اثنين X زائد واحد لما X تقول لصفر
+
+70
+00:08:35,650 --> 00:08:44,170
+فهذه عبارة عن كثير حدود P of X فحسب
+
+71
+00:08:44,170 --> 00:08:48,390
+الجزء الأول من النظرية السابقة Limit لـ P of X يطلع
+
+72
+00:08:48,390 --> 00:08:51,230
+P عند صفر صح؟
+
+73
+00:08:53,080 --> 00:08:56,160
+عوض عن X بصفر مباشرة يعني يعني عوض عن X بصفر
+
+74
+00:08:56,160 --> 00:08:58,540
+مباشرة يعني يعني عوض عن X بصفر مباشرة يعني يعني
+
+75
+00:08:58,540 --> 00:08:58,640
+عوض عن X بصفر مباشرة يعني يعني عوض عن X بصفر
+
+76
+00:08:58,640 --> 00:09:03,040
+مباشرة يعني عوض
+
+77
+00:09:03,040 --> 00:09:03,220
+عن X بصفر مباشرة يعني عوض عن X بصفر مباشرة يعني
+
+78
+00:09:03,220 --> 00:09:03,960
+عوض عن X بصفر مباشرة يعني عوض عن X بصفر مباشرة
+
+79
+00:09:03,960 --> 00:09:09,000
+يعني عوض عن X بصفر مباشرة يعني عوض عن X بصفر
+
+80
+00:09:09,000 --> 00:09:14,540
+مباشرة يعني عوض عن X بصفر مباشرة يعني عوض عن X
+
+81
+00:09:14,540 --> 00:09:19,920
+بصفر مباشرة يعني عوض عن X بصفر مباشرة يعني عوض
+
+82
+00:09:24,080 --> 00:09:28,500
+طبعًا هذه كثيرة حدود على كثيرة حدود ونلاحظ أن
+
+83
+00:09:28,500 --> 00:09:32,320
+Limit كثيرة الحدود اللي في المقام لما X تقول لسالب
+
+84
+00:09:32,320 --> 00:09:37,320
+واحد يطلع اثنين لا يساوي صفر إذًا حسب الـ part
+
+85
+00:09:37,320 --> 00:09:44,960
+الثاني بقدر أعوض عن X يساوي سالب واحد مباشرة يعني
+
+86
+00:09:44,960 --> 00:09:51,040
+لو سميت الـ function هذه R of X فالمفروض هذا يطلع R
+
+87
+00:09:51,040 --> 00:09:57,660
+عند سالب واحد يعني عوض عن X يساوي سالب واحد فيطلع
+
+88
+00:09:57,660 --> 00:10:02,300
+عندي خمسة في سالب واحد تربيع يطلع خمسة سالب سالب
+
+89
+00:10:02,300 --> 00:10:09,360
+واحد يطلع واحد زائد اثنين على واحد زائد واحد و
+
+90
+00:10:09,360 --> 00:10:14,240
+يطلع ثمانية على اثنين يطلع أربعة
+
+91
+00:10:20,930 --> 00:10:26,470
+أو ممكن نستخدم حل ثاني نستخدم قوانين نهايات نقول
+
+92
+00:10:26,470 --> 00:10:31,970
+نهاية دالة كسرية زي هذه تساوي Limit البسط على
+
+93
+00:10:31,970 --> 00:10:35,730
+Limit المقام بشرط أن Limit المقام ما يساوي صفر Limit
+
+94
+00:10:35,730 --> 00:10:39,490
+المقام طلعت ما يساوي صفر يساوي هي اثنين إذًا بقدر
+
+95
+00:10:39,490 --> 00:10:42,150
+استخدم القانون لكن لو كانت Limit المقام يساوي صفر
+
+96
+00:10:42,150 --> 00:10:45,130
+إذًا ما أستطيع استخدم هذا القانون
+
+97
+00:10:50,800 --> 00:10:55,080
+في كثير من النظريات الخاصة بنهايات الـ sequences
+
+98
+00:10:55,080 --> 00:11:00,380
+أيضًا في قبلها نظريات أخرى خاصة بنهايات الـ
+
+99
+00:11:00,380 --> 00:11:04,620
+functions فمن
+
+100
+00:11:04,620 --> 00:11:05,880
+النظريات هذه
+
+101
+00:11:18,220 --> 00:11:25,280
+من هذه النظريات النظرية التالية let f be a
+
+102
+00:11:25,280 --> 00:11:38,240
+function from a to r and c be a cluster point
+
+103
+00:11:38,240 --> 00:11:40,320
+of A
+
+104
+00:11:50,350 --> 00:12:00,110
+suppose أن الـ function F of X قيمها محصورة من
+
+105
+00:12:00,110 --> 00:12:08,770
+العدد A والعدد B لكل X ينتمي إلى A حيث X لا تساوي
+
+106
+00:12:08,770 --> 00:12:18,600
+C إذا كان ال limit لـ F of X as X tends to C exist
+
+107
+00:12:18,600 --> 00:12:27,120
+موجودة then ال limit أيضًا لل function هتطلع محصورة
+
+108
+00:12:27,120 --> 00:12:34,560
+بين العددين A و B وهي
+
+109
+00:12:34,560 --> 00:12:38,640
+البرهان proof say دعنا
+
+110
+00:12:41,100 --> 00:12:44,140
+نسمي ال limit لـ F of X
+
+111
+00:12:55,430 --> 00:13:03,430
+العدد الحقيقي المطلوب إثبات ال claim التالي المطلوب
+
+112
+00:13:03,430 --> 00:13:08,870
+إثبات أن العدد L أكبر من أو يساوي A أصغر من أو
+
+113
+00:13:08,870 --> 00:13:18,170
+يساوي B لبرهان ذلك to see this we
+
+114
+00:13:18,170 --> 00:13:22,690
+use sequential criterion let
+
+115
+00:13:26,050 --> 00:13:33,090
+X N be a sequence in A حدودها مختلفة عن الـ C such
+
+116
+00:13:33,090 --> 00:13:40,430
+that Limit X N as N tends to infinity يساوي الـ C
+
+117
+00:13:40,430 --> 00:13:44,870
+طيب،
+
+118
+00:13:44,870 --> 00:13:57,640
+since ال Limit لـ F of X as X tends to C يساوي L by
+
+119
+00:13:57,640 --> 00:14:05,680
+sequential ... sequential criterion إذا
+
+120
+00:14:05,680 --> 00:14:09,620
+كانت ال limit لدالة F of X لما X تقول C يساوي L
+
+121
+00:14:09,620 --> 00:14:15,120
+فهذا يقدّي أن لأي sequence في ال domain تبعت الـ
+
+122
+00:14:15,120 --> 00:14:19,940
+function F مختلفة عن ... حدودها مختلفة عن الـ C و
+
+123
+00:14:19,940 --> 00:14:21,280
+نهايتها تساوي C
+
+124
+00:14:24,670 --> 00:14:33,530
+يطلع Limit لصورة ال sequence X N under F تساوي
+
+125
+00:14:33,530 --> 00:14:40,250
+العدد ال ... هذا by sequential criterion تمام؟
+
+126
+00:15:01,600 --> 00:15:07,780
+by hypothesis star هذا
+
+127
+00:15:07,780 --> 00:15:15,580
+الفرض احنا فرضنا أن دي قيمها محصورة بين A و B لكل
+
+128
+00:15:15,580 --> 00:15:22,020
+X في A مختلفة عن C ف by hypothesis star عندي F of X
+
+129
+00:15:22,020 --> 00:15:28,600
+N يطلع أصغر من أو يساوي B أكبر من أو يساوي الـ A
+
+130
+00:15:28,600 --> 00:15:36,200
+وهذا صحيح لكل N ينتمي إلى A لأن كل X N في الـ
+
+131
+00:15:36,200 --> 00:15:42,840
+sequence هذه هو عنصر في A ومختلف عن الـ C فمن الفرض
+
+132
+00:15:42,840 --> 00:15:48,380
+star يطلع هذا الكلام صحيح now
+
+133
+00:15:48,380 --> 00:15:51,980
+by
+
+134
+00:15:59,090 --> 00:16:03,410
+النظرية الخاصة بالـ sequences بتقول إنه لو في عندي
+
+135
+00:16:03,410 --> 00:16:07,870
+sequence كل حدودها محصورة بين A و B وال limit للـ
+
+136
+00:16:07,870 --> 00:16:12,410
+sequence exist فلازم ال limit تكون محصورة أيضًا بين
+
+137
+00:16:12,410 --> 00:16:16,410
+A و B هذه هي ال previous theorem أخذناها قبلك و
+
+138
+00:16:16,410 --> 00:16:25,360
+برهناها by previous theorem ل F of X N as N tends to
+
+139
+00:16:25,360 --> 00:16:30,020
+infinity يطلع أصغر من أو يساوي B أكبر من أو يساوي
+
+140
+00:16:30,020 --> 00:16:34,500
+A طب ما هذا ال limit هذه حسب ال sequential
+
+141
+00:16:34,500 --> 00:16:40,240
+criterion بساوي L إذا بيطلع لدي A أصغر من أو يساوي
+
+142
+00:16:40,240 --> 00:16:47,680
+L أصغر من أو يساوي B وهذا اللي بدنا نثبته ان ال L
+
+143
+00:16:47,680 --> 00:16:52,360
+اللي هي ال limit ده F of X أما X تقترب من C محصورة بين
+
+144
+00:16:52,360 --> 00:17:02,120
+A و B okay تمام هنا هيك أثبتنا النظرية تمام واضح؟
+
+145
+00:17:02,120 --> 00:17:06,900
+في أي استفسار؟ في أي سؤال؟
+
+146
+00:17:12,010 --> 00:17:17,050
+في كمان نظرية ال sandwich theorem أو ال squeeze 
+
+147
+00:17:17,050 --> 00:17:27,130
+theorem لل functions squeeze
+
+148
+00:17:27,130 --> 00:17:32,850
+theorem for 
+
+149
+00:17:32,850 --> 00:17:33,710
+functions
+
+150
+00:17:43,800 --> 00:17:54,960
+فا let f و g و h be functions from a to r be
+
+151
+00:17:54,960 --> 00:17:59,340
+functions و
+
+152
+00:17:59,340 --> 00:18:11,160
+c a cluster point of a and
+
+153
+00:18:13,430 --> 00:18:20,070
+f of x أصغر من أو يساوي h of x أصغر من أو يساوي g of
+
+154
+00:18:20,070 --> 00:18:31,510
+x for every x تنتمي إلى a و x different from c إذا
+
+155
+00:18:31,510 --> 00:18:37,650
+كان ال limit لل functions على الأطراف اللي هي f of
+
+156
+00:18:37,650 --> 00:18:46,670
+x as x tends to c بساوي L وكذلك ال limit لل
+
+157
+00:18:46,670 --> 00:18:51,530
+function g of x as x tends to c بساوي L حيث L 
+
+158
+00:18:51,530 --> 00:18:59,660
+عدد حقيقي يعني limit ل F و limit ل G exist and c و
+
+159
+00:18:59,660 --> 00:19:04,980
+كلهما بساوي نفس العدد الحقيقي L then limit لدالة
+
+160
+00:19:04,980 --> 00:19:10,720
+المحزورة في الوسط اللي هي H of X as X tends to C
+
+161
+00:19:10,720 --> 00:19:16,520
+أيضا بساوي العدد L و
+
+162
+00:19:16,520 --> 00:19:21,160
+البرهان أزاي برهان 
+
+163
+00:19:23,410 --> 00:19:28,750
+النظرية السابقة هنستخدم
+
+164
+00:19:28,750 --> 00:19:37,170
+الـ sequential criterion زائد to use
+
+165
+00:19:37,170 --> 00:19:45,610
+sequential criterion and to use the squeeze theorem
+
+166
+00:19:45,610 --> 00:19:50,570
+for 
+
+167
+00:19:58,560 --> 00:20:05,040
+sequences فال let
+
+168
+00:20:05,040 --> 00:20:12,340
+xn بي sequence in a حدودها مختلفة عن c such that
+
+169
+00:20:12,340 --> 00:20:16,020
+limit xn بساوي c
+
+170
+00:20:20,090 --> 00:20:26,950
+then عندي بطلع f of xn أصغر من أو يساوي h of xn
+
+171
+00:20:26,950 --> 00:20:34,930
+أصغر من أو يساوي g of xn لكل n وعندي
+
+172
+00:20:34,930 --> 00:20:38,270
+by
+
+173
+00:20:38,270 --> 00:20:49,380
+sequential criterion أنا عندي ال limitF of X لما
+
+174
+00:20:49,380 --> 00:20:57,930
+x تقترب من C بساوي L بيقودى ان ال limitلـ f of x n
+
+175
+00:20:57,930 --> 00:21:06,670
+as n tends to infinity بساوي L and ال limit ل g of
+
+176
+00:21:06,670 --> 00:21:12,010
+x as x tends to c بساوي L احنا فرضين ان ال limit ل
+
+177
+00:21:12,010 --> 00:21:17,030
+g and c بساوي L فهذا بيقودى by sequential criterion
+
+178
+00:21:17,030 --> 00:21:23,830
+ان ال limit ل sequence x n اللي نهايتها c نهاية 
+
+179
+00:21:23,830 --> 00:21:24,550
+صورتها
+
+180
+00:21:27,560 --> 00:21:34,660
+هتطلع أيضا بساوي L وبالتالي
+
+181
+00:21:34,660 --> 00:21:46,520
+so by squeeze theorem for sequences يعني 
+
+182
+00:21:46,520 --> 00:21:51,800
+عندي تلات متتاليات هذه 
+
+183
+00:21:51,800 --> 00:21:57,840
+ال limit بتاعتها بتطلع L وهذه ال limit تبعتها
+
+184
+00:21:57,840 --> 00:22:05,080
+بيطلع لعدد L لما N تقول infinity إذا
+
+185
+00:22:05,080 --> 00:22:10,960
+ال sequence اللي في وسط ال limit تبعتها limit ال
+
+186
+00:22:10,960 --> 00:22:15,860
+sequence H of X N as N tends to infinity بتطلع
+
+187
+00:22:15,860 --> 00:22:21,060
+بساوي L وبالتالي
+
+188
+00:22:21,060 --> 00:22:22,340
+therefore
+
+189
+00:22:24,080 --> 00:22:31,520
+by sequential criterion مرة أخرى ال sequential
+
+190
+00:22:31,520 --> 00:22:38,660
+criterion بتقول عشان أثبت أن ال limit لل function
+
+191
+00:22:38,660 --> 00:22:46,360
+h of x لما x تقترب إلى c بساوي العدد L عشان أثبت
+
+192
+00:22:46,360 --> 00:22:52,380
+limit ال function h لما x تقترب إلى c بساوي L لازم
+
+193
+00:22:52,380 --> 00:22:56,260
+هذا بكافئ حسب الـ sequential criterion هذا بكافئ
+
+194
+00:22:56,260 --> 00:23:00,580
+إننا نثبت لو أخدت أي sequence في مجال الدالة كل
+
+195
+00:23:00,580 --> 00:23:04,000
+حدودها مختلفة عن الـ C و ال sequence نهايتها C
+
+196
+00:23:04,000 --> 00:23:09,400
+فلازم أثبت نهاية صورة ال sequence بالساوية لعدد L
+
+197
+00:23:09,400 --> 00:23:13,640
+وهذا أثبتناه، اذا by sequential criterion بطلع
+
+198
+00:23:13,640 --> 00:23:20,950
+limit H عن C بساوية L وهو المطلوب Okay تمام إذا هذا 
+
+199
+00:23:20,950 --> 00:23:25,230
+برهان الـ sequential criteria الـ squeeze theorem
+
+200
+00:23:25,230 --> 00:23:31,070
+for functions تمام و في كتير من نظريات الأخرى اللي
+
+201
+00:23:31,070 --> 00:23:36,370
+أثبتناها بالنسبة لل limits of sequences فيه
+
+202
+00:23:36,370 --> 00:23:38,290
+بيقابلها أو بيوزيها
+
+203
+00:23:40,890 --> 00:23:45,890
+نظريات خاصة بال functions هتشوفوا بعض النظريات هذه 
+
+204
+00:23:45,890 --> 00:23:53,470
+في التمرين فهينبرنتلكم بعضهم ناخد الآن بعض الأمثلة 
+
+205
+00:23:53,470 --> 00:23:58,010
+على النظريات هذه وخاصة النظرية الأخيرة
+
+206
+00:24:20,680 --> 00:24:30,940
+examples أمثلة واحد show أن ال limit لل function x 
+
+207
+00:24:30,940 --> 00:24:41,220
+أس تلاتة على اتنين لما x تقترب من صفر بساوي صفر و
+
+208
+00:24:41,220 --> 00:24:43,540
+طبعا هنا ال x موجبة
+
+209
+00:25:03,540 --> 00:25:07,520
+الفترة المفتوحة من 0 إلى ما لا نهاية
+
+210
+00:25:12,860 --> 00:25:15,920
+طبعا مافيش ولا قانون من القوانين النهائية اللي
+
+211
+00:25:15,920 --> 00:25:19,680
+اخدناها سابقا بيعطيني ان ال limit هذي بالساوية صفر
+
+212
+00:25:19,680 --> 00:25:24,520
+يعني ماجدرش اقول عوض عن x بساوية صفر بطلع صفر
+
+213
+00:25:24,520 --> 00:25:29,640
+مافيش ماكان لاش قانون زي هذا هذي ليست polynomial x
+
+214
+00:25:29,640 --> 00:25:33,740
+أُس 3 over 2 is not polynomial لو كانت polynomial
+
+215
+00:25:33,740 --> 00:25:39,420
+بنعود على طول مباشرة عن x بالساوية صفر ولكنها ليست 
+
+216
+00:25:39,420 --> 00:25:45,270
+polynomial فلبرهان النظرية هذه بناخد الدالة f of x
+
+217
+00:25:45,270 --> 00:25:49,970
+بساوي x أس تلاتة على اتنين بنلاحظ أولا، لاحظي
+
+218
+00:25:49,970 --> 00:25:56,550
+note ان ال x for
+
+219
+00:25:56,550 --> 00:26:02,610
+x أكبر من صفر أصغر من أو يساوي الواحد بطلع عندي x
+
+220
+00:26:02,610 --> 00:26:07,330
+تقريبا أصغر من أو يساوي x أصغر من أو يساوي الواحد
+
+221
+00:26:11,890 --> 00:26:21,370
+أو X أصغر من أو يساوي X أس نص أصغر من أو يساوي
+
+222
+00:26:21,370 --> 00:26:29,290
+الواحد كلمة
+
+223
+00:26:29,290 --> 00:26:35,050
+X الجذر التربيعي ل X بتطلع أكبر من أو يساوي ال X
+
+224
+00:26:35,050 --> 00:26:39,630
+إذا ال X محصورة من صفر لواحد وبالتالي
+
+225
+00:26:41,250 --> 00:26:51,130
+لو ضربت .. لو ضربت هنا في .. في x فهذا هيقودى ان x
+
+226
+00:26:51,130 --> 00:26:57,150
+تربيع أصغر من أو يساوي x أس تلاتة على اتنين أصغر من
+
+227
+00:26:57,150 --> 00:27:01,430
+أو يساوي x إذن هذا الكلام صحيح
+
+228
+00:27:05,500 --> 00:27:13,040
+هذا كلام صحيح لكل x أكبر من صفر أصغر من أو يساوي
+
+229
+00:27:13,040 --> 00:27:21,180
+الواحد الآن لما x تقترب إلى صفر x تربيعها كثيرة حدود
+
+230
+00:27:21,180 --> 00:27:26,760
+ال limit بتاعتها صفر وال x هذه polynomial لما x 
+
+231
+00:27:26,760 --> 00:27:32,820
+تقترب لسفر تطلع نهايتها صفر تمام؟ إذا by squeeze
+
+232
+00:27:32,820 --> 00:27:33,400
+theorem
+
+233
+00:27:52,880 --> 00:28:01,340
+المثال التاني show أن limit ال function sin x لما
+
+234
+00:28:01,340 --> 00:28:03,980
+x تقترب من صفر بساوي صفر
+
+235
+00:28:08,850 --> 00:28:18,290
+لبرهان ذلك فيه متباينة معروفة it is known in
+
+236
+00:28:18,290 --> 00:28:21,870
+chapter
+
+237
+00:28:21,870 --> 00:28:32,370
+8 that sin x دائما أصغر من أو يساوي x أكبر من أو 
+
+238
+00:28:32,370 --> 00:28:44,050
+يساوي سالب x لكل X ينتمي إلى R معروف وهذا .. هيتم ..
+
+239
+00:28:44,050 --> 00:28:49,290
+هذا له برهان في chapter 8 في نفس الكتاب تبعنا اللي 
+
+240
+00:28:49,290 --> 00:28:52,470
+هياخده منكم تحليل حقيقة 2 هياخده ال chapter هذا
+
+241
+00:28:52,470 --> 00:28:57,910
+ففيه برهان هناك للمتباينة هذه او الحقيقة هذه ان
+
+242
+00:28:57,910 --> 00:29:02,350
+sin X دائما أصغر من أو يساوي X أكبر من أو يساوي
+
+243
+00:29:02,350 --> 00:29:03,270
+negative X
+
+244
+00:29:08,790 --> 00:29:12,770
+هذا الرسم ما توضح يعني مش حاكيه بأن برهان زي ما
+
+245
+00:29:12,770 --> 00:29:18,250
+أقول ان هذا موجود في chapter ثمانية فالناس هياخدوا
+
+246
+00:29:18,250 --> 00:29:23,290
+real analysis اتنين هيشوفوه هاي ال .. ال function
+
+247
+00:29:23,290 --> 00:29:34,130
+y بساوي x و هاي ال function و
+
+248
+00:29:34,130 --> 00:29:38,170
+هاد ال function y بساوي negative x و الـ sine
+
+249
+00:29:38,170 --> 00:29:42,170
+function الموجة
+
+250
+00:29:42,170 --> 00:29:52,790
+أو ال wave تبعتها زي هيك إذا
+
+251
+00:29:52,790 --> 00:30:00,090
+هذه ال function y بساوي sin x فلاحظوا ان ال sign
+
+252
+00:30:00,090 --> 00:30:05,330
+function محصورة بين y بساوي x و y بساوي سالب x
+
+253
+00:30:05,330 --> 00:30:11,510
+تمام؟ هذا طبعا من الرسم لكن الرسم ليس برهان أنا
+
+254
+00:30:11,510 --> 00:30:17,230
+مجرد توضيح الآن أنا عندي ال function هذه لما x 
+
+255
+00:30:17,230 --> 00:30:22,850
+تقترب للصفر لما x تقترب للصفر بتروح للصفر وهذه برضه 
+
+256
+00:30:22,850 --> 00:30:28,600
+ال function لما x تقترب للصفر بتروح للصفر إذا by 
+
+257
+00:30:28,600 --> 00:30:36,760
+squeeze theorem .. إذا by squeeze
+
+258
+00:30:36,760 --> 00:30:45,240
+theorem ال limit لل function sin x اللي هي محصورة 
+
+259
+00:30:45,240 --> 00:30:51,640
+في النص and السفر مساوي صفر okay تمام؟ لأن هذه 
+
+260
+00:30:51,640 --> 00:30:56,760
+حقيقة برهانها بتم هكذا ماذا؟ في أي سؤال؟
+
+261
+00:31:00,750 --> 00:31:05,610
+كمان في مثال
+
+262
+00:31:05,610 --> 00:31:12,310
+تالت ممكن إثبات ان ال cosine ل X لما X تقترب ل 0
+
+263
+00:31:12,310 --> 00:31:23,490
+بساوي 1 proof we
+
+264
+00:31:23,490 --> 00:31:31,480
+use المتباينة في متباينة اللي هي واحد سالب X تربيع
+
+265
+00:31:31,480 --> 00:31:37,240
+على اتنين أصغر من أو يساوي cosine X أصغر من أو يساوي 
+
+266
+00:31:37,240 --> 00:31:42,920
+واحد for every X ينتمي إلى R المتباينة هذه صحيحة
+
+267
+00:31:42,920 --> 00:31:45,680
+لكل الأعداد الحقيقية
+
+268
+00:31:50,520 --> 00:32:01,100
+و هذه برضه ممكن إثباتها في chapter 8 
+
+269
+00:32:01,100 --> 00:32:07,460
+في real analysis 2 الآن ال function هذه لما x تقترب
+
+270
+00:32:07,460 --> 00:32:15,280
+ل 0 بتروح ل 1 tends to one as x tends to zero صح؟
+
+271
+00:32:15,280 --> 00:32:20,990
+هذه كثيرة حدود عوض عن x بساوية صفر وهذه دالة ثابتة،
+
+272
+00:32:20,990 --> 00:32:27,310
+نهايتها واحد لما x تقترب إلى أي حاجة، إذا ممكن نطبق
+
+273
+00:32:27,310 --> 00:32:32,870
+ال squeeze theorem، okay؟ إذا by squeeze theorem،
+
+274
+00:32:32,870 --> 00:32:39,210
+ال limit ل ال function cosine
+
+275
+00:32:39,210 --> 00:32:47,780
+x as x tends to zero تطلع بساوية واحد تمام؟ okay
+
+276
+00:32:47,780 --> 00:32:53,000
+واضح؟ في كتير من ال limits المعروفة ممكن إثباتها
+
+277
+00:32:53,000 --> 00:33:01,140
+بنفس الطرق هذه خلينا ناخد كمان مثال limit ل
+
+278
+00:33:01,140 --> 00:33:09,120
+function x ضرب sin 1 على x as x tends to zero
+
+279
+00:33:09,120 --> 00:33:12,320
+exists و بساوي صفر
+
+280
+00:33:14,890 --> 00:33:21,610
+فالـ function هنا الـ 
+
+281
+00:33:21,610 --> 00:33:25,610
+function اللي بدي أخدلها ال limit هي عبارة عن الـ
+
+282
+00:33:25,610 --> 00:33:33,670
+function f of x بساوي x ضرب sign واحد على x طبعا
+
+283
+00:33:33,670 --> 00:33:38,410
+هنا x لا تساوي صفر ال domain للـ function هذه كل
+
+284
+00:33:38,410 --> 00:33:45,650
+الأعداد الحقيقية ماعدا الصفر لأن اسمه على صفر هنا مش
+
+285
+00:33:45,650 --> 00:33:54,170
+معرفة تمام؟ بنثبت أن ال limit للدالة هذه عند الصفر
+
+286
+00:33:54,170 --> 00:33:59,230
+بساوي صفر بالمناسبة الدالة هذه هي نفس الدالة
+
+287
+00:33:59,230 --> 00:34:07,570
+المرسومة على ال cover تبع الكتاب هذه رسمة الدالة
+
+288
+00:34:09,910 --> 00:34:17,470
+فلن متدالة f of x بساوي x في sin واحد على x و
+
+289
+00:34:17,470 --> 00:34:19,910
+الدالة هذه زي ما أنتم شايفين المرحلة تبعها كل ما
+
+290
+00:34:19,910 --> 00:34:25,450
+قربنا من الصفر كل ما قربنا من الصفر سواء من اليمين
+
+291
+00:34:25,450 --> 00:34:30,810
+أو اليسار لكن لإثبات ذلك نستخدم ال squeeze theorem
+
+292
+00:34:30,810 --> 00:34:35,030
+طيب
+
+293
+00:34:39,940 --> 00:34:49,380
+لتكن x لا تساوي صفر، خلّيني أخد x بساوي صفر، إذا by
+
+294
+00:34:49,380 --> 00:34:53,320
+trichotomy property باستخدام الخاصية الثلاثية، x
+
+295
+00:34:53,320 --> 00:34:59,740
+أكبر من صفر أو x أصغر من صفر، فحناخد في الحالتين
+
+296
+00:35:09,860 --> 00:35:14,980
+طيب، لنثبت أنه
+
+297
+00:35:14,980 --> 00:35:20,940
+لأي x لا تساوي 0 زي هذه سالب absolute x أصغر من أو 
+
+298
+00:35:20,940 --> 00:35:28,000
+يساوي x في sin 1 على x أصغر من أو يساوي absolute x،
+
+299
+00:35:28,000 --> 00:35:30,900
+هذا صحيح لكل x لا تساوي 0
+
+300
+00:35:39,610 --> 00:35:44,550
+أنا بدي أثبت أنه لكل x لا تساوي صفر المتباينة هذا
+
+301
+00:35:44,550 --> 00:35:54,970
+صحيح ف to see this لإثبات ذلك to see this fix x لا
+
+302
+00:35:54,970 --> 00:35:59,930
+يساوي صفر خلّيني أخد arbitrary x مختلفة عن الصفر و
+
+303
+00:35:59,930 --> 00:36:04,570
+نثبت إن المتباينة هي دي صحيحة لها then إذا أخدت
+
+304
+00:36:04,570 --> 00:36:09,850
+أنا x لا تساوي صفر فإما x أكبر من صفر or x أصغر من
+
+305
+00:36:09,850 --> 00:36:18,760
+صفر فإذا في عندي أنا حالتين نشوف الحالة الأولى لما
+
+306
+00:36:18,760 --> 00:36:24,480
+تكون الـ X أكبر من الصفر هذا بيقدي أن absolute X
+
+307
+00:36:24,480 --> 00:36:37,620
+بساوي X صح؟ طيب هنا note that sign
+
+308
+00:36:39,330 --> 00:36:45,010
+ثيتا دائما أكبر من أو يساوي سالب واحد أصغر من أو يساوي
+
+309
+00:36:45,010 --> 00:36:49,690
+واحد for all ثيتا تنتمي إلى R هذه المتباينة
+
+310
+00:36:49,690 --> 00:36:53,690
+الصحيحة ال absolute maximum value ل sin واحد وال
+
+311
+00:36:53,690 --> 00:36:59,710
+absolute minimum value سالب واحد خلينا
+
+312
+00:36:59,710 --> 00:37:08,180
+نضرب المتباينة هذه في absolute X و ال theta هذه
+
+313
+00:37:08,180 --> 00:37:13,580
+نبدلها
+
+314
+00:37:13,580 --> 00:37:20,420
+ب 1 على x ف since sin 1 على x أكبر من أو يساوي سالب
+
+315
+00:37:20,420 --> 00:37:26,020
+واحد أصغر من أو يساوي واحد لكل x لا تساوي صفر هذا
+
+316
+00:37:26,020 --> 00:37:26,660
+بيقدي
+
+317
+00:37:29,420 --> 00:37:33,100
+لو ضربت المتباينة هذه في absolute x اللي هي
+
+318
+00:37:33,100 --> 00:37:40,160
+بساوي x و absolute x هنا هتطلع موجبة بقى فهيطلع
+
+319
+00:37:40,160 --> 00:37:45,300
+عندي سالب absolute x أصغر من أو يساوي x في sin 1
+
+320
+00:37:45,300 --> 00:37:49,760
+على x أصغر من أو يساوي absolute x وبالتالي
+
+321
+00:37:49,760 --> 00:37:52,320
+المتباينة اللي أنا عايز أثبتها دي هي صحيحة
+
+322
+00:37:56,180 --> 00:38:00,780
+أضرب هنا في absolute x و absolute x بساوي x عدد
+
+323
+00:38:00,780 --> 00:38:07,200
+موجب فالمتباينة تبقى إشارتها زي ما هي و هي ضربت في
+
+324
+00:38:07,200 --> 00:38:10,600
+سالب absolute x ال x هنا هي برضه سالب absolute x
+
+325
+00:38:10,600 --> 00:38:14,460
+فهذه المتباينة اللي إحنا عايزين نثبتها في الإدعاء
+
+326
+00:38:14,460 --> 00:38:20,200
+تبعنا الحالة التالية لو كانت x سالب
+
+327
+00:38:31,100 --> 00:38:37,440
+كيس اتنين لو كانت x سالبة فهذا بيقدي أن absolute
+
+328
+00:38:37,440 --> 00:38:45,580
+x بساوي سالب x طبعا هذا موجب بقى وبالتالي multiply
+
+329
+00:38:49,120 --> 00:38:54,100
+multiply المتباينة سالب واحد اصغر من أو يساوي sin
+
+330
+00:38:54,100 --> 00:39:00,200
+واحد على x اصغر من أو يساوي واحد by absolute x
+
+331
+00:39:00,200 --> 00:39:05,240
+بساوي سالب x اكبر من الصفر هذا يعني أنه
+
+332
+00:39:12,420 --> 00:39:19,180
+فهذا بيقدي أنه إيه؟ أنه سالب absolute x أصغر من أو
+
+333
+00:39:19,180 --> 00:39:26,300
+يساوي سالب
+
+334
+00:39:26,300 --> 00:39:33,080
+x في sin 1 على x أصغر من أو يساوي absolute ال x
+
+335
+00:39:46,010 --> 00:39:54,230
+نضرب في سالب واحد نضرب في سالب واحد نضرب في سالب
+
+336
+00:39:54,230 --> 00:39:58,910
+واحد نضرب في سالب واحد نضرب في سالب واحد نضرب في
+
+337
+00:39:58,910 --> 00:39:58,970
+سالب واحد نضرب في سالب واحد نضرب في سالب واحد نضرب
+
+338
+00:39:58,970 --> 00:39:59,210
+في سالب واحد نضرب في سالب واحد نضرب في سالب واحد
+
+339
+00:39:59,210 --> 00:40:02,030
+نضرب في سالب واحد نضرب في سالب واحد نضرب في سالب
+
+340
+00:40:02,030 --> 00:40:06,590
+واحد نضرب في سالب واحد نضرب في سالب واحد نضرب في
+
+341
+00:40:06,590 --> 00:40:13,670
+سالب واحد و هذه هي المتباينة اللي إحنا عايزين
+
+342
+00:40:13,670 --> 00:40:21,370
+نثبتها صح؟ إذا this completes
+
+343
+00:40:21,370 --> 00:40:24,770
+the
+
+344
+00:40:24,770 --> 00:40:34,330
+proof of the claim إذا هنا برهنة ل claim وهو أن
+
+345
+00:40:34,330 --> 00:40:37,970
+المتباينة هذه صحيحة تمام؟
+
+346
+00:40:40,820 --> 00:40:50,560
+الآن من ال claim now
+
+347
+00:40:50,560 --> 00:41:02,560
+by above .. by above the claim and squeeze the
+
+348
+00:41:02,560 --> 00:41:06,560
+theorem since
+
+349
+00:41:06,560 --> 00:41:12,450
+بما أنه ال limit الـ absolute x as x tends to 0
+
+350
+00:41:12,450 --> 00:41:18,310
+بساوي ال limit لسالب absolute x as x tends to 0
+
+351
+00:41:18,310 --> 00:41:26,170
+بساوي 0 we have نحصل على أن ال limit المحصورة في
+
+352
+00:41:26,170 --> 00:41:32,710
+الوسط اللي هي x في sin 1 على x as x tends to 0
+
+353
+00:41:32,710 --> 00:41:39,760
+بساوي أيضا الصفر وهو المطلوب إذا هنا استخدمنا الـ
+
+354
+00:41:39,760 --> 00:41:45,780
+squeeze theorem تمام؟ لأن هذا برهان أن ال limit لل
+
+355
+00:41:45,780 --> 00:41:51,380
+function x sin 1 على x عندما x تقوى للصفر بيساوي
+
+356
+00:41:51,380 --> 00:41:56,680
+صفر تمام؟ واضح؟ في أي سفسار؟ أي سؤال؟
+
+357
+00:42:03,260 --> 00:42:10,860
+Okay باقي نظرية واحدة في ال section 4-2 فاتحقلوا
+
+358
+00:42:10,860 --> 00:42:17,060
+تقرؤوها تفهموا البرهانة و تبدؤوا في حل التمرين و
+
+359
+00:42:17,060 --> 00:42:21,360
+إن شاء الله المرة الجاية بنكمل نشرح النظرية هذه و
+
+360
+00:42:21,360 --> 00:42:25,820
+بنبدأ section جديد اللي هو section 4-3
+
+361
+00:42:28,450 --> 00:42:33,790
+شكرا لحضوركم و شوفكم إن شاء الله في المحاضرة
+
+362
+00:42:33,790 --> 00:42:34,250
+الجاية 
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/unjPK5-iKR8_raw.json b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/unjPK5-iKR8_raw.json
new file mode 100644
index 0000000000000000000000000000000000000000..00e2aa214c4f09fa7c2fb6a100927feae288dd3b
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/unjPK5-iKR8_raw.json
@@ -0,0 +1 @@
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X tends to C بساوي قيمة ال polynomial and C then if", "tokens": [47, 401, 2534, 47429, 2445, 295, 1783, 550, 337, 604, 383, 957, 1230, 4948, 430, 295, 1783, 382, 1783, 12258, 281, 383, 4724, 3794, 995, 45865, 12174, 32640, 3660, 2423, 26110, 293, 383, 550, 498], "avg_logprob": -0.3621961838669247, "compression_ratio": 1.1304347826086956, "no_speech_prob": 5.960464477539063e-08, "words": [{"start": 85.48, "end": 86.68, "word": "Polynomial", "probability": 0.5862503051757812}, {"start": 86.68, "end": 87.22, "word": " function", "probability": 0.8359375}, {"start": 87.22, "end": 87.9, "word": " of", "probability": 0.87744140625}, {"start": 87.9, "end": 89.28, "word": " X", "probability": 0.60546875}, {"start": 89.28, "end": 93.06, "word": " then", "probability": 0.275634765625}, {"start": 93.06, "end": 94.98, "word": " for", "probability": 0.84130859375}, {"start": 94.98, "end": 95.62, "word": " any", "probability": 0.9111328125}, {"start": 95.62, "end": 97.82, "word": " C", 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{"start": 198.14, "end": 198.3, "word": " the", "probability": 0.9111328125}, {"start": 198.3, "end": 198.66, "word": " power", "probability": 0.89990234375}, {"start": 198.66, "end": 199.2, "word": " k", "probability": 0.479248046875}, {"start": 199.2, "end": 200.52, "word": " as", "probability": 0.408447265625}, {"start": 200.52, "end": 200.96, "word": " x", "probability": 0.9775390625}, {"start": 200.96, "end": 201.32, "word": " tends", "probability": 0.7529296875}, {"start": 201.32, "end": 201.5, "word": " to", "probability": 0.98876953125}, {"start": 201.5, "end": 201.86, "word": " c", "probability": 0.80322265625}, {"start": 201.86, "end": 202.76, "word": " بساوي", "probability": 0.8546142578125}, {"start": 202.76, "end": 203.26, "word": " c", "probability": 0.884765625}, {"start": 203.26, "end": 203.6, "word": " to", "probability": 0.9248046875}, {"start": 203.6, "end": 204.08, "word": " k", "probability": 0.97119140625}, {"start": 204.08, "end": 205.24, "word": " for", 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نظريات النهايات", "tokens": [10721, 12984, 3555, 2655, 8315, 12174, 36150, 29910, 23275, 14739, 2423, 39184, 4948, 24976, 6575, 2445, 5296, 15042, 2031, 6055, 4587, 12610, 5296, 269, 4724, 3794, 995, 2407, 3660, 269, 5551, 12984, 3555, 2655, 8315, 29910, 23275, 4724, 995, 14851, 9778, 3215, 10943, 37279, 16572, 5172, 17889, 8289, 12174, 1211, 8315, 5296, 10721, 1829, 2212, 17889, 2826, 8289, 4724, 3794, 995, 2407, 3660, 17889, 6024, 48506, 11331, 35457, 8717, 19913, 16572, 9307, 28239, 11296, 1829, 9307], "avg_logprob": -0.29159901668499044, "compression_ratio": 1.4918032786885247, "no_speech_prob": 0.0, "words": [{"start": 213.85999999999999, "end": 214.7, "word": "أثبتنا", "probability": 0.828564453125}, {"start": 214.7, "end": 215.0, "word": " قبل", "probability": 0.6524658203125}, {"start": 215.0, "end": 215.2, "word": " ذلك", "probability": 0.554107666015625}, {"start": 215.2, "end": 215.4, "word": " أن", "probability": 0.67138671875}, {"start": 215.4, "end": 215.58, "word": " 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tends to C بساوي ال limit هي عندي ال polynomial", "tokens": [2322, 4948, 538, 4948, 20904, 8717, 19913, 2288, 10632, 28239, 11296, 1829, 9307, 4724, 9566, 1211, 3615, 18871, 16254, 2423, 4948, 5296, 430, 295, 1783, 382, 1783, 12258, 281, 383, 4724, 3794, 995, 45865, 2423, 4948, 39896, 18871, 16254, 2423, 26110], "avg_logprob": -0.2801339377959569, "compression_ratio": 1.3008130081300813, "no_speech_prob": 0.0, "words": [{"start": 268.4, "end": 269.52, "word": "by", "probability": 0.1370849609375}, {"start": 269.52, "end": 270.64, "word": " limit", "probability": 0.87353515625}, {"start": 270.64, "end": 272.1, "word": " by", "probability": 0.32080078125}, {"start": 272.1, "end": 275.78, "word": " limit", "probability": 0.97900390625}, {"start": 275.78, "end": 276.54, "word": " theorem", "probability": 0.90869140625}, {"start": 276.54, "end": 279.04, "word": " نظرية", "probability": 0.8802490234375}, {"start": 279.04, "end": 279.8, "word": " النهايات", "probability": 0.88134765625}, 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384.18, "end": 384.82, "word": " بثبت", "probability": 0.9237060546875}, {"start": 384.82, "end": 386.08, "word": " الجزء", "probability": 0.9661458333333334}, {"start": 386.08, "end": 386.6, "word": " الأول", "probability": 0.97705078125}, {"start": 386.6, "end": 387.52, "word": " من", "probability": 0.98779296875}, {"start": 387.52, "end": 388.28, "word": " النظرية", "probability": 0.7874755859375}], "temperature": 1.0}, {"id": 17, "seek": 41559, "start": 388.53, "end": 415.59, "text": "لإثبات الجزء التاني Assume أن الـ Q عند الـ C لا يساوي سفر ثم Limit لR of X لما X تقول إلى C بساوي Limit البصر", "tokens": [1211, 28814, 12984, 3555, 9307, 25724, 11622, 38207, 16712, 7649, 1829, 6281, 2540, 14739, 2423, 39184, 1249, 43242, 2423, 39184, 383, 20193, 7251, 3794, 995, 45865, 8608, 5172, 2288, 38637, 2304, 16406, 270, 5296, 49, 295, 1783, 5296, 15042, 1783, 6055, 39648, 30731, 383, 4724, 3794, 995, 45865, 16406, 270, 29739, 9381, 2288], "avg_logprob": -0.4123263993748912, 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كسرية زي هذه بساوي limit ال bus علي limit المقام بشرط ان limit المقام ماساوي سفر limit المقام طلعت ماساوي سفر بساوي هي اتنين اذا بقدر استخدم القانون لكن لو كانت limit المقام بساوي سفر اذا مابستطيع استخدم هذا القانون", "tokens": [10721, 2407, 3714, 43020, 8717, 14851, 9778, 40448, 11331, 1211, 6055, 7649, 1829, 8717, 14851, 9778, 40448, 12174, 2407, 7649, 9957, 8717, 11296, 1829, 9307, 8717, 39648, 8717, 11296, 10632, 11778, 6027, 3660, 9122, 3794, 2288, 10632, 30767, 1829, 29538, 4724, 3794, 995, 45865, 4948, 2423, 1255, 25894, 4948, 9673, 4587, 10943, 4724, 46309, 9566, 16472, 4948, 9673, 4587, 10943, 19446, 3794, 995, 45865, 8608, 5172, 2288, 4948, 9673, 4587, 10943, 23032, 1211, 34268, 19446, 3794, 995, 45865, 8608, 5172, 2288, 4724, 3794, 995, 45865, 39896, 1975, 2655, 1863, 9957, 1975, 15730, 4724, 28543, 2288, 44713, 9778, 40448, 25062, 7649, 11536, 44381, 45164, 25961, 2655, 4948, 9673, 4587, 10943, 4724, 3794, 995, 45865, 8608, 5172, 2288, 1975, 15730, 19446, 3555, 14851, 9566, 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{"start": 657.3, "end": 657.96, "word": " نظريات", "probability": 0.984619140625}, {"start": 657.96, "end": 658.64, "word": " أخرى", "probability": 0.9441731770833334}, {"start": 658.64, "end": 659.42, "word": " خاصة", "probability": 0.9775390625}, {"start": 659.42, "end": 660.14, "word": " بنهايات", "probability": 0.9322509765625}, {"start": 660.14, "end": 660.38, "word": " الـ", "probability": 0.7376708984375}, {"start": 660.38, "end": 660.88, "word": " functions", "probability": 0.845703125}, {"start": 660.88, "end": 664.62, "word": " فمن", "probability": 0.929443359375}, {"start": 664.62, "end": 665.46, "word": " النظريات", "probability": 0.9599609375}, {"start": 665.46, "end": 665.88, "word": " هذه", "probability": 0.91796875}], "temperature": 1.0}, {"id": 27, "seek": 70032, "start": 678.22, "end": 700.32, "text": "من هذه النظريات النظرية التالية let f be a function from a to r and c be a cluster point of A", "tokens": [27842, 29538, 28239, 19913, 16572, 9307, 28239, 19913, 2288, 10632, 16712, 6027, 10632, 718, 283, 312, 257, 2445, 490, 257, 281, 367, 293, 269, 312, 257, 13630, 935, 295, 316], "avg_logprob": -0.18724798194823727, "compression_ratio": 1.2121212121212122, "no_speech_prob": 0.0, "words": [{"start": 678.22, "end": 678.68, "word": "من", "probability": 0.7041015625}, {"start": 678.68, "end": 679.12, "word": " هذه", "probability": 0.9130859375}, {"start": 679.12, "end": 679.98, "word": " النظريات", "probability": 0.93505859375}, {"start": 679.98, "end": 680.58, "word": " النظرية", "probability": 0.8553466796875}, {"start": 680.58, "end": 681.28, "word": " التالية", "probability": 0.9964192708333334}, {"start": 681.28, "end": 683.08, "word": " let", "probability": 0.3935546875}, {"start": 683.08, "end": 684.78, "word": " f", "probability": 0.5478515625}, {"start": 684.78, "end": 685.1, "word": " be", "probability": 0.81640625}, {"start": 685.1, "end": 685.28, "word": " a", "probability": 0.63818359375}, {"start": 685.28, "end": 685.82, "word": " function", "probability": 0.97705078125}, {"start": 685.82, "end": 686.36, "word": " from", "probability": 0.88671875}, {"start": 686.36, "end": 686.88, "word": " a", "probability": 0.5341796875}, {"start": 686.88, "end": 687.22, "word": " to", "probability": 0.97314453125}, {"start": 687.22, "end": 687.76, "word": " r", "probability": 0.78759765625}, {"start": 687.76, "end": 690.1, "word": " and", "probability": 0.91162109375}, {"start": 690.1, "end": 691.18, "word": " c", "probability": 0.94287109375}, {"start": 691.18, "end": 693.34, "word": " be", "probability": 0.91357421875}, {"start": 693.34, "end": 693.84, "word": " a", "probability": 0.9453125}, {"start": 693.84, "end": 694.38, "word": " cluster", "probability": 0.92333984375}, {"start": 694.38, "end": 698.24, "word": " point", "probability": 0.95947265625}, {"start": 698.24, "end": 699.56, "word": " of", "probability": 0.95703125}, {"start": 699.56, "end": 700.32, "word": " A", "probability": 0.56591796875}], "temperature": 1.0}, {"id": 28, "seek": 72925, "start": 710.35, "end": 729.25, "text": "suppose أن الـ function f of x قيمها محصورة من العدد a والعدد b لكل x ينتمي إلى a حيث x لا تساوي sc", "tokens": [36622, 541, 14739, 2423, 39184, 2445, 283, 295, 2031, 12174, 32640, 11296, 3714, 5016, 9381, 13063, 3660, 9154, 18863, 3215, 3215, 257, 16070, 22488, 3215, 272, 5296, 28820, 2031, 7251, 29399, 2304, 1829, 30731, 257, 11331, 1829, 12984, 2031, 20193, 6055, 3794, 995, 45865, 795], "avg_logprob": -0.1635529862149902, "compression_ratio": 1.2113821138211383, "no_speech_prob": 0.0, "words": [{"start": 710.35, "end": 711.37, "word": "suppose", "probability": 0.56329345703125}, {"start": 711.37, "end": 712.39, "word": " أن", "probability": 0.492919921875}, {"start": 712.39, "end": 712.77, "word": " الـ", "probability": 0.748779296875}, {"start": 712.77, "end": 713.29, "word": " function", "probability": 0.8701171875}, {"start": 713.29, "end": 713.73, "word": " f", "probability": 0.70263671875}, {"start": 713.73, "end": 713.97, "word": " of", "probability": 0.28515625}, {"start": 713.97, "end": 714.39, "word": " x", "probability": 0.86279296875}, {"start": 714.39, "end": 715.63, "word": " قيمها", "probability": 0.9339192708333334}, {"start": 715.63, "end": 716.71, "word": " محصورة", "probability": 0.96298828125}, {"start": 716.71, "end": 720.11, "word": " من", "probability": 0.904296875}, {"start": 720.11, "end": 720.71, "word": " العدد", "probability": 0.95947265625}, {"start": 720.71, "end": 721.07, "word": " a", "probability": 0.796875}, {"start": 721.07, "end": 721.77, "word": " والعدد", "probability": 0.8616536458333334}, {"start": 721.77, "end": 722.23, "word": " b", "probability": 0.9052734375}, {"start": 722.23, "end": 723.67, "word": " لكل", "probability": 0.922119140625}, {"start": 723.67, "end": 724.11, "word": " x", "probability": 0.9716796875}, {"start": 724.11, "end": 724.95, "word": " ينتمي", "probability": 0.973388671875}, {"start": 724.95, "end": 725.29, "word": " إلى", "probability": 0.9736328125}, {"start": 725.29, "end": 725.79, "word": " a", "probability": 0.97802734375}, {"start": 725.79, "end": 727.37, "word": " حيث", "probability": 0.9153645833333334}, {"start": 727.37, "end": 727.77, "word": " x", "probability": 0.95556640625}, {"start": 727.77, "end": 728.03, "word": " لا", "probability": 0.70849609375}, {"start": 728.03, "end": 728.77, "word": " تساوي", "probability": 0.9403076171875}, {"start": 728.77, "end": 729.25, "word": " sc", "probability": 0.34375}], "temperature": 1.0}, {"id": 29, "seek": 75864, "start": 730.42, "end": 758.64, "text": "إذا كان ال limit ل f of x as x tends to c exist موجودة then ال limit أيضا لل function هتطلع محصورة بين العددين a و b وهي البرهان proof say دعنا", "tokens": [28814, 15730, 25961, 2423, 4948, 5296, 283, 295, 2031, 382, 2031, 12258, 281, 269, 2514, 3714, 29245, 23328, 3660, 550, 2423, 4948, 36632, 11242, 995, 24976, 2445, 8032, 2655, 9566, 1211, 3615, 3714, 5016, 9381, 13063, 3660, 49374, 18863, 3215, 3215, 9957, 257, 4032, 272, 37037, 1829, 2423, 26890, 3224, 7649, 8177, 584, 11778, 3615, 8315], "avg_logprob": -0.21436403717911034, "compression_ratio": 1.294871794871795, "no_speech_prob": 0.0, "words": [{"start": 730.42, "end": 730.82, "word": "إذا", "probability": 0.82763671875}, {"start": 730.82, "end": 731.36, "word": " كان", "probability": 0.994140625}, {"start": 731.36, "end": 732.56, "word": " ال", "probability": 0.459716796875}, {"start": 732.56, "end": 732.94, "word": " limit", "probability": 0.560546875}, {"start": 732.94, "end": 734.34, "word": " ل", "probability": 0.78369140625}, {"start": 734.34, "end": 734.68, "word": " f", "probability": 0.287353515625}, {"start": 734.68, "end": 734.94, "word": " of", "probability": 0.4072265625}, {"start": 734.94, "end": 735.36, "word": " x", "probability": 0.912109375}, {"start": 735.36, "end": 735.9, "word": " as", "probability": 0.82958984375}, {"start": 735.9, "end": 736.36, "word": " x", "probability": 0.94287109375}, {"start": 736.36, "end": 736.78, "word": " tends", "probability": 0.8076171875}, {"start": 736.78, "end": 736.96, "word": " to", "probability": 0.98388671875}, {"start": 736.96, "end": 737.28, "word": " c", "probability": 0.875}, {"start": 737.28, "end": 738.6, "word": " exist", "probability": 0.448974609375}, {"start": 738.6, "end": 739.5, "word": " موجودة", "probability": 0.91943359375}, {"start": 739.5, "end": 741.2, "word": " then", "probability": 0.525390625}, {"start": 741.2, "end": 741.58, "word": " ال", "probability": 0.73486328125}, {"start": 741.58, "end": 741.9, "word": " limit", "probability": 0.9775390625}, {"start": 741.9, "end": 742.54, "word": " أيضا", "probability": 0.9002278645833334}, {"start": 742.54, "end": 742.76, "word": " لل", "probability": 0.75439453125}, {"start": 742.76, "end": 743.38, "word": " function", "probability": 0.8828125}, {"start": 743.38, "end": 745.94, "word": " هتطلع", "probability": 0.92529296875}, {"start": 745.94, "end": 747.12, "word": " محصورة", "probability": 0.89541015625}, {"start": 747.12, "end": 747.56, "word": " بين", "probability": 0.9619140625}, {"start": 747.56, "end": 749.78, "word": " العددين", "probability": 0.978271484375}, {"start": 749.78, "end": 750.26, "word": " a", "probability": 0.8388671875}, {"start": 750.26, "end": 750.44, "word": " و", "probability": 0.9521484375}, {"start": 750.44, "end": 750.78, "word": " b", "probability": 0.54248046875}, {"start": 750.78, "end": 754.56, "word": " وهي", "probability": 0.7509765625}, {"start": 754.56, "end": 755.14, "word": " البرهان", "probability": 0.913818359375}, {"start": 755.14, "end": 755.7, "word": " proof", "probability": 0.1396484375}, {"start": 755.7, "end": 758.02, "word": " say", "probability": 0.66845703125}, {"start": 758.02, "end": 758.64, "word": " دعنا", "probability": 0.9264322916666666}], "temperature": 1.0}, {"id": 30, "seek": 76414, "start": 761.1, "end": 764.14, "text": "نسمي الـ limit لـ f of x", "tokens": [1863, 38251, 1829, 2423, 39184, 4948, 5296, 39184, 283, 295, 2031], "avg_logprob": -0.7851562251647314, "compression_ratio": 0.8461538461538461, "no_speech_prob": 5.960464477539063e-08, "words": [{"start": 761.1, "end": 761.86, "word": "نسمي", "probability": 0.6107177734375}, {"start": 761.86, "end": 762.24, "word": " الـ", "probability": 0.2255859375}, {"start": 762.24, "end": 762.56, "word": " limit", "probability": 0.7158203125}, {"start": 762.56, "end": 763.2, "word": " لـ", "probability": 0.594482421875}, {"start": 763.2, "end": 763.44, "word": " f", "probability": 0.47509765625}, {"start": 763.44, "end": 763.7, "word": " of", "probability": 0.64453125}, {"start": 763.7, "end": 764.14, "word": " x", "probability": 0.9267578125}], "temperature": 1.0}, {"id": 31, "seek": 80269, "start": 775.43, "end": 802.69, "text": "العدد حقيقي المطلوب اثبات ال claim التالي المطلوب اثبات ان العدد L أكبر من أو يساوي A أصغر من أو يساوي B لبرهان ذلك to see this we use sequential criterion let", "tokens": [6027, 22488, 3215, 11331, 38436, 38436, 9673, 9566, 1211, 37746, 1975, 12984, 3555, 9307, 2423, 3932, 16712, 6027, 1829, 9673, 9566, 1211, 37746, 1975, 12984, 3555, 9307, 16472, 18863, 3215, 3215, 441, 5551, 4117, 26890, 9154, 34051, 7251, 3794, 995, 45865, 316, 5551, 9381, 17082, 2288, 9154, 34051, 7251, 3794, 995, 45865, 363, 5296, 26890, 3224, 7649, 29910, 23275, 281, 536, 341, 321, 764, 42881, 46691, 718], "avg_logprob": -0.15153952315449715, "compression_ratio": 1.588235294117647, "no_speech_prob": 5.960464477539063e-08, "words": [{"start": 775.43, "end": 776.53, "word": "العدد", "probability": 0.7996419270833334}, {"start": 776.53, "end": 777.27, "word": " حقيقي", "probability": 0.8279622395833334}, {"start": 777.27, "end": 779.67, "word": " المطلوب", "probability": 0.877197265625}, {"start": 779.67, "end": 780.31, "word": " اثبات", "probability": 0.84649658203125}, {"start": 780.31, "end": 780.85, "word": " ال", "probability": 0.7880859375}, {"start": 780.85, "end": 781.35, "word": " claim", "probability": 0.77587890625}, {"start": 781.35, "end": 782.23, "word": " التالي", "probability": 0.94091796875}, {"start": 782.23, "end": 783.43, "word": " المطلوب", "probability": 0.9366455078125}, {"start": 783.43, "end": 783.85, "word": " اثبات", "probability": 0.97216796875}, {"start": 783.85, "end": 784.07, "word": " ان", "probability": 0.7841796875}, {"start": 784.07, "end": 784.63, "word": " العدد", "probability": 0.8829752604166666}, {"start": 784.63, "end": 784.89, "word": " L", "probability": 0.339111328125}, {"start": 784.89, "end": 785.47, "word": " أكبر", "probability": 0.8689778645833334}, {"start": 785.47, "end": 785.71, "word": " من", "probability": 0.95068359375}, {"start": 785.71, "end": 785.91, "word": " أو", "probability": 0.8828125}, {"start": 785.91, "end": 786.49, "word": " يساوي", "probability": 0.853515625}, {"start": 786.49, "end": 786.93, "word": " A", "probability": 0.53857421875}, {"start": 786.93, "end": 788.25, "word": " أصغر", "probability": 0.9295654296875}, {"start": 788.25, "end": 788.55, "word": " من", "probability": 0.99462890625}, {"start": 788.55, "end": 788.87, "word": " أو", "probability": 0.82421875}, {"start": 788.87, "end": 789.41, "word": " يساوي", "probability": 0.9827880859375}, {"start": 789.41, "end": 789.81, "word": " B", "probability": 0.94384765625}, {"start": 789.81, "end": 791.85, "word": " لبرهان", "probability": 0.8187255859375}, {"start": 791.85, "end": 792.29, "word": " ذلك", "probability": 0.992919921875}, {"start": 792.29, "end": 792.51, "word": " to", "probability": 0.72509765625}, {"start": 792.51, "end": 792.83, "word": " see", "probability": 0.927734375}, {"start": 792.83, "end": 794.45, "word": " this", "probability": 0.96826171875}, {"start": 794.45, "end": 798.17, "word": " we", "probability": 0.79736328125}, {"start": 798.17, "end": 798.57, "word": " use", "probability": 0.86962890625}, {"start": 798.57, "end": 799.39, "word": " sequential", "probability": 0.939453125}, {"start": 799.39, "end": 800.25, "word": " criterion", "probability": 0.94580078125}, {"start": 800.25, "end": 802.69, "word": " let", "probability": 0.85595703125}], "temperature": 1.0}, {"id": 32, "seek": 83539, "start": 806.05, "end": 835.39, "text": "x in B sequence in A هدولها مختلفة عن الـ C such that limit x in as N tends to infinity بساوي الـ C طيب، since ال limit ل F of X as X tends to C بساوي L", "tokens": [87, 294, 363, 8310, 294, 316, 8032, 3215, 12610, 11296, 3714, 46456, 46538, 3660, 18871, 2423, 39184, 383, 1270, 300, 4948, 2031, 294, 382, 426, 12258, 281, 13202, 4724, 3794, 995, 45865, 2423, 39184, 383, 23032, 1829, 3555, 12399, 1670, 2423, 4948, 5296, 479, 295, 1783, 382, 1783, 12258, 281, 383, 4724, 3794, 995, 45865, 441], "avg_logprob": -0.24629935047082735, "compression_ratio": 1.2857142857142858, "no_speech_prob": 0.0, "words": [{"start": 806.05, "end": 806.59, "word": "x", "probability": 0.2490234375}, {"start": 806.59, "end": 807.05, "word": " in", "probability": 0.380126953125}, {"start": 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أخناها قبلك و برحلناها by previous theorem", "tokens": [6027, 1863, 19913, 2288, 10632, 33962, 33546, 3660, 20666, 39184, 22978, 39894, 39648, 36145, 3224, 45164, 8978, 18871, 16254, 8310, 28242, 11331, 3215, 23328, 11296, 3714, 5016, 9381, 13063, 3660, 49374, 257, 4032, 272, 4032, 2423, 4948, 24976, 8310, 2514, 6156, 1211, 31377, 2304, 2423, 4948, 6055, 30544, 3714, 5016, 9381, 13063, 3660, 36632, 11242, 995, 49374, 257, 4032, 272, 29538, 39896, 2423, 3894, 20904, 5551, 9778, 8315, 11296, 12174, 36150, 4117, 4032, 4724, 2288, 5016, 1211, 8315, 11296, 538, 3894, 20904], "avg_logprob": -0.2808734911033906, "compression_ratio": 1.645320197044335, "no_speech_prob": 5.960464477539063e-08, "words": [{"start": 959.0899999999999, "end": 959.81, "word": "النظرية", "probability": 0.6849365234375}, {"start": 959.81, "end": 960.33, "word": " الخاصة", "probability": 0.8226725260416666}, {"start": 960.33, "end": 960.61, "word": " بالـ", "probability": 0.570068359375}, {"start": 960.61, "end": 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limit لل functions على الأطراف اللي هي f of x as x tends to c بساوي L", "tokens": [69, 295, 2031, 5551, 9381, 17082, 2288, 9154, 34051, 8608, 995, 45865, 276, 295, 2031, 5551, 9381, 17082, 2288, 9154, 34051, 8608, 995, 45865, 290, 295, 2031, 337, 633, 2031, 6055, 29399, 2304, 1829, 30731, 257, 4032, 2031, 819, 490, 269, 11933, 15730, 25961, 2423, 4948, 24976, 6828, 15844, 16247, 9566, 23557, 5172, 13672, 1829, 39896, 283, 295, 2031, 382, 2031, 12258, 281, 269, 4724, 3794, 995, 45865, 441], "avg_logprob": -0.11484374648758343, "compression_ratio": 1.475609756097561, "no_speech_prob": 0.0, "words": [{"start": 1093.43, "end": 1093.87, "word": "f", "probability": 0.3486328125}, {"start": 1093.87, "end": 1094.19, "word": " of", "probability": 0.3662109375}, {"start": 1094.19, "end": 1094.63, "word": " x", "probability": 0.94189453125}, {"start": 1094.63, "end": 1095.23, "word": " أصغر", "probability": 0.9166259765625}, {"start": 1095.23, "end": 1095.39, "word": " من", "probability": 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as", "probability": 0.91064453125}, {"start": 1119.45, "end": 1119.77, "word": " x", "probability": 0.9765625}, {"start": 1119.77, "end": 1120.01, "word": " tends", "probability": 0.79638671875}, {"start": 1120.01, "end": 1120.13, "word": " to", "probability": 0.98779296875}, {"start": 1120.13, "end": 1120.37, "word": " c", "probability": 0.84912109375}, {"start": 1120.37, "end": 1121.15, "word": " بساوي", "probability": 0.9320068359375}, {"start": 1121.15, "end": 1121.57, "word": " L", "probability": 0.6865234375}], "temperature": 1.0}, {"id": 43, "seek": 113375, "start": 1123.31, "end": 1133.75, "text": "وكذلك ال limit لل function g of x as x instances of c بساوي L حيث L عدد حقيقي", "tokens": [2407, 4117, 8848, 23275, 2423, 4948, 24976, 2445, 290, 295, 2031, 382, 2031, 14519, 295, 269, 4724, 3794, 995, 45865, 441, 11331, 1829, 12984, 441, 6225, 3215, 3215, 11331, 38436, 38436], "avg_logprob": -0.3515625111758709, "compression_ratio": 1.0408163265306123, "no_speech_prob": 0.0, 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1190.57, "text": "النظرية السابقة هنستخدم الـ sequential criterion زائد to use sequential criterion and to squeeze the theorem for", "tokens": [6027, 1863, 19913, 2288, 10632, 21136, 16758, 28671, 8032, 1863, 14851, 9778, 40448, 2423, 39184, 42881, 46691, 30767, 16373, 3215, 281, 764, 42881, 46691, 293, 281, 13578, 264, 20904, 337], "avg_logprob": -0.19468245967741934, "compression_ratio": 1.2173913043478262, "no_speech_prob": 5.960464477539063e-08, "words": [{"start": 1163.4099999999999, "end": 1164.81, "word": "النظرية", "probability": 0.88837890625}, {"start": 1164.81, "end": 1165.55, "word": " السابقة", "probability": 0.9855143229166666}, {"start": 1165.55, "end": 1168.75, "word": " هنستخدم", "probability": 0.9103515625}, {"start": 1168.75, "end": 1169.69, "word": " الـ", "probability": 0.6614990234375}, {"start": 1169.69, "end": 1170.13, "word": " sequential", "probability": 0.90576171875}, {"start": 1170.13, "end": 1170.87, "word": " criterion", "probability": 0.9794921875}, 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"text": "sequences فال let xn بي sequence in a حدودها مختلفة عن c such that limit xn بساوي c", "tokens": [11834, 2667, 6156, 6027, 718, 2031, 77, 4724, 1829, 8310, 294, 257, 11331, 3215, 23328, 11296, 3714, 46456, 46538, 3660, 18871, 269, 1270, 300, 4948, 2031, 77, 4724, 3794, 995, 45865, 269], "avg_logprob": -0.27107007575757575, "compression_ratio": 1.07, "no_speech_prob": 0.0, "words": [{"start": 1198.5599999999997, "end": 1199.9599999999998, "word": "sequences", "probability": 0.466339111328125}, {"start": 1199.9599999999998, "end": 1201.36, "word": " فال", "probability": 0.627197265625}, {"start": 1201.36, "end": 1205.04, "word": " let", "probability": 0.439453125}, {"start": 1205.04, "end": 1205.94, "word": " xn", "probability": 0.5999755859375}, {"start": 1205.94, "end": 1206.62, "word": " بي", "probability": 0.584716796875}, {"start": 1206.62, "end": 1207.22, "word": " sequence", "probability": 0.73486328125}, {"start": 1207.22, "end": 1207.74, "word": " in", "probability": 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"text": "then عندي بطلع f of xn أصغر من أو ساوي h of xn أصغر من أو ساوي g of xn لكل n وعندي by sequential criterion أنا عندي ال limit", "tokens": [19096, 18871, 16254, 4724, 9566, 1211, 3615, 283, 295, 2031, 77, 5551, 9381, 17082, 2288, 9154, 34051, 8608, 995, 45865, 276, 295, 2031, 77, 5551, 9381, 17082, 2288, 9154, 34051, 8608, 995, 45865, 290, 295, 2031, 77, 5296, 28820, 297, 4032, 3615, 1863, 16254, 538, 42881, 46691, 41850, 18871, 16254, 2423, 4948], "avg_logprob": -0.17216981019613878, "compression_ratio": 1.4297520661157024, "no_speech_prob": 0.0, "words": [{"start": 1220.09, "end": 1220.63, "word": "then", "probability": 0.139404296875}, {"start": 1220.63, "end": 1221.23, "word": " عندي", "probability": 0.76318359375}, {"start": 1221.23, "end": 1221.69, "word": " بطلع", "probability": 0.78155517578125}, {"start": 1221.69, "end": 1222.03, "word": " f", "probability": 0.63525390625}, {"start": 1222.03, "end": 1222.31, "word": " of", "probability": 0.642578125}, {"start": 1222.31, "end": 1223.25, "word": " xn", "probability": 0.6090087890625}, {"start": 1223.25, "end": 1224.39, "word": " أصغر", "probability": 0.93994140625}, {"start": 1224.39, "end": 1224.57, "word": " من", "probability": 0.9775390625}, {"start": 1224.57, "end": 1224.75, "word": " أو", "probability": 0.95263671875}, {"start": 1224.75, "end": 1225.39, "word": " ساوي", "probability": 0.9091796875}, {"start": 1225.39, "end": 1225.77, "word": " h", "probability": 0.6669921875}, {"start": 1225.77, "end": 1226.05, "word": " of", "probability": 0.95068359375}, {"start": 1226.05, "end": 1226.95, "word": " xn", "probability": 0.987548828125}, {"start": 1226.95, "end": 1227.49, "word": " أصغر", "probability": 0.98291015625}, {"start": 1227.49, "end": 1227.61, "word": " من", "probability": 0.99560546875}, {"start": 1227.61, "end": 1227.77, "word": " أو", "probability": 0.98974609375}, {"start": 1227.77, "end": 1228.41, "word": " ساوي", "probability": 0.9630533854166666}, {"start": 1228.41, "end": 1230.01, "word": " g", "probability": 0.8955078125}, {"start": 1230.01, "end": 1230.25, "word": " of", "probability": 0.9638671875}, {"start": 1230.25, "end": 1230.93, "word": " xn", "probability": 0.983154296875}, {"start": 1230.93, "end": 1231.41, "word": " لكل", "probability": 0.986572265625}, {"start": 1231.41, "end": 1231.77, "word": " n", "probability": 0.8173828125}, {"start": 1231.77, "end": 1234.93, "word": " وعندي", "probability": 0.8192138671875}, {"start": 1234.93, "end": 1238.27, "word": " by", "probability": 0.92578125}, {"start": 1238.27, "end": 1240.87, "word": " sequential", "probability": 0.94775390625}, {"start": 1240.87, "end": 1242.01, "word": " criterion", "probability": 0.97314453125}, {"start": 1242.01, "end": 1244.33, "word": " أنا", "probability": 0.5966796875}, {"start": 1244.33, "end": 1244.79, "word": " عندي", "probability": 0.929931640625}, {"start": 1244.79, "end": 1245.29, "word": " ال", "probability": 0.693359375}, {"start": 1245.29, "end": 1245.71, "word": " limit", "probability": 0.95703125}], "temperature": 1.0}, {"id": 48, "seek": 125424, "start": 1247.84, "end": 1254.24, "text": "F of X لما اكسطر O لـ C بساوي L بيقدي ان ال limit", "tokens": [37, 295, 1783, 5296, 15042, 1975, 4117, 3794, 9566, 2288, 422, 5296, 39184, 383, 4724, 3794, 995, 45865, 441, 4724, 1829, 4587, 16254, 16472, 2423, 4948], "avg_logprob": -0.5876736287717466, "compression_ratio": 0.9864864864864865, "no_speech_prob": 0.0, "words": [{"start": 1247.84, "end": 1248.42, "word": "F", "probability": 0.04901123046875}, {"start": 1248.42, "end": 1248.76, "word": " of", "probability": 0.42626953125}, {"start": 1248.76, "end": 1249.12, "word": " X", "probability": 0.82177734375}, {"start": 1249.12, "end": 1249.38, "word": " لما", "probability": 0.4776611328125}, {"start": 1249.38, "end": 1249.88, "word": " اكسطر", "probability": 0.546484375}, {"start": 1249.88, "end": 1250.1, "word": " O", "probability": 0.475341796875}, {"start": 1250.1, "end": 1250.46, "word": " لـ", "probability": 0.771484375}, {"start": 1250.46, "end": 1250.68, "word": " C", "probability": 0.451171875}, {"start": 1250.68, "end": 1251.46, "word": " بساوي", "probability": 0.8258056640625}, {"start": 1251.46, "end": 1251.8, "word": " L", "probability": 0.677734375}, {"start": 1251.8, "end": 1252.48, "word": " بيقدي", "probability": 0.55560302734375}, {"start": 1252.48, "end": 1253.7, "word": " ان", "probability": 0.76171875}, {"start": 1253.7, "end": 1253.86, "word": " ال", "probability": 0.56298828125}, {"start": 1253.86, "end": 1254.24, "word": " limit", "probability": 0.7607421875}], "temperature": 1.0}, {"id": 49, "seek": 128455, "start": 1255.99, "end": 1284.55, "text": "لـ f of x n as n tends to infinity بساوي L and ال limit ل g of x as x tends to c بساوي L احنا فرضين ان ال limit ل g and c بساوي L فهذا بيقدي by sequential criterion ان ال limit ل sequence x n اللي نهايتها c نهاية صورتها", "tokens": [1211, 39184, 283, 295, 2031, 297, 382, 297, 12258, 281, 13202, 4724, 3794, 995, 45865, 441, 293, 2423, 4948, 5296, 290, 295, 2031, 382, 2031, 12258, 281, 269, 4724, 3794, 995, 45865, 441, 1975, 5016, 8315, 6156, 43042, 9957, 16472, 2423, 4948, 5296, 290, 293, 269, 4724, 3794, 995, 45865, 441, 6156, 3224, 15730, 4724, 1829, 4587, 16254, 538, 42881, 46691, 16472, 2423, 4948, 5296, 8310, 2031, 297, 13672, 1829, 8717, 11296, 36081, 11296, 269, 8717, 11296, 10632, 20328, 13063, 2655, 11296], "avg_logprob": -0.27899097749985846, "compression_ratio": 1.6514285714285715, "no_speech_prob": 0.0, "words": [{"start": 1255.99, "end": 1256.43, "word": "لـ", "probability": 0.250274658203125}, {"start": 1256.43, "end": 1256.67, "word": " f", "probability": 0.3134765625}, {"start": 1256.67, "end": 1256.97, "word": " of", "probability": 0.427978515625}, {"start": 1256.97, "end": 1257.37, "word": " x", "probability": 0.89794921875}, {"start": 1257.37, "end": 1257.93, "word": " n", "probability": 0.176025390625}, {"start": 1257.93, "end": 1258.65, "word": " as", "probability": 0.82275390625}, {"start": 1258.65, "end": 1259.05, "word": " n", "probability": 0.80810546875}, {"start": 1259.05, "end": 1259.35, "word": " tends", "probability": 0.615234375}, {"start": 1259.35, "end": 1259.51, "word": " to", "probability": 0.89453125}, {"start": 1259.51, "end": 1260.07, "word": " infinity", "probability": 0.8720703125}, {"start": 1260.07, "end": 1261.07, "word": " بساوي", "probability": 0.8017578125}, {"start": 1261.07, "end": 1261.43, "word": " L", "probability": 0.7158203125}, {"start": 1261.43, "end": 1262.77, "word": " and", "probability": 0.5048828125}, {"start": 1262.77, "end": 1263.89, "word": " ال", "probability": 0.53515625}, {"start": 1263.89, "end": 1264.27, "word": " limit", "probability": 0.41015625}, {"start": 1264.27, "end": 1266.05, "word": " ل", "probability": 0.87744140625}, {"start": 1266.05, "end": 1266.37, "word": " g", "probability": 0.41943359375}, {"start": 1266.37, "end": 1266.67, "word": " of", "probability": 0.92529296875}, {"start": 1266.67, "end": 1267.15, "word": " x", "probability": 0.96337890625}, {"start": 1267.15, "end": 1267.67, "word": " as", "probability": 0.97607421875}, {"start": 1267.67, "end": 1268.09, "word": " x", "probability": 0.97314453125}, {"start": 1268.09, "end": 1268.47, "word": " tends", "probability": 0.78271484375}, {"start": 1268.47, "end": 1268.71, "word": " to", "probability": 0.9755859375}, {"start": 1268.71, "end": 1268.99, "word": " c", "probability": 0.82373046875}, {"start": 1268.99, "end": 1269.71, "word": " بساوي", "probability": 0.95556640625}, {"start": 1269.71, "end": 1270.07, "word": " L", "probability": 0.98388671875}, {"start": 1270.07, "end": 1270.93, "word": " احنا", "probability": 0.7598470052083334}, {"start": 1270.93, "end": 1271.35, "word": " فرضين", "probability": 0.9622395833333334}, {"start": 1271.35, "end": 1271.49, "word": " ان", "probability": 0.79638671875}, {"start": 1271.49, "end": 1271.61, "word": " ال", "probability": 0.8017578125}, {"start": 1271.61, "end": 1271.83, "word": " limit", "probability": 0.9443359375}, {"start": 1271.83, "end": 1272.01, "word": " ل", "probability": 0.9189453125}, {"start": 1272.01, "end": 1272.27, "word": " g", "probability": 0.7802734375}, {"start": 1272.27, "end": 1272.51, "word": " and", "probability": 0.45947265625}, {"start": 1272.51, "end": 1272.83, "word": " c", "probability": 0.9453125}, {"start": 1272.83, "end": 1273.37, "word": " بساوي", "probability": 0.946533203125}, {"start": 1273.37, "end": 1273.63, "word": " L", "probability": 0.9765625}, {"start": 1273.63, "end": 1274.13, "word": " فهذا", "probability": 0.93505859375}, {"start": 1274.13, "end": 1274.73, "word": " بيقدي", "probability": 0.68023681640625}, {"start": 1274.73, "end": 1275.67, "word": " by", "probability": 0.77197265625}, {"start": 1275.67, "end": 1276.21, "word": " sequential", "probability": 0.78515625}, {"start": 1276.21, "end": 1277.03, "word": " criterion", "probability": 0.92724609375}, {"start": 1277.03, "end": 1277.41, "word": " ان", "probability": 0.7978515625}, {"start": 1277.41, "end": 1277.61, "word": " ال", "probability": 0.8740234375}, {"start": 1277.61, "end": 1277.95, "word": " limit", "probability": 0.96923828125}, {"start": 1277.95, "end": 1279.97, "word": " ل", "probability": 0.257080078125}, {"start": 1279.97, "end": 1280.41, "word": " sequence", "probability": 0.60498046875}, {"start": 1280.41, "end": 1280.85, "word": " x", "probability": 0.97216796875}, {"start": 1280.85, "end": 1281.21, "word": " n", "probability": 0.47900390625}, {"start": 1281.21, "end": 1281.37, "word": " اللي", "probability": 0.92041015625}, {"start": 1281.37, "end": 1281.91, "word": " نهايتها", "probability": 0.862548828125}, {"start": 1281.91, "end": 1282.23, "word": " c", "probability": 0.794921875}, {"start": 1282.23, "end": 1283.83, "word": " نهاية", "probability": 0.7980143229166666}, {"start": 1283.83, "end": 1284.55, "word": " صورتها", "probability": 0.9923095703125}], "temperature": 1.0}, {"id": 50, "seek": 131542, "start": 1287.56, "end": 1315.42, "text": "هتطلع أيضا بساوي L وبالتالي so by squeeze theorem for sequences يعني عندي تلات متتاليات هذه ال limit بتاعتها بتطلع L", "tokens": [3224, 2655, 9566, 1211, 3615, 36632, 11242, 995, 4724, 3794, 995, 45865, 441, 46599, 6027, 2655, 6027, 1829, 370, 538, 13578, 20904, 337, 22978, 37495, 22653, 18871, 16254, 6055, 1211, 9307, 44650, 2655, 6027, 1829, 9307, 29538, 2423, 4948, 39894, 995, 34268, 11296, 39894, 9566, 1211, 3615, 441], "avg_logprob": -0.2606823979591837, "compression_ratio": 1.2773722627737227, "no_speech_prob": 0.0, "words": [{"start": 1287.56, "end": 1288.24, "word": "هتطلع", "probability": 0.773291015625}, {"start": 1288.24, "end": 1288.72, "word": " أيضا", "probability": 0.7353515625}, {"start": 1288.72, "end": 1289.66, "word": " بساوي", "probability": 0.756103515625}, {"start": 1289.66, "end": 1290.16, "word": " L", "probability": 0.580078125}, {"start": 1290.16, "end": 1294.66, "word": " وبالتالي", "probability": 0.876953125}, {"start": 1294.66, "end": 1295.2, "word": " so", "probability": 0.1937255859375}, {"start": 1295.2, "end": 1296.18, "word": " by", "probability": 0.92138671875}, {"start": 1296.18, "end": 1298.26, "word": " squeeze", "probability": 0.70556640625}, {"start": 1298.26, "end": 1300.48, "word": " theorem", "probability": 0.78369140625}, {"start": 1300.48, "end": 1302.4, "word": " for", "probability": 0.91552734375}, {"start": 1302.4, "end": 1303.46, "word": " sequences", "probability": 0.8525390625}, {"start": 1303.46, "end": 1306.52, "word": " يعني", "probability": 0.57659912109375}, {"start": 1306.52, "end": 1306.8, "word": " عندي", "probability": 0.7724609375}, {"start": 1306.8, "end": 1307.12, "word": " تلات", "probability": 0.74462890625}, {"start": 1307.12, "end": 1308.1, "word": " متتاليات", "probability": 0.96376953125}, {"start": 1308.1, "end": 1311.8, "word": " هذه", "probability": 0.6796875}, {"start": 1311.8, "end": 1311.98, "word": " ال", "probability": 0.826171875}, {"start": 1311.98, "end": 1312.22, "word": " limit", "probability": 0.951171875}, {"start": 1312.22, "end": 1312.92, "word": " بتاعتها", "probability": 0.96484375}, {"start": 1312.92, "end": 1315.12, "word": " بتطلع", "probability": 0.8699951171875}, {"start": 1315.12, "end": 1315.42, "word": " L", "probability": 0.95166015625}], "temperature": 1.0}, {"id": 51, "seek": 134234, "start": 1316.32, "end": 1342.34, "text": "و هذه ال limit تبعتها بيطلع لعدد L لما N تقول infinity إذا ال sequence اللي في وسط ال limit تبعتها limit ال sequence H of X N as N tends to infinity بتطلع بساوي L وبالتالي therefore", "tokens": [2407, 29538, 2423, 4948, 6055, 3555, 34268, 11296, 4724, 1829, 9566, 1211, 3615, 5296, 22488, 3215, 441, 5296, 15042, 426, 6055, 39648, 13202, 11933, 15730, 2423, 8310, 13672, 1829, 8978, 46952, 9566, 2423, 4948, 6055, 3555, 34268, 11296, 4948, 2423, 8310, 389, 295, 1783, 426, 382, 426, 12258, 281, 13202, 39894, 9566, 1211, 3615, 4724, 3794, 995, 45865, 441, 46599, 6027, 2655, 6027, 1829, 4412], "avg_logprob": -0.27367422978083294, "compression_ratio": 1.485207100591716, "no_speech_prob": 0.0, "words": [{"start": 1316.32, "end": 1316.56, "word": "و", "probability": 0.72314453125}, {"start": 1316.56, "end": 1316.92, "word": " هذه", "probability": 0.296875}, {"start": 1316.92, "end": 1317.04, "word": " ال", "probability": 0.3837890625}, {"start": 1317.04, "end": 1317.26, "word": " limit", "probability": 0.568359375}, {"start": 1317.26, "end": 1317.84, "word": " تبعتها", "probability": 0.79852294921875}, {"start": 1317.84, "end": 1318.18, "word": " بيطلع", "probability": 0.7989990234375}, {"start": 1318.18, "end": 1318.66, "word": " لعدد", "probability": 0.9054361979166666}, {"start": 1318.66, "end": 1318.94, "word": " L", "probability": 0.685546875}, {"start": 1318.94, "end": 1319.34, "word": " لما", "probability": 0.747802734375}, {"start": 1319.34, "end": 1319.72, "word": " N", "probability": 0.6005859375}, {"start": 1319.72, "end": 1320.38, "word": " تقول", "probability": 0.856689453125}, {"start": 1320.38, "end": 1321.42, "word": " infinity", "probability": 0.427001953125}, {"start": 1321.42, "end": 1325.08, "word": " إذا", "probability": 0.589599609375}, {"start": 1325.08, "end": 1325.26, "word": " ال", "probability": 0.712890625}, {"start": 1325.26, "end": 1325.68, "word": " sequence", "probability": 0.97314453125}, {"start": 1325.68, "end": 1325.88, "word": " اللي", "probability": 0.7939453125}, {"start": 1325.88, "end": 1325.98, "word": " في", "probability": 0.89013671875}, {"start": 1325.98, "end": 1326.38, "word": " وسط", "probability": 0.96435546875}, {"start": 1326.38, "end": 1326.5, "word": " ال", "probability": 0.60302734375}, {"start": 1326.5, "end": 1326.7, "word": " limit", "probability": 0.97607421875}, {"start": 1326.7, "end": 1327.52, "word": " تبعتها", "probability": 0.9595947265625}, {"start": 1327.52, "end": 1329.8, "word": " limit", "probability": 0.7451171875}, {"start": 1329.8, "end": 1330.96, "word": " ال", "probability": 0.64208984375}, {"start": 1330.96, "end": 1331.44, "word": " sequence", "probability": 0.9912109375}, {"start": 1331.44, "end": 1331.84, "word": " H", "probability": 0.44287109375}, {"start": 1331.84, "end": 1332.1, "word": " of", "probability": 0.54638671875}, {"start": 1332.1, "end": 1332.48, "word": " X", "probability": 0.9462890625}, {"start": 1332.48, "end": 1332.96, "word": " N", "probability": 0.41552734375}, {"start": 1332.96, "end": 1334.18, "word": " as", "probability": 0.87109375}, {"start": 1334.18, "end": 1334.52, "word": " N", "probability": 0.8896484375}, {"start": 1334.52, "end": 1334.8, "word": " tends", "probability": 0.282470703125}, {"start": 1334.8, "end": 1334.92, "word": " to", "probability": 0.9345703125}, {"start": 1334.92, "end": 1335.32, "word": " infinity", "probability": 0.853515625}, {"start": 1335.32, "end": 1335.86, "word": " بتطلع", "probability": 0.9249267578125}, {"start": 1335.86, "end": 1336.64, "word": " بساوي", "probability": 0.772705078125}, {"start": 1336.64, "end": 1337.0, "word": " L", "probability": 0.93994140625}, {"start": 1337.0, "end": 1341.06, "word": " وبالتالي", "probability": 0.92578125}, {"start": 1341.06, "end": 1342.34, "word": " therefore", "probability": 0.57861328125}], "temperature": 1.0}, {"id": 52, "seek": 136996, "start": 1344.08, "end": 1369.96, "text": "by sequential criterion مرة أخرى ال sequential criterion بتقول عشان أثبت أن ال limit لل function h of x لما x تقول إلى c بساوي العدد L عشان أثبت limit ال function h لما x تقول إلى c بساوي L", "tokens": [2322, 42881, 46691, 3714, 25720, 5551, 34740, 7578, 2423, 42881, 46691, 39894, 39648, 6225, 8592, 7649, 5551, 12984, 3555, 2655, 14739, 2423, 4948, 24976, 2445, 276, 295, 2031, 5296, 15042, 2031, 6055, 39648, 30731, 269, 4724, 3794, 995, 45865, 18863, 3215, 3215, 441, 6225, 8592, 7649, 5551, 12984, 3555, 2655, 4948, 2423, 2445, 276, 5296, 15042, 2031, 6055, 39648, 30731, 269, 4724, 3794, 995, 45865, 441], "avg_logprob": 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"word": " أثبت", "probability": 0.96142578125}, {"start": 1355.06, "end": 1355.26, "word": " أن", "probability": 0.5537109375}, {"start": 1355.26, "end": 1355.44, "word": " ال", "probability": 0.4794921875}, {"start": 1355.44, "end": 1355.84, "word": " limit", "probability": 0.935546875}, {"start": 1355.84, "end": 1358.08, "word": " لل", "probability": 0.58642578125}, {"start": 1358.08, "end": 1358.66, "word": " function", "probability": 0.90283203125}, {"start": 1358.66, "end": 1359.24, "word": " h", "probability": 0.76171875}, {"start": 1359.24, "end": 1359.5, "word": " of", "probability": 0.443115234375}, {"start": 1359.5, "end": 1360.0, "word": " x", "probability": 0.92138671875}, {"start": 1360.0, "end": 1360.4, "word": " لما", "probability": 0.857177734375}, {"start": 1360.4, "end": 1360.78, "word": " x", "probability": 0.94140625}, {"start": 1360.78, "end": 1361.48, "word": " تقول", "probability": 0.634521484375}, {"start": 1361.48, "end": 1362.0, "word": " إلى", "probability": 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0.98193359375}, {"start": 1368.46, "end": 1368.74, "word": " تقول", "probability": 0.97998046875}, {"start": 1368.74, "end": 1368.92, "word": " إلى", "probability": 0.7783203125}, {"start": 1368.92, "end": 1369.08, "word": " c", "probability": 0.9423828125}, {"start": 1369.08, "end": 1369.66, "word": " بساوي", "probability": 0.9705810546875}, {"start": 1369.66, "end": 1369.96, "word": " L", "probability": 0.9521484375}], "temperature": 1.0}, {"id": 53, "seek": 139826, "start": 1371.62, "end": 1398.26, "text": "لازم هذا بكافئ حسب الـ sequential criterion هذا بكافئ إننا نثبت لو أخدت أي sequence في مجال الدالة كل حدودها مختلفة عن الـ C و ال sequence نهايتها C فلازم أثبت نهاية صورة ال sequence بالساوية لعدد L وهذا أثبتناه، اذا by sequential criterion بطلع limit H عن C بساوية L وهو المطلوب", "tokens": [1211, 31377, 2304, 23758, 4724, 4117, 31845, 19986, 11331, 35457, 2423, 39184, 42881, 46691, 23758, 4724, 4117, 31845, 19986, 36145, 8315, 8717, 12984, 3555, 2655, 45164, 5551, 9778, 3215, 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مثال limit ل function x غرب sin 1 على x as x tends to zero exists و بالساوي سفر", "tokens": [39237, 10943, 22807, 1392, 4032, 46958, 5016, 22807, 8978, 9122, 2655, 13546, 9154, 2423, 10406, 9673, 3615, 32887, 5172, 3660, 3714, 43020, 1975, 12984, 3555, 9307, 11296, 44945, 36178, 41950, 2288, 4587, 29538, 16490, 1211, 9957, 995, 8717, 47283, 3215, 9122, 2304, 7649, 50113, 6027, 4948, 5296, 2445, 2031, 32771, 25513, 3343, 502, 15844, 2031, 382, 2031, 12258, 281, 4018, 8198, 4032, 20666, 3794, 995, 45865, 8608, 5172, 2288], "avg_logprob": -0.25625001234667644, "compression_ratio": 1.3473684210526315, "no_speech_prob": 0.0, "words": [{"start": 1965.6200000000001, "end": 1966.7, "word": "تمام؟", "probability": 0.6464029947916666}, {"start": 1966.7, "end": 1967.78, "word": " okay", "probability": 0.1622314453125}, {"start": 1967.78, "end": 1969.78, "word": " واضح؟", "probability": 0.89111328125}, {"start": 1969.78, "end": 1970.36, "word": " في", "probability": 0.491455078125}, {"start": 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for all ثيتا تنتمي إلى R هذه المتباينة الصحيحة ال absolute maximum value ل سايل واحد وال absolute minimum value سالب واحد خلينا نضرب المتباينة هذه في absolute X", "tokens": [12984, 36081, 995, 11778, 16373, 15042, 5551, 4117, 26890, 9154, 34051, 3794, 995, 2407, 8608, 6027, 3555, 36764, 24401, 5551, 9381, 17082, 2288, 9154, 34051, 3794, 995, 2407, 36764, 24401, 337, 439, 38637, 36081, 995, 6055, 29399, 2304, 1829, 30731, 497, 29538, 9673, 2655, 3555, 995, 9957, 3660, 31767, 5016, 1829, 5016, 3660, 2423, 8236, 6674, 2158, 5296, 8608, 995, 26895, 36764, 24401, 16070, 8236, 7285, 2158, 8608, 6027, 3555, 36764, 24401, 16490, 1211, 9957, 995, 8717, 11242, 25513, 9673, 2655, 3555, 995, 9957, 3660, 29538, 8978, 8236, 1783], "avg_logprob": -0.21093750496705374, "compression_ratio": 1.7807486631016043, "no_speech_prob": 0.0, "words": [{"start": 2199.33, "end": 2200.13, "word": "ثيتا", "probability": 0.68310546875}, {"start": 2200.13, "end": 2200.93, "word": " دائما", "probability": 0.71728515625}, {"start": 2200.93, "end": 2201.57, "word": " أكبر", "probability": 0.8134765625}, {"start": 2201.57, "end": 2201.77, "word": " من", "probability": 0.966796875}, {"start": 2201.77, "end": 2202.31, "word": " أوساو", "probability": 0.7760009765625}, {"start": 2202.31, "end": 2202.77, "word": " سالب", "probability": 0.9002278645833334}, {"start": 2202.77, "end": 2203.19, "word": " واحد", "probability": 0.984375}, {"start": 2203.19, "end": 2204.41, "word": " أصغر", "probability": 0.804443359375}, {"start": 2204.41, "end": 2204.53, "word": " من", "probability": 0.9921875}, {"start": 2204.53, "end": 2205.01, "word": " أوساو", "probability": 0.96484375}, {"start": 2205.01, "end": 2205.35, "word": " واحد", "probability": 0.988525390625}, {"start": 2205.35, "end": 2205.63, "word": " for", "probability": 0.81884765625}, {"start": 2205.63, "end": 2206.01, "word": " all", "probability": 0.94775390625}, {"start": 2206.01, "end": 2206.67, "word": " ثيتا", "probability": 0.6846923828125}, {"start": 2206.67, "end": 2207.39, "word": " تنتمي", "probability": 0.934814453125}, {"start": 2207.39, "end": 2207.61, "word": " إلى", "probability": 0.8974609375}, {"start": 2207.61, "end": 2207.95, "word": " R", "probability": 0.66357421875}, {"start": 2207.95, "end": 2209.01, "word": " هذه", "probability": 0.61767578125}, {"start": 2209.01, "end": 2209.69, "word": " المتباينة", "probability": 0.9256184895833334}, {"start": 2209.69, "end": 2210.73, "word": " الصحيحة", "probability": 0.894921875}, {"start": 2210.73, "end": 2211.03, "word": " ال", "probability": 0.78125}, {"start": 2211.03, "end": 2211.45, "word": " absolute", "probability": 0.72119140625}, {"start": 2211.45, "end": 2212.03, "word": " maximum", "probability": 0.93408203125}, {"start": 2212.03, "end": 2212.49, "word": " value", "probability": 0.99169921875}, {"start": 2212.49, "end": 2212.67, "word": " ل", "probability": 0.56787109375}, {"start": 2212.67, "end": 2213.09, "word": " سايل", "probability": 0.6085103352864584}, {"start": 2213.09, "end": 2213.55, "word": " واحد", "probability": 0.97802734375}, {"start": 2213.55, "end": 2213.69, "word": " وال", "probability": 0.705078125}, {"start": 2213.69, "end": 2214.11, "word": " absolute", "probability": 0.9189453125}, {"start": 2214.11, "end": 2214.57, "word": " minimum", "probability": 0.96142578125}, {"start": 2214.57, "end": 2215.03, "word": " value", "probability": 0.9931640625}, {"start": 2215.03, "end": 2215.49, "word": " سالب", "probability": 0.9597981770833334}, {"start": 2215.49, "end": 2215.85, "word": " واحد", "probability": 0.990234375}, {"start": 2215.85, "end": 2219.71, "word": " خلينا", "probability": 0.639434814453125}, {"start": 2219.71, "end": 2220.27, "word": " نضرب", "probability": 0.9793294270833334}, {"start": 2220.27, "end": 2222.47, "word": " المتباينة", "probability": 0.978515625}, {"start": 2222.47, "end": 2222.89, "word": " هذه", "probability": 0.716796875}, {"start": 2222.89, "end": 2223.07, "word": " في", "probability": 0.95361328125}, {"start": 2223.07, "end": 2223.51, "word": " absolute", "probability": 0.8916015625}, {"start": 2223.51, "end": 2224.07, "word": " X", "probability": 0.57080078125}], "temperature": 1.0}, {"id": 84, "seek": 224666, "start": 2226.06, "end": 2246.66, "text": "و ال theta هذه تبدلها ب 1 على x ف since sin 1 على x أكبر من أو ساوي سالب واحد أصغر من أو ساوي واحد لكل x لا تساوي سفر هذا بيقدر", "tokens": [2407, 2423, 9725, 29538, 6055, 44510, 1211, 11296, 4724, 502, 15844, 2031, 6156, 1670, 3343, 502, 15844, 2031, 5551, 4117, 26890, 9154, 34051, 8608, 995, 45865, 8608, 6027, 3555, 36764, 24401, 5551, 9381, 17082, 2288, 9154, 34051, 8608, 995, 45865, 36764, 24401, 5296, 28820, 2031, 20193, 6055, 3794, 995, 45865, 8608, 5172, 2288, 23758, 4724, 1829, 28543, 2288], "avg_logprob": -0.3230932233697277, "compression_ratio": 1.489051094890511, "no_speech_prob": 2.384185791015625e-07, "words": [{"start": 2226.06, "end": 2226.9, "word": "و", "probability": 0.21240234375}, {"start": 2226.9, "end": 2227.34, "word": " ال", "probability": 0.285400390625}, {"start": 2227.34, "end": 2227.64, "word": " theta", "probability": 0.386962890625}, {"start": 2227.64, "end": 2228.18, "word": " هذه", "probability": 0.5771484375}, {"start": 2228.18, "end": 2233.58, "word": " تبدلها", "probability": 0.5482177734375}, {"start": 2233.58, "end": 2233.72, "word": " ب", "probability": 0.90380859375}, {"start": 2233.72, "end": 2233.92, "word": " 1", "probability": 0.346923828125}, {"start": 2233.92, "end": 2234.18, "word": " على", "probability": 0.5966796875}, {"start": 2234.18, "end": 2234.52, "word": " x", "probability": 0.7177734375}, {"start": 2234.52, "end": 2235.74, "word": " ف", "probability": 0.79150390625}, {"start": 2235.74, "end": 2236.3, "word": " since", "probability": 0.703125}, {"start": 2236.3, "end": 2236.98, "word": " sin", "probability": 0.4453125}, {"start": 2236.98, "end": 2238.08, "word": " 1", "probability": 0.6201171875}, {"start": 2238.08, "end": 2238.42, "word": " على", "probability": 0.74658203125}, {"start": 2238.42, "end": 2238.76, "word": " x", "probability": 0.947265625}, {"start": 2238.76, "end": 2239.26, "word": " أكبر", "probability": 0.9462890625}, {"start": 2239.26, "end": 2239.46, "word": " من", "probability": 0.9384765625}, {"start": 2239.46, "end": 2239.66, "word": " أو", "probability": 0.90185546875}, {"start": 2239.66, "end": 2240.0, "word": " ساوي", "probability": 0.7666015625}, {"start": 2240.0, "end": 2240.42, "word": " سالب", "probability": 0.6852213541666666}, {"start": 2240.42, "end": 2240.82, "word": " واحد", "probability": 0.946044921875}, {"start": 2240.82, "end": 2241.88, "word": " أصغر", "probability": 0.8818359375}, {"start": 2241.88, "end": 2242.02, "word": " من", "probability": 0.99267578125}, {"start": 2242.02, "end": 2242.2, "word": " أو", "probability": 0.9501953125}, {"start": 2242.2, "end": 2242.6, "word": " ساوي", "probability": 0.96875}, {"start": 2242.6, "end": 2243.04, "word": " واحد", "probability": 0.98876953125}, {"start": 2243.04, "end": 2243.52, "word": " لكل", "probability": 0.974609375}, {"start": 2243.52, "end": 2243.84, "word": " x", "probability": 0.884765625}, {"start": 2243.84, "end": 2244.06, "word": " لا", "probability": 0.74365234375}, {"start": 2244.06, "end": 2244.58, "word": " تساوي", "probability": 0.97021484375}, {"start": 2244.58, "end": 2245.12, "word": " سفر", "probability": 0.75830078125}, {"start": 2245.12, "end": 2246.02, "word": " هذا", "probability": 0.697265625}, {"start": 2246.02, "end": 2246.66, "word": " بيقدر", "probability": 0.6817626953125}], "temperature": 1.0}, {"id": 85, "seek": 227232, "start": 2249.42, "end": 2272.32, "text": "لو ضربت المتباينة هذه في absolute x اللي هي بالساوي x و absolute x هنا هتطلع موجة بقى فهيطلع عندي سالب absolute x أصغر من أو ساوي x في sin 1 على x أصغر من أو ساوي absolute x وبالتالي المتباينة اللي انا عايز اثبتها ده هي صحيحة", "tokens": [1211, 2407, 48812, 25513, 2655, 9673, 2655, 3555, 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2256.86, "word": " هتطلع", "probability": 0.94775390625}, {"start": 2256.86, "end": 2257.32, "word": " موجة", "probability": 0.9430338541666666}, {"start": 2257.32, "end": 2257.6, "word": " بقى", "probability": 0.9378255208333334}, {"start": 2257.6, "end": 2260.16, "word": " فهيطلع", "probability": 0.8828938802083334}, {"start": 2260.16, "end": 2260.44, "word": " عندي", "probability": 0.66845703125}, {"start": 2260.44, "end": 2260.88, "word": " سالب", "probability": 0.8492838541666666}, {"start": 2260.88, "end": 2261.38, "word": " absolute", "probability": 0.94091796875}, {"start": 2261.38, "end": 2261.98, "word": " x", "probability": 0.98828125}, {"start": 2261.98, "end": 2262.42, "word": " أصغر", "probability": 0.9681396484375}, {"start": 2262.42, "end": 2262.56, "word": " من", "probability": 0.9765625}, {"start": 2262.56, "end": 2262.68, "word": " أو", "probability": 0.9384765625}, {"start": 2262.68, "end": 2263.12, "word": " ساوي", "probability": 0.9654947916666666}, {"start": 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"word": " وبالتالي", "probability": 0.9205078125}, {"start": 2269.76, "end": 2270.52, "word": " المتباينة", "probability": 0.9856770833333334}, {"start": 2270.52, "end": 2270.64, "word": " اللي", "probability": 0.948486328125}, {"start": 2270.64, "end": 2270.76, "word": " انا", "probability": 0.783447265625}, {"start": 2270.76, "end": 2270.96, "word": " عايز", "probability": 0.9444986979166666}, {"start": 2270.96, "end": 2271.46, "word": " اثبتها", "probability": 0.91435546875}, {"start": 2271.46, "end": 2271.64, "word": " ده", "probability": 0.862060546875}, {"start": 2271.64, "end": 2271.76, "word": " هي", "probability": 0.6220703125}, {"start": 2271.76, "end": 2272.32, "word": " صحيحة", "probability": 0.9546875}], "temperature": 1.0}, {"id": 86, "seek": 230019, "start": 2276.18, "end": 2300.2, "text": "أضرب هنا في absolute x و absolute x بالساوية x عدد موجب فالمتباينة تبقى أشرتها زي ما هي و هي ضربت في سالب absolute x ال x هنا هي برضه سالب absolute x فهذه المتباينة اللي احنا عايزين نثبتها في الإدعاء تبعنا الحالة التالية لو كانت x سالب", "tokens": [10721, 11242, 25513, 34105, 8978, 8236, 2031, 4032, 8236, 2031, 20666, 3794, 995, 2407, 10632, 2031, 6225, 3215, 3215, 3714, 29245, 3555, 6156, 45340, 2655, 3555, 995, 9957, 3660, 6055, 3555, 4587, 7578, 5551, 8592, 43500, 11296, 30767, 1829, 19446, 39896, 4032, 39896, 48812, 25513, 2655, 8978, 8608, 6027, 3555, 8236, 2031, 2423, 2031, 34105, 39896, 4724, 43042, 3224, 8608, 6027, 3555, 8236, 2031, 6156, 3224, 24192, 9673, 2655, 3555, 995, 9957, 3660, 13672, 1829, 1975, 5016, 8315, 6225, 47302, 11622, 9957, 8717, 12984, 3555, 2655, 11296, 8978, 33688, 3215, 3615, 16606, 6055, 3555, 3615, 8315, 21542, 6027, 3660, 16712, 6027, 10632, 45164, 25961, 2655, 2031, 8608, 6027, 3555], "avg_logprob": -0.1781250002709302, "compression_ratio": 1.8731707317073172, "no_speech_prob": 0.0, "words": [{"start": 2276.18, "end": 2276.66, "word": "أضرب", "probability": 0.6693522135416666}, {"start": 2276.66, "end": 2276.96, "word": " هنا", 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claim", "probability": 0.30712890625}, {"start": 2433.01, "end": 2433.65, "word": " وهو", "probability": 0.7763671875}, {"start": 2433.65, "end": 2434.33, "word": " ان", "probability": 0.76904296875}, {"start": 2434.33, "end": 2435.17, "word": " المتباينة", "probability": 0.9918619791666666}, {"start": 2435.17, "end": 2435.59, "word": " هذه", "probability": 0.85986328125}, {"start": 2435.59, "end": 2436.65, "word": " صحيحة", "probability": 0.969140625}, {"start": 2436.65, "end": 2437.97, "word": " تمام؟", "probability": 0.9488932291666666}], "temperature": 1.0}, {"id": 92, "seek": 246814, "start": 2440.82, "end": 2468.14, "text": "الان من ال claim now by above .. by above the claim and squeeze the theorem since بما أنه ال limit", "tokens": [6027, 7649, 9154, 2423, 3932, 586, 538, 3673, 4386, 538, 3673, 264, 3932, 293, 13578, 264, 20904, 1670, 4724, 15042, 14739, 3224, 2423, 4948], "avg_logprob": -0.2717187595367432, "compression_ratio": 1.163265306122449, "no_speech_prob": 0.0, "words": [{"start": 2440.82, "end": 2441.5, "word": "الان", "probability": 0.7279052734375}, {"start": 2441.5, "end": 2442.22, "word": " من", "probability": 0.9501953125}, {"start": 2442.22, "end": 2442.46, "word": " ال", "probability": 0.92578125}, {"start": 2442.46, "end": 2442.9, "word": " claim", "probability": 0.7783203125}, {"start": 2442.9, "end": 2450.56, "word": " now", "probability": 0.360107421875}, {"start": 2450.56, "end": 2452.66, "word": " by", "probability": 0.6669921875}, {"start": 2452.66, "end": 2453.34, "word": " above", "probability": 0.95361328125}, {"start": 2453.34, "end": 2453.74, "word": " ..", "probability": 0.328369140625}, {"start": 2453.74, "end": 2455.74, "word": " by", "probability": 0.8935546875}, {"start": 2455.74, "end": 2456.22, "word": " above", "probability": 0.96875}, {"start": 2456.22, "end": 2456.46, "word": " the", "probability": 0.70361328125}, {"start": 2456.46, "end": 2456.92, "word": " claim", "probability": 0.96484375}, {"start": 2456.92, 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السفر وهو المطلوب", "tokens": [6027, 39184, 8236, 2031, 382, 2031, 12258, 281, 1958, 4724, 3794, 995, 45865, 2423, 4948, 5296, 3794, 6027, 3555, 8236, 2031, 382, 2031, 12258, 281, 1958, 4724, 3794, 995, 45865, 1958, 321, 362, 8717, 5016, 36520, 15844, 16472, 2423, 4948, 9673, 5016, 9381, 13063, 3660, 8978, 2423, 41779, 9566, 13672, 1829, 39896, 2031, 8978, 3343, 502, 15844, 2031, 382, 2031, 12258, 281, 1958, 4724, 3794, 995, 45865, 36632, 11242, 995, 21136, 5172, 2288, 37037, 2407, 9673, 9566, 1211, 37746], "avg_logprob": -0.22421875316649675, "compression_ratio": 1.7530864197530864, "no_speech_prob": 5.960464477539063e-08, "words": [{"start": 2469.33, "end": 2469.73, "word": "الـ", "probability": 0.28216552734375}, {"start": 2469.73, "end": 2470.11, "word": " absolute", "probability": 0.474609375}, {"start": 2470.11, "end": 2470.81, "word": " x", "probability": 0.8857421875}, {"start": 2470.81, "end": 2471.27, "word": " as", "probability": 0.89111328125}, {"start": 2471.27, "end": 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المحصورة", "probability": 0.96806640625}, {"start": 2486.07, "end": 2486.17, "word": " في", "probability": 0.9326171875}, {"start": 2486.17, "end": 2487.21, "word": " الوسط", "probability": 0.9187825520833334}, {"start": 2487.21, "end": 2487.63, "word": " اللي", "probability": 0.84375}, {"start": 2487.63, "end": 2487.85, "word": " هي", "probability": 0.6357421875}, {"start": 2487.85, "end": 2488.21, "word": " x", "probability": 0.89794921875}, {"start": 2488.21, "end": 2488.51, "word": " في", "probability": 0.8173828125}, {"start": 2488.51, "end": 2489.01, "word": " sin", "probability": 0.4736328125}, {"start": 2489.01, "end": 2489.57, "word": " 1", "probability": 0.5751953125}, {"start": 2489.57, "end": 2489.95, "word": " على", "probability": 0.56298828125}, {"start": 2489.95, "end": 2490.33, "word": " x", "probability": 0.98681640625}, {"start": 2490.33, "end": 2491.33, "word": " as", "probability": 0.97607421875}, {"start": 2491.33, "end": 2491.69, "word": " x", "probability": 0.99267578125}, {"start": 2491.69, "end": 2492.05, "word": " tends", "probability": 0.810546875}, {"start": 2492.05, "end": 2492.23, "word": " to", "probability": 0.9765625}, {"start": 2492.23, "end": 2492.71, "word": " 0", "probability": 0.97412109375}, {"start": 2492.71, "end": 2494.49, "word": " بساوي", "probability": 0.95849609375}, {"start": 2494.49, "end": 2495.07, "word": " أيضا", "probability": 0.853515625}, {"start": 2495.07, "end": 2495.73, "word": " السفر", "probability": 0.7910970052083334}, {"start": 2495.73, "end": 2496.29, "word": " وهو", "probability": 0.8486328125}, {"start": 2496.29, "end": 2496.85, "word": " المطلوب", "probability": 0.9879150390625}], "temperature": 1.0}, {"id": 94, "seek": 251667, "start": 2498.04, "end": 2516.68, "text": "اذا هنا استخدمنا الـ squeeze theorem تمام؟ لأن هذا برهان أن ال limit لل function x sin 1 على x عندما x تقوى للصفر بيساوي صفر تمام؟ واضح؟ في أي سفسار؟ أي سؤال؟", "tokens": [995, 15730, 34105, 44713, 9778, 40448, 8315, 2423, 39184, 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{"start": 2500.94, "end": 2502.38, "word": " تمام؟", "probability": 0.6172688802083334}, {"start": 2502.38, "end": 2503.96, "word": " لأن", "probability": 0.63916015625}, {"start": 2503.96, "end": 2504.18, "word": " هذا", "probability": 0.86669921875}, {"start": 2504.18, "end": 2504.82, "word": " برهان", "probability": 0.9512939453125}, {"start": 2504.82, "end": 2505.1, "word": " أن", "probability": 0.54541015625}, {"start": 2505.1, "end": 2505.24, "word": " ال", "probability": 0.45703125}, {"start": 2505.24, "end": 2505.52, "word": " limit", "probability": 0.64990234375}, {"start": 2505.52, "end": 2505.78, "word": " لل", "probability": 0.69482421875}, {"start": 2505.78, "end": 2506.18, "word": " function", "probability": 0.72216796875}, {"start": 2506.18, "end": 2506.52, "word": " x", "probability": 0.8076171875}, {"start": 2506.52, "end": 2506.9, "word": " sin", "probability": 0.44384765625}, {"start": 2506.9, "end": 2507.74, "word": " 1", "probability": 0.580078125}, {"start": 2507.74, "end": 2508.02, "word": " على", "probability": 0.46142578125}, {"start": 2508.02, "end": 2508.28, "word": " x", "probability": 0.96240234375}, {"start": 2508.28, "end": 2508.52, "word": " عندما", "probability": 0.62646484375}, {"start": 2508.52, "end": 2508.76, "word": " x", "probability": 0.958984375}, {"start": 2508.76, "end": 2508.92, "word": " تقوى", "probability": 0.5064010620117188}, {"start": 2508.92, "end": 2509.52, "word": " للصفر", "probability": 0.67864990234375}, {"start": 2509.52, "end": 2511.38, "word": " بيساوي", "probability": 0.866015625}, {"start": 2511.38, "end": 2511.98, "word": " صفر", "probability": 0.787109375}, {"start": 2511.98, "end": 2514.2, "word": " تمام؟", "probability": 0.78076171875}, {"start": 2514.2, "end": 2514.82, "word": " واضح؟", "probability": 0.9268798828125}, {"start": 2514.82, "end": 2514.96, "word": " في", "probability": 0.72314453125}, {"start": 2514.96, "end": 2515.1, "word": " أي", "probability": 0.81298828125}, {"start": 2515.1, "end": 2515.78, "word": " سفسار؟", "probability": 0.702880859375}, {"start": 2515.78, "end": 2515.96, "word": " أي", "probability": 0.677734375}, {"start": 2515.96, "end": 2516.68, "word": " سؤال؟", "probability": 0.983642578125}], "temperature": 1.0}, {"id": 95, "seek": 254582, "start": 2523.26, "end": 2545.82, "text": "Okay باقي نظرية واحدة في ال section 4-2 فاتحقلوا تقرؤوها تفهموا البرهانة و تبدووا في حال التمرين و ان شاء الله المرة جاية بنكمل نشرح النظرية هذه و بنبدأ section جديد اللي هو section 4-3", "tokens": [8297, 4724, 995, 38436, 8717, 19913, 2288, 10632, 36764, 24401, 3660, 8978, 2423, 3541, 1017, 12, 17, 6156, 9307, 5016, 4587, 1211, 14407, 6055, 4587, 2288, 33604, 2407, 11296, 6055, 5172, 16095, 14407, 2423, 26890, 3224, 7649, 3660, 4032, 6055, 44510, 2407, 14407, 8978, 11331, 6027, 16712, 29973, 9957, 4032, 16472, 13412, 16606, 21984, 9673, 25720, 10874, 995, 10632, 44945, 24793, 1211, 8717, 46309, 5016, 28239, 19913, 2288, 10632, 29538, 4032, 44945, 44510, 10721, 3541, 10874, 16254, 3215, 13672, 1829, 31439, 3541, 1017, 12, 18], "avg_logprob": -0.18677326309126477, "compression_ratio": 1.5885416666666667, "no_speech_prob": 1.1920928955078125e-07, "words": [{"start": 2523.26, "end": 2524.26, "word": "Okay", "probability": 0.75732421875}, {"start": 2524.26, "end": 2525.26, "word": " باقي", "probability": 0.701416015625}, {"start": 2525.26, "end": 2525.72, "word": " نظرية", "probability": 0.9814453125}, {"start": 2525.72, "end": 2526.18, "word": " واحدة", "probability": 0.9708658854166666}, {"start": 2526.18, "end": 2526.5, "word": " في", "probability": 0.93115234375}, {"start": 2526.5, "end": 2526.8, "word": " ال", "probability": 0.59619140625}, {"start": 2526.8, "end": 2527.34, "word": " section", "probability": 0.79150390625}, {"start": 2527.34, "end": 2527.72, "word": " 4", "probability": 0.73876953125}, {"start": 2527.72, "end": 2528.22, "word": "-2", "probability": 0.68017578125}, {"start": 2528.22, "end": 2530.86, "word": " فاتحقلوا", "probability": 0.809814453125}, {"start": 2530.86, "end": 2531.48, "word": " تقرؤوها", "probability": 0.8807779947916666}, {"start": 2531.48, "end": 2532.06, "word": " تفهموا", "probability": 0.85101318359375}, {"start": 2532.06, "end": 2532.84, "word": " البرهانة", "probability": 0.8443359375}, {"start": 2532.84, "end": 2533.66, "word": " و", "probability": 0.66357421875}, {"start": 2533.66, "end": 2534.14, "word": " تبدووا", "probability": 0.8387451171875}, {"start": 2534.14, "end": 2534.24, "word": " في", "probability": 0.85888671875}, {"start": 2534.24, "end": 2534.44, "word": " حال", "probability": 0.87548828125}, {"start": 2534.44, "end": 2535.1, "word": " التمرين", "probability": 0.7504069010416666}, {"start": 2535.1, "end": 2537.06, "word": " و", "probability": 0.95654296875}, {"start": 2537.06, "end": 2537.24, "word": " ان", "probability": 0.55908203125}, {"start": 2537.24, "end": 2537.54, "word": " شاء", "probability": 0.990478515625}, {"start": 2537.54, "end": 2537.7, "word": " الله", "probability": 0.939453125}, {"start": 2537.7, "end": 2538.22, "word": " المرة", "probability": 0.83984375}, {"start": 2538.22, "end": 2538.5, "word": " جاية", "probability": 0.5142415364583334}, {"start": 2538.5, "end": 2539.12, "word": " بنكمل", "probability": 0.88427734375}, {"start": 2539.12, "end": 2539.48, "word": " نشرح", "probability": 0.8855794270833334}, {"start": 2539.48, "end": 2540.0, "word": " النظرية", "probability": 0.947509765625}, {"start": 2540.0, "end": 2540.4, "word": " هذه", "probability": 0.748046875}, {"start": 2540.4, "end": 2541.36, "word": " و", "probability": 0.9248046875}, {"start": 2541.36, "end": 2542.1, "word": " بنبدأ", "probability": 0.9249674479166666}, {"start": 2542.1, "end": 2542.66, "word": " section", "probability": 0.90283203125}, {"start": 2542.66, "end": 2544.02, "word": " جديد", "probability": 0.9251302083333334}, {"start": 2544.02, "end": 2544.12, "word": " اللي", "probability": 0.859130859375}, {"start": 2544.12, "end": 2544.3, "word": " هو", "probability": 0.98779296875}, {"start": 2544.3, "end": 2544.76, "word": " section", "probability": 0.83837890625}, {"start": 2544.76, "end": 2545.26, "word": " 4", "probability": 0.9892578125}, {"start": 2545.26, "end": 2545.82, "word": "-3", "probability": 0.971923828125}], "temperature": 1.0}, {"id": 96, "seek": 255425, "start": 2548.45, "end": 2554.25, "text": "شكرا لإصداركم وشوفكم ان شاء الله في المحاضرة الجاية", "tokens": [8592, 4117, 23557, 5296, 28814, 9381, 3215, 9640, 24793, 4032, 8592, 38688, 24793, 16472, 13412, 16606, 21984, 8978, 9673, 5016, 46958, 25720, 25724, 995, 10632], "avg_logprob": -0.2504507223000893, "compression_ratio": 1.1898734177215189, "no_speech_prob": 0.0, "words": [{"start": 2548.45, "end": 2549.45, "word": "شكرا", "probability": 0.9534505208333334}, {"start": 2549.45, "end": 2550.45, "word": " لإصداركم", "probability": 0.6548665364583334}, {"start": 2550.45, "end": 2551.73, "word": " وشوفكم", "probability": 0.8212890625}, {"start": 2551.73, "end": 2551.87, "word": " ان", "probability": 0.72314453125}, {"start": 2551.87, "end": 2552.07, "word": " شاء", "probability": 0.9912109375}, {"start": 2552.07, "end": 2552.29, "word": " الله", "probability": 0.96337890625}, {"start": 2552.29, "end": 2553.29, "word": " في", "probability": 0.51318359375}, {"start": 2553.29, "end": 2553.79, "word": " المحاضرة", "probability": 0.9827880859375}, {"start": 2553.79, "end": 2554.25, "word": " الجاية", "probability": 0.6971028645833334}], "temperature": 1.0}], "language": "ar", "language_probability": 1.0, "duration": 2555.449, "duration_after_vad": 2261.8559687499896}
\ No newline at end of file
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/wn5D8jbScjs.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/wn5D8jbScjs.srt
new file mode 100644
index 0000000000000000000000000000000000000000..1e91110d7f13adeada67939ca2e9a4f67ef0b430
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/wn5D8jbScjs.srt
@@ -0,0 +1,1755 @@
+1
+00:00:20,670 --> 00:00:25,090
+تحدثنا عن ال absolute value و عرفنا ال absolute
+
+2
+00:00:25,090 --> 00:00:29,810
+value لأي real number إما a إذا كان a غير سالب أو 
+
+3
+00:00:29,810 --> 00:00:33,310
+سالب العدد نفسه إذا كان العدد سالب نفس التعريف
+
+4
+00:00:33,310 --> 00:00:38,090
+اللي أخدناه في ال calculus وبكل من التعريف
+
+5
+00:00:38,090 --> 00:00:43,740
+باستخدام التعريف و ال logic أو مبادئ الرياضيات ممكن
+
+6
+00:00:43,740 --> 00:00:47,220
+بسهولة نبرهن أن ال absolute value ل negative a
+
+7
+00:00:47,220 --> 00:00:51,900
+بساوي absolute a ال absolute value لحاصل الضرب
+
+8
+00:00:51,900 --> 00:00:58,720
+بساوي حاصل ضرب ال absolute values لو كان C عدد غير 
+
+9
+00:00:58,720 --> 00:01:04,240
+سالب ف ال inequality هذه بتكافئ ال inequality هذه
+
+10
+00:01:06,320 --> 00:01:11,960
+لأي عدد حقيقي a دائما أي عدد حقيقي بيكون أصغر من 
+
+11
+00:01:11,960 --> 00:01:15,320
+أو يساوي ال absolute value له وأكبر من أو يساوي
+
+12
+00:01:15,320 --> 00:01:19,860
+سالب ال absolute value هذا دائما صحيح في عند ال
+
+13
+00:01:19,860 --> 00:01:24,000
+triangle inequality هذه متباينة كتير مهمة بتقول أن 
+
+14
+00:01:24,000 --> 00:01:28,560
+ال absolute value لمجموعة أو الفرق بين عددين
+
+15
+00:01:28,560 --> 00:01:33,000
+الحقيقيين less than or equal لمجموعة ال absolute
+
+16
+00:01:33,000 --> 00:01:38,990
+values للعددين ومن المتباينة هذه ممكن نستنتج
+
+17
+00:01:38,990 --> 00:01:44,970
+متباينة أخرى لا تقل عنها أهمية وهي المتباينة 
+
+18
+00:01:44,970 --> 00:01:45,730
+التالية
+
+19
+00:01:52,010 --> 00:01:56,450
+اللي هي المتباينة هذه لو أخدت ال absolute value ل
+
+20
+00:01:56,450 --> 00:02:01,010
+A و طرحت منها ال absolute value ل B و أخدت ال
+
+21
+00:02:01,010 --> 00:02:06,830
+absolute value للفرق فهذا دائما أصغر من أو يساوي 
+
+22
+00:02:06,830 --> 00:02:14,490
+absolute value للفرق و البرهان تبع ال inequality 
+
+23
+00:02:14,490 --> 00:02:19,730
+هذه نتيجة مباشرة على ال triangle inequality
+
+24
+00:02:21,020 --> 00:02:26,900
+فالبراهين سهلة وبسيطة ويعني موجودة عندكم أي حد
+
+25
+00:02:26,900 --> 00:02:31,600
+فيكم بيقدر يقرأها ويفهمها لو مافيش .. في أي مشكلة
+
+26
+00:02:31,600 --> 00:02:36,520
+ممكن يعني اترجعوني خلال الساعات المكتبية أو خلال
+
+27
+00:02:36,520 --> 00:02:44,220
+ساعة المناقشة في
+
+28
+00:02:44,220 --> 00:02:55,550
+اللي هو ملاحظة متباينة المثلث ممكن تعميمها بدل
+
+29
+00:02:55,550 --> 00:03:02,850
+.. احنا قلنا أن ال absolute value ل a أو a1 زائد 
+
+30
+00:03:02,850 --> 00:03:12,440
+a2 اصغر من أو يساوي absolute a1 زائد absolute a2 أو 
+
+31
+00:03:12,440 --> 00:03:16,100
+absolute A زي بي less than or equal to absolute A 
+
+32
+00:03:16,100 --> 00:03:22,180
+زي absolute B المتباينة هذه ممكن تعميمها مش بس ناخد 
+
+33
+00:03:22,180 --> 00:03:24,780
+ال absolute value لعددين ممكن ناخد ال absolute
+
+34
+00:03:24,780 --> 00:03:30,320
+value ل N من الأعداد الحقيقية و نجمعهم فهذا أصغر 
+
+35
+00:03:30,320 --> 00:03:34,490
+من أو يساوي مجموع ال absolute values وطبعا هذا ممكن
+
+36
+00:03:34,490 --> 00:03:39,710
+برهانها .. المتباينة هذا ممكن برهانها بسهولة by
+
+37
+00:03:39,710 --> 00:03:46,930
+induction وطبعا ومتباينة في المثلث okay أعتقد
+
+38
+00:03:46,930 --> 00:03:56,130
+يعني تقريبا هذا اللي شرحنا اليوم 
+
+39
+00:03:56,130 --> 00:04:02,320
+بدنا ناخد تعريف ال epsilon neighborhood فالـ
+
+40
+00:04:02,320 --> 00:04:09,680
+epsilon neighborhood أو جوار epsilon لأي عدد حقيقي هي
+
+41
+00:04:09,680 --> 00:04:19,100
+خط العداد الحقيقية وهي العدد a وهي المسافة هذه أو
+
+42
+00:04:19,100 --> 00:04:25,800
+النقطة هذه a سالب epsilon والنقطة هذه a موجب
+
+43
+00:04:25,800 --> 00:04:30,580
+epsilon والepsilon هذا عدد موجب
+
+44
+00:04:34,530 --> 00:04:39,430
+إذا الفترة المفتوحة هذه الفترة المفتوحة هذه اللي 
+
+45
+00:04:39,430 --> 00:04:44,710
+هي a سالب إبسلون و a موجب إبسلون اللي هي الفترة
+
+46
+00:04:44,710 --> 00:04:49,770
+هذه الفترة
+
+47
+00:04:49,770 --> 00:04:57,230
+المفتوحة هذه بنسميها لحظة هذه الفترة مركزها a ونصف 
+
+48
+00:04:57,230 --> 00:05:02,490
+قطرها إبسلون عدد موجب فالفترة المفتوحة دي اللي 
+
+49
+00:05:02,490 --> 00:05:06,510
+مركزها a ونصف قطرها epsilon بنسميها epsilon
+
+50
+00:05:06,510 --> 00:05:12,510
+neighbourhood of a جوار epsilon أو جوار بأمق
+
+51
+00:05:12,510 --> 00:05:18,270
+epsilon للنقطة a وهكذا
+
+52
+00:05:18,270 --> 00:05:22,270
+تعريفه هاي v epsilon ومصدره بالرمز v epsilon ل a
+
+53
+00:05:22,270 --> 00:05:27,920
+جوار epsilon ل a فهو كل الأعداد الحقيقية اللي 
+
+54
+00:05:27,920 --> 00:05:33,840
+المسافة بينها وبين ال A أصغر من Epsilon أو هي كل
+
+55
+00:05:33,840 --> 00:05:38,040
+الأعداد الحقيقية اللي بتحقق المتباينة هذه اللي هي 
+
+56
+00:05:38,040 --> 00:05:41,780
+بالظبط كل الأعداد الحقيقية X اللي هي كل الأعداد
+
+57
+00:05:41,780 --> 00:05:47,460
+الحقيقية X اللي هي المسافة بينها وبين ال A أصغر من
+
+58
+00:05:47,460 --> 00:05:53,700
+المسافة هذه اللي هي Epsilon واضح تمام فهذا بنسمي
+
+59
+00:05:53,700 --> 00:05:56,800
+ابسلون neighborhood of A في حاجة اسمها
+
+60
+00:05:56,800 --> 00:06:05,340
+neighborhood of A بدون ابسلون فأي
+
+61
+00:06:05,340 --> 00:06:10,280
+.. طبعا اختصار neighborhood اللي بنختصرها NBD
+
+62
+00:06:10,280 --> 00:06:17,000
+neighborhood في تعريف تاني neighborhood of A is
+
+63
+00:06:17,000 --> 00:06:23,380
+any set V of A بحيث بقدر ألاقي داخلها epsilon
+
+64
+00:06:23,380 --> 00:06:28,340
+neighborhood ل A بقدر ألاقي أنه عبارة عن set V of
+
+65
+00:06:28,340 --> 00:06:32,680
+A بقدر ألاقي داخلها epsilon neighborhood of A كمان 
+
+66
+00:06:32,680 --> 00:06:36,720
+مرة يعني ممكن يكون عندي مثلا حاجة زي هذه مثلا set
+
+67
+00:06:36,720 --> 00:06:41,180
+زي هذه وهي
+
+68
+00:06:41,180 --> 00:06:48,860
+النقطة A فترة مثلا نصف مفتوحة زي هذه في هذه الفترة 
+
+69
+00:06:50,180 --> 00:06:54,820
+ممكن تكون مثلا a سالب واحد و a موجب باتنين من
+
+70
+00:06:54,820 --> 00:07:01,060
+النقطة هذه ممكن ولا مش ممكن؟ وهذه فترة نص مفتوحة
+
+71
+00:07:01,060 --> 00:07:06,640
+هذه الفترة حسب التعريف b هي عبارة عن epsilon 
+
+72
+00:07:06,640 --> 00:07:10,560
+neighborhood ل a لإيه؟ لأشان تكون الفترة هذه
+
+73
+00:07:10,560 --> 00:07:16,460
+epsilon عشان الفترة هذه أو ال set هذه اللي بنسميها
+
+74
+00:07:16,460 --> 00:07:22,600
+v of a عشان تكون neighborhood جوار ل A لازم ألاقي
+
+75
+00:07:22,600 --> 00:07:29,780
+داخلها epsilon neighborhood لل A وهذا موجود هاي
+
+76
+00:07:29,780 --> 00:07:37,080
+باخد المسافة هذه مثلا هاي A سالب نص وهاي A موجب نص 
+
+77
+00:07:37,080 --> 00:07:48,080
+مثلا فالفترة هذه الفترة المفتوحة هذه هي مركزها a و
+
+78
+00:07:48,080 --> 00:07:53,200
+نص كترها epsilon بساوي نص عدد موجب إذن هذه نجحت في
+
+79
+00:07:53,200 --> 00:08:01,320
+إيجاد v نص ل a و هذه الفترة المفتوحة واقع كليا
+
+80
+00:08:01,320 --> 00:08:07,080
+عبارة عن مجموعة جزئية من الفترة نص المفتوحة هذه 
+
+81
+00:08:07,080 --> 00:08:13,880
+اللي هي a سالب واحد و a موجب اتنين okay تمام إذن
+
+82
+00:08:13,880 --> 00:08:20,840
+هذه الفترة عبارة عن neighborhood لـA إذا ال
+
+83
+00:08:20,840 --> 00:08:25,980
+neighborhood هو أي مجموعة تحتوي
+
+84
+00:08:25,980 --> 00:08:30,880
+A بحيث أن أنا أقدر ألاقي داخلها epsilon 
+
+85
+00:08:30,880 --> 00:08:36,200
+neighborhood للـA طيب
+
+86
+00:08:36,200 --> 00:08:41,760
+وبالتالي من التعريفين هذول بنلاحظ أن كل epsilon
+
+87
+00:08:41,760 --> 00:08:46,120
+neighborhood كل y neighborhood ل a هو neighborhood
+
+88
+00:08:46,120 --> 00:09:02,320
+لكن العكس مش صحيح في نظرية هنا بتقول أن لو
+
+89
+00:09:02,320 --> 00:09:10,500
+كان عندي عدد حقيقي a فالعدد x العدد الحقيقي x بكون 
+
+90
+00:09:10,500 --> 00:09:13,140
+موجود في كل neighborhoods ل a
+
+91
+00:09:16,470 --> 00:09:21,330
+لو كان X موجود في كل ال neighborhoods ل A فلازم ال
+
+92
+00:09:21,330 --> 00:09:26,510
+X هذه تساوي A لازم ال X تساوي A كمان مرة النظرية
+
+93
+00:09:26,510 --> 00:09:35,470
+هذه بتقول نعمل رسمة هاي
+
+94
+00:09:35,470 --> 00:09:42,130
+A لو كل Y neighborhood ل X
+
+95
+00:09:48,500 --> 00:09:53,240
+لو .. لو .. لو كل .. لو كل epsilon neighborhood أو
+
+96
+00:09:53,240 --> 00:09:58,800
+كل neighborhood هد عبارة عن neighborhood لإيه؟ لو 
+
+97
+00:09:58,800 --> 00:10:03,140
+كل neighborhood لإيه يحتوي ال X؟ لو أي
+
+98
+00:10:03,140 --> 00:10:08,080
+neighborhood لإيه يحتوي ال X؟
+
+99
+00:10:08,080 --> 00:10:12,900
+أو كل الجوارات لل إيه تحتوي ال X؟ فلازم ال X هدي 
+
+100
+00:10:12,900 --> 00:10:17,650
+تكون هي نفس الإيه وهذا البرهان بسيط بيعتمد على 
+
+101
+00:10:17,650 --> 00:10:23,290
+خاصية أخدناها قبل هيك و قلتلكم عنها مهمة و
+
+102
+00:10:23,290 --> 00:10:28,450
+هنستخدمها لحظة وهي هنشوفها الجيت موجودة في نظرية
+
+103
+00:10:28,450 --> 00:10:35,850
+واحد تمانية البرهان بما 
+
+104
+00:10:35,850 --> 00:10:39,970
+أن كل epsilon neighborhood ل a هو عبارة عن 
+
+105
+00:10:39,970 --> 00:10:46,180
+neighborhood ل a فمن الفرض كل neighborhood ل A
+
+106
+00:10:46,180 --> 00:10:47,080
+يحتوي X
+
+107
+00:11:03,320 --> 00:11:07,140
+هذا معناه أنه من تعريف الـ epsilon neighborhood هذا
+
+108
+00:11:07,140 --> 00:11:11,920
+معناه أن x المسافة بين x و a المسافة بين ال
+
+109
+00:11:11,920 --> 00:11:16,680
+absolute value بين x و a اللي هي المسافة بين x و a 
+
+110
+00:11:16,680 --> 00:11:20,760
+أصغر من epsilon وهذا صحيح لكل epsilon أكبر من
+
+111
+00:11:20,760 --> 00:11:27,980
+الصفر طب في نظرية واحد تمانية أخدنا المتباينة هذه
+
+112
+00:11:27,980 --> 00:11:34,740
+اللي بتقول لو كان العدد بي أكبر من أو يساوي صفر 
+
+113
+00:11:34,740 --> 00:11:41,480
+أصغر من إبسلون لكل إبسلون أكبر من الصفر فهذا بيدي
+
+114
+00:11:41,480 --> 00:11:47,040
+أن بي بيساوي صفر، بظبط؟ هذه نظرية واحد تمانية أو
+
+115
+00:11:47,040 --> 00:11:53,440
+جزء منها فأنا هي عندي .. هي عندي ال بي هذا عبارة
+
+116
+00:11:53,440 --> 00:11:58,950
+عن عدد غير سالب القيمة المطلقة لأي عدد هو عدد غير
+
+117
+00:11:58,950 --> 00:12:07,350
+سالب فهذا هو ال B هذا هو العدد B هذا هو هذا
+
+118
+00:12:07,350 --> 00:12:14,070
+ال B هيعندي بي أكبر من أو يساوي صفر هذا صحيح و أصغر
+
+119
+00:12:14,070 --> 00:12:17,910
+من إبسلون لكل إبسلون أكبر من الصفر إذا حسب النظرية 
+
+120
+00:12:17,910 --> 00:12:22,550
+هذه الـ b اللي هو absolute x minus a لازم بيساوي
+
+121
+00:12:22,550 --> 00:12:27,830
+صفر وبالتالي إذا x سالب a بيساوي صفر ومنها بطلع x 
+
+122
+00:12:27,830 --> 00:12:35,270
+بيساوى a هذا هو برهان النظرية تمام؟ في أي سؤال؟
+
+123
+00:12:39,260 --> 00:12:44,480
+برهان سهل واضح شوف بعض الأمثلة على ال
+
+124
+00:12:44,480 --> 00:12:49,880
+neighborhoods و ال epsilon neighborhoods يتوضح
+
+125
+00:12:49,880 --> 00:12:57,540
+برسمة كل مثال نعمله رسمة لو أخدت المجموعة U لو 
+
+126
+00:12:57,540 --> 00:13:03,540
+أخدت U عبارة عن المجموعة أو الفترة المفتوحة من 0
+
+127
+00:13:03,540 --> 00:13:04,340
+إلى 1
+
+128
+00:13:10,710 --> 00:13:19,490
+هذه عبارة عن المجموعة U فالمجموعة 
+
+129
+00:13:19,490 --> 00:13:23,610
+U هذه أو الفترة المفتوحة تعتبر neighborhood لأي
+
+130
+00:13:23,610 --> 00:13:32,450
+نقطة داخلها يعني لو أخدت أي نقطة A داخل
+
+131
+00:13:32,450 --> 00:13:39,810
+المجموعة A فالمجموعة U هي عبارة عن neighborhood لأي
+
+132
+00:13:39,810 --> 00:13:49,690
+نقطة داخلها فلبرهان ذلك .. لبرهان ذلك بدنا أن ..
+
+133
+00:13:49,690 --> 00:13:54,710
+حسب التعريف عشان U تكون neighborhood للنقطة A اللي
+
+134
+00:13:54,710 --> 00:13:59,090
+داخلها لازم نلاقي epsilon neighborhood حوالين الـA
+
+135
+00:13:59,090 --> 00:14:07,420
+ويكون داخل المجموعة U مظبوط، هذا حسب التعريف طيب ال
+
+136
+00:14:07,420 --> 00:14:16,080
+.. بنيجي ناخد epsilon بيساوي ال minimum الأصغر بين
+
+137
+00:14:16,080 --> 00:14:22,360
+المسافتين المسافة هذه .. المسافة هذه ايه؟ المسافة
+
+138
+00:14:22,360 --> 00:14:27,660
+هذه ايه؟ طيب و المسافة التانية هذه؟ واحد نقص ايه؟
+
+139
+00:14:27,660 --> 00:14:34,520
+واحد سالب ايه؟ فالـ minimum بين العددين a و 1 سالب
+
+140
+00:14:34,520 --> 00:14:41,000
+a بناخده هو ال epsilon تمام؟ في الرسمة .. في
+
+141
+00:14:41,000 --> 00:14:47,360
+الرسمة هذه ال .. الأصغر بين العددين a و 1 سالب a
+
+142
+00:14:47,360 --> 00:14:52,320
+هو ال a فبقى أجي بكون حوالين ال a epsilon
+
+143
+00:14:52,320 --> 00:14:58,940
+neighborhood هي فهذا عبارة عن epsilon neighborhood
+
+144
+00:15:01,920 --> 00:15:08,720
+فالفترة المفتوحة هذه السودا بالتأكيد إذا هنا ال a
+
+145
+00:15:08,720 --> 00:15:14,320
+سالب ابسلون و a زائد ابسلون هذه بالتأكيد واقع
+
+146
+00:15:14,320 --> 00:15:23,590
+داخل ال U حسب الرسم وبالتالي هيني نجحت في إيجاد V
+
+147
+00:15:23,590 --> 00:15:27,670
+Epsilon أو Epsilon neighborhood لـ A و هذا ال
+
+148
+00:15:27,670 --> 00:15:34,310
+Epsilon neighborhood لـ A واقع داخل ال U إذن حسب
+
+149
+00:15:34,310 --> 00:15:39,450
+التعريف الـ U أبارع ال neighborhood للنقطة A و لما
+
+150
+00:15:39,450 --> 00:15:44,850
+كانت A نقطة عشوائية إذن U أبارع ال neighborhood
+
+151
+00:15:44,850 --> 00:15:49,990
+لأي نقطة داخلها تمام؟
+
+152
+00:15:53,400 --> 00:15:58,160
+Okay مثال ثاني لو
+
+153
+00:15:58,160 --> 00:16:01,360
+أخدنا الفترة المغلقة هنا أخدنا U بالساوية الفترة
+
+154
+00:16:01,360 --> 00:16:07,900
+المفتوحة لو أخدنا هذه
+
+155
+00:16:07,900 --> 00:16:14,600
+خط الأعداد وهذه الفترة المغلقة من 0 إلى 1 هذه
+
+156
+00:16:14,600 --> 00:16:16,440
+الفترة نسميها I
+
+157
+00:16:21,110 --> 00:16:28,110
+I بتساوي الفترة المغلقة من 0 إلى 1 طبعا
+
+158
+00:16:28,110 --> 00:16:33,230
+زي ما في المثال السابق الفترة I هذه ممكن إثبات
+
+159
+00:16:33,230 --> 00:16:38,710
+أنها neighborhood لأي نقطة داخلها معدا 0 والواحد
+
+160
+00:16:38,710 --> 00:16:46,210
+نفس البرهان لكن الفترة I هذه أو المجموعة I ليست
+
+161
+00:16:46,210 --> 00:16:52,410
+neighborhood ليست I is not neighborhood للعنصر صفر
+
+162
+00:16:52,410 --> 00:16:58,930
+وكذلك ليست neighborhood للعنصر واحد ليه؟ لأنه بقدر
+
+163
+00:16:58,930 --> 00:17:08,670
+ألاقي لأن كل epsilon كل epsilon neighborhood للصفر
+
+164
+00:17:08,670 --> 00:17:13,170
+هذه لو أخدت أي epsilon مهما كانت صغيرة
+
+165
+00:17:15,840 --> 00:17:20,480
+يعني هي epsilon عدد موجب لأن هذا هيصير صفر زائد
+
+166
+00:17:20,480 --> 00:17:25,740
+epsilon وهي كمان المسافة هذه epsilon فهيصير هذه
+
+167
+00:17:25,740 --> 00:17:33,700
+طبعا سالب epsilon فلو أخدت أي فترة مفتوحة بعمق
+
+168
+00:17:33,700 --> 00:17:39,100
+epsilon هيثم epsilon عدد موجب إذا لأي إبسلون عدد
+
+169
+00:17:39,100 --> 00:17:42,900
+موجب لو كونت epsilon neighborhood للصفر إذا هذه
+
+170
+00:17:42,900 --> 00:17:51,100
+الفترة الحمرة عبارة عن V epsilon للصفر فـ V epsilon
+
+171
+00:17:51,100 --> 00:17:59,640
+للصفر هذا ليس ليس مجموعة جزئية من I هذا ليس مجموعة
+
+172
+00:17:59,640 --> 00:18:09,360
+جزئية من I لكل إبسلون لكل إبسلون أكبر من الصفر لكل
+
+173
+00:18:09,360 --> 00:18:13,560
+إبسلون أكبر من الصفر لكل إبسلون أكبر من الصفر لكل
+
+174
+00:18:13,560 --> 00:18:14,260
+إبسلون أكبر من الصفر لكل إبسلون أكبر من الصفر لكل
+
+175
+00:18:14,260 --> 00:18:14,560
+إبسلون أكبر من الصفر لكل إبسلون أكبر من الصفر لكل
+
+176
+00:18:14,560 --> 00:18:16,940
+إبسلون أكبر من الصفر لكل إبسلون أكبر من الصفر لكل
+
+177
+00:18:16,940 --> 00:18:17,100
+إبسلون أكبر من الصفر لكل إبسلون أكبر من الصفر لكل
+
+178
+00:18:17,100 --> 00:18:18,560
+إبسلون أكبر من الصفر لكل إبسلون أكبر من الصفر لكل
+
+179
+00:18:18,560 --> 00:18:20,920
+إبسلون أكبر من الصفر لكل إبسلون أكبر من الصفر لكل
+
+180
+00:18:20,920 --> 00:18:26,910
+إبسلون أكبر من الصفر بالمثل الفترة I ليست نبرهود
+
+181
+00:18:26,910 --> 00:18:33,150
+للواحد لأن أي جوار بعمق إبسلون للواحد لا يقع كله
+
+182
+00:18:33,150 --> 00:18:37,410
+داخل الفترة I الجزء هذا خارج الفترة I، الجزء هذا
+
+183
+00:18:37,410 --> 00:18:40,010
+خارج الفترة I، الجزء هذا خارج الفترة I، الجزء هذا
+
+184
+00:18:40,010 --> 00:18:40,990
+خارج الفترة I، الجزء هذا خارج الفترة I، الجزء هذا
+
+185
+00:18:40,990 --> 00:18:41,330
+خارج الفترة I، الجزء هذا خارج الفترة I، الجزء هذا
+
+186
+00:18:41,330 --> 00:18:42,670
+خارج الفترة I، الجزء هذا خارج الفترة I، الجزء هذا
+
+187
+00:18:42,670 --> 00:18:44,910
+خارج الفترة I، الجزء هذا خارج الفترة I، الجزء هذا
+
+188
+00:18:44,910 --> 00:18:46,930
+خارج الفترة I، الجزء هذا خارج الفترة I، الجزء هذا
+
+189
+00:18:46,930 --> 00:18:55,140
+خارج الفترة I، الجزء هذا خارج الفتمثال الثالث مثال
+
+190
+00:18:55,140 --> 00:19:01,220
+ثالث لو
+
+191
+00:19:01,220 --> 00:19:07,000
+أخدت أي ثلاثة أعداد حقيقية x, y, z لو أخدت أي ثلاثة
+
+192
+00:19:07,000 --> 00:19:18,240
+أعداد حقيقية x و y و z بحيث أن هذه ال x أصغر من z
+
+193
+00:19:18,240 --> 00:19:28,110
+أصغر من أو يساوي zو X أصغر من أو يساوي Y و Y أصغر
+
+194
+00:19:28,110 --> 00:19:36,110
+من Z فهذا يبدي لازم يبدي أن المسافة المسافة بين X
+
+195
+00:19:36,110 --> 00:19:43,070
+و Y زائد المسافة بين Y و Z بتساوي المسافة بين X و Z
+
+196
+00:19:43,070 --> 00:19:51,590
+هذا واضح هذا واضح من الرسم لكن طبعا الرسم أمره ما
+
+197
+00:19:51,590 --> 00:19:57,930
+كان برهان أما لكن ممكن يوضح في فكرة البرهان إذا
+
+198
+00:19:57,930 --> 00:20:02,470
+المجموعة القيم المطلقة هذه بساوي القيم المطلقة هذه
+
+199
+00:20:02,470 --> 00:20:07,090
+بس في حالة لما ال X أصغر من Y أو ال Y تكون بين X و
+
+200
+00:20:07,090 --> 00:20:13,290
+Z و X أصغر من أو يساوي Z البرهان بكل بساطة كالتالي
+
+201
+00:20:13,290 --> 00:20:19,630
+أنا عندي الفرض تبعي X أصغر من أو يساوي Z و Y بين X و
+
+202
+00:20:19,630 --> 00:20:27,690
+Z هذا يجعل القيمة المطلقة لـ x-y .. الآن x أصغر من
+
+203
+00:20:27,690 --> 00:20:31,570
+y، فهذا الفرق سالب، فالقيمة المطلقة لعدد سالب سالب
+
+204
+00:20:31,570 --> 00:20:38,920
+نفسه بالمثل هذا y سالب z و y أصغر من z إذا العدد
+
+205
+00:20:38,920 --> 00:20:41,360
+هذا اللي جوه ال absolute value سالب هذا حسب تعريف
+
+206
+00:20:41,360 --> 00:20:47,840
+القيمة المطلقة هذا بساوي سالبه و كذلك absolute x
+
+207
+00:20:47,840 --> 00:20:52,600
+ناقص z بساوي سالب العدد نفسه
+
+208
+00:20:57,610 --> 00:21:03,050
+بنجمع هذه absolute x ناقص y و absolute y ناقص z
+
+209
+00:21:03,050 --> 00:21:11,070
+فهي عندي بيطلع سالب x موجب y زائد سالب y موجب z و
+
+210
+00:21:11,070 --> 00:21:16,350
+هدول طبعا بيطلع بساوي المجموع هذا و هذا بيساوي
+
+211
+00:21:16,350 --> 00:21:22,350
+absolute x ناقص z لأن x أصغر من z okay تمام إذا
+
+212
+00:21:22,350 --> 00:21:30,220
+هذا هو البرهان هذا هو البرامج نعم نعم X أقل من أو
+
+213
+00:21:30,220 --> 00:21:38,740
+يساوي Z نعم شوف هذا تبين ساوي هو يعني سواء X أصغر
+
+214
+00:21:38,740 --> 00:21:43,120
+من أو يساوي Z أو بتساوي في هذه المتباينة صحيحة
+
+215
+00:21:43,120 --> 00:21:52,370
+يعني لو X بتساوي Z يعني في الحالة الخاصة اللي X
+
+216
+00:21:52,370 --> 00:21:57,490
+فيها بتنطبق على Z فـ Y هتنطبق على X وعلى Z يعني
+
+217
+00:21:57,490 --> 00:22:00,970
+الثلاثة هدول هيطلعوا متساويين وبالتالي هذه ال
+
+218
+00:22:00,970 --> 00:22:05,350
+absolute value صفر وهذه صفر صفر زائد صفر بساوي صفر
+
+219
+00:22:05,350 --> 00:22:12,090
+وبالتالي العلاقة أو المعادلة هذه صحيحة okay هذه
+
+220
+00:22:12,090 --> 00:22:16,350
+الحالة الخاصة اللي فيها بيكون X بساوي Z في الحالة
+
+221
+00:22:16,350 --> 00:22:21,690
+هذه لازم يطلع طبعا أي عدد بين X و Z سيكون بيساويهم
+
+222
+00:22:21,690 --> 00:22:28,670
+وبالتالي هذا صفر وهذا صفر وهذا صفر في أي سؤال
+
+223
+00:22:28,670 --> 00:22:37,310
+ثاني؟ طيب في كمان مثال رابع
+
+224
+00:22:40,090 --> 00:22:44,170
+برضه كل الأمثلة هذه على التطبيقات على ال absolute
+
+225
+00:22:44,170 --> 00:22:47,490
+value لو في اندي فانكشن يعني افرض انك أنت في
+
+226
+00:22:47,490 --> 00:22:51,810
+calculus ولا في course calculus أو algebra وطلب
+
+227
+00:22:51,810 --> 00:22:58,150
+منك عرف الدالة f of x علي أنها 2x تربيع زائد 3x زائد 1
+
+228
+00:22:58,150 --> 00:23:03,630
+على 2x ناقص 1 والمجال تبع الدالة هذه الفترة
+
+229
+00:23:03,630 --> 00:23:09,210
+المغلقة من 2 لـ 3 برهن أن القيمة المطلقة للدالة هذا
+
+230
+00:23:09,210 --> 00:23:14,650
+أصغر من أو يساوي 28 على 3 لكل x في المجال تبعها
+
+231
+00:23:14,650 --> 00:23:21,550
+فكيف نبرهن هذا؟ نستخدم خواص القيمة المطلقة بدي أخد
+
+232
+00:23:21,550 --> 00:23:24,850
+.. بدي أثبت absolute f of x أصغر من أو يساوي العدد
+
+233
+00:23:24,850 --> 00:23:32,610
+هذا باخد absolute f of x فـ قيمة
+
+234
+00:23:32,610 --> 00:23:36,590
+f of x بساوي قيمة المطلقة للبسط على قيمة المطلقة
+
+235
+00:23:36,590 --> 00:23:40,230
+للمقام هذه خاصية من خواص ال absolute value لأن
+
+236
+00:23:40,230 --> 00:23:44,290
+القسمة هي ضرب وقلنا absolute value للضرب بساوي
+
+237
+00:23:44,290 --> 00:23:48,590
+حاصل ضرب ال absolute value الآن تعالوا نشتغل على
+
+238
+00:23:48,590 --> 00:23:51,790
+الـ بسط هذه الـ بسط هو عبارة عن القيمة المطلقة
+
+239
+00:23:51,790 --> 00:23:56,310
+دي باستخدام ال generalized triangle inequality ال
+
+240
+00:23:56,310 --> 00:23:59,730
+absolute value لمجموعة ثلاثة أعداد أصغر من أو يساوي
+
+241
+00:23:59,730 --> 00:24:06,300
+مجموعة ال absolute values لهم ابسلوت 2 X
+
+242
+00:24:06,300 --> 00:24:10,680
+تربيع زائد ابسلوت 3 X بيطلع 3 ابسلوت X
+
+243
+00:24:10,680 --> 00:24:12,580
+زائد ابسلوت 1 بيطلع 1
+
+244
+00:24:34,520 --> 00:24:39,900
+فنجمع الأرقام هذه بيطلع 28 إذا ال absolute
+
+245
+00:24:39,900 --> 00:24:45,520
+value للبسط هذا ال absolute value للبسط هذا طلع
+
+246
+00:24:45,520 --> 00:24:51,080
+أصغر من أو يساوي 28 الآن تعالوا نشوف ال absolute
+
+247
+00:24:51,080 --> 00:24:52,220
+value للمقام
+
+248
+00:24:54,600 --> 00:24:59,580
+absolute 2x ناقص 1 نستخدم المرة هذه ال triangle
+
+249
+00:24:59,580 --> 00:25:04,480
+inequality هذه اللي هي absolute absolute a ناقص
+
+250
+00:25:04,480 --> 00:25:19,040
+absolute b أكبر من أو يساوي absolute a ناقص b أقل
+
+251
+00:25:19,040 --> 00:25:22,500
+أقل صحيح
+
+252
+00:25:27,790 --> 00:25:34,990
+فأنا عندي هنا ال .. أنا عندي ال A بساوي 2 X
+
+253
+00:25:34,990 --> 00:25:42,010
+و ال B بساوي 1 لأن 
+
+254
+00:25:42,010 --> 00:25:46,590
+هتكسلي سؤال بدون إذن واتكلمتي بدون بس أسمع عليه فمش
+
+255
+00:25:46,590 --> 00:25:53,380
+هجاوب على سؤال كلأ خرجت القاعدة هذه هي المتباينة
+
+256
+00:25:53,380 --> 00:25:58,000
+هذه أخذناها حكينا عنها اليوم صح أظبط لأ بدي أستخدم
+
+257
+00:25:58,000 --> 00:26:02,780
+المتباينة هذه يعني عندي absolute 2x ناقص 1 فهي
+
+258
+00:26:02,780 --> 00:26:11,490
+absolute 2x a باخد a بساوي 2x و b بساوي 1 فهذا
+
+259
+00:26:11,490 --> 00:26:17,750
+المفروض يصير أكبر
+
+260
+00:26:17,750 --> 00:26:20,170
+من أو يساوي أكبر من أو يساوي أكبر من أو يساوي أكبر من
+
+261
+00:26:20,170 --> 00:26:20,810
+أو يساوي أكبر من أو يساوي أكبر من أو يساوي أكبر من أو
+
+262
+00:26:20,810 --> 00:26:29,470
+يساوي أكبر من أو يساوي أكبر من أو يساوي أكبر من
+
+263
+00:26:29,470 --> 00:26:36,150
+أو يساوي أكبر من أو يساوي أكبر من أو يساوي أكبر من أو
+
+264
+00:26:36,150 --> 00:26:37,690
+يساوي أكبر من أو يساوي أكبر من أو يساوي أكبر من أو
+
+265
+00:26:37,690 --> 00:26:37,910
+يساوي أكبر من أو يساوي أكبر من أو يساوي أكبر من أو
+
+266
+00:26:37,910 --> 00:26:38,110
+يساوي أكبر من أو يساوي أكبر من أو يساوي أكبر من أو
+
+267
+00:26:38,110 --> 00:26:39,310
+يساوي أكبر من أو يساوي أكبر من أو يساوي أكبر من أو
+
+268
+00:26:39,310 --> 00:26:44,120
+يساوي أكبر من أو يساوي أكبر من أو س-1 و أنا عندي x
+
+269
+00:26:44,120 --> 00:26:50,440
+أكبر من أو يساوي لـ 2 إذا absolute x أكبر من أو
+
+270
+00:26:50,440 --> 00:26:57,020
+يساوي 2 إذا هذا أكبر من أو يساوي 2 في 2 ناقص 1
+
+271
+00:26:57,020 --> 00:27:05,130
+اللي هو 3 كمان مرة باستخدام المتباينة هذه absolute
+
+272
+00:27:05,130 --> 00:27:09,470
+a ناقص b أكبر من أو يساوي absolute a اللي هي 2 في
+
+273
+00:27:09,470 --> 00:27:14,050
+absolute x ناقص absolute b اللي هو absolute 1 بيطلع
+
+274
+00:27:14,050 --> 00:27:18,630
+1 الآن ال X هذه أكبر من أو يساوي 2 إذا القيمة
+
+275
+00:27:18,630 --> 00:27:24,710
+المطلقة برضه أكبر من أو يساوي لـ 2 إذا هذه أكبر من
+
+276
+00:27:24,710 --> 00:27:28,130
+أو يساوي 2 في 2 ناقص 1 بيطلع 3
+
+277
+00:27:30,990 --> 00:27:38,890
+الآن إذا هذا عبارة عن المقام هنا فإذا كان العدد
+
+278
+00:27:38,890 --> 00:27:43,290
+احنا أخذنا قبل هيك خاصية ما تنساش الخواص اللي
+
+279
+00:27:43,290 --> 00:27:51,370
+أخذناها في بتقول لو كان عندي أنا عدد A غير سالب أو
+
+280
+00:27:51,370 --> 00:27:55,290
+أكبر من الصفر فمقلوب 
+
+281
+00:28:00,600 --> 00:28:10,640
+أو لو كان a أكبر من b أكبر من 0 فهذا بيقدي أن 1
+
+282
+00:28:10,640 --> 00:28:17,520
+على a أصغر من 1 على b، مظبوط؟ فأنا إذا مقلوب ..
+
+283
+00:28:17,520 --> 00:28:22,480
+إذا مقلوب المقدار هذا أصغر من أو يساوي مقلوب
+
+284
+00:28:22,480 --> 00:28:23,140
+الثلاثة
+
+285
+00:28:25,760 --> 00:28:30,340
+إذاً هذا بيطلع .. إذاً هذا الـ gate نساوي بيطلع
+
+286
+00:28:30,340 --> 00:28:36,820
+أصغر من أو خلينا نكتبه على صورة هاي .. هذا عبارة
+
+287
+00:28:36,820 --> 00:28:48,520
+عن absolute 2x تربيع زائد 3x زائد 1 ضرب ضرب 1 على
+
+288
+00:28:48,520 --> 00:28:51,820
+absolute 2x ناقص 1
+
+289
+00:28:57,850 --> 00:29:02,550
+يعني أنا حولت القسمة لضرب الآن الحد الأول أصغر من
+
+290
+00:29:02,550 --> 00:29:06,330
+أو يساوي ثمانية وعشرين والحد الثاني اللي هو
+
+291
+00:29:06,330 --> 00:29:09,810
+المقلوب absolute of x ناقص واحد أصغر من أو يساوي
+
+292
+00:29:09,810 --> 00:29:16,530
+ثلاثة عفوا ثلث وبالتالي بيطلع إذا absolute f of x
+
+293
+00:29:16,530 --> 00:29:22,950
+أصغر من أو يساوي ثمانية وعشرين على ثلاثة okay تمام
+
+294
+00:29:24,140 --> 00:29:29,260
+في أي سؤال؟ إذا هنا يعني احنا استخدمنا متباينة
+
+295
+00:29:29,260 --> 00:29:34,320
+المثلث العادية ومتباينة المثلث الأخرى اللي هي هذه
+
+296
+00:29:34,320 --> 00:29:47,660
+في إثبات اللي احنا عايزينه في السؤال هذا طيب
+
+297
+00:29:47,660 --> 00:29:53,530
+ننتقل لموضوع ثاني اللي هو الـ completeness property
+
+298
+00:29:53,530 --> 00:30:02,450
+of R خاصية تمام للأعداد الحقيقية بالمناسبة
+
+299
+00:30:02,450 --> 00:30:07,470
+في مرجع أنا يمكن مش مكتوب في ال syllabus عندكم لكن
+
+300
+00:30:07,470 --> 00:30:15,650
+خليني أنا أحكي لكم عنه المرجع هذا مكتوب باللغة
+
+301
+00:30:15,650 --> 00:30:20,630
+العربية ويمكن يكون سهل بالنسبة لكم ومش غلط إنكم
+
+302
+00:30:20,630 --> 00:30:29,490
+تقتنوه أو تقتنوها اللي هو كتاب كتاب
+
+303
+00:30:29,490 --> 00:30:40,330
+التحليل الحقيقي لجامعة
+
+304
+00:30:40,330 --> 00:30:43,570
+الخدّام
+
+305
+00:30:43,570 --> 00:30:46,970
+المفتوحة
+
+306
+00:30:49,530 --> 00:30:57,930
+أنا درست الـ course من الكتاب هذا وكتاب رائع جدا و
+
+307
+00:30:57,930 --> 00:31:01,910
+مكتوب من قبل نخبة من المدرسين وأنا رجعت الكتاب و
+
+308
+00:31:01,910 --> 00:31:08,410
+نقحته وعملنا فيه بعض التعديلات فيعني لو اطلعتوا 
+
+309
+00:31:08,410 --> 00:31:14,050
+عليه هيكون مفيد جدا بالإضافة لمرجعتنا اللي احنا
+
+310
+00:31:14,050 --> 00:31:15,790
+قررنا عليه
+
+311
+00:31:23,230 --> 00:31:29,730
+تمام؟ Yes sir إذا الـ completeness property of R
+
+312
+00:31:29,730 --> 00:31:34,830
+أو خاصية التمام أو الكمال للأعداد الحقيقية نشوف شو
+
+313
+00:31:34,830 --> 00:31:39,410
+معناها عشان نصل للخاصية هذه بدنا شوية تعريفات وشوية
+
+314
+00:31:39,410 --> 00:31:45,250
+تمهيل فلو أخذت أي مجموعة جزئية من R
+
+315
+00:31:54,920 --> 00:31:59,360
+هذه خط الأعداد الحقيقية وهذه R وهو محور X ممكن
+
+316
+00:31:59,360 --> 00:32:05,480
+نعتبره R صح؟ ومحور Y برضه R شو الفرق بينهم فنأخذ
+
+317
+00:32:05,480 --> 00:32:10,100
+الأعداد الحقيقية هيك رأسية وهذه المجموعة هذه هيك
+
+318
+00:32:10,100 --> 00:32:19,680
+مجموعة الأعداد هذه S هي مجموعة جزئية من R تمام؟
+
+319
+00:32:19,680 --> 00:32:27,650
+هذه المجموعة جزئية من R فلو أخذت عدد U زي هذا U عدد
+
+320
+00:32:27,650 --> 00:32:33,090
+حقيقي فالعدد هذا بنسميه upper bound للمجموعة 
+
+321
+00:32:33,090 --> 00:32:39,910
+الجزئية S إذا كان كل X في
+
+322
+00:32:39,910 --> 00:32:43,530
+المجموعة 
+
+323
+00:32:43,530 --> 00:32:48,390
+S أصغر من أو يساوي U إذا مرة ثانية العدد U upper
+
+324
+00:32:48,390 --> 00:32:54,220
+bound حد أعلى للمجموعة S إذا كان كل X في S أصغر من 
+
+325
+00:32:54,220 --> 00:33:01,840
+أو يساوي الـ U والعدد W بنسميه lower bound أو حد
+
+326
+00:33:01,840 --> 00:33:10,360
+أدنى للمجموعة S إذا كان كل X إذا كان أي عدد X في S
+
+327
+00:33:10,360 --> 00:33:16,020
+أكبر من أو يساوي الـ W أو W أصغر من أو يساوي كل
+
+328
+00:33:16,020 --> 00:33:24,290
+عناصر الـ S ففي الحالة هذه بنسمي w lower bound طيب
+
+329
+00:33:24,290 --> 00:33:30,650
+في شوية ملاحظات عن التعريفين هدول except may or
+
+330
+00:33:30,650 --> 00:33:34,130
+may not have an upper or lower bound على سبيل
+
+331
+00:33:34,130 --> 00:33:37,770
+المثال for example على سبيل المثال الأعداد
+
+332
+00:33:37,770 --> 00:33:45,600
+الحقيقية الأعداد الحقيقية هذا هي المجموعة ممكن يكون
+
+333
+00:33:45,600 --> 00:33:49,420
+لها upper bound وممكن ما يكونش لها upper bound
+
+334
+00:33:49,420 --> 00:33:53,380
+فمثلا
+
+335
+00:33:53,380 --> 00:34:02,320
+الفترة يعني هذه الأعداد هي من صفر إلى واحد أو
+
+336
+00:34:02,320 --> 00:34:09,160
+الفترة المغلقة من صفر إلى واحد هذه الواحد الواحد أو
+
+337
+00:34:09,160 --> 00:34:14,520
+أي U أكبر من أو يساوي الواحد أي عدد حقيقي U أكبر من
+
+338
+00:34:14,520 --> 00:34:19,060
+أو يساوي الواحد بيطلع upper bound لها ليه؟ لأن كل 
+
+339
+00:34:19,060 --> 00:34:24,920
+العناصر اللي هنا أصغر من أو يساوي الواحد، صح؟ وكذلك
+
+340
+00:34:24,920 --> 00:34:32,800
+لو أخذت W عدد أصغر من أو يساوي صفر فكل X في الفترة
+
+341
+00:34:32,800 --> 00:34:33,720
+المغلقة هذه
+
+342
+00:34:36,600 --> 00:34:40,120
+كل x في الفترة المغلقة هذه أصغر من أو يساوي صفر
+
+343
+00:34:40,120 --> 00:34:46,360
+صح؟ وبالتالي أصغر من أو يساوي w إذا أي عدد حقيقي
+
+344
+00:34:46,360 --> 00:34:50,940
+بساوي صفر w بساوي صفر أو أصغر من صفر عبارة عن
+
+345
+00:34:50,940 --> 00:34:57,120
+lower bound للست S هذه كذلك أي عدد حقيقي U بساوي
+
+346
+00:34:57,120 --> 00:35:01,540
+واحد أو أكبر من واحد بيكون upper bound للمجموعة 
+
+347
+00:35:01,540 --> 00:35:06,660
+هذه لأن كل X في المجموعة S هذه أكبر من أو يساوي 
+
+348
+00:35:06,660 --> 00:35:13,140
+الواحد، تمام؟ لكن مجموعة الأعداد الحقيقية R أو 
+
+349
+00:35:13,140 --> 00:35:18,300
+المجموعة الأعداد الحقيقية R هذه ما لهاش upper bound
+
+350
+00:35:18,300 --> 00:35:20,060
+وما لهاش lower bound
+
+351
+00:35:23,110 --> 00:35:28,250
+الأعداد الحقيقية ما فيش عدد حقيقي أكبر من أو يساوي كل
+
+352
+00:35:28,250 --> 00:35:32,390
+الأعداد الحقيقية ولا فيه نقدر نحط أصبعنا على العدد
+
+353
+00:35:32,390 --> 00:35:38,210
+وما فيش ولا عدد حقيقي أصغر من أو يساوي كل الأعداد
+
+354
+00:35:38,210 --> 00:35:43,970
+الحقيقيةكتور؟ نعم مش الفترة من صفر لواحد مغلقة
+
+355
+00:35:43,970 --> 00:35:48,450
+تكسّبها؟ أو مفتوحة مش مشكلة طب إذا كانت مغلقة كيف
+
+356
+00:35:48,450 --> 00:35:55,130
+دي يكون الـ lower bound أقل من صفر؟ لو كانت مغلقة ف
+
+357
+00:35:55,130 --> 00:36:01,850
+.. ف w بساوي صفر عبارة عن lower bound أو الـ u
+
+358
+00:36:01,850 --> 00:36:06,470
+بتساوي واحد الـ upper bound و u بساوي واحد upper
+
+359
+00:36:06,470 --> 00:36:10,670
+bound أو حينما كانت مغلقة مش هيكوا بس كمان لو أخذت
+
+360
+00:36:10,670 --> 00:36:16,390
+u بساوي اثنين يعني أكبر من واحد برضه بيطلع upper
+
+361
+00:36:16,390 --> 00:36:22,720
+bound صح؟ حتى حتى لو مش مغلقة يعني اثنين هذا العدد
+
+362
+00:36:22,720 --> 00:36:26,760
+اثنين أكبر من العناصر اللي هنا كلها ولا لأ حتى لو
+
+363
+00:36:26,760 --> 00:36:34,860
+مش مغلقة سواء 
+
+364
+00:36:34,860 --> 00:36:39,580
+الفترة هذه مغلقة أو مش مغلقة أي عدد حقيقي أكبر من
+
+365
+00:36:39,580 --> 00:36:44,940
+أو يساوي الواحد بيكون upper bound يعني الـ upper
+
+366
+00:36:44,940 --> 00:36:47,740
+bound ممكن إنه ما ينتمش للفترة ممكن ينتمي إلى 
+
+367
+00:36:47,740 --> 00:36:54,080
+المجموعة وممكن لا ينتمي إليها يعني
+
+368
+00:36:54,080 --> 00:37:01,980
+لو أخذت الفترة مغلقة الفترة
+
+369
+00:37:01,980 --> 00:37:07,140
+مغلقة U بساوي اثنين upper bound لها في شك في ذلك؟
+
+370
+00:37:07,140 --> 00:37:11,060
+لكن الـ upper bound هذا مش شرط يكون ينتمي إلى
+
+371
+00:37:11,060 --> 00:37:18,850
+المجموعة الواحد برضه upper bound للمجموعة دي و
+
+372
+00:37:18,850 --> 00:37:21,470
+ينتمي ل .. ممكن ينتمي إذا الـ upper bound ممكن 
+
+373
+00:37:21,470 --> 00:37:27,410
+ينتمي ل .. للمجموعة ممكن ما ينتميش لو أخدت U
+
+374
+00:37:27,410 --> 00:37:32,210
+بساوي عشرة برضه هذا upper bound لمجموعة S لو
+
+375
+00:37:32,210 --> 00:37:39,560
+أخدت U بساوي مية برضه هذا upper bound للستو الـ u
+
+376
+00:37:39,560 --> 00:37:43,760
+بساوي واحد upper bound لل 6 الآن هذا الـ u بساوي
+
+377
+00:37:43,760 --> 00:37:48,080
+واحد هو أصغر upper bound هذا الـ u بساوي واحد هو 
+
+378
+00:37:48,080 --> 00:37:53,420
+أصغر upper bound كذلك لو أخذت w بساوي صفر هذا
+
+379
+00:37:53,420 --> 00:37:58,240
+عبارة عن lower bound لل 6 هذه لأن كل عنصر في ال 6 
+
+380
+00:37:58,240 --> 00:38:04,850
+أكبر منها بساوي صفر لو أخذت w بساوي سالب نص، برضه
+
+381
+00:38:04,850 --> 00:38:09,190
+هذا lower bound للست هذه، لو u أخدتها بساوي سالب
+
+382
+00:38:09,190 --> 00:38:16,450
+عشرة، برضه هذا lower bound آخر، وكذا، الآن w بساوي
+
+383
+00:38:16,450 --> 00:38:21,710
+صفر هذا عبارة عن أكبر lower bound للمجموعة هذه أكبر
+
+384
+00:38:21,710 --> 00:38:27,990
+حد أدنى، تمام؟ إذن في حد أدنى وفي أكبر حد أدنى، في 
+
+385
+00:38:27,990 --> 00:38:33,770
+حد أعلى وفي أصغر حد أعلى، تمام؟ هدول مهمين، أصغر 
+
+386
+00:38:33,770 --> 00:38:38,510
+حد أعلى وأكبر حد أدنى مهمين وهنطلق .. هنعطيهم
+
+387
+00:38:38,510 --> 00:38:45,090
+أسماء كمان شوية، لكن قبل خلينا نكمل الملاحظات إذا 
+
+388
+00:38:45,090 --> 00:38:48,650
+شفنا إحنا أن الـ upper bound أو المجموعة ممكن يكون
+
+389
+00:38:48,650 --> 00:38:53,050
+لها upper bound أو ما يكونش المجموعة الـ upper bound
+
+390
+00:38:53,050 --> 00:38:55,750
+أو الـ lower bound ممكن ينتمي للمجموعة وممكن 
+
+391
+00:38:55,750 --> 00:39:01,850
+ما ينتميش لها الملاحظة الثانية لو كان U upper bound
+
+392
+00:39:01,850 --> 00:39:06,930
+لست S فأي عدد أكبر منه بيطلع upper bound أيضا لنفس
+
+393
+00:39:06,930 --> 00:39:11,820
+الست يعني هي عندي إذا إحنا الآن في الملاحظة الثانية
+
+394
+00:39:11,820 --> 00:39:18,140
+لو كان U upper bound لل 6 وأخذت V عدد أكبر من U
+
+395
+00:39:18,140 --> 00:39:24,060
+فالـ V أيضا هذا بيطلع upper bound لل 6 طيب السؤال 
+
+396
+00:39:24,060 --> 00:39:32,760
+اللي بيفرح نفسه كم عدد V ممكن ألاقي أكبر من U عدد 
+
+397
+00:39:32,760 --> 00:39:39,010
+لانهائي إذا لو كان الست S لها upper bound واحد 
+
+398
+00:39:39,010 --> 00:39:42,990
+فممكن ألاقي عدد لانهائي لها من الـ upper bounds
+
+399
+00:39:42,990 --> 00:39:48,810
+طبعا؟ إذا قلنا كمان مرة لو كان الـ U لها upper
+
+400
+00:39:48,810 --> 00:39:53,450
+bound أو الست لها upper bound U فأي عدد V أكبر من 
+
+401
+00:39:53,450 --> 00:39:58,170
+U بيطلع upper bound وبالتالي لو كان الست لها upper 
+
+402
+00:39:58,170 --> 00:40:01,450
+bound واحد فبيكون لها infinitely many upper bounds
+
+403
+00:40:02,480 --> 00:40:07,420
+بالمثل لو كان الست لها lower bound فبالتالي بيكون 
+
+404
+00:40:07,420 --> 00:40:11,300
+لها infinitely many lower bounds عدد لانهائي في الـ
+
+405
+00:40:11,300 --> 00:40:15,280
+lower bounds طيب المجموعة الخالية المجموعة الخالية 
+
+406
+00:40:15,280 --> 00:40:23,000
+بتمتاز بإنه أي عدد حقيقي بيطلع upper bound و lower 
+
+407
+00:40:23,000 --> 00:40:29,580
+bound في نفس الوقت هذا بده برهان هذا مش واضح اللي
+
+408
+00:40:29,580 --> 00:40:32,640
+هو انتبهوا ويرجعوا لمبادئ الرياضيات وقوانين الـ 
+
+409
+00:40:32,640 --> 00:40:41,060
+logic هاي برهان بالتناقض أنا عندي S مجموعة خالية 
+
+410
+00:40:41,060 --> 00:40:47,860
+مجموعة محترمة هاي S بساوي فاي مجموعة جزئية من R
+
+411
+00:40:47,860 --> 00:40:54,760
+الفاية مجموعة جزئية محترمة من R هذه المجموعة أي عدد
+
+412
+00:40:54,760 --> 00:40:59,850
+حقيقي مهما كان هو lower bound و upper bound اللي في 
+
+413
+00:40:59,850 --> 00:41:06,350
+نفس الوجود خلينا نثبت إنه لو أخذت أي عدد حقيقي R0 
+
+414
+00:41:06,350 --> 00:41:16,970
+يعني تاني لـ R فبدنا نثبت أن هذا is upper bound of 
+
+415
+00:41:16,970 --> 00:41:25,050
+Phi بدنا نثبت إنه أي عدد حقيقي R0 بيطلع upper bound
+
+416
+00:41:25,050 --> 00:41:34,030
+لـ Phi البرهان ذلك بالتناقض افترض
+
+417
+00:41:34,030 --> 00:41:39,290
+أن ولا عدد حقيقي هو 
+
+418
+00:41:39,290 --> 00:41:48,570
+upper bound للفاية يعني
+
+419
+00:41:48,570 --> 00:41:55,030
+يوجد عدد حقيقي R0 ليس upper bound 
+
+420
+00:41:58,090 --> 00:42:02,030
+ففي الحالة هذه هذا معناه من تعريف الـ upper bound 
+
+421
+00:42:02,030 --> 00:42:06,990
+إذا كان R0 ليس upper bound للـ set S التي هي Phi
+
+422
+00:42:06,990 --> 00:42:13,110
+معناته بقدر ألاقي عنصر في الـ set أكبر من الـ R0 
+
+423
+00:42:13,110 --> 00:42:20,110
+R0 لا يحد العناصر من أعلى، وهذا بدّيني تناقض، ليش 
+
+424
+00:42:20,110 --> 00:42:25,810
+تناقض؟ لأن هذا أدى إلى أن يوجد عنصر S في S 
+
+425
+00:42:25,810 --> 00:42:31,250
+S اللي هي Phi، و Phi خالية، لا يوجد فيها أي عنصر.
+
+426
+00:42:31,250 --> 00:42:39,910
+لا يوجد فيها أو داخلها أي عنصر اسمه الـ S، إذا الفرض 
+
+427
+00:42:39,910 --> 00:42:46,990
+هذا خطأ، وبالتالي كل الأعداد الحقيقية هي upper
+
+428
+00:42:46,990 --> 00:42:53,590
+bounds للمجموعة Phi، بالمثل بطريقة مماثلة ممكن نثبت
+
+429
+00:42:53,590 --> 00:42:59,570
+أن نفس العدد أو أي عدد R0 هو lower bound للمجموعة 
+
+430
+00:42:59,570 --> 00:43:00,050
+Phi
+
+431
+00:43:04,940 --> 00:43:11,500
+طيب نرجع نيجي الآن لـ ..
+
+432
+00:43:11,500 --> 00:43:23,360
+شوية تعريفات ثانية للمجموعة
+
+433
+00:43:23,360 --> 00:43:30,640
+طيب
+
+434
+00:43:30,640 --> 00:43:42,980
+المحاضرة يعني انتهت تقريبا، المرة الجاية هناخد تعريف
+
+435
+00:43:42,980 --> 00:43:50,820
+الـ bounded set و تعريف الـ supremum اللي هو least
+
+436
+00:43:50,820 --> 00:43:55,360
+upper bound أو أصغر حد أعلى، و الـ infimum اللي هو
+
+437
+00:43:55,360 --> 00:44:01,250
+أكبر حد أدنى، و ندرس خواصه وبعدين نشوف ما هي الـ
+
+438
+00:44:01,250 --> 00:44:05,670
+completeness property أو خاصية التمام اللي كانت
+
+439
+00:44:05,670 --> 00:44:08,590
+عنوان جانبي للجزء هذا من المحاضرة 
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/wn5D8jbScjs_raw.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/wn5D8jbScjs_raw.srt
new file mode 100644
index 0000000000000000000000000000000000000000..ff2733ae42bbcc7890cb27a6877d838de06de141
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/wn5D8jbScjs_raw.srt
@@ -0,0 +1,1756 @@
+1
+00:00:20,670 --> 00:00:25,090
+تحدثنا عن ال absolute value و عرفنا ال absolute
+
+2
+00:00:25,090 --> 00:00:29,810
+value لأي real number إما a إذا كان a غير سالب أو
+
+3
+00:00:29,810 --> 00:00:33,310
+سالب العدد نفسه إذا كان العدد سالب نفس التعريف
+
+4
+00:00:33,310 --> 00:00:38,090
+اللي أخدناه في ال calculus و بكل من التعريف
+
+5
+00:00:38,090 --> 00:00:43,740
+باستخدام التعريف و ال logicأو مبادئ الرياضيات ممكن
+
+6
+00:00:43,740 --> 00:00:47,220
+بسهولة نبره ان ال absolute value ل negative a
+
+7
+00:00:47,220 --> 00:00:51,900
+بساوي absolute a ال absolute value لحاصل الضرب
+
+8
+00:00:51,900 --> 00:00:58,720
+بساوي حاصل ضرب ال absolute values لو كان C عدد غير
+
+9
+00:00:58,720 --> 00:01:04,240
+سالب ف ال inequality هذه بتكافئ ال inequality هذه
+
+10
+00:01:06,320 --> 00:01:11,960
+لأي عدد حقيقي ايه دائما أي عدد حقيقي بيكون أصغر من
+
+11
+00:01:11,960 --> 00:01:15,320
+أو يساوي ال absolute value له وأكبر من أو يساوي
+
+12
+00:01:15,320 --> 00:01:19,860
+سالب ال absolute value هذا دائما صحيح في عند ال
+
+13
+00:01:19,860 --> 00:01:24,000
+triangle inequality هذه متباينة كتير مهمة بتقول ان
+
+14
+00:01:24,000 --> 00:01:28,560
+ال absolute value لمجموعة أو الفرق بين عددين
+
+15
+00:01:28,560 --> 00:01:33,000
+الحقيقيين less than or equal لمجموعة ال absolute
+
+16
+00:01:33,000 --> 00:01:38,990
+values للعددينومن المتباينة هذه ممكن نستنتج
+
+17
+00:01:38,990 --> 00:01:44,970
+متباينة أخرى لاتقل عنها أهمية وهي المتباينة
+
+18
+00:01:44,970 --> 00:01:45,730
+التالية
+
+19
+00:01:52,010 --> 00:01:56,450
+اللي هي المتباينة هذه لو أخدت ال absolute value ل
+
+20
+00:01:56,450 --> 00:02:01,010
+A و طرحت منها ال absolute value ل B و أخدت ال
+
+21
+00:02:01,010 --> 00:02:06,830
+absolute value للفرق فهذا دائما أصغر من أو ساوي
+
+22
+00:02:06,830 --> 00:02:14,490
+absolute value للفرق و البرهان تبع ال inequality
+
+23
+00:02:14,490 --> 00:02:19,730
+هذه نتيجة مباشرة على ال triangle inequality
+
+24
+00:02:21,020 --> 00:02:26,900
+فالبراهين سهلة وبسيطة ويعني موجودة عندكم أي حد
+
+25
+00:02:26,900 --> 00:02:31,600
+فيكم بيقدر يقرأها ويفهمها لو مافيش .. في اي مشكلة
+
+26
+00:02:31,600 --> 00:02:36,520
+ممكن يعني اترجعوني خلال الساعات المكتبية او خلال
+
+27
+00:02:36,520 --> 00:02:44,220
+ساعة المناخشة في
+
+28
+00:02:44,220 --> 00:02:55,550
+اللي هو ملاحظة متباينة المثلثممكن تعميمها بدل
+
+29
+00:02:55,550 --> 00:03:02,850
+.. احنا قلنا ان ال absolute value ل a او a1 زائد
+
+30
+00:03:02,850 --> 00:03:12,440
+a2 اصغر من او ساوي absolute a1 زايد absolute a2أو
+
+31
+00:03:12,440 --> 00:03:16,100
+absolute A زي بي less than or equal to absolute A
+
+32
+00:03:16,100 --> 00:03:22,180
+زي absolute B المتباين هذه ممكن تعميمها مش بس ناخد
+
+33
+00:03:22,180 --> 00:03:24,780
+ال absolute value لعددين ممكن ناخد ال absolute
+
+34
+00:03:24,780 --> 00:03:30,320
+value ل N من الأعداد الحقيقية و نجمعهم فهذا أصغر
+
+35
+00:03:30,320 --> 00:03:34,490
+من أو ساوي مجموع ال absolute valuesوطبعا هذا ممكن
+
+36
+00:03:34,490 --> 00:03:39,710
+برهانة .. المتباين هذا ممكن برهانها بسهولة by
+
+37
+00:03:39,710 --> 00:03:46,930
+induction وطبعا ومتباينة في المثلثة okay اعتقد
+
+38
+00:03:46,930 --> 00:03:56,130
+يعني تقريبا هذا اللي شرحنا اليوم
+
+39
+00:03:56,130 --> 00:04:02,320
+بدنا ناخد تعريف ال epsilon neighborhoodفالـ
+
+40
+00:04:02,320 --> 00:04:09,680
+epsilon neverhood أو جوار epsilon لأي عدد حقيقي هي
+
+41
+00:04:09,680 --> 00:04:19,100
+خط العداد الحقيقية وهي العدد a وهي المسافة هذه او
+
+42
+00:04:19,100 --> 00:04:25,800
+النقطة هذه a سالب epsilon والنقطة هذه a موجب
+
+43
+00:04:25,800 --> 00:04:30,580
+epsilon والepsilon هذا عدد موجب
+
+44
+00:04:34,530 --> 00:04:39,430
+أذا الفترة المفتوحة هذه الفترة المفتوحة هذه اللي
+
+45
+00:04:39,430 --> 00:04:44,710
+هي a سالب إبسلون و a موجب إبسلون اللي هي الفترة
+
+46
+00:04:44,710 --> 00:04:49,770
+هذه الفترة
+
+47
+00:04:49,770 --> 00:04:57,230
+المفتوحة هذه بنسميها لحظة هذه الفترة مركزها a و نص
+
+48
+00:04:57,230 --> 00:05:02,490
+قطرها إبسلون عدد موجبفالفترة المفتوحة دي اللي
+
+49
+00:05:02,490 --> 00:05:06,510
+مركزها a و نص قطرة epsilon بنسميها epsilon
+
+50
+00:05:06,510 --> 00:05:12,510
+neighbourhood of a جوار epsilon أو جوار بأمق
+
+51
+00:05:12,510 --> 00:05:18,270
+epsilon للنقطة a وهكذا
+
+52
+00:05:18,270 --> 00:05:22,270
+تعريفه هاي v epsilon ومصدره بالرمز v epsilon ل a
+
+53
+00:05:22,270 --> 00:05:27,920
+جوار epsilon ل aفهو كل الأعداد الحقيقية اللي
+
+54
+00:05:27,920 --> 00:05:33,840
+المسافة بينها وبين ال A أصغر من Epsilon أو هي كل
+
+55
+00:05:33,840 --> 00:05:38,040
+الأعداد الحقيقية اللي بتحقق المتباينة هذه اللي هي
+
+56
+00:05:38,040 --> 00:05:41,780
+بالظبط كل الأعداد الحقيقية X اللي هي كل الأعداد
+
+57
+00:05:41,780 --> 00:05:47,460
+الحقيقية X اللي هي المسافة بينها وبين ال A أصغر من
+
+58
+00:05:47,460 --> 00:05:53,700
+المسافة هذه اللي هي Epsilonواضح تمام فهذا بنسمي
+
+59
+00:05:53,700 --> 00:05:56,800
+ابسلون neighborhood of A في حاجة اسمها
+
+60
+00:05:56,800 --> 00:06:05,340
+neighborhood of A بدون ابسلون فأي
+
+61
+00:06:05,340 --> 00:06:10,280
+.. طبعا اختصار neighborhood اللي بنختصرها NBD
+
+62
+00:06:10,280 --> 00:06:17,000
+neighborhood في تعريف تاني neighborhood of A is
+
+63
+00:06:17,000 --> 00:06:23,380
+any set V of Aبحيث بقدر ألاقي داخلها epsilon
+
+64
+00:06:23,380 --> 00:06:28,340
+neighborhood ل A بقدر ألاقي أنه عبارة عن set V of
+
+65
+00:06:28,340 --> 00:06:32,680
+A بقدر ألاقي داخلها epsilon neighborhood of A كمان
+
+66
+00:06:32,680 --> 00:06:36,720
+مرة يعني ممكن يكون عندي مثلا حاجة زي هذه مثلا set
+
+67
+00:06:36,720 --> 00:06:41,180
+زي هذه وهي
+
+68
+00:06:41,180 --> 00:06:48,860
+النقطة A فترة مثلا نصف مفتوحة زي هذه في هذه الفترة
+
+69
+00:06:50,180 --> 00:06:54,820
+ممكن تكون مثلا a سالب واحد و a موجة باتنين من
+
+70
+00:06:54,820 --> 00:07:01,060
+النقطة هذه ممكن ولا مش ممكن؟ وهذه فترة نص مفتوحة
+
+71
+00:07:01,060 --> 00:07:06,640
+هذه الفترة حسب التعريف b هي عبارة عن epsilon
+
+72
+00:07:06,640 --> 00:07:10,560
+neighborhood ل a لإيه؟ لأشان تكون الفترة هذه
+
+73
+00:07:10,560 --> 00:07:16,460
+epsilon عشان الفترة هذه أو ال set هذه اللي بنسميها
+
+74
+00:07:16,460 --> 00:07:22,600
+v of aعشان تكون neighborhood جوار ل A لازم ألاقي
+
+75
+00:07:22,600 --> 00:07:29,780
+داخلها epsilon neighborhood لل A وهذا موجود هاي
+
+76
+00:07:29,780 --> 00:07:37,080
+باخد المسافة هذه مثلا هاي A سالب نص وهاي A موجب نص
+
+77
+00:07:37,080 --> 00:07:48,080
+مثلا فالفترة هذه الفترة المفتوحة هذههي مركزها a و
+
+78
+00:07:48,080 --> 00:07:53,200
+نص كترها epsilon بساوة نص عدد موجب إذن هذه نجحت في
+
+79
+00:07:53,200 --> 00:08:01,320
+إيجاد v نص ل a و هذه الفترة المفتوحة واقع كليا
+
+80
+00:08:01,320 --> 00:08:07,080
+عبارة عن مجموعة جزئية من الفترة نص المفتوحة هذه
+
+81
+00:08:07,080 --> 00:08:13,880
+اللي هي a سالب واحد و a موجب اتنين okay تمام إذن
+
+82
+00:08:13,880 --> 00:08:20,840
+هذه الفترةعبارة عن neighborhood لـA إذا ال
+
+83
+00:08:20,840 --> 00:08:25,980
+neighborhood هو أي مجموعة تحتوي
+
+84
+00:08:25,980 --> 00:08:30,880
+A بحيث أن أنا أقدر ألاقي داخلها epsilon
+
+85
+00:08:30,880 --> 00:08:36,200
+neighborhood للـA طيب
+
+86
+00:08:36,200 --> 00:08:41,760
+وبالتالي من التعريفين هذول بنلاحظ أنه كل epsilon
+
+87
+00:08:41,760 --> 00:08:46,120
+neighborhoodكل y neighborhood ل a هو neighborhood
+
+88
+00:08:46,120 --> 00:09:02,320
+لكن العكس مش صحيح في نظرية هنا بتقول انه لو
+
+89
+00:09:02,320 --> 00:09:10,500
+كان عندي عدد حقيقي a فالعدد x العدد الحقيقي x بكون
+
+90
+00:09:10,500 --> 00:09:13,140
+موجود في كل neighborhoods ل a
+
+91
+00:09:16,470 --> 00:09:21,330
+لو كان X موجود في كل ال neighborhoods ل A فلازم ال
+
+92
+00:09:21,330 --> 00:09:26,510
+X هذه تساوي A لازم ال X تساوي A كمان مرة النظرية
+
+93
+00:09:26,510 --> 00:09:35,470
+هذه بتقول نعمل رسمة هاي
+
+94
+00:09:35,470 --> 00:09:42,130
+A لو كل Y neighborhood ل X
+
+95
+00:09:48,500 --> 00:09:53,240
+لو .. لو .. لو كل .. لو كل epsilon neighborhood أو
+
+96
+00:09:53,240 --> 00:09:58,800
+كل neighborhood هد عبارة عن neighborhood لإيه؟ لو
+
+97
+00:09:58,800 --> 00:10:03,140
+كل neighborhood لإيه يحتوي ال X؟ لو أي
+
+98
+00:10:03,140 --> 00:10:08,080
+neighborhood لإيه يحتوي ال X؟
+
+99
+00:10:08,080 --> 00:10:12,900
+أو كل الجوارات لل إيه تحتوي ال X؟ فلازم ال X هدي
+
+100
+00:10:12,900 --> 00:10:17,650
+تكون هي نفس الإيهوهذا البرهان بسيط بيعتمد على
+
+101
+00:10:17,650 --> 00:10:23,290
+خاصية أخدناها قبل هيك و قلتلكم عنها مهمة و
+
+102
+00:10:23,290 --> 00:10:28,450
+هنستخدمها لحظة وهي هنشوفها الجيت موجودة في نظرية
+
+103
+00:10:28,450 --> 00:10:35,850
+واحد تمانية البرهان بما
+
+104
+00:10:35,850 --> 00:10:39,970
+أن كل epsilon neighborhood ل a هو عبارة عن
+
+105
+00:10:39,970 --> 00:10:46,180
+neighborhood ل a فمن الفرضكل neighborhood ل A
+
+106
+00:10:46,180 --> 00:10:47,080
+يحتوى X
+
+107
+00:11:03,320 --> 00:11:07,140
+هذا معناه أنه من تعريف الـ epsilon neverhood هذا
+
+108
+00:11:07,140 --> 00:11:11,920
+معناه أن x المسافة بين x و a المسافة بين ال
+
+109
+00:11:11,920 --> 00:11:16,680
+absolute value بين x و a اللي هي المسافة بين x و a
+
+110
+00:11:16,680 --> 00:11:20,760
+أصغر من epsilon وهذا صحيح لكل epsilon أكبر من
+
+111
+00:11:20,760 --> 00:11:27,980
+السفر طب في نظرية واحد تمانية أخدنا المتباينة هذه
+
+112
+00:11:27,980 --> 00:11:34,740
+اللي بتقوللو كان العدد بي أكبر من أو يساوي صفر
+
+113
+00:11:34,740 --> 00:11:41,480
+أصغر من إبسلون لكل إبسلون أكبر من الصفر فهذا بيدّي
+
+114
+00:11:41,480 --> 00:11:47,040
+أن بي بيساوي صفر، بظبط؟ هذه نظرية واحد تمانية أو
+
+115
+00:11:47,040 --> 00:11:53,440
+جزء منها فأنا هي عندي .. هي عندي ال بي هذا عبارة
+
+116
+00:11:53,440 --> 00:11:58,950
+عن عدد غير سالمالقيمة المطلقة لأي عدد هو عدد غير
+
+117
+00:11:58,950 --> 00:12:07,350
+سالب فهذا هو ال B هذا هو العدد B هذا هو هذا
+
+118
+00:12:07,350 --> 00:12:14,070
+ال Bهيعندي بي أكبر من أو ساوي سفر هذا صحيح و أصغر
+
+119
+00:12:14,070 --> 00:12:17,910
+من إبسلون لكل إبسلون أكبر من السفر إذا حسب النظرية
+
+120
+00:12:17,910 --> 00:12:22,550
+هذه الـ b اللي هو absolute x minus a لازم بساوي
+
+121
+00:12:22,550 --> 00:12:27,830
+سفر وبالتالي إذا x سالب a بساوي سفر ومنها بطلع x
+
+122
+00:12:27,830 --> 00:12:35,270
+بساوي a هذا هو برهان النظرية تمام؟ في أي سؤال؟
+
+123
+00:12:39,260 --> 00:12:44,480
+برهان سهل واضح شوف بعض الأمثلة على ال
+
+124
+00:12:44,480 --> 00:12:49,880
+neighborhoods و ال epsilon neighborhoods يتوضح
+
+125
+00:12:49,880 --> 00:12:57,540
+برسمة كل مثال نعمله رسمة لو أخدت المجموعة U لو
+
+126
+00:12:57,540 --> 00:13:03,540
+أخدت U عبارة عن المجموعة أو الفترة المفتوحة من 0
+
+127
+00:13:03,540 --> 00:13:04,340
+إلى 1
+
+128
+00:13:10,710 --> 00:13:19,490
+هذه عبارة عن المجموعة U فالمجموعة
+
+129
+00:13:19,490 --> 00:13:23,610
+U هذه أو الفترة المفتوحة تعتبر neighborhood لأي
+
+130
+00:13:23,610 --> 00:13:32,450
+نقطة داخلها يعني لو أخدت أي نقطة A داخل
+
+131
+00:13:32,450 --> 00:13:39,810
+المجموعة Aفالمجموعة U هي عبارة عن neighborhood لأي
+
+132
+00:13:39,810 --> 00:13:49,690
+نقطة داخلها فلبرهان ذلك .. لبرهان ذلك بدنا ان ..
+
+133
+00:13:49,690 --> 00:13:54,710
+حسب التعريف عشان U تكون neighborhood للنقطة A اللي
+
+134
+00:13:54,710 --> 00:13:59,090
+داخلها لازم نلاقي epsilon neighborhood حوالين الـA
+
+135
+00:13:59,090 --> 00:14:07,420
+ويكون داخل المجموعة U مظبوط، هذا حسب التعريفطيب ال
+
+136
+00:14:07,420 --> 00:14:16,080
+.. بنيجي ناخد epsilon بساوي ال minimum الأصغر بين
+
+137
+00:14:16,080 --> 00:14:22,360
+المسافتين المسافة هذه .. المسافة هذه ايه؟ المسافة
+
+138
+00:14:22,360 --> 00:14:27,660
+هذه ايه؟ طيب و المسافة التانية هذه؟ واحد نقص ايه؟
+
+139
+00:14:27,660 --> 00:14:34,520
+واحد سالب ايه؟فالـ minimum بين العددين a و 1 سالب
+
+140
+00:14:34,520 --> 00:14:41,000
+a بناخده هو ال epsilon تمام؟ في الرسمة .. في
+
+141
+00:14:41,000 --> 00:14:47,360
+الرسمة هذه ال .. الأصغر بين العددين a و 1 سالب a
+
+142
+00:14:47,360 --> 00:14:52,320
+هو ال a فبقى أجي بكون حوالين ال a epsilon
+
+143
+00:14:52,320 --> 00:14:58,940
+neighborhood هي فهذا عبارة عن epsilon neighborhood
+
+144
+00:15:01,920 --> 00:15:08,720
+فالفترة المفتوحة هذه السودا بالتأكيد اذا هنا ال a
+
+145
+00:15:08,720 --> 00:15:14,320
+سالب ابسلون و a موجة بابسلون هذه بالتأكيد واقع
+
+146
+00:15:14,320 --> 00:15:23,590
+داخل ال U حسب الرسم وبالتالي هيني نجحتفي إيجاد V
+
+147
+00:15:23,590 --> 00:15:27,670
+Epsilon أو Epsilon neighborhood ل A و هذا ال
+
+148
+00:15:27,670 --> 00:15:34,310
+Epsilon neighborhood ل A واقع داخل ال U إذن حسب
+
+149
+00:15:34,310 --> 00:15:39,450
+التعريف ال U أبارع ال neighborhood للنقطة A و لما
+
+150
+00:15:39,450 --> 00:15:44,850
+كانت A نقطة عشوائية إذن U أبارع ال neighborhood
+
+151
+00:15:44,850 --> 00:15:49,990
+لأي نقطة داخلها تمام؟
+
+152
+00:15:53,400 --> 00:15:58,160
+Okay مثال تاني لو
+
+153
+00:15:58,160 --> 00:16:01,360
+أخدنا الفترة المغلقة هنا أخدنا U بالساوية الفترة
+
+154
+00:16:01,360 --> 00:16:07,900
+المفتوحة لو أخدنا هذه
+
+155
+00:16:07,900 --> 00:16:14,600
+خط الأعداد وهذه الفترة المغلقة من 0 إلى 1 هذه
+
+156
+00:16:14,600 --> 00:16:16,440
+الفترة نسميها I
+
+157
+00:16:21,110 --> 00:16:28,110
+I بتساوي الفترة المغلفة من 0 إلى 1 طبعا
+
+158
+00:16:28,110 --> 00:16:33,230
+زي ما في المثال السابق الفترة I هذه ممكن اثبات
+
+159
+00:16:33,230 --> 00:16:38,710
+انها neighborhood لأي نقطة داخلها معدى 0 والواحد
+
+160
+00:16:38,710 --> 00:16:46,210
+نفس البرهن لكن الفترة I هذه أو الست I ليست
+
+161
+00:16:46,210 --> 00:16:52,410
+neighborhoodليست I is not neighborhood للعنصر سفر
+
+162
+00:16:52,410 --> 00:16:58,930
+وكذلك ليست neighborhood للعنصر واحد ليه؟ لأنه بقدر
+
+163
+00:16:58,930 --> 00:17:08,670
+ألاقي لأن كل epsilon كل epsilon neighborhood للسفر
+
+164
+00:17:08,670 --> 00:17:13,170
+هاي لو أخدت أي epsilon مهما كانت صغيرة
+
+165
+00:17:15,840 --> 00:17:20,480
+يعني هي epsilon عدد موجب لأن هذا هيصير سفر زاد
+
+166
+00:17:20,480 --> 00:17:25,740
+epsilon وهي كمان المسافة هذه epsilon فهيصير هذه
+
+167
+00:17:25,740 --> 00:17:33,700
+طبعا سالب epsilon فلو أخدت أي فترة مفتوحة بأمق
+
+168
+00:17:33,700 --> 00:17:39,100
+epsilon هيثم epsilon عدد موجبإذا لأي إبسلون عدد
+
+169
+00:17:39,100 --> 00:17:42,900
+موجب لو كونت إبسلون neighborhood للسفر إذا هذه
+
+170
+00:17:42,900 --> 00:17:51,100
+الفترة الحمرة عبارة عن V إبسلون للسفر ف V إبسلون
+
+171
+00:17:51,100 --> 00:17:59,640
+للسفر هذا ليس ليس مجموعة جزئية من I هذا ليس مجموعة
+
+172
+00:17:59,640 --> 00:18:09,360
+جزئية من I لكل إبسلونلكل إبسلون أكبر من السفر لكل
+
+173
+00:18:09,360 --> 00:18:13,560
+إبسلون أكبر من السفر لكل إبسلون أكبر من السفر لكل
+
+174
+00:18:13,560 --> 00:18:14,260
+إبسلون أكبر من السفر لكل إبسلون أكبر من السفر لكل
+
+175
+00:18:14,260 --> 00:18:14,560
+إبسلون أكبر من السفر لكل إبسلون أكبر من السفر لكل
+
+176
+00:18:14,560 --> 00:18:16,940
+إبسلون أكبر من السفر لكل إبسلون أكبر من السفر لكل
+
+177
+00:18:16,940 --> 00:18:17,100
+إبسلون أكبر من السفر لكل إبسلون أكبر من السفر لكل
+
+178
+00:18:17,100 --> 00:18:18,560
+إبسلون أكبر من السفر لكل إبسلون أكبر من السفر لكل
+
+179
+00:18:18,560 --> 00:18:20,920
+إبسلون أكبر من السفر لكل إبسلون أكبر من السفر لكل
+
+180
+00:18:20,920 --> 00:18:26,910
+إبسلون أكبر من السفربالمثل الفترة I ليست نبرهود
+
+181
+00:18:26,910 --> 00:18:33,150
+للواحد لأن أي جوار بعمق إبسلون للواحد لا يقع كله
+
+182
+00:18:33,150 --> 00:18:37,410
+داخل الفترة I الجزء هذا خارج الفترة I، الجزء هذا
+
+183
+00:18:37,410 --> 00:18:40,010
+خارج الفترة I، الجزء هذا خارج الفترة I، الجزء هذا
+
+184
+00:18:40,010 --> 00:18:40,990
+خارج الفترة I، الجزء هذا خارج الفترة I، الجزء هذا
+
+185
+00:18:40,990 --> 00:18:41,330
+خارج الفترة I، الجزء هذا خارج الفترة I، الجزء هذا
+
+186
+00:18:41,330 --> 00:18:42,670
+خارج الفترة I، الجزء هذا خارج الفترة I، الجزء هذا
+
+187
+00:18:42,670 --> 00:18:44,910
+خارج الفترة I، الجزء هذا خارج الفترة I، الجزء هذا
+
+188
+00:18:44,910 --> 00:18:46,930
+خارج الفترة I، الجزء هذا خارج الفترة I، الجزء هذا
+
+189
+00:18:46,930 --> 00:18:55,140
+خارج الفترة I، الجزء هذا خارج الفتمثال تالت مثال
+
+190
+00:18:55,140 --> 00:19:01,220
+تالت لو
+
+191
+00:19:01,220 --> 00:19:07,000
+أخدت أي تلت أعداد حقيقية x, y, z لو أخدت أي تلت
+
+192
+00:19:07,000 --> 00:19:18,240
+أعداد حقيقية x و y و z بحيث أنه هاي ال x أصغر من z
+
+193
+00:19:18,240 --> 00:19:28,110
+أصغر من أو يساوي zو X أصغر من أو يساوي Y و Y أصغر
+
+194
+00:19:28,110 --> 00:19:36,110
+من Z فهذا بيقدي لازم يقدي أن المسافة المسافة بين X
+
+195
+00:19:36,110 --> 00:19:43,070
+و Y زاد المسافة بين Y و Z بتساوي المسافة بين X و Z
+
+196
+00:19:43,070 --> 00:19:51,590
+هذا واضح هذا واضح من الرسم لكن طبعا الرسم أمره ما
+
+197
+00:19:51,590 --> 00:19:57,930
+كان برهانأما لكن ممكن يوضح في فكرة البرهان إذا
+
+198
+00:19:57,930 --> 00:20:02,470
+المجموعة القيم المطلقة هذه بساوة القيم المطلقة هذه
+
+199
+00:20:02,470 --> 00:20:07,090
+بس في حالة لما ال X أصغر من Y أو ال Y تكون بين X و
+
+200
+00:20:07,090 --> 00:20:13,290
+Z و X أصغر من أو ساو Z البرهان بكل بساطة كالتالي
+
+201
+00:20:13,290 --> 00:20:19,630
+أنا عندي الفرض تبعي X أصغر من أو ساو Z و Y بين X و
+
+202
+00:20:19,630 --> 00:20:27,690
+Zهذا يجعل القيمة المطلقة لـ x-y .. الآن x أصغر من
+
+203
+00:20:27,690 --> 00:20:31,570
+y، فهذا الفرق سالب، فالقيمة المطلقة لعدد سالب سالب
+
+204
+00:20:31,570 --> 00:20:38,920
+نفسهبالمثل هذا y سالب z و y أصغر من z إذا العدد
+
+205
+00:20:38,920 --> 00:20:41,360
+هذا اللي جوه ال absolute value سالب هذا حسب تعريف
+
+206
+00:20:41,360 --> 00:20:47,840
+القيمة المطلقة هذا بساوي سالبه و كذلك absolute x
+
+207
+00:20:47,840 --> 00:20:52,600
+minus z بساوي سالب العدد نفسه
+
+208
+00:20:57,610 --> 00:21:03,050
+بنجمع هاي absolute x minus y و absolute y minus z
+
+209
+00:21:03,050 --> 00:21:11,070
+فهي عندي بيطلع سالب x موجب y زائد سالب y موجب z و
+
+210
+00:21:11,070 --> 00:21:16,350
+هدوله طبعا بيطلع بساوي المجموع هذا و هذا بيساوي
+
+211
+00:21:16,350 --> 00:21:22,350
+absolute x minus z لأن x أصغر من z okay تمام إذا
+
+212
+00:21:22,350 --> 00:21:30,220
+هذا هو البرهانهذا هو البرامج نعم نعم X أقل من أو
+
+213
+00:21:30,220 --> 00:21:38,740
+يساوي Z نعم شوف هذا تبين ساوي هو يعني سواء X أصغر
+
+214
+00:21:38,740 --> 00:21:43,120
+من أو يساوي Z أو بتساوي في هذه المتباينة صحيحة
+
+215
+00:21:43,120 --> 00:21:52,370
+يعني لو X بتساوي Zيعني في الحالة الخاصة اللي X
+
+216
+00:21:52,370 --> 00:21:57,490
+فيها بتنطبق على Z ف Y هتنطبق على X وعلى Z يعني
+
+217
+00:21:57,490 --> 00:22:00,970
+التلتة هدول هيطلعوا متساويين وبالتالي هذه ال
+
+218
+00:22:00,970 --> 00:22:05,350
+absolute value سفر وهذه سفر سفر زاد سفر بساوي سفر
+
+219
+00:22:05,350 --> 00:22:12,090
+وبالتالي العلاقة أو المعادلة هذه صحيحة okay هذه
+
+220
+00:22:12,090 --> 00:22:16,350
+الحالة الخاصة اللي فيها بيكون X بساوي Z في الحالة
+
+221
+00:22:16,350 --> 00:22:21,690
+هذه لازم يطلع طبعاأي عدد بين X و Z سيكون بيسويهم
+
+222
+00:22:21,690 --> 00:22:28,670
+وبالتالي هذا سفر وهذا سفر وهذا سفر في أي سؤال
+
+223
+00:22:28,670 --> 00:22:37,310
+تاني؟ طيب في كمان مثال رابع
+
+224
+00:22:40,090 --> 00:22:44,170
+برضه كل الأمثلة هذه على التطبيقات على ال absolute
+
+225
+00:22:44,170 --> 00:22:47,490
+value لو في اندي فانكشن يعني افرض انك انت في
+
+226
+00:22:47,490 --> 00:22:51,810
+calculus ولا في course calculus او algebra وطلب
+
+227
+00:22:51,810 --> 00:22:58,150
+منك عرف الدالة f of x علي انها 2x تربيه زي 3x زي 1
+
+228
+00:22:58,150 --> 00:23:03,630
+على 2x زالف 1 والمجال تبع الدالة هذه الفترة
+
+229
+00:23:03,630 --> 00:23:09,210
+المغلقة من 2 ل 3برهن أن القيمة المطلقة لدالة هذا
+
+230
+00:23:09,210 --> 00:23:14,650
+أصغر من أو ساوي 28 على 3 لكل x في المجال تبعها
+
+231
+00:23:14,650 --> 00:23:21,550
+فكيف نبرهن هذا؟ نستخدم خواص القيمة المطلقة بدي أخد
+
+232
+00:23:21,550 --> 00:23:24,850
+.. بدي أثبت absolute f of x أصغر من أو ساوي العدد
+
+233
+00:23:24,850 --> 00:23:32,610
+هذا باخد absolute xبـ absolute f of x فقيمة
+
+234
+00:23:32,610 --> 00:23:36,590
+f of x بساوية قيمة المطلقة للبسط على قيمة المطلقة
+
+235
+00:23:36,590 --> 00:23:40,230
+للمقرنة هذه خاصية من قواصة ال absolute value لأن
+
+236
+00:23:40,230 --> 00:23:44,290
+القسمة هي ضرب وقلنا absolute value للضرب بساوية
+
+237
+00:23:44,290 --> 00:23:48,590
+حاصل ضرب ال absolute valueالان تعالوا نشتغل على
+
+238
+00:23:48,590 --> 00:23:51,790
+الـ BEST هاي الـ BEST هو عبارة عن القيمة المطلقة
+
+239
+00:23:51,790 --> 00:23:56,310
+دي باستخدام ال generalized triangle inequality ال
+
+240
+00:23:56,310 --> 00:23:59,730
+absolute value لمجموعة تلت أعداد أصغر من أو ساوي
+
+241
+00:23:59,730 --> 00:24:06,300
+مجموعة ال absolute values لهمأبسل يوت اتنين X
+
+242
+00:24:06,300 --> 00:24:10,680
+تربية زاد أبسل يوت تلاتة X بيطلع تلاتة أبسل يوت X
+
+243
+00:24:10,680 --> 00:24:12,580
+زاد أبسل يوت واحد بيطلع واحد
+
+244
+00:24:34,520 --> 00:24:39,900
+فنجمع الأرقام هذه بيطلع MD 28 إذا ال absolute
+
+245
+00:24:39,900 --> 00:24:45,520
+value لل bus هذا ال absolute value لل bus هذا طلع
+
+246
+00:24:45,520 --> 00:24:51,080
+أصغر من أو ساوي 28 الآن تعالوا نشوف ال absolute
+
+247
+00:24:51,080 --> 00:24:52,220
+value للمقام
+
+248
+00:24:54,600 --> 00:24:59,580
+absolute 2x سالب واحد نستخدم المرة هذه ال triangle
+
+249
+00:24:59,580 --> 00:25:04,480
+inequality هذه اللي هي absolute absolute a minus
+
+250
+00:25:04,480 --> 00:25:19,040
+absolute b أكبر من أو يساوي absolute a minus b أقل
+
+251
+00:25:19,040 --> 00:25:22,500
+أقل صحيح
+
+252
+00:25:27,790 --> 00:25:34,990
+فانا عندي هنا ال .. انا عندي ال A بساوي اتنين X
+
+253
+00:25:34,990 --> 00:25:42,010
+وال B بساوي واحد لانت
+
+254
+00:25:42,010 --> 00:25:46,590
+هكسلتي سؤال بدون أذن واتكلمتي بدون بسمعليه فمش
+
+255
+00:25:46,590 --> 00:25:53,380
+هجاوب على سؤالكلأ خرجت القاعدة هذه هي المتباينة
+
+256
+00:25:53,380 --> 00:25:58,000
+هذه أخدناها حكينا عنها اليوم صح أظبط لأ بدي أستخدم
+
+257
+00:25:58,000 --> 00:26:02,780
+المتباينة هذه يعني عندي absolute 2x سالب واحد فهي
+
+258
+00:26:02,780 --> 00:26:11,490
+absolute 2x a باخد a بساوي 2x و b بساوي واحدفهذا
+
+259
+00:26:11,490 --> 00:26:17,750
+المفروض يصير أكبر
+
+260
+00:26:17,750 --> 00:26:20,170
+من أو ساوي أكبر من أو ساوي أكبر من أو ساوي أكبر من
+
+261
+00:26:20,170 --> 00:26:20,810
+أو ساوي أكبر من أو ساوي أكبر من أو ساوي أكبر من أو
+
+262
+00:26:20,810 --> 00:26:29,470
+ساوي أكبر من أو ساوي أكبر من أو ساوي أكبر من
+
+263
+00:26:29,470 --> 00:26:36,150
+أو ساوي أكبر من أو ساوي أكبر من أو ساوي أكبر من أو
+
+264
+00:26:36,150 --> 00:26:37,690
+ساوي أكبر من أو ساوي أكبر من أو ساوي أكبر من أو
+
+265
+00:26:37,690 --> 00:26:37,910
+ساوي أكبر من أو ساوي أكبر من أو ساوي أكبر من أو
+
+266
+00:26:37,910 --> 00:26:38,110
+ساوي أكبر من أو ساوي أكبر من أو ساوي أكبر من أو
+
+267
+00:26:38,110 --> 00:26:39,310
+ساوي أكبر من أو ساوي أكبر من أو ساوي أكبر من أو
+
+268
+00:26:39,310 --> 00:26:44,120
+ساوي أكبر من أو ساوي أكبر من أو س-1 وانا عندي x
+
+269
+00:26:44,120 --> 00:26:50,440
+أكبر من أو يساوي ل 2 إذا absolute x أكبر من أو
+
+270
+00:26:50,440 --> 00:26:57,020
+يساوي 2 إذا هذا أكبر من أو يساوي 2 في 2 سالب 1
+
+271
+00:26:57,020 --> 00:27:05,130
+اللي هو 3كمان مرة باستخدام المتباينة هذه absolute
+
+272
+00:27:05,130 --> 00:27:09,470
+a minus b أكبر من أو ساوي absolute a اللي هي 2 في
+
+273
+00:27:09,470 --> 00:27:14,050
+absolute x سالب absolute b اللي هو absolute 1 بطلع
+
+274
+00:27:14,050 --> 00:27:18,630
+1 الآن ال X هذه أكبر من أو ساوي 2 إذا القيمة
+
+275
+00:27:18,630 --> 00:27:24,710
+المطلقة برضه أكبر من أو ساوي ل 2 إذا هذه أكبر من
+
+276
+00:27:24,710 --> 00:27:28,130
+أو ساوي 2 في 2 سالب 1 بطلع 3
+
+277
+00:27:30,990 --> 00:27:38,890
+الان اذا هذا عبارة عن المقام هنا فإذا كان العدد
+
+278
+00:27:38,890 --> 00:27:43,290
+احنا خدنا قبل هيك خاصية ما تنساش الخواص اللي
+
+279
+00:27:43,290 --> 00:27:51,370
+خدناها في بتقول لو كان عندي أنا عدد A غير سالب او
+
+280
+00:27:51,370 --> 00:27:55,290
+اكبر من السفر فمقلوب
+
+281
+00:28:00,600 --> 00:28:10,640
+أو لو كان a أكبر من b أكبر من 0 فهذا بيقدي أن 1
+
+282
+00:28:10,640 --> 00:28:17,520
+على a أصغر من 1 على b، مظبوط؟ فأنا إذا مقلوب ..
+
+283
+00:28:17,520 --> 00:28:22,480
+إذا مقلوب المقدار هذا أصغر من أو ساوي مقلوب
+
+284
+00:28:22,480 --> 00:28:23,140
+التلاتة
+
+285
+00:28:25,760 --> 00:28:30,340
+إذاً هذا بيطلع .. إذاً هذا الـ gate نساوي بيطلع
+
+286
+00:28:30,340 --> 00:28:36,820
+أصغر من أو خلّينا نكتبه على صورة هاي .. هذا عبارة
+
+287
+00:28:36,820 --> 00:28:48,520
+عن absolute 2x تربيع زاد 3x زاد 1 ضرب ضرب 1 على
+
+288
+00:28:48,520 --> 00:28:51,820
+absolute 2x سالب 1
+
+289
+00:28:57,850 --> 00:29:02,550
+يعني انا حولت القسمة لضرب الان الحد الاول اصغر من
+
+290
+00:29:02,550 --> 00:29:06,330
+او ساوي تمانية و عشرين و الحد التاني اللي هو
+
+291
+00:29:06,330 --> 00:29:09,810
+المقلوب absolutely of x سالب واحد اصغر من او ساوي
+
+292
+00:29:09,810 --> 00:29:16,530
+تلاتة عفوا تلت وبالتالي بيطلع اذا absolute f of x
+
+293
+00:29:16,530 --> 00:29:22,950
+اصغر من او ساوي تمانية و عشرين على تلاتة okay تمام
+
+294
+00:29:24,140 --> 00:29:29,260
+في أي سؤال؟ اذا هنا يعني احنا استخدمنا متباينة
+
+295
+00:29:29,260 --> 00:29:34,320
+المثلث العادية ومتباينة المثلث الأخرى اللي هي هذه
+
+296
+00:29:34,320 --> 00:29:47,660
+في اثبات اللي احنا عايزينه في السؤال هذا طيب
+
+297
+00:29:47,660 --> 00:29:53,530
+ننتقل لموضوع تانياللي هو الـ completeness property
+
+298
+00:29:53,530 --> 00:30:02,450
+of R خاصة تمام للأعداد الحقيقية بالمناسبة
+
+299
+00:30:02,450 --> 00:30:07,470
+في مرجع انا يمكن مش مكتوب في ال syllabus عندكم لكن
+
+300
+00:30:07,470 --> 00:30:15,650
+خليني انا احكيلكم عنه المرجع هذا مكتوب باللغة
+
+301
+00:30:15,650 --> 00:30:20,630
+العربية ويمكن يكون سهل بالنسبالكمو مش غلط انكم
+
+302
+00:30:20,630 --> 00:30:29,490
+تقتلوه او تقتلينه اللي هو كتاب كتاب
+
+303
+00:30:29,490 --> 00:30:40,330
+التحليل الحقيقي لجامعة
+
+304
+00:30:40,330 --> 00:30:43,570
+الخدس
+
+305
+00:30:43,570 --> 00:30:46,970
+المفتوحة
+
+306
+00:30:49,530 --> 00:30:57,930
+أنا درست ال course من الكتاب هذا و كتاب رائع جدا و
+
+307
+00:30:57,930 --> 00:31:01,910
+مكتوب من قبل نخبة من المدرسين و أنا رجعت الكتاب و
+
+308
+00:31:01,910 --> 00:31:08,410
+نقحته و عملنا فيه بعض التعديلات فيعني لو اطلعتوا
+
+309
+00:31:08,410 --> 00:31:14,050
+عليه هيكون مفيد جدا بالاضافة لمرجعتنا اللي احنا
+
+310
+00:31:14,050 --> 00:31:15,790
+قررين عليه
+
+311
+00:31:23,230 --> 00:31:29,730
+Tamam؟ Yes sir إذا ال completeness property of R
+
+312
+00:31:29,730 --> 00:31:34,830
+أو خاصية التمام أو الكمال للعداد الحقيقية نشوف شو
+
+313
+00:31:34,830 --> 00:31:39,410
+معناه عشان نصل للخاصية هذه بدنا شوية تعريفات و
+
+314
+00:31:39,410 --> 00:31:45,250
+شوية تمهيل فلو أخدت أي مجموعة جزئية من R
+
+315
+00:31:54,920 --> 00:31:59,360
+هذه خط الأعداد الحقيقية وهذه R وهو محور X ممكن
+
+316
+00:31:59,360 --> 00:32:05,480
+نعتبره R صح؟ ومحور Y برضه R شو الفرق بينهم فنأخد
+
+317
+00:32:05,480 --> 00:32:10,100
+الأعداد الحقيقية هيك رأسية وهذه المجموعة هذه هيك
+
+318
+00:32:10,100 --> 00:32:19,680
+مجموعة الأعداد هذه S هي مجموعة جزئية من R تمام؟
+
+319
+00:32:19,680 --> 00:32:27,650
+هذه المجموعة جزئية من Rفلو أخدت عدد U زي هذا U عدد
+
+320
+00:32:27,650 --> 00:32:33,090
+حقيقي فالعدد هذا بنسميه upper bound للمجموعة
+
+321
+00:32:33,090 --> 00:32:39,910
+الجزئية S إذا كان كل X في
+
+322
+00:32:39,910 --> 00:32:43,530
+المجموعة
+
+323
+00:32:43,530 --> 00:32:48,390
+S أصغر من أو يساوي U إذا مرة تانية العدد U upper
+
+324
+00:32:48,390 --> 00:32:54,220
+bound حد أعلىللمجموعة S إذا كان كل X في S أصغر من
+
+325
+00:32:54,220 --> 00:33:01,840
+أو يساوي الـ U و العدد W بنسميه lower bound أو حد
+
+326
+00:33:01,840 --> 00:33:10,360
+أدنى للمجموعة S إذا كان كل X إذا كان أي عدد X في S
+
+327
+00:33:10,360 --> 00:33:16,020
+أكبر من أو يساوي الـ W أو W أصغر من أو يساوي كل
+
+328
+00:33:16,020 --> 00:33:24,290
+عناصر الـ Sففي الحالة هذه بنسمي w lower bound طيب
+
+329
+00:33:24,290 --> 00:33:30,650
+في شوية ملاحظات عن التعريفين هدول except may or
+
+330
+00:33:30,650 --> 00:33:34,130
+may not have an upper or lower bound على سبيل
+
+331
+00:33:34,130 --> 00:33:37,770
+المثال for example على سبيل المثال الأعداد
+
+332
+00:33:37,770 --> 00:33:45,600
+الحقيقية الأعداد الحقيقية هدا هيالمجموعة ممكن يكون
+
+333
+00:33:45,600 --> 00:33:49,420
+لها upper bound وممكن مايكونش لها upper bound
+
+334
+00:33:49,420 --> 00:33:53,380
+فمثلا
+
+335
+00:33:53,380 --> 00:34:02,320
+الفترة يعني هذه الأعداد هي من صفر إلى واحد أو
+
+336
+00:34:02,320 --> 00:34:09,160
+الفترة المغلقة من صفر إلى واحدهذه الواحد الواحد أو
+
+337
+00:34:09,160 --> 00:34:14,520
+أي U أكبر من أو ساوء الواحد أي عدد حقيقي U أكبر من
+
+338
+00:34:14,520 --> 00:34:19,060
+أو ساوء الواحد بيطلع upper bound لها ليه؟ لأن كل
+
+339
+00:34:19,060 --> 00:34:24,920
+العناصر اللي هنا أصغر من أو ساوء الواحد، صح؟ وكذلك
+
+340
+00:34:24,920 --> 00:34:32,800
+لو أخدت W عدد أصغر من أو ساوء سفر فكل X في الفترة
+
+341
+00:34:32,800 --> 00:34:33,720
+المغلقة هذه
+
+342
+00:34:36,600 --> 00:34:40,120
+كل x في الفترة المغلقة هذه أصغر من أوي ساوي سفر
+
+343
+00:34:40,120 --> 00:34:46,360
+صح؟ وبالتالي أصغر من أوي ساوي w إذا أي عدد حقيقي
+
+344
+00:34:46,360 --> 00:34:50,940
+بساوي سفر w بساوي سفر أو أصغر من سفر عبارة عن
+
+345
+00:34:50,940 --> 00:34:57,120
+lower bound للست S هذهكذلك أي عدد حقيقي U بساوي
+
+346
+00:34:57,120 --> 00:35:01,540
+واحد أو أكبر من واحد بيكون upper bound للمجموعة
+
+347
+00:35:01,540 --> 00:35:06,660
+هذه لأن كل X في المجموعة S هذه أكبر من أو ساوي
+
+348
+00:35:06,660 --> 00:35:13,140
+الواحد، تمام؟ لكن مجموعة الأعداد الحقيقية R أو
+
+349
+00:35:13,140 --> 00:35:18,300
+المجموعة الأعداد الحقيقية R هذه مالهاش upper bound
+
+350
+00:35:18,300 --> 00:35:20,060
+ومالهاش lower bound
+
+351
+00:35:23,110 --> 00:35:28,250
+العداد الحقيقية مافيش عدد حقيقي أكبر من أو ساوي كل
+
+352
+00:35:28,250 --> 00:35:32,390
+العداد الحقيقية ولا فيه نقدر نحط أصبعنا على العدد
+
+353
+00:35:32,390 --> 00:35:38,210
+ومافيش ولا عدد حقيقي أصغر من أو ساوي كل العداد
+
+354
+00:35:38,210 --> 00:35:43,970
+الحقيقية كتور؟ نعم مش الفترة من صفر لواحد مغلقة
+
+355
+00:35:43,970 --> 00:35:48,450
+تكسبها؟ او مفتوحة مش مشكلة طب إذا كانت مغلقة كيف
+
+356
+00:35:48,450 --> 00:35:55,130
+دي يكون ال lower bound أقل من صفر؟لو كانت مغلقة ف
+
+357
+00:35:55,130 --> 00:36:01,850
+.. ف w بساوي سفر عبارة عن lower bound او ال u
+
+358
+00:36:01,850 --> 00:36:06,470
+بتساوي واحد ال upper bound و u بساوي واحد upper
+
+359
+00:36:06,470 --> 00:36:10,670
+bound او حينما كانت مغلقة مش هيكوا بس كمان لو أخدت
+
+360
+00:36:10,670 --> 00:36:16,390
+u بساوي اتنين يعني اكبر من واحد برضه بطلع upper
+
+361
+00:36:16,390 --> 00:36:22,720
+bound صح؟حتى حتى لو مش مغلقة يعني اتنين هذا العدد
+
+362
+00:36:22,720 --> 00:36:26,760
+اتنين اكبر من العناصر اللي هنا كلها ولا لأ حتى لو
+
+363
+00:36:26,760 --> 00:36:34,860
+مش مغلقة سواء
+
+364
+00:36:34,860 --> 00:36:39,580
+الفترة هذه مغلقة او مش مغلقة اي عدد حقيقي اكبر من
+
+365
+00:36:39,580 --> 00:36:44,940
+او يساوي الواحد بيكون upper bound يعني ال upper
+
+366
+00:36:44,940 --> 00:36:47,740
+bound ممكن انه ماينتمش الفترةممكن ينتمي إلى
+
+367
+00:36:47,740 --> 00:36:54,080
+المجموعة وممكن لا ينتمي إليها يعني
+
+368
+00:36:54,080 --> 00:37:01,980
+لو أخدت الفترة مغلقة الفترة
+
+369
+00:37:01,980 --> 00:37:07,140
+مغلقة U بساوة اتنين upper bound لها في شك في ذلك؟
+
+370
+00:37:07,140 --> 00:37:11,060
+لكن ال upper bound هذا مش شرط يكون ينتمي إلى
+
+371
+00:37:11,060 --> 00:37:18,850
+المجموعةالواحد برضه upper bound للمجموعة دي و
+
+372
+00:37:18,850 --> 00:37:21,470
+ينتمي ل .. ممكن ينتمي إذا ال upper bound ممكن
+
+373
+00:37:21,470 --> 00:37:27,410
+ينتمي ل .. للمجموعة ممكن ماينتميش لو أخدت U
+
+374
+00:37:27,410 --> 00:37:32,210
+بالساوي عشرة برضه هذا upper bound لمجموعة S لو
+
+375
+00:37:32,210 --> 00:37:39,560
+أخدت U بالساوي مية برضه هذا upper bound للستو ال u
+
+376
+00:37:39,560 --> 00:37:43,760
+بساوية واحد upper bound لل 6 الان هذا ال u بساوية
+
+377
+00:37:43,760 --> 00:37:48,080
+واحد هو اصغر upper bound هذا ال u بساوية واحد هو
+
+378
+00:37:48,080 --> 00:37:53,420
+اصغر upper bound كذلك لو أخدت w بساوية سفر هذا
+
+379
+00:37:53,420 --> 00:37:58,240
+عبارة عن lower bound لل 6 هذه لأن كل عنصر في ال 6
+
+380
+00:37:58,240 --> 00:38:04,850
+أكبر منها بساوية سفرلو أخدت w بساوي سالب نص، برضه
+
+381
+00:38:04,850 --> 00:38:09,190
+هذا lower bound للست هذه، لو u أخدتها بساوي سالب
+
+382
+00:38:09,190 --> 00:38:16,450
+عشرة، برضه هذا lower bound آخر، وكذا، الان w بساوي
+
+383
+00:38:16,450 --> 00:38:21,710
+سفرهذا عبارة عن أكبر lower bound للمجموعة هذه أكبر
+
+384
+00:38:21,710 --> 00:38:27,990
+حد أدنى، تمام؟ إذن في حد أدنى وفي أكبر حد أدنى، في
+
+385
+00:38:27,990 --> 00:38:33,770
+حد أعلى وفي أصغر حد أعلى، تمام؟ هدولة مهمين، أصغر
+
+386
+00:38:33,770 --> 00:38:38,510
+حد أعلى و أكبر حد أدنى مهمين و هنطلق .. هنعطيهم
+
+387
+00:38:38,510 --> 00:38:45,090
+أسماء كمان شوية، لكن قبلخلّينا نكمل الملاحظات إذا
+
+388
+00:38:45,090 --> 00:38:48,650
+شفنا إحنا أن ال upper bound أو المجموعة ممكن يكون
+
+389
+00:38:48,650 --> 00:38:53,050
+لها upper bound أو مايكونش المجموعة ال upper bound
+
+390
+00:38:53,050 --> 00:38:55,750
+أو ال lower bound ممكن ينتمي للمجموعة و ممكن
+
+391
+00:38:55,750 --> 00:39:01,850
+ماينتميش لها الملاحظة التانية لو كان U upper bound
+
+392
+00:39:01,850 --> 00:39:06,930
+لست S فأي عدد أكبر منه بيطلع upper bound أيضا لنفس
+
+393
+00:39:06,930 --> 00:39:11,820
+الست يعني هي عنديإذا إحنا الآن في الملاحظة التانية
+
+394
+00:39:11,820 --> 00:39:18,140
+لو كان U upper bound لل 6 و أخدت V عدد أكبر من U
+
+395
+00:39:18,140 --> 00:39:24,060
+فال V أيضا هذا بيطلع upper bound لل 6 طيب السؤال
+
+396
+00:39:24,060 --> 00:39:32,760
+اللي بفرح نفسه كام عدد V ممكن ألاقي أكبر من U عدد
+
+397
+00:39:32,760 --> 00:39:39,010
+لانهائيإذا لو كان الست S لها upper bound واحد
+
+398
+00:39:39,010 --> 00:39:42,990
+فممكن ألاقي عدد لانهائي لها من ال upper bounds
+
+399
+00:39:42,990 --> 00:39:48,810
+طبعا؟ إذا قلنا كمان مرة لو كان ال U لها upper
+
+400
+00:39:48,810 --> 00:39:53,450
+bound أو الست لها upper bound U فأي عدد V أكبر من
+
+401
+00:39:53,450 --> 00:39:58,170
+U بيطلع upper bound وبالتالي لو كان الست لها upper
+
+402
+00:39:58,170 --> 00:40:01,450
+bound واحد فبيكون لها infinitely many upper bounds
+
+403
+00:40:02,480 --> 00:40:07,420
+بالمثل لو كان الست لها lower bound فبالتالي بيكون
+
+404
+00:40:07,420 --> 00:40:11,300
+لها infinitely many lower bounds عدد لانها in ال
+
+405
+00:40:11,300 --> 00:40:15,280
+lower bounds طيب المجموعة الخالية المجموعة الخالية
+
+406
+00:40:15,280 --> 00:40:23,000
+بتمتاز بإنه أي عدد حقيقي بيطلع upper bound و lower
+
+407
+00:40:23,000 --> 00:40:29,580
+bound في نفس الوقت هذا بده برهانة هذا مش واضحاللي
+
+408
+00:40:29,580 --> 00:40:32,640
+هو انتبهوا ويرجعوا لمبادئ الرياضيات وقوانين ال
+
+409
+00:40:32,640 --> 00:40:41,060
+logic هاي برهان بالتناقض انا عندي S مجموعة خالية
+
+410
+00:40:41,060 --> 00:40:47,860
+مجموعة محترمة هاي S بساوي فاي مجموعة جزئية من R
+
+411
+00:40:47,860 --> 00:40:54,760
+الفاي مجموعة جزئية محترمة من R هذه المجموعة اي عدد
+
+412
+00:40:54,760 --> 00:40:59,850
+حقيقي مهما كانهو lower bound و upper bound اللي في
+
+413
+00:40:59,850 --> 00:41:06,350
+نفس الوجود خلّينا نثبت أنه لو خدت أي عدد حقيقي R0
+
+414
+00:41:06,350 --> 00:41:16,970
+يعني تاني لـR فبدا أثبت أن هذا is upper bound of
+
+415
+00:41:16,970 --> 00:41:25,050
+Phi بدا أثبت أنه أي عدد حقيقي R0 بيطلع upper bound
+
+416
+00:41:25,050 --> 00:41:34,030
+لـPhiالبرهان ذلك بالتناقض افرض
+
+417
+00:41:34,030 --> 00:41:39,290
+ان ولا عدد حقيقي هو
+
+418
+00:41:39,290 --> 00:41:48,570
+upper bound لفاية يعني
+
+419
+00:41:48,570 --> 00:41:55,030
+يوجد عدد حقيقي R0 ليس upper bound
+
+420
+00:41:58,090 --> 00:42:02,030
+ففي الحالة هذه هذا معناه من تعريف ال upper bound
+
+421
+00:42:02,030 --> 00:42:06,990
+إذا كان R0 ليس upper bound لل set S التي هي Phi
+
+422
+00:42:06,990 --> 00:42:13,110
+معناته بقدر ألاقي أنصر في ال set أكبر من ال R0 ال
+
+423
+00:42:13,110 --> 00:42:20,110
+R0 لا يحد الأنصر من أعلىوهذا بدّيني تناقض، ليش
+
+424
+00:42:20,110 --> 00:42:25,810
+تناقض؟ لأن هذا أدى إلى أن يوجد S أنصر S في capital
+
+425
+00:42:25,810 --> 00:42:31,250
+S اللي هي Phi و Phi خالية، لا يوجد Phi خالية، لا
+
+426
+00:42:31,250 --> 00:42:39,910
+يوجدفيها او داخلها اي عنصر اسمه الـ S اذا الفرض
+
+427
+00:42:39,910 --> 00:42:46,990
+هذا خطأ وبالتالي كل العداد الحقيقية هي upper
+
+428
+00:42:46,990 --> 00:42:53,590
+bounds للمجموعة Phi بالمثل بطريقة مماثلة ممكن نثبت
+
+429
+00:42:53,590 --> 00:42:59,570
+ان نفس العدد او اي عدد R0 هو lower bound للمجموعة
+
+430
+00:42:59,570 --> 00:43:00,050
+Phi
+
+431
+00:43:04,940 --> 00:43:11,500
+طيب نرجع نيجي الأن ل ..
+
+432
+00:43:11,500 --> 00:43:23,360
+شوية تعريفات تانية المجموعة
+
+433
+00:43:23,360 --> 00:43:30,640
+طيب
+
+434
+00:43:30,640 --> 00:43:42,980
+المحاضرة يعنيانتهت تقريبا المرة الجاية هناخد تعريف
+
+435
+00:43:42,980 --> 00:43:50,820
+ال bounded set و تعريف ال supremum اللي هو least
+
+436
+00:43:50,820 --> 00:43:55,360
+upper bound او اصغر حد اعلى و ال optimum اللي هو
+
+437
+00:43:55,360 --> 00:44:01,250
+اكبر حد ادنىوندرس خاصة وبعدين نشوف ما هي ال
+
+438
+00:44:01,250 --> 00:44:05,670
+completeness property أو خاصية التمام اللي كانت
+
+439
+00:44:05,670 --> 00:44:08,590
+عنوان جانبي للجزء هذا من المحاضرة
+
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/xZkuM2ApdRM.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/xZkuM2ApdRM.srt
new file mode 100644
index 0000000000000000000000000000000000000000..d455fc7961ad4448cc5c5ce49bef286e98a32f1a
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/xZkuM2ApdRM.srt
@@ -0,0 +1,1603 @@
+1
+00:00:21,190 --> 00:00:25,470
+في المحاضرة السابقة تعرفنا على أنواع الفترات أو ال
+
+2
+00:00:25,470 --> 00:00:28,950
+intervals وشوفنا 
+
+3
+00:00:28,950 --> 00:00:33,290
+أن أول تلت أنواع من الفترات اللي هي ال open و ال
+
+4
+00:00:33,290 --> 00:00:37,110
+closed و ال half open أو half closed intervals كل
+
+5
+00:00:37,110 --> 00:00:43,790
+الفترات هذه كانت bounded intervals أو فترات محدودة
+
+6
+00:00:45,730 --> 00:00:50,070
+الأنواع الأربعة الأخرى من النوع الرابع إلى النوع
+
+7
+00:00:50,070 --> 00:00:55,750
+الثامن كلها فترات غير محدودة أو unbounded
+
+8
+00:00:55,750 --> 00:01:02,710
+intervals و الفترات هذه عبارة عن النوع الرابع open
+
+9
+00:01:02,710 --> 00:01:10,410
+left ray شعاع أيصر مفتوح أو open right ray شعاع
+
+10
+00:01:10,410 --> 00:01:16,650
+أيمن مفتوح أو closed left ray شعاع أيمن مغلق
+
+11
+00:01:16,650 --> 00:01:22,890
+closed right ray شعاع أيمن مغلق أو شعاع الجبلة شعاع
+
+12
+00:01:22,890 --> 00:01:27,140
+أيصر مغلق وطبعا هذه عبارة عن فترات زي ما احنا
+
+13
+00:01:27,140 --> 00:01:31,660
+شايفين النوع الأخير اللي هو مجموعة كل الأعداد اللي
+
+14
+00:01:31,660 --> 00:01:35,320
+حصلت فيها the set of all real numbers اللي هي
+
+15
+00:01:35,320 --> 00:01:39,160
+الفترة من سالب ما لا نهاية للملا نهاية وهذه طبعا
+
+16
+00:01:39,160 --> 00:01:45,560
+أيضا unbounded interval فترة غير محصورة أو غير
+
+17
+00:01:45,560 --> 00:01:49,640
+محدودة لو
+
+18
+00:01:49,640 --> 00:01:54,710
+بصينا لأي نوع من أنواع الثمانية هذه من الفترات لو
+
+19
+00:01:54,710 --> 00:02:02,590
+أخذنا أي نوع من الأنواع الثمانية فبنلاحظ
+
+20
+00:02:02,590 --> 00:02:10,850
+وسمينا الفترة هذه I لأن لو أخذنا one of the eight
+
+21
+00:02:10,850 --> 00:02:17,250
+types of intervals سمناها I فبنلاحظ أن الفترة I هذه
+
+22
+00:02:17,250 --> 00:02:26,690
+بتحقق الخاصية اللي هي أمامنا هذه وهي أنه لو كان في
+
+23
+00:02:26,690 --> 00:02:32,050
+عندي يعني
+
+24
+00:02:32,050 --> 00:02:38,130
+أي فترة زي هذه مثلا
+
+25
+00:02:38,130 --> 00:02:45,290
+الفترة
+
+26
+00:02:45,290 --> 00:02:50,130
+من A إلى مالنهاية لو النقطة هذه A
+
+27
+00:02:57,030 --> 00:03:01,490
+فهذا open right ray لو أخدت أي نقطتين في الفترة
+
+28
+00:03:01,490 --> 00:03:07,810
+هذه x و y واقعت
+
+29
+00:03:07,810 --> 00:03:13,790
+x و y ينتموا للفترة هذه فبنلاحظ أن الفترة المغلقة
+
+30
+00:03:13,790 --> 00:03:19,690
+ما بين x و y the closed interval from x و y is
+
+31
+00:03:19,690 --> 00:03:27,120
+contained الفترة المغلقة هذه بتكون دائما موجودة في
+
+32
+00:03:27,120 --> 00:03:35,620
+الفترة I لأن لو أخدت أي فترة I احنا أخدنا مثال ال
+
+33
+00:03:35,620 --> 00:03:41,100
+open right ray فبنلاحظ أن هذه الفترة أو أي فترة
+
+34
+00:03:41,100 --> 00:03:47,940
+ثانية من الأنواع الثمانية بتحقق الخاصية هذه اللي فوق
+
+35
+00:03:47,940 --> 00:03:53,080
+طبعا هذه احنا عطينا رقم أربعة بس عشان كبرنا الخط
+
+36
+00:03:55,050 --> 00:04:01,070
+فرقم أربعة مش ظاهر على الشاشة فأي فترة زي هذه
+
+37
+00:04:01,070 --> 00:04:05,310
+بتحقق خاصية أربعة وهو لو أخدت أي نقطتين في الفترة
+
+38
+00:04:05,310 --> 00:04:12,470
+I فالفترة اللي واقعة بين X وY أيضا بتكون موجودة
+
+39
+00:04:12,470 --> 00:04:20,290
+داخل الفترة I الآن العكس تبع الكلام هذا صحيح كما
+
+40
+00:04:20,290 --> 00:04:24,510
+هو موضح في نظرية واحد وعشرين
+
+41
+00:04:27,170 --> 00:04:34,990
+بمعنى أنه لو كان في عندي فترة لو كان في عندي أخدت
+
+42
+00:04:34,990 --> 00:04:39,990
+مجموعة جزئية من R المجموعة هذه عدد عناصرها أكبر من
+
+43
+00:04:39,990 --> 00:04:42,950
+أو يساوي اتنين ال cardinal number تبعها أكبر من أو
+
+44
+00:04:42,950 --> 00:04:46,890
+يساوي اتنين يعني المجموعة هذه فيها على الأقل عنصرين
+
+45
+00:04:46,890 --> 00:04:51,550
+أو أكثر ممكن تكون finite set ممكن تكون infinite
+
+46
+00:04:53,020 --> 00:04:55,760
+يكون فيها على الأقل عنصرين فلو كان في عندي
+
+47
+00:04:55,760 --> 00:04:58,980
+مجموعة جزئية من الـ R فيها على الأقل عنصرين و
+
+48
+00:04:58,980 --> 00:05:03,600
+بتحقق الخاصية أربعة فلازم المجموعة الجزئية هذه
+
+49
+00:05:03,600 --> 00:05:11,820
+تطلع فترة و البرهان تبع النظرية هذه مش صعب و ممكن
+
+50
+00:05:11,820 --> 00:05:15,840
+يعني احنا كاتبينه بالتفصيل لكم
+
+51
+00:05:18,790 --> 00:05:26,110
+وقلنا أنتم بمكانكم تقرؤوه تفهموه البرهان
+
+52
+00:05:26,110 --> 00:05:32,150
+بنقسّمه لأربع حالات المجموعة S هذه اللي احنا عايزين
+
+53
+00:05:32,150 --> 00:05:36,190
+نثبت أنها interval فيها أربع احتمالات إما بتكون
+
+54
+00:05:36,190 --> 00:05:40,150
+bounded يعني محدودة من الجهتين
+
+55
+00:05:42,590 --> 00:05:48,630
+أو بتكون bounded الاحتمال الثاني تكون bounded above
+
+56
+00:05:48,630 --> 00:05:54,090
+but not below أو احتمال ثالث أنها تكون bounded
+
+57
+00:05:54,090 --> 00:05:58,330
+below but not above الاحتمال الرابع أنها is
+
+58
+00:05:58,330 --> 00:06:02,250
+neither bounded above nor below يعني لا محدودة من
+
+59
+00:06:02,250 --> 00:06:06,770
+أسفل ولا من أعلى وفي
+
+60
+00:06:06,770 --> 00:06:11,110
+كل حالة من الحالات الأربع في كل حالة من الحالات
+
+61
+00:06:11,110 --> 00:06:18,580
+الأربع هي الحالة الأولى S bounded في كل حالة من
+
+62
+00:06:18,580 --> 00:06:24,580
+الحالات الأربع بنحاول نثبت أن الـ S هي فترة ما
+
+63
+00:06:24,580 --> 00:06:32,320
+فترة معينة فمثلا لنأخذ نشوف الحالة الأولى الحالة
+
+64
+00:06:32,320 --> 00:06:36,260
+الأولى أن S تكون bounded وبالتالي bounded above
+
+65
+00:06:36,260 --> 00:06:39,940
+and bounded below بما أنها bounded below إذا ال
+
+66
+00:06:39,940 --> 00:06:44,720
+infimum تبعها exists سميّه a بما أنها bounded above
+
+67
+00:06:44,720 --> 00:06:48,440
+إذا بالـ supremum property ال supremum تبعها exists
+
+68
+00:06:48,440 --> 00:06:56,460
+سميّه b وبالتالي S بتصير محصورة داخل أو subset من
+
+69
+00:06:56,460 --> 00:07:02,800
+الفترة المغلقة AB طبعا لو قرأته أنتم التفاصيل
+
+70
+00:07:02,800 --> 00:07:07,980
+هتطلع الفترة الـ S هذه في نهاية البرهان هتطلع فترة
+
+71
+00:07:07,980 --> 00:07:19,540
+half open زي هذه أو half open زي هذه هو
+
+72
+00:07:19,540 --> 00:07:26,700
+بالتالي فترة وهو المطلوب في الحالة الثانية اللي
+
+73
+00:07:26,700 --> 00:07:32,300
+هنأخذ الحالة الثانية أن الـ set S is bounded above
+
+74
+00:07:32,300 --> 00:07:37,270
+but not bounded below محدودة من أعلى لكن مش محدودة
+
+75
+00:07:37,270 --> 00:07:43,390
+من أسفل في الحالة هذه هتطلع الـ S هذه فترة وهتطلع
+
+76
+00:07:43,390 --> 00:07:48,770
+هتطلع left ray left ray لأنها مش محدودة من أسفل
+
+77
+00:07:48,770 --> 00:07:56,290
+فيعني لو أنتم تابعتوا البرهان هتطلع الفترة أو ال
+
+78
+00:07:56,290 --> 00:07:59,250
+S هي عبارة عن إما
+
+79
+00:08:08,630 --> 00:08:14,990
+هتطلع الـ set is عبارة عن closed left ray زي هذا
+
+80
+00:08:14,990 --> 00:08:20,870
+أو open left ray زي هذا فهنسيبكم
+
+81
+00:08:20,870 --> 00:08:25,130
+تقراوا البرهان مش صعب أكيد كل واحد منكم هيتفهمه
+
+82
+00:08:25,130 --> 00:08:30,450
+الآن الحالات الثالثة والرابعة شبيهة بالحالات
+
+83
+00:08:30,450 --> 00:08:36,270
+الأولى والثانية وبالتالي هنسيبها إليكم كتمرين
+
+84
+00:08:36,270 --> 00:08:40,940
+exercise تتدربوا على كتابة أو تحاولوا تكتبوا
+
+85
+00:08:40,940 --> 00:08:46,720
+البرهان هيكون شبيه بالحالتين السابقتين okay أي
+
+86
+00:08:46,720 --> 00:08:52,900
+طالبة أُسِيب تمرين زي هذا مطالبة أن تكتب يعني تحل
+
+87
+00:08:52,900 --> 00:08:59,040
+التمرين هذا على الأقل عشان تفهم المادة إذا في عندك
+
+88
+00:08:59,040 --> 00:09:04,340
+أي تساؤل أو استفسار تتصل فيا أو ممكن تكتب البرهان
+
+89
+00:09:04,340 --> 00:09:10,150
+على ورقة وتعطيها لي في أقرب فرصة وأنا بأراجع لكِ في
+
+90
+00:09:10,150 --> 00:09:17,510
+المحاضرة اللي بعدها تمام؟ إذا الحالات الثالثة و
+
+91
+00:09:17,510 --> 00:09:20,970
+الرابعة شبيهة بالحالات الأولى والثانية وبالتالي
+
+92
+00:09:20,970 --> 00:09:28,910
+هنسيبها .. هنتركها للطالب كتمرين في عندي هنا تعريف
+
+93
+00:09:43,000 --> 00:09:47,580
+لو أخدت sequence of intervals I N هذه عبارة عن
+
+94
+00:09:47,580 --> 00:09:52,920
+فترة الآن هذه عبارة عن sequence على سرها فترات I N
+
+95
+00:09:52,920 --> 00:09:58,640
+فالـ sequence هذه بنسميها nested الـ sequence of
+
+96
+00:09:58,640 --> 00:10:04,690
+intervals هذه بنسميها nested إذا كانت بتحقق الخاصية
+
+97
+00:10:04,690 --> 00:10:08,670
+هذه وهي أن الفترة الأولى تحتوي الثانية والثانية
+
+98
+00:10:08,670 --> 00:10:13,690
+تحتوي الثالثة وهكذا أو بمعنى آخر الفترة رقم n I n
+
+99
+00:10:13,690 --> 00:10:18,410
+contains الفترة اللي بعدها مباشرة I n زائد واحد لكل
+
+100
+00:10:18,410 --> 00:10:24,420
+n إذا الفترات أو sequence of intervals that
+
+101
+00:10:24,420 --> 00:10:29,280
+satisfies this property بنسميها nested it's called
+
+102
+00:10:29,280 --> 00:10:35,600
+nested sequence of intervals هذه مثال مثال على
+
+103
+00:10:35,600 --> 00:10:42,500
+sequence of nested intervals لو أخدت الفترة I N هي
+
+104
+00:10:42,500 --> 00:10:46,400
+الفترة المغلقة من صفر لواحد على N حيث N أي عدد
+
+105
+00:10:46,400 --> 00:10:55,860
+طبيعي فالفترات هذه واضح جدا أنها nested هاي خط
+
+106
+00:10:55,860 --> 00:11:10,380
+الاعداد هاي خط الاعداد هاي صفر واحد هاي فترة هادي I
+
+107
+00:11:10,380 --> 00:11:16,200
+واحد خد إن بس I واحد I واحد فترة مغلقة من صفر ل
+
+108
+00:11:16,200 --> 00:11:23,640
+واحد I اثنين خد in بساوي اثنين معناته هذا نص يعني في
+
+109
+00:11:23,640 --> 00:11:34,520
+عندي فترة الفترة هذه I اثنين I I اثنين I واحد
+
+110
+00:11:34,520 --> 00:11:40,000
+تحتوي I اثنين لو أخدت in ثلاثة بيصير في عندي هنا
+
+111
+00:11:40,000 --> 00:11:43,180
+ثلث وفي الفترة هذه
+
+112
+00:11:45,730 --> 00:11:53,470
+هذه الفترة I ثلاثة و I ثلاثة واضح هنا contained I
+
+113
+00:11:53,470 --> 00:11:59,530
+اثنين وهكذا إذا واضح هنا أن I one contains I two
+
+114
+00:11:59,530 --> 00:12:07,270
+contains I three and so on وهكذا وبالتالي ال
+
+115
+00:12:07,270 --> 00:12:16,350
+sequence of intervals هذه تطلع nested تمام؟ طيب ال
+
+116
+00:12:16,350 --> 00:12:21,270
+sequence هذه مش بس nested كمان التقاطع تبعها لا
+
+117
+00:12:21,270 --> 00:12:27,430
+يساوي فاي يعني لو قطعت الفترات هذه كلها فهنُجد أنه
+
+118
+00:12:27,430 --> 00:12:31,230
+التقاطع تبعها لا يساوي فاي يعني ممكن ألاقي على
+
+119
+00:12:31,230 --> 00:12:36,410
+الأقل عنصر واحد في التقاطع in fact في حقيقة الأمر
+
+120
+00:12:36,410 --> 00:12:42,810
+التقاطع هذا بيساوي singleton set zero يعني هنا في
+
+121
+00:12:42,810 --> 00:12:48,530
+الصفر هذا ينتمي للتقاطع هذا طبعا الكلام مش واضح
+
+122
+00:12:48,530 --> 00:12:54,750
+هذا ليس واضحا فعشان نوضحه أو نبرهن له to see this
+
+123
+00:12:54,750 --> 00:13:03,170
+لإثبات أن التقاطع هذا الفترات هذه بيساوي single
+
+124
+00:13:03,170 --> 00:13:09,790
+ton consist zero فبنثبت
+
+125
+00:13:09,790 --> 00:13:19,090
+أن مجموعة بِتسَاوِي مجموعة عندي الاحتواء هذا واضح واضح أن
+
+126
+00:13:19,090 --> 00:13:30,410
+الصفر ينتمي للفترة صفر واحد على n اللي هي I n
+
+127
+00:13:30,410 --> 00:13:39,690
+لكل n في n وبالتالي هذا بيقدم صفر ينتمي لتقاطع
+
+128
+00:13:39,690 --> 00:13:45,490
+كل الـ I n وبالتالي المجموعة اللي فيها العنصر الوحيد
+
+129
+00:13:45,490 --> 00:13:55,690
+صفر subset من تقاطع المجموعات I n تمام هذا واضح لكن
+
+130
+00:13:55,690 --> 00:14:01,970
+مش واضح أنه نثبت الآن العكس أو ال reverse
+
+131
+00:14:01,970 --> 00:14:06,270
+inclusion عشان نثبت مساواة أن الـ set هذه بِتسَاوِي 
+
+132
+00:14:06,270 --> 00:14:13,480
+هذه باقي نثبت الـ inclusion هذه تمام فخلّينا ناخد
+
+133
+00:14:13,480 --> 00:14:22,080
+let x تنتمي لتقاطع كل الـ I n هذا معناه أن X
+
+134
+00:14:22,080 --> 00:14:27,960
+تنتمي لـ I n لكل N طيب ما الـ I n هي عبارة عن صفر
+
+135
+00:14:27,960 --> 00:14:34,380
+فترة مغلقة من صفر ل 1 على N هذا معناه أن X أكبر من
+
+136
+00:14:34,380 --> 00:14:39,500
+أو يساوي صفر أصغر من أو يساوي 1 على N لكل N
+
+137
+00:14:44,000 --> 00:14:50,260
+تمام الآن أنا بُثبت أن
+
+138
+00:14:50,260 --> 00:14:56,060
+الـ X هذا بيساوي صفر إيش بُثبت أنا بُثبت الاحتواء
+
+139
+00:14:56,060 --> 00:15:00,900
+المعاكس أخذت X عنصر في التقاطع بتثبت أن X ينتمي
+
+140
+00:15:00,900 --> 00:15:04,900
+للمجموعة عادي يعني X بيساوي صفر لأن هذه المجموعة
+
+141
+00:15:04,900 --> 00:15:08,640
+عاملة ما فيها غير صفر إذا لو أثبتت أن X بيساوي
+
+142
+00:15:08,640 --> 00:15:15,780
+صفر بكون أثبتت الاحتواء المعاكس، طيب الآن نعمل برهان
+
+143
+00:15:15,780 --> 00:15:21,800
+بالتناقض، assume نفترض
+
+144
+00:15:21,800 --> 00:15:27,880
+أن x لا يساوي صفر طب
+
+145
+00:15:27,880 --> 00:15:31,780
+أنا عندي الـ x هنا أكبر من أو يساوي صفر والآن 
+
+146
+00:15:31,780 --> 00:15:34,860
+بيستويش صفر، إذا الـ x أكبر من صفر
+
+147
+00:15:40,290 --> 00:15:44,590
+طيب by Archimedean property by Archimedean
+
+148
+00:15:44,590 --> 00:15:47,810
+property 
+
+149
+00:15:47,810 --> 00:15:53,990
+هذه اللي حاطين عليها علامة النجمة، إيش الـ
+
+150
+00:15:53,990 --> 00:15:58,830
+Archimedean property بتقول لأي عدد موجب زي X هذا
+
+151
+00:15:58,830 --> 00:16:09,170
+بنقدر نلاقي يوجد عدد طبيعي نسميه N0 بحيث إنه مقلوب
+
+152
+00:16:09,170 --> 00:16:18,310
+العدد الطبيعي N0 أصغر من الـ X من العدد الموجب، وهذا 
+
+153
+00:16:18,310 --> 00:16:23,710
+بيؤدي لتناقض، contradiction هذا بيؤدي لتناقض
+
+154
+00:16:23,710 --> 00:16:28,770
+contradiction، ليه؟ لأنه أنا عندي من هنا هذا الـ
+
+155
+00:16:28,770 --> 00:16:32,770
+statement الأخير، هذا الـ statement الأخير هذا
+
+156
+00:16:32,770 --> 00:16:36,960
+بتناقض مع الـ statement اللي هنا الآن هذا الـ
+
+157
+00:16:36,960 --> 00:16:41,780
+statement بيقول بما أن N0 هذا عدد طبيعي إذا لازم X
+
+158
+00:16:41,780 --> 00:16:47,580
+تكون أصغر من أو تساوي واحد على N0، إذا X أصغر من أو 
+
+159
+00:16:47,580 --> 00:16:51,360
+تساوي واحد على N0، X أكبر من واحد على N0، هذا تناقض
+
+160
+00:16:51,360 --> 00:17:00,340
+إذا التناقض هذا سببه مين؟ الـ assumption تبعنا هذا إن
+
+161
+00:17:00,340 --> 00:17:05,960
+X لا تساوي صفر، إذا الصح إن X تساوي صفر كما هو
+
+162
+00:17:05,960 --> 00:17:09,660
+مطلوب، وبالتالي هيك منكون أثبتنا إن كل X في التقاطع
+
+163
+00:17:09,660 --> 00:17:15,260
+بيساوي صفر، يعني كل عنصر في التقاطع موجود في
+
+164
+00:17:15,260 --> 00:17:20,740
+المجموعة هذه، إذا هيك منكون أثبتنا المساواة إن
+
+165
+00:17:20,740 --> 00:17:26,320
+المجموعة هذه فعلاً أو التقاطع تبع الفترة I N بيساوي
+
+166
+00:17:26,320 --> 00:17:28,200
+Singleton Zero
+
+167
+00:17:33,320 --> 00:17:51,320
+هذا المثال بيأسس أو هو حافظ لنظرية رقم 22، بنلاحظ
+
+168
+00:17:51,320 --> 00:17:56,220
+في المثال هذا إن الفترات هذه، طبعاً شفنا إن هي nested
+
+169
+00:17:56,220 --> 00:18:03,170
+متداخلة، يعني nested معناها متداخلة بالعربي، بعدين
+
+170
+00:18:03,170 --> 00:18:09,630
+واضح إن الفترات هذه bounded محصورة، وكذلك الفترات
+
+171
+00:18:09,630 --> 00:18:16,830
+هذه closed فترات مغلقة صح، فالنظرية هذه بتعمم
+
+172
+00:18:16,830 --> 00:18:21,510
+المعلومات اللي هنا أو النتيجة اللي هنا اللي شفناها
+
+173
+00:18:21,510 --> 00:18:26,350
+في المثال، فالخاصية هذه بتسميها nested interval
+
+174
+00:18:26,350 --> 00:18:27,010
+property
+
+175
+00:18:30,610 --> 00:18:34,550
+ماهي النظرية اللي بتقول لو في اندي سيكوانس of
+
+176
+00:18:34,550 --> 00:18:39,890
+closed, bounded و nested intervals nested
+
+177
+00:18:39,890 --> 00:18:45,430
+intervals و closed و bounded، تلت إيه؟ تلت صفات
+
+178
+00:18:45,430 --> 00:18:50,890
+فلازم التقاطع تبع الفترات هذه يكون غير خالي، non
+
+179
+00:18:50,890 --> 00:18:54,990
+-empty يعني نقدر نلاقي عنصر واحد على الأقل صي
+
+180
+00:18:54,990 --> 00:18:56,970
+ينتمي للتقاطع
+
+181
+00:18:59,530 --> 00:19:06,510
+إضافة لذلك، moreover لو الـ infimum للأعداد غير
+
+182
+00:19:06,510 --> 00:19:13,690
+السالبة هذه، هذه أعداد غير سالبة إذا 
+
+183
+00:19:13,690 --> 00:19:17,290
+كان الـ infimum للـ non-negative real numbers هذه للـ
+
+184
+00:19:17,290 --> 00:19:20,450
+sequence of non-negative real numbers بيساوي صفر
+
+185
+00:19:21,930 --> 00:19:26,170
+فالعدد ساي هذا بيكون وحيد، يعني التقاطع هذا ما فيش
+
+186
+00:19:26,170 --> 00:19:31,730
+فيه إلا عنصر واحد بالضبط زي ما شوفنا هنا، إذا إن
+
+187
+00:19:31,730 --> 00:19:38,330
+الجزء الثاني، الجزء الثاني من النظرية لو فرضنا إن
+
+188
+00:19:38,330 --> 00:19:42,890
+الـ infimum للعداد غير السالب بهذا بيساوي صفر فالـ
+
+189
+00:19:42,890 --> 00:19:46,110
+ساي بتكون هي نقطة الوحيدة الموجودة في التقاطع تبع
+
+190
+00:19:46,110 --> 00:19:49,770
+الفترات زي ما في المثال بالظبط، الصفر هي النقطة 
+
+191
+00:19:49,770 --> 00:19:57,090
+الوحيدة في تقاطع الفترات المتداخلة، نثبت النظرية هذه
+
+192
+00:20:25,320 --> 00:20:30,080
+فلبرهان النظرية، أنا عندي الفترات هذه nested
+
+193
+00:20:30,080 --> 00:20:38,560
+وبالتالي I1 هتكون أكبر فترة تحتوي I2 و I3 و تحتوي
+
+194
+00:20:38,560 --> 00:20:44,100
+كل الـ IN لكل الأعداد الطبيعي، هذا أكيد يعني
+
+195
+00:20:44,100 --> 00:20:49,920
+وبالتالي هاي عندي هاي الفترة I1
+
+196
+00:20:53,730 --> 00:21:05,150
+هذه الفترة I1 تحتوي الفترة I N وهذه الفترة I N
+
+197
+00:21:05,150 --> 00:21:13,070
+وهذه I 1، فواضح 
+
+198
+00:21:13,070 --> 00:21:18,870
+أن A N هتكون أصغر من أو تساوي B 1 لكل N في M
+
+199
+00:21:21,540 --> 00:21:27,040
+وبالتالي B 1 عبارة عن upper bound لكل العناصر
+
+200
+00:21:27,040 --> 00:21:33,280
+AN، إذا الـ set هذه AN حيث إن عدد الطبيعي is bounded
+
+201
+00:21:33,280 --> 00:21:39,520
+above by B 1 وبالتالي حسب الـ supremum property
+
+202
+00:21:39,520 --> 00:21:44,500
+الـ set هذه بما أنها bounded above، إذا الـ supremum
+
+203
+00:21:44,500 --> 00:21:50,000
+تبعها exist، إذا يوجد عدد حقيقي R هو supremum للـ
+
+204
+00:21:50,000 --> 00:21:56,950
+set أو الـ set هذه لها supremum سميه ساي، بما أن هذا
+
+205
+00:21:56,950 --> 00:22:01,530
+الـ ساي supremum لكل
+
+206
+00:22:01,530 --> 00:22:07,570
+عناصر الـ set AN فهو upper bound، لذلك الـ ساي أكبر من
+
+207
+00:22:07,570 --> 00:22:14,930
+أو تساوي كل عناصر الـ set الآن
+
+208
+00:22:28,600 --> 00:22:36,720
+طبعاً هذه الـ .. لأن أنا في عندي a n، أثبتنا أن a n
+
+209
+00:22:36,720 --> 00:22:44,080
+أصغر من أو تساوي ساي لكل n في N، هذه
+
+210
+00:22:44,080 --> 00:22:51,700
+رقمها خمسة، متبينة طبعاً الرقم مش غير الآن
+
+211
+00:22:51,700 --> 00:22:52,660
+لو أثبتنا
+
+212
+00:22:58,370 --> 00:23:01,830
+احنا عايزين نثبت إنه الـ .. احنا عايزين في النهاية
+
+213
+00:23:01,830 --> 00:23:08,270
+نثبت إن الـ ساي هذا ينتمي لـ I N اللي هو بيساوي
+
+214
+00:23:08,270 --> 00:23:16,850
+الفترة المغلقة من A N لـ B N لكل N في N وبالتالي
+
+215
+00:23:16,850 --> 00:23:21,930
+تقاطع الفترات هذه يحتوي ساي صح؟ هذا اللي احنا
+
+216
+00:23:21,930 --> 00:23:27,050
+عايزين نثبته، طيب، بنهينا أثبتنا إن ساي أكبر من أو
+
+217
+00:23:27,050 --> 00:23:36,150
+تساوي A N، إذا عايزين نثبت إذا عشان نكمل البرهان في 
+
+218
+00:23:36,150 --> 00:23:43,390
+الجزء الأول إذا هنا to prove عشان
+
+219
+00:23:43,390 --> 00:23:56,250
+نثبت الكلام هذا اه we need to show إنه
+
+220
+00:23:56,250 --> 00:24:09,930
+الأصغر من أو تساوي BN لكل N عدد طبيعي، نسمي هذه 6 لو
+
+221
+00:24:09,930 --> 00:24:16,430
+أثبتنا المتباينة 6 لكل N فمعناته الـ Psi أكبر من
+
+222
+00:24:16,430 --> 00:24:20,270
+أو تساوي AN أصغر من أو تساوي BN يعني تنتمي للفترة
+
+223
+00:24:20,270 --> 00:24:27,300
+المغلقة هذه، وبالتالي هذا بقدّي أن التقاطع على كل آي
+
+224
+00:24:27,300 --> 00:24:36,460
+ان بيساوي ساي لأ أو يحتوي ساي، تنتمي للتقاطع على كل
+
+225
+00:24:36,460 --> 00:24:44,120
+آي ان وهذا بيكون برهان، هذا بيكمل برهان الجزء الأول
+
+226
+00:24:45,450 --> 00:24:50,850
+إذا احنا عايزين نثبت المتباينة ستة هذه طلعت خمسة 
+
+227
+00:24:50,850 --> 00:24:55,590
+وهذه طلعت لوحدها، باقي نثبت المتباينة هذه ستة
+
+228
+00:24:55,590 --> 00:25:06,630
+فالإثبات المتباينة الستة هذه يكفي لإثبات
+
+229
+00:25:06,630 --> 00:25:08,230
+المتباينة ستة هذه
+
+230
+00:25:13,060 --> 00:25:23,440
+يكفي إثبات المتباينة هذه لكل K، لو
+
+231
+00:25:23,440 --> 00:25:32,040
+أنا ثبتت الـ N وأثبتت أن BN هذه أكبر من أو تساوي كل
+
+232
+00:25:32,040 --> 00:25:37,760
+العناصر A K لكل K عدد طبيعي معناته BN هذه upper
+
+233
+00:25:37,760 --> 00:25:44,110
+bound، Upper bound للمجموعة هذه، طب احنا قبل شوية قلنا
+
+234
+00:25:44,110 --> 00:25:49,930
+إن الـ supremum للمجموعة هذه هو Psi، أنا عندي Psi
+
+235
+00:25:49,930 --> 00:25:57,450
+بيساوي الـ supremum، الـ supremum للمجموعة هذه بيساوي
+
+236
+00:25:57,450 --> 00:26:05,330
+Psi، فلو أثبتت إن B N upper bound للمجموعة هذه والـ 
+
+237
+00:26:05,330 --> 00:26:10,230
+Psi هو أصغر upper bound، معناته الـ Psi هيطلع أصغر 
+
+238
+00:26:10,230 --> 00:26:13,930
+من أو يساوي الـ upper bound BN، وبالتالي هيكم نكون 
+
+239
+00:26:13,930 --> 00:26:20,160
+أثبتنا ستة، واضحة النقطة هذه كمان مرة لإثبات أن 
+
+240
+00:26:20,160 --> 00:26:26,180
+المتباينة 6 هذه يكفي أن احنا نثبت إنه لكل fixed 
+
+241
+00:26:26,180 --> 00:26:30,840
+لكل fixed N الـ BN هذه upper bound للـ 6 هذه
+
+242
+00:26:30,840 --> 00:26:35,100
+وبالتالي الـ least upper bound للـ 6 هذه اللي هو
+
+243
+00:26:35,100 --> 00:26:40,900
+صغير بيطلع أصغر من أو يساوي BN، طبعاً إذا باقي نثبت A
+
+244
+00:26:40,900 --> 00:26:47,600
+المتباينة اللي هنا، لثبات المتباينة هذه بنثبت N و
+
+245
+00:26:47,600 --> 00:26:53,740
+بناخد أي arbitrary natural number K فبنلاحظ إن الـ
+
+246
+00:26:53,740 --> 00:26:58,720
+N أما هتكون أصغر من أو تساوي K أو N أكبر من K by 
+
+247
+00:26:58,720 --> 00:27:04,040
+trichotomy property إن K عدد طبيعي إذا أما N أصغر
+
+248
+00:27:04,040 --> 00:27:10,520
+من أو تساوي K أو N أكبر من K، طيب في الحالة الأولى
+
+249
+00:27:10,520 --> 00:27:16,920
+لو كانت الـ N أصغر من أو تساوي K، احنا عارفين إن 
+
+250
+00:27:16,920 --> 00:27:20,840
+الفترات هذه nested وبالتالي الفترة اللي الـ index
+
+251
+00:27:20,840 --> 00:27:26,380
+تبعها أكبر تطلع هي الأصغر تمام؟ 
+
+252
+00:27:26,380 --> 00:27:34,690
+إذا أنا بطلع عندي I K subset من I N وبالتالي هاي 
+
+253
+00:27:34,690 --> 00:27:45,030
+نرسم هاي IK هاي الفترة IK، نقاط
+
+254
+00:27:45,030 --> 00:27:52,770
+أطرافها AK BK وهاي الفترة وهذه
+
+255
+00:27:52,770 --> 00:28:00,330
+محتوى داخل الفترة IN، هاي الفترة IN، كبّر
+
+256
+00:28:00,330 --> 00:28:14,950
+شوية، هذه الفترة IN هذه، نقاط أطرافها، فبما 
+
+257
+00:28:14,950 --> 00:28:20,520
+أن الفترة IK subset من الفترة IN، إذا اللي بطلع عندي
+
+258
+00:28:20,520 --> 00:28:26,840
+AK أصغر من أو تساوي BK و BK أصغر من أو تساوي BN
+
+259
+00:28:26,840 --> 00:28:34,060
+وبالتالي هيني أثبتت أن AK أصغر من أو تساوي BN كما
+
+260
+00:28:34,060 --> 00:28:40,540
+هو مطلوب، طيب في الحالة الثانية افرض أن N هي اللي
+
+261
+00:28:40,540 --> 00:28:46,110
+أكبر من K، ففي الحالة هذه بما إن الـ sequence in
+
+262
+00:28:46,110 --> 00:29:02,270
+nested إذا IN محتوى داخل IK، إذا
+
+263
+00:29:02,270 --> 00:29:06,150
+هاي عندي IN
+
+264
+00:29:11,080 --> 00:29:20,360
+الفترة IN هذا هي محتوى داخل الفترة IK
+
+265
+00:29:20,360 --> 00:29:23,660
+يعني
+
+266
+00:29:23,660 --> 00:29:28,880
+نقاط أطرافها AK BK، وبالتالي في الحالة هذه بيطلع AK
+
+267
+00:29:28,880 --> 00:29:36,400
+أصغر من أو تساوي AN أصغر من أو تساوي BN مظبوط؟ إذا 
+
+268
+00:29:36,400 --> 00:29:39,780
+هنا في الحالة الثانية برضه أثبتنا إن AK أصغر من أو
+
+269
+00:29:39,780 --> 00:29:45,680
+تساوي BN، إذا في الحالتين أنا أثبتت إن AK أصغر من أو
+
+270
+00:29:45,680 --> 00:29:57,480
+تساوي BN وهذا صحيح لكل for any K إذا هذا معناه إنه
+
+271
+00:29:57,480 --> 00:30:02,440
+من هنا B زي ما قلنا BN is upper bound للـ set هذه
+
+272
+00:30:03,020 --> 00:30:07,480
+وبالتالي الـ supremum تبع الـ 6 اللي هو Psi بيطلع
+
+273
+00:30:07,480 --> 00:30:14,440
+أصغر من أو تساوي BN، إذن هذا بيثبت اللي هو المتباينة
+
+274
+00:30:14,440 --> 00:30:20,740
+6 الأقل من 6 .. من 5 و 6 زي ما شرحنا بنحصل على إن
+
+275
+00:30:20,740 --> 00:30:21,580
+الـ Psi
+
+276
+00:30:29,990 --> 00:30:35,610
+من خمسة وستة بنحصل على إن الـ Psi هذا هي أكبر من 
+
+277
+00:30:35,610 --> 00:30:39,050
+أو التساوي AN أصغر من أو التساوي BN، يعني الـ Psi 
+
+278
+00:30:39,050 --> 00:30:45,330
+تنتمي لتقاطع الفترات كما هو مطلوب، هذا بيثبت أو
+
+279
+00:30:45,330 --> 00:30:51,210
+بيبرهن الجزء الأول من النظرية، لبرهان الجزء الثاني 
+
+280
+00:31:03,930 --> 00:31:10,910
+الجزء الثاني بإن
+
+281
+00:31:10,910 --> 00:31:15,210
+نثبت أن التقاطة هذا الـ psi هذه اللي طلعناها هي
+
+282
+00:31:15,210 --> 00:31:17,470
+النقطة الوحيدة في التقاطة ما فيش غيرها
+
+283
+00:31:20,460 --> 00:31:25,380
+طبعًا هنا الهدف نحصل عليه تحت الشرط الإضافي أن ال
+
+284
+00:31:25,380 --> 00:31:30,000
+minimum للأعداد الغير سالبة هذه لاحظوا أن أن أن أن
+
+285
+00:31:30,000 --> 00:31:33,040
+أن أن أن أن أن أن أن أن أن أن أن أن أن أن أن أن
+
+286
+00:31:33,040 --> 00:31:34,180
+أن أن أن أن أن أن أن أن أن أن أن أن أن أن أن أن
+
+287
+00:31:34,180 --> 00:31:39,160
+أن أن أن أن
+
+288
+00:31:39,160 --> 00:31:39,180
+أن أن أن أن أن أن أن أن أن أن أن أن أن أن أن أن
+
+289
+00:31:39,180 --> 00:31:41,160
+أن أن أن أن أن أن أن أن أن أن أن أن أن أن أن أن
+
+290
+00:31:41,160 --> 00:31:41,360
+أن أن أن أن أن أن أن أن أن أن أن أن أن أن أن أن
+
+291
+00:31:41,360 --> 00:31:41,480
+أن أن أن أن أن أن أن أن أن أن أن أن أن أن أن أن
+
+292
+00:31:49,520 --> 00:31:54,560
+إذا لو كان الـ infimum من الأعداد السالبة دي بيساوي صفر
+
+293
+00:31:54,560 --> 00:32:00,660
+فبنثبت أنه التقاطع في نقطة واحدة هي psi طيب
+
+294
+00:32:00,660 --> 00:32:06,260
+لبراهين ذلك بنلاحظ أن الـ set هذه غير خالية طبعًا لأن
+
+295
+00:32:06,260 --> 00:32:10,200
+أنا عندي الـ sequence of intervals اللي لها نقاط
+
+296
+00:32:10,200 --> 00:32:14,840
+أطراف فالـ sequence هذه فيها على الأقل فترة واحدة
+
+297
+00:32:16,200 --> 00:32:23,600
+هذه المجموعة غير خالية وبعدها bounded below by a1
+
+298
+00:32:23,600 --> 00:32:33,700
+يعني احنا عارفين أن a1 هي a1 أو
+
+299
+00:32:33,700 --> 00:32:38,460
+الفترة I1 هذه تحتوي كل الفترات
+
+300
+00:32:42,320 --> 00:32:49,760
+تحتوي كل الفترات I N وبالتالي
+
+301
+00:32:49,760 --> 00:32:59,020
+أنا عندي A1 من هنا بتطلع عندي A1 أصغر من أو يساوي
+
+302
+00:32:59,020 --> 00:33:09,420
+B N لكل N في N وبالتالي a1 is a lower bound أو ال
+
+303
+00:33:09,420 --> 00:33:15,880
+set of all b in is bounded below by a1 وبالتالي by
+
+304
+00:33:15,880 --> 00:33:23,160
+the infimum property يوجد عدد حقيقي eta وهذا هو ال
+
+305
+00:33:23,160 --> 00:33:31,030
+infimum لـ set of all b in حيث أن عدد طبيعي إن هنا ال
+
+306
+00:33:31,030 --> 00:33:36,110
+set هذي الـ set of all b in حيث in natural number
+
+307
+00:33:36,110 --> 00:33:40,270
+bounded below وبالتالي الـ infimum تبعها exist
+
+308
+00:33:40,270 --> 00:33:42,790
+خلينا نسميه إيتا
+
+309
+00:33:45,470 --> 00:33:52,890
+الآن by an argument by an argument similar to the
+
+310
+00:33:52,890 --> 00:33:57,190
+proof of six احنا شوفنا قبل شوية متباينة six
+
+311
+00:33:57,190 --> 00:34:07,030
+أثبتناها فباستخدام برهان شبيه ببرهان المتباينة six
+
+312
+00:34:07,030 --> 00:34:13,770
+ممكن نصل إلى متباينة زي المتباينة six اللي هي هذه
+
+313
+00:34:13,770 --> 00:34:20,330
+أن a n تطلع أصغر من أو يساوي إيتا لكل n لأن هذه
+
+314
+00:34:20,330 --> 00:34:27,250
+برهانها زي برهان ستة خلينا نسميها ستة prime
+
+315
+00:34:27,250 --> 00:34:33,690
+برهان هذه نرجع لبرهان ستة ونشوف كيف برهنها ونعمل
+
+316
+00:34:33,690 --> 00:34:34,570
+برهان مشابه
+
+317
+00:34:37,400 --> 00:34:41,240
+Okay، إذا الـ ETA هنا، العدد ETA ده اللي هو الـ
+
+318
+00:34:41,240 --> 00:34:47,440
+infimum للـ set of all BN هذا إيش بيطلع من هنا، من
+
+319
+00:34:47,440 --> 00:34:51,260
+المتباينة الجديدة six prime؟ بيطلع عبارة عن upper
+
+320
+00:34:51,260 --> 00:34:58,300
+bound لكل العناصر A N، هذه ETA upper bound
+
+321
+00:34:58,300 --> 00:35:02,890
+للمجموعة هذه وبالتالي الـ supremum للمجموعة هذه
+
+322
+00:35:02,890 --> 00:35:07,190
+بيطلع أصغر من أو يساوي الـ upper bound هذا اللي هو
+
+323
+00:35:07,190 --> 00:35:14,130
+Psi إذا بيطلع عندي هنا Psi بيطلع أصغر من أو يساوي
+
+324
+00:35:14,130 --> 00:35:19,010
+Eta إذا 
+
+325
+00:35:19,010 --> 00:35:25,870
+طلع عندي أنا هنا Psi أصغر من أو يساوي Eta
+
+326
+00:35:31,240 --> 00:35:39,040
+الآن بنلاحظ أنه لو أخدت أي x في الفترة رقم in لكل
+
+327
+00:35:39,040 --> 00:35:45,900
+in لو كان x موجود هنا هذا بكافي أن x أكبر من أو 
+
+328
+00:35:45,900 --> 00:35:54,080
+يساوي psi أصغر من أو يساوي eta لتوضيح
+
+329
+00:35:54,080 --> 00:36:04,120
+ذلك أو لبرهان ذلك الـ ..لو كانت X موجودة هنا فالـ X
+
+330
+00:36:04,120 --> 00:36:08,480
+تطلع upper bound للـ set هذه هنا خلّيني أوضح بالرسم
+
+331
+00:36:08,480 --> 00:36:13,480
+لو 
+
+332
+00:36:13,480 --> 00:36:21,520
+كانت الـ X تنتبه لـ I N الفترة
+
+333
+00:36:21,520 --> 00:36:32,530
+I N وهي الفترة I N وهي الـ X فلو كانت X موجودة في
+
+334
+00:36:32,530 --> 00:36:42,450
+الفترة I N الفترة هذه فهذا معناه أن الـ A N أصغر من 
+
+335
+00:36:42,450 --> 00:36:49,890
+أو يساوي X لكل N وبالتالي X upper bound للمجموعة
+
+336
+00:36:49,890 --> 00:36:56,770
+هذه وبالتالي الـ supremum للمجموعة هذه اللي هو Psi
+
+337
+00:36:56,770 --> 00:37:05,220
+بيطلع أصغر من أو يساوي X، مظبوط؟ كذلك من هنا الـ X
+
+338
+00:37:05,220 --> 00:37:13,100
+أصغر من أو يساوي BN لكل N وبالتالي الـ X أبارع ال
+
+339
+00:37:13,100 --> 00:37:19,340
+lower bound لمجموعة الأعداد BN وبالتالي الـ infimum
+
+340
+00:37:19,340 --> 00:37:26,780
+لمجموعة الأعداد BN الـ infimum للمجموعة هذه بيطلع
+
+341
+00:37:26,780 --> 00:37:30,620
+أكبر من أو يساوي الـ X اللي هو lower bound لها صح؟
+
+342
+00:37:31,090 --> 00:37:35,010
+إذا الـ infimum المجموعة هذه اللي هو إيتا أكبر من أو
+
+343
+00:37:35,010 --> 00:37:40,750
+ساوي X اللي هو lower bound للمجموعة هذه okay إذا
+
+344
+00:37:40,750 --> 00:37:44,430
+لو هذا الكلام صح لو كانت X موجودة في الفترة I N
+
+345
+00:37:44,430 --> 00:37:51,140
+لكل N فهذا بيؤدي أن الـ X أكبر من أو يساوي psi و أصغر
+
+346
+00:37:51,140 --> 00:37:55,840
+من أو يساوي إيتا يعني X محصورة بين psi و إيتا و
+
+347
+00:37:55,840 --> 00:38:00,740
+طبعًا العكس لو هذا الكلام صحيح بيؤدي أن الـ X موجودة
+
+348
+00:38:00,740 --> 00:38:07,280
+هنا تمام؟ إذا هذا برهان الـ statement اللي هنا الآن
+
+349
+00:38:07,280 --> 00:38:11,100
+أنا
+
+350
+00:38:11,100 --> 00:38:11,500
+عندي 
+
+351
+00:38:29,400 --> 00:38:33,340
+أنا عندي الـ infimum للمجموعة هذه احنا فرضنا أن ال
+
+352
+00:38:33,340 --> 00:38:40,520
+infimum لمجموعة الأعداد الغير سالبة هذه الـ infimum
+
+353
+00:38:40,520 --> 00:38:45,300
+لها بيساوي صفر احنا هذا فرضناه عشان نثبت أن التقاطة
+
+354
+00:38:45,300 --> 00:38:52,240
+في نقطة واحدة صح هذا فرض قائم واحنا أخذنا لمّة
+
+355
+00:38:52,240 --> 00:38:58,490
+واحد اثنا عشر في بيجي كان بعديها exercise بيجي بعديها
+
+356
+00:38:58,490 --> 00:39:06,370
+exercise بيقول عشان المجموعة S لو كانت المجموعة في
+
+357
+00:39:06,370 --> 00:39:07,490
+لها lower bound
+
+358
+00:39:10,250 --> 00:39:16,210
+فعشان الـ lower bound هذا يكون هو الـ infimum هذا
+
+359
+00:39:16,210 --> 00:39:21,290
+بكافي أنه لكل epsilon أكبر من الصفر يوجد عنصر S
+
+360
+00:39:21,290 --> 00:39:25,870
+epsilon في المجموعة S بحيث أن العنصر أصغر من ال
+
+361
+00:39:25,870 --> 00:39:27,150
+infimum زائد epsilon
+
+362
+00:39:30,620 --> 00:39:34,020
+طيب إذا أنا بدي أستخدم الـ exercise هذا اللي هو
+
+363
+00:39:34,020 --> 00:39:39,760
+شبيه لمّة واحد اثنا عشر هي عندي المجموعة هذه وهي
+
+364
+00:39:39,760 --> 00:39:44,620
+عندي الـ infimum تبعها بيساوي صفر الصفر هذا اللي هو
+
+365
+00:39:44,620 --> 00:39:51,300
+lower bound للمجموعة هذه أنتم لاحظوا أن a b n أو
+
+366
+00:39:51,300 --> 00:39:53,860
+a n b n
+
+367
+00:39:57,350 --> 00:40:03,870
+بن ماينص ان أكبر من أو يساوي صفر لكل n فالصفر هذا
+
+368
+00:40:03,870 --> 00:40:08,710
+lower bound لمجموعة الأعداد هذه وهو مش lower
+
+369
+00:40:08,710 --> 00:40:13,030
+bound بس احنا معطى .. معطى من الفرض أنه هو ال
+
+370
+00:40:13,030 --> 00:40:16,510
+infimum هو أكبر lower bound إذا حسب التمرين اللي
+
+371
+00:40:16,510 --> 00:40:20,670
+بيجي بعد لما واحد اثنا عشر بما أنه الصفر هو الـ infimum
+
+372
+00:40:20,670 --> 00:40:25,670
+للمجموعة هذه إذا لأي epsilon أو لكل epsilon أكبر
+
+373
+00:40:25,670 --> 00:40:30,730
+من الصفر يوجد عنصر في المجموعة هذه هذا هو
+
+374
+00:40:33,930 --> 00:40:40,010
+الـ index تبعه المؤشر تبعه n هسميه ن إبسلون يعتمد
+
+375
+00:40:40,010 --> 00:40:44,550
+على الـ epsilon لكل إبسلون فيه عدد طبيعي ن إبسلون
+
+376
+00:40:44,550 --> 00:40:51,110
+وبالتالي فيه يوجد عنصر في الـ set هذه أصغر من ال
+
+377
+00:40:51,110 --> 00:40:56,770
+infimum زائد إبسلون هذا حسب الـ exercise الآن هذا
+
+378
+00:40:56,770 --> 00:41:06,750
+العدد هذا b n إبسلون أكبر من أو يساوي إيتا و 
+
+379
+00:41:06,750 --> 00:41:11,230
+epsilon أصغر من أو يساوي psi فالفرق بين هدول أصغر من
+
+380
+00:41:11,230 --> 00:41:19,970
+أو يساوي الفرق بين هدول وأنا عندي psi أصغر من أو
+
+381
+00:41:19,970 --> 00:41:25,990
+يساوي إيتا فالفرق بين psi و إيتا أكبر من أو يساوي صفر
+
+382
+00:41:26,920 --> 00:41:33,420
+تمام؟ إذا المتباينة الأخيرة هذه بحصل منها على إيه؟
+
+383
+00:41:33,420 --> 00:41:37,560
+المتباينة الأخيرة منها هذه بحصل على إنه صفر أصغر من
+
+384
+00:41:37,560 --> 00:41:45,560
+أو يساوي eta minus epsilon أصغر من epsilon لأ عفوا
+
+385
+00:41:45,560 --> 00:41:48,280
+هذه eta minus psi
+
+386
+00:41:51,930 --> 00:41:58,010
+وهذا صحيح لكل epsilon أكبر من الصفر أي لأي epsilon
+
+387
+00:41:58,010 --> 00:42:01,830
+أكبر من الصفر وصلنا إلى أنه صفر أصغر من أو يساوي
+
+388
+00:42:01,830 --> 00:42:09,810
+eta minus psi و eta minus psi أصغر من epsilon طيب
+
+389
+00:42:09,810 --> 00:42:13,390
+احنا قلنا قبل هيك لمّة بتقول لو في عندي عدد حقيقي
+
+390
+00:42:13,390 --> 00:42:18,610
+زي هذا غير سالب وأصغر من epsilon لكل epsilon أكبر
+
+391
+00:42:18,610 --> 00:42:23,330
+من الصفر هذا بيؤدي أن إيتا العدد هذا بيساوي صفر
+
+392
+00:42:23,330 --> 00:42:32,090
+وبالتالي إذا بيطلع عندي إيتا بيساوي psi okay تمام
+
+393
+00:42:32,090 --> 00:42:38,410
+إذا هيك بنكون احنا أثبتنا أن إيتا بيساوي psi
+
+394
+00:42:38,410 --> 00:42:41,610
+وبالتالي psi هي النقطة الوحيدة
+
+395
+00:42:45,050 --> 00:42:50,830
+إذا psi هي النقطة الوحيدة اللي موجودة في التقاطة
+
+396
+00:42:50,830 --> 00:42:59,450
+وهذا يكمل برهان النظرية إذا احنا خلصنا .. يعني
+
+397
+00:42:59,450 --> 00:43:07,010
+انتهينا من برهان النظرية الطويلة هذه وفي بعض
+
+398
+00:43:07,010 --> 00:43:10,410
+الملاحظات على النظرية هنشوفها إن شاء الله المرة
+
+399
+00:43:10,410 --> 00:43:17,450
+القادمة فهنوقف لأن الوقت خلص هننهي المحاضرة الآن و
+
+400
+00:43:17,450 --> 00:43:21,990
+نشوف الملاحظات والحاجات الخاصة بالنظرية في اللقاء
+
+401
+00:43:21,990 --> 00:43:22,990
+القادم إن شاء الله
diff --git a/PL9fwy3NUQKwaf4aPlmtIQXiASpDxVLEnO/0MG_WyJdBaA_postprocess.srt b/PL9fwy3NUQKwaf4aPlmtIQXiASpDxVLEnO/0MG_WyJdBaA_postprocess.srt
new file mode 100644
index 0000000000000000000000000000000000000000..145e44a4dc93f14dfde1a747e09d44f82eb9eb37
--- /dev/null
+++ b/PL9fwy3NUQKwaf4aPlmtIQXiASpDxVLEnO/0MG_WyJdBaA_postprocess.srt
@@ -0,0 +1,1600 @@
+1
+00:00:01,430 --> 00:00:05,370
+بسم الله الرحمن الرحيم مرة تانية ان شاء الله هنبدأ
+
+2
+00:00:05,370 --> 00:00:10,610
+الآن نكمل محاضرة الـ hypertensionhigh blood
+
+3
+00:00:10,610 --> 00:00:17,210
+pressure part 2 الجزء التاني كنا اتحدثنا وحكينا عن
+
+4
+00:00:17,210 --> 00:00:21,470
+ال statistics وقلنا عن اهم definition of
+
+5
+00:00:21,470 --> 00:00:25,250
+hypertension sustained elevation of the systolic
+
+6
+00:00:25,250 --> 00:00:30,850
+blood pressure أكتر من 139 سيستوليك وديستوليك اكتر
+
+7
+00:00:30,850 --> 00:00:35,210
+من 89 وطبعا ماينفعش اجيس مرة واحدة واجول الناس انت
+
+8
+00:00:35,210 --> 00:00:41,720
+في عندك ضغط وماينفعش اجيس اتحاد الظروفوانه المريض
+
+9
+00:00:41,720 --> 00:00:47,320
+الانسان اللى بده يجيس ماخد كفائين وعمل exercise و
+
+10
+00:00:47,320 --> 00:00:51,420
+smoking في اخر تلاتين دقيقة وانه ال cuff تبع ال
+
+11
+00:00:51,420 --> 00:00:57,360
+مجياز الدغط لابد انه يكون مناسب وانجيس اكتر من مرة
+
+12
+00:00:57,360 --> 00:01:01,580
+وكنا عملنا classification اذا فاكرين لل blood
+
+13
+00:01:01,580 --> 00:01:04,980
+pressure normal pre hypertension stage one
+
+14
+00:01:04,980 --> 00:01:08,480
+hypertension و stage two hypertension قولنا ايش ال
+
+15
+00:01:08,480 --> 00:01:13,460
+pre hypertensionوإنه ما بحتجش علاج إلا في بعض
+
+16
+00:01:13,460 --> 00:01:23,760
+الحالات معينة وقلنا لـ Stage 1 و Stage 2 ماذا
+
+17
+00:01:23,760 --> 00:01:29,820
+تعني؟ وقلنا إن الـ hypertension من أهم الأمراض
+
+18
+00:01:29,820 --> 00:01:33,920
+اللي بتمثل risk factors إلى جانب risk factors أخرى
+
+19
+00:01:33,920 --> 00:01:40,790
+عند مرضى القلب والجهاز الوعائيوكنا تحددنا كمان
+
+20
+00:01:40,790 --> 00:01:44,390
+وقلنا في نوبات بتصير اسمها hypertensive crisis
+
+21
+00:01:44,390 --> 00:01:47,510
+وفرجنا الـ hypertensive crisis قلنا في منها
+
+22
+00:01:47,510 --> 00:01:52,270
+emergency وفي urgency قلنا ال hypertensive urgency
+
+23
+00:01:52,270 --> 00:01:56,510
+هو الـ no progressive target organ dysfunction
+
+24
+00:01:56,510 --> 00:01:59,450
+بنقول عنه accelerated hypertension وفي ال
+
+25
+00:01:59,450 --> 00:02:02,670
+hypertensive emergency بيصير عندنا progressive end
+
+26
+00:02:02,670 --> 00:02:07,920
+organ dysfunction زي headache زي فتائبوزي حاجات زي
+
+27
+00:02:07,920 --> 00:02:12,980
+هيك وبنسميه كمان Malignant Hypertension قولنا الـ
+
+28
+00:02:12,980 --> 00:02:18,380
+Hypertensive Emergency غالباً
+
+29
+00:02:18,380 --> 00:02:22,360
+سبب تبعه Under-controlled Hypertension والـ
+
+30
+00:02:22,360 --> 00:02:29,220
+Hypertensive Emergency في الغالب بيبقى نتيجة مشكلة
+
+31
+00:02:29,220 --> 00:02:34,040
+أكبر وقلنا إنه ممكن يسببنا مشاكل تانية زي
+
+32
+00:02:34,040 --> 00:02:38,020
+EncephalopathyLeft Ventricle Failure، Acute MI
+
+33
+00:02:38,020 --> 00:02:43,340
+وDescending Aortic Aneurysm وطلبنا من حضراتكوا إيش
+
+34
+00:02:43,340 --> 00:02:46,600
+معناه Descending Aortic Aneurysm وإيش فيه له
+
+35
+00:02:46,600 --> 00:02:52,460
+تفصيلات وقلنا إن الـHypertensive Crisis ممكن ييجي
+
+36
+00:02:52,460 --> 00:02:56,300
+كصورة Hypertensive Encephalopathy، Confusion مثلا،
+
+37
+00:02:56,300 --> 00:02:59,870
+Renal Insufficiencyأورينا الـ Failure، Heart
+
+38
+00:02:59,870 --> 00:03:03,310
+Failure، قلمونة ريئة ديما، Diaphoresis كنا طلبنا
+
+39
+00:03:03,310 --> 00:03:06,410
+يا جماعة حاولوا تعرفونا أيش معناه الـ Diaphoresis
+
+40
+00:03:06,410 --> 00:03:14,260
+ومعظمكم بعتلي التصور عن الـ Diaphoresisوتحدثنا في
+
+41
+00:03:14,260 --> 00:03:16,920
+الآخر في الآخر عن اللي هي الـ Types of
+
+42
+00:03:16,920 --> 00:03:20,120
+Hypertension قلنا فيه Hypertension Primary
+
+43
+00:03:20,120 --> 00:03:22,960
+Hypertension اللي هو الـ Disease الـ Hypertension
+
+44
+00:03:22,960 --> 00:03:27,700
+itself is the disease وده بمثل 95% من حالات الـ
+
+45
+00:03:27,700 --> 00:03:31,260
+Hypertension وفيه secondary هو less common حوالي 5
+
+46
+00:03:31,260 --> 00:03:36,230
+%وعادة بيبقى فيه سبب تاني هو اللي أدى
+
+47
+00:03:36,230 --> 00:03:40,110
+للـhypertension بيقول عنه سكندرهيبرتنشن بيقولنا
+
+48
+00:03:40,110 --> 00:03:44,270
+فيه causes وكل كو انشغلته شوية الأسبوع اللي في
+
+49
+00:03:44,270 --> 00:03:49,570
+الأيام اللي فاتته حاولته اتفصله في sleep apnea إيش
+
+50
+00:03:49,570 --> 00:03:53,450
+يعني sleep apnea أو obstructive apnea drug-induced
+
+51
+00:03:53,450 --> 00:03:55,710
+causes chronic kidney disease
+
+52
+00:03:58,540 --> 00:04:05,120
+مع كاشنغ سندروم كوركوتشن of the aorta and hyper
+
+53
+00:04:05,120 --> 00:04:09,260
+and hypothyroidism هذه من الأسباب إلى جانب أسباب
+
+54
+00:04:09,260 --> 00:04:14,520
+أخرى زي الـ hyper-aldosteronism على سبيل المثال
+
+55
+00:04:14,520 --> 00:04:21,710
+ويقولنا الأسباب الـ secondaryعلى رأسها الـ Renal
+
+56
+00:04:21,710 --> 00:04:24,690
+Parenchymal Disease أيش يعني Parenchymal Disease؟
+
+57
+00:04:24,690 --> 00:04:29,870
+يعني من الـ Kidney itself، مش من الـ Vascularity
+
+58
+00:04:29,870 --> 00:04:33,550
+تبعها، من الـ Renal، من الـ Kidney نفسها و Common
+
+59
+00:04:33,550 --> 00:04:38,130
+Cause يُعتبر بمثل حوالي 5% من أسباب secondary
+
+60
+00:04:40,580 --> 00:04:45,220
+هيبرتنشن وممكن يكون هو ناتج عن مشكلة يعني واحد
+
+61
+00:04:45,220 --> 00:04:48,040
+عنده hypertension تعمله مشاكل في الكيلة لكن مشاكل
+
+62
+00:04:48,040 --> 00:04:54,400
+الكيلة ممكن تعمل برضه hypertensionوطبعاً الأسباب
+
+63
+00:04:54,400 --> 00:04:59,360
+الكتيرة في الموضوع هذا وأسباب جذد الـ Renal
+
+64
+00:04:59,360 --> 00:05:02,540
+Disease إلها إثيولوجيا كتير زي الـ Lumeronephritis
+
+65
+00:05:02,540 --> 00:05:06,180
+زي الـ Nephrotic Syndrome زي الـ Pielonephritis،
+
+66
+00:05:06,180 --> 00:05:10,080
+التهيبات الكلة زي الـ Obstructive Stones كتير
+
+67
+00:05:10,080 --> 00:05:13,740
+حاجات ممكن تعملنا Renal Disease وهذه ممكن تعملنا
+
+68
+00:05:13,740 --> 00:05:16,940
+بالتالي Hypertension وفيه إن الـ Renovascular
+
+69
+00:05:16,940 --> 00:05:24,330
+Hypertensionوهذا يعني رينو فاسكولر رين يعني كيلية
+
+70
+00:05:24,330 --> 00:05:28,790
+فاسكولر أو عيط الكيلية الشراين والـ vein اللي
+
+71
+00:05:28,790 --> 00:05:33,990
+طالعين ودخلين على الكيلة ولمّا يصير فيه تضيقلسبب
+
+72
+00:05:33,990 --> 00:05:37,470
+من الأسباب، هنحكي أيش أسباب التضيق، هذا ممكن يعملي
+
+73
+00:05:37,470 --> 00:05:40,790
+hypertension زي أيش التضيق الـ atherosclerosis،
+
+74
+00:05:40,790 --> 00:05:44,070
+كلكم عارفين الـ atherosclerosis اللي هو تيبّس
+
+75
+00:05:44,070 --> 00:05:49,010
+الشرايين في 95% لـ 90% زي ما بنشوفه عند الـ Old
+
+76
+00:05:49,010 --> 00:05:52,470
+Ages، فينا الـ fibromuscular dysplasia وهذا بيصير
+
+77
+00:05:52,470 --> 00:05:55,910
+أجل في الـ young patient، خاصة الفيميل، 10% لـ 25
+
+78
+00:05:55,910 --> 00:06:00,950
+%، جلال يعنيوممكن طبعاً الـ Aortic Dissection أو
+
+79
+00:06:00,950 --> 00:06:04,190
+الـ Renal Dissection أو الـ Tromboembolic Emboli
+
+80
+00:06:04,190 --> 00:06:08,510
+أو الـ Post Transplantation Stenosis أو الـ Post
+
+81
+00:06:08,510 --> 00:06:13,170
+Radiation أو الـ CVD هذه
+
+82
+00:06:13,170 --> 00:06:16,810
+الأمراض كلها ممكن تديني وتعملي Renovascular
+
+83
+00:06:16,810 --> 00:06:19,970
+Hypertension أو تعملي مشكلة في الـ Renovascular
+
+84
+00:06:19,970 --> 00:06:25,150
+وبالتالي Hypertension طب وإحنا ليش خايفين من الـ
+
+85
+00:06:25,150 --> 00:06:29,240
+Hypertension؟يا سيدي وضغطه عالي ضغطه عالي هو يعني
+
+86
+00:06:29,240 --> 00:06:35,760
+إيش هيعمل مثلا سؤال كتير مهم ومحق ليش أنا بخاف من
+
+87
+00:06:35,760 --> 00:06:41,140
+ال hypertension يا جماعة ال hypertension هو لو لو
+
+88
+00:06:41,140 --> 00:06:45,340
+لو مرة واحدة ارتفعت ضغط طبعا مابقصدش عن ال
+
+89
+00:06:45,340 --> 00:06:48,700
+emergency و ال urgency لكن لمرة مش هيعمللي مشكلة
+
+90
+00:06:48,700 --> 00:06:55,740
+لكن بيكونوا تتصوروا أنهالضغط هو عبارة عن تأثير
+
+91
+00:06:55,740 --> 00:07:05,640
+مباشر شديد على الشرايين طول الوجد هذا إيش بيقدّي؟
+
+92
+00:07:05,640 --> 00:07:08,940
+بيقدّي إنه يصير عندنا changes في ال vessels
+
+93
+00:07:08,940 --> 00:07:13,740
+وطبعاً،
+
+94
+00:07:13,740 --> 00:07:17,480
+وهذا بيقدّي في الآخر لـfailure للـorgans هذا
+
+95
+00:07:17,480 --> 00:07:21,910
+المشكلة تبقى أتنا كلهاعلى سبيل المثال لو جينا
+
+96
+00:07:21,910 --> 00:07:27,050
+قولنا في الـ Retina في العين على سبيل المثال ممكن
+
+97
+00:07:27,050 --> 00:07:30,030
+تعملي مشكلة وهذا بإمكاننا نشوفه في حاجة اسمها الـ
+
+98
+00:07:30,030 --> 00:07:34,070
+Funduscope الـ blood vessels ممكن تتأثر اللي في
+
+99
+00:07:34,070 --> 00:07:39,110
+العين فأنا ممكن من خلال فحص Funduscope للعين، لقاع
+
+100
+00:07:39,110 --> 00:07:42,530
+العين أعرف إن فلان عنده hypertension ليش؟ لأن
+
+101
+00:07:42,530 --> 00:07:46,610
+الشرايين اللي في قاع العين زي ما إحنا هنشوفها
+
+102
+00:07:46,610 --> 00:07:48,710
+الـgate أتصور إنه هي الصورة
+
+103
+00:07:52,140 --> 00:07:59,840
+الرتينة أو الشبكية والشرايين اللي فيها تظهر تغيرات
+
+104
+00:07:59,840 --> 00:08:03,400
+نابعة أو نادجة عن ال hypertension فبتدلني أنه في
+
+105
+00:08:03,400 --> 00:08:06,400
+hypertension تغيرات هذه أنه ممكن أنا ألاقي
+
+106
+00:08:06,400 --> 00:08:09,620
+hemorrhages عسم المثال النقطة البيضة هي اللي
+
+107
+00:08:09,620 --> 00:08:14,580
+موجودة على العالـ Retina ممكن ألاقي Exudate وهذه
+
+108
+00:08:14,580 --> 00:08:17,980
+عبارة عن Fatted Deposits وممكن ألاقي حاجة اسمها
+
+109
+00:08:17,980 --> 00:08:23,300
+Cotton Wool يعني زي القطن زي كوام القطن Spots هذه
+
+110
+00:08:23,300 --> 00:08:28,000
+نتيجة Micro Strokes بتسير في وين؟ الصورة C عندنا
+
+111
+00:08:28,000 --> 00:08:30,100
+هنا بتسير في ال Retina
+
+112
+00:08:33,060 --> 00:08:39,100
+الشرايين في كل مكان ممكن تتأثر بارتفاع ضغط الدم من
+
+113
+00:08:39,100 --> 00:08:42,860
+الأمثلة الأخرى غير جذية الـ I أو الـ Retinopathy
+
+114
+00:08:42,860 --> 00:08:48,240
+اللي حكينا عنها اللي هو nervous systemالـ nervous
+
+115
+00:08:48,240 --> 00:08:52,200
+system أنا مش هتوسع الآن لأن المفروض أنه أنتوا
+
+116
+00:08:52,200 --> 00:08:55,320
+أخدته في موضوع ال stroke أو هتاخده في موضوع ال
+
+117
+00:08:55,320 --> 00:08:58,600
+stroke اللي هو ال hemorrhagic و ال ischemic stroke
+
+118
+00:08:58,600 --> 00:09:04,040
+فممكن نتيجة ارتفاع ضغط الدم يصير عندنا stroke و
+
+119
+00:09:04,040 --> 00:09:07,060
+يصير عندنا بعد هيك cerebral atrophy and dementia
+
+120
+00:09:09,370 --> 00:09:12,530
+وذكرنا قبل شوية إن الـ brain stroke في منه
+
+121
+00:09:12,530 --> 00:09:15,770
+hemorrhagic يعني بيصير عندنا انفجار لشريان من
+
+122
+00:09:15,770 --> 00:09:20,870
+الشراين أو ischemic بيصير عندنا thrombus أو
+
+123
+00:09:20,870 --> 00:09:23,670
+embolus وهذا بيسكتر للشريان اللي رايح على المنطقة
+
+124
+00:09:23,670 --> 00:09:28,650
+اللي فلانية في الدماغ فبيعمللي إيش؟ بيعمللي
+
+125
+00:09:28,650 --> 00:09:34,370
+ischemic stroke طبعا من الحاجات اللي بتنعمل كمان
+
+126
+00:09:34,370 --> 00:09:39,180
+إنه ال effects on the kidneysبمعنى، زي ما احنا
+
+127
+00:09:39,180 --> 00:09:41,520
+شايفين أو الـ «effects on the cardiovascular
+
+128
+00:09:41,520 --> 00:09:45,840
+system» بمعنى إن الـ hypertension بيأثرلي على
+
+129
+00:09:45,840 --> 00:09:49,160
+القلب، بيأثرلي على الـ kidney، بيأثرلي على الـ
+
+130
+00:09:49,160 --> 00:09:53,340
+nervous system، بيأثرلي على الـ eyes، بيأثرلي على
+
+131
+00:09:53,340 --> 00:09:55,980
+حاجات كتير، احنا بعض الأحيان كذلك، مش كلها كذلك،
+
+132
+00:09:55,980 --> 00:10:02,140
+بس احنا كذا كلنا، أهم أربع أعضاء بيتأثروا بارتفاع
+
+133
+00:10:02,140 --> 00:10:08,180
+ضغط الدم، heart، brain، kidney،الـ Arteries and
+
+134
+00:10:08,180 --> 00:10:13,540
+Retinopathy أو الـ Retina أو الشبكية كلها بتتأثر
+
+135
+00:10:13,540 --> 00:10:19,020
+هذه هي الـ Target Organs وطبعاً أي حد بتبع دكتور
+
+136
+00:10:19,020 --> 00:10:22,380
+الـ Hypertension لابد أنه يعمل Consultations
+
+137
+00:10:22,380 --> 00:10:26,480
+للمرضى هدول بشأن الحاجات هذه و الانتباه اللي لها
+
+138
+00:10:26,480 --> 00:10:29,480
+طيب إيه بيعمل في الـ cardiovascular؟ بتديني
+
+139
+00:10:29,480 --> 00:10:33,160
+Ventricular Hypertrophy؟ مع dysfunction؟ مع
+
+140
+00:10:33,160 --> 00:10:39,280
+failure؟ كيف يعني؟ تصوروا أن الجلب بضخ؟وبضغط ضد
+
+141
+00:10:39,280 --> 00:10:46,480
+ضغط عالي فبضطر الجلب يعمل عشان يتغلب على الضغط هذا
+
+142
+00:10:46,480 --> 00:10:50,180
+يعمل dilation في الأول وبعدين hypertrophy فبصير
+
+143
+00:10:50,180 --> 00:10:54,520
+تضخم وبعد التضخم طبعاً بيصير failure بيبطل يعني
+
+144
+00:10:54,520 --> 00:10:58,620
+ممكن يمشي شهر شهرين لكن بعدها بيقدرش الجلب يكمل
+
+145
+00:10:58,980 --> 00:11:01,780
+طبعاً نتيجته التضخم بيصير عنا حاجة اسمها
+
+146
+00:11:01,780 --> 00:11:04,540
+Arrhythmias أش يعني Arrhythmias؟ Arrhythmias
+
+147
+00:11:04,540 --> 00:11:09,280
+معناته تغير
+
+148
+00:11:09,280 --> 00:11:15,070
+في شكل النبضات وعدد النبضاتالنبضات القلب ممكن يصير
+
+149
+00:11:15,070 --> 00:11:18,610
+عندي MI هنحكي عنها الـ Ischemic heart disease بعد
+
+150
+00:11:18,610 --> 00:11:22,950
+هيك وطبعا ممكن يعمللي arterial aneurysm مع
+
+151
+00:11:22,950 --> 00:11:28,010
+dissection مع rupture وحكينا عن الكلة وحكينا عن ال
+
+152
+00:11:28,010 --> 00:11:32,670
+nervous system وحكينا عن ال eye و اللي ممكن يعملها
+
+153
+00:11:32,670 --> 00:11:36,930
+مش بس احنا كنا حكينا تو انه بيصير عندنا زي النزف
+
+154
+00:11:36,930 --> 00:11:41,210
+في ال retinaبعض الأحيان وبيعمللي impairment للـ
+
+155
+00:11:41,210 --> 00:11:45,590
+vision لكن كمان الـ vitreous العين في جلبها ميّه
+
+156
+00:11:45,590 --> 00:11:48,730
+من جوا، هذا بيقول أنا vitreous body بيعمللي
+
+157
+00:11:48,730 --> 00:11:52,270
+vitreous hemorrhage وممكن يعمللي انفصال شبكي
+
+158
+00:11:52,270 --> 00:11:56,870
+retinal detachment الموضوع مش بيظلش لحدها ممكن
+
+159
+00:11:56,870 --> 00:12:00,690
+يصير عندي neuropathy of the nerves leading to
+
+160
+00:12:00,690 --> 00:12:04,710
+extraocular muscle paralysis and dysfunction انتشت
+
+161
+00:12:04,710 --> 00:12:05,330
+الموضوع
+
+162
+00:12:08,020 --> 00:12:13,080
+طب أحنا هو أنا يعني الآن كويس ومنيح لكن أنا إيش
+
+163
+00:12:13,080 --> 00:12:18,360
+بدي من البريض
+
+164
+00:12:18,360 --> 00:12:25,740
+أجاني عنده ضبط إيش اللي بديه منه أول حاجة بدي هذا
+
+165
+00:12:25,740 --> 00:12:29,220
+اللي بنقول عنه الـobjective of the evaluation إيش
+
+166
+00:12:29,220 --> 00:12:33,490
+بدي منه؟to assess lifestyle and identify other
+
+167
+00:12:33,490 --> 00:12:36,210
+cardiovascular risk factors أدور إذا في أي برض
+
+168
+00:12:36,210 --> 00:12:41,910
+تاني عنده و أشوف الـ lifestyle تبعه that may
+
+169
+00:12:41,910 --> 00:12:45,150
+affect prognosis and guide treatment و اللي ممكن
+
+170
+00:12:45,150 --> 00:12:49,950
+تأثر على الـ prognosis و العلاج الحاجة التانية،
+
+171
+00:12:49,950 --> 00:12:54,850
+بدي أحاول أعرف و الله إذا كان السبب secondary أو
+
+172
+00:12:54,850 --> 00:12:57,210
+إذا كان الـ hypertension secondary، بدي أعرف السبب
+
+173
+00:12:57,210 --> 00:13:05,370
+مهمالأمر التالت بدي أشوف إذا كان الأعضاء اتأثرت
+
+174
+00:13:05,370 --> 00:13:08,350
+يعني أنا ما يكفي إن أنا فلاني عرفت إنه في عنده ضغط
+
+175
+00:13:08,350 --> 00:13:12,070
+بدي أشوف هل عينه فيها مشكلة بدي أبعته على دكتور
+
+176
+00:13:12,070 --> 00:13:16,310
+العيون يعمل ليه فندس فندسكوب بدي أبعته على دكتور
+
+177
+00:13:16,310 --> 00:13:21,110
+الجلب يعمللي تخطيط قلب ويشوف إذا فيه Arrhythmias
+
+178
+00:13:21,110 --> 00:13:25,030
+يعمللي echocardiography ويشوف إذا في عنده أي
+
+179
+00:13:25,030 --> 00:13:30,040
+hypertrophyأردت أن أرسله إلى دكتور الفاطنة لأرى
+
+180
+00:13:30,040 --> 00:13:34,420
+إذا كان هناك مشاكل في الكلة وهل هو مجرم؟ هذه
+
+181
+00:13:34,420 --> 00:13:41,600
+أساسياته لماذا أريد أن أصل مع العيّان هذه؟ لازم
+
+182
+00:13:41,600 --> 00:13:45,200
+أبحث عن سبب إذا كان هناك سبب أرى هل أثرت على أي
+
+183
+00:13:45,200 --> 00:13:48,780
+أعضاء عنده وأرى إذا كان عنده مشاكل أخرى، فهي تصعب
+
+184
+00:13:48,780 --> 00:13:54,080
+الأمر الـ Goals of Treatment طبعاً أنا أريد أن
+
+185
+00:13:54,080 --> 00:14:01,460
+أنزل ضغطه في الآخربنزل دغته سيستوليك ودياستوليك
+
+186
+00:14:01,460 --> 00:14:08,200
+لأجل من 140 ل 90 يعني من 139 ل 89 هذه الأساس تبع
+
+187
+00:14:08,200 --> 00:14:11,880
+الموضوع لكن في ناس تانية بدأ نزلها أجل من هيك
+
+188
+00:14:17,330 --> 00:14:20,870
+زي على سبيل المثال، الناس اللي عندها Diabetes سكري
+
+189
+00:14:20,870 --> 00:14:24,730
+أو عندها Renal Disease الـBlood Pressure Goal أقل
+
+190
+00:14:24,730 --> 00:14:31,470
+من 130 لـ 80، هنزله وطبعاً
+
+191
+00:14:31,470 --> 00:14:36,810
+أحاول جدر الإمكان أنظم أمور المريض مع مشاكل
+
+192
+00:14:36,810 --> 00:14:42,490
+الـKiller ومع مشاكل القلب طيب، ونزلنا الضغط، يعني
+
+193
+00:14:42,490 --> 00:14:48,300
+وإحنا هل بتنزلنا الضغط ممكنإنه يتحسن وضع المريض
+
+194
+00:14:48,300 --> 00:14:54,700
+ممكن؟ إيش رأيكوا أنتوا؟ أه، ما هو هذا الهدف في
+
+195
+00:14:54,700 --> 00:14:58,480
+الموضوع وبتحسنوا، لاجوا من خلال الإحصائيات والـ
+
+196
+00:14:58,480 --> 00:15:03,780
+Statistics Studies يعني، لاجوا إنه إذا أنا أديت
+
+197
+00:15:03,780 --> 00:15:08,620
+علاج وتحسن المريض Reduction in stroke incidence
+
+198
+00:15:08,620 --> 00:15:15,060
+تقريباً من 35% ل40%، هذا رقم مهول، رقم ممتازمش رقم
+
+199
+00:15:15,060 --> 00:15:20,040
+بسيط، أنا تقريباً فوق الـ Tilt نجست الـ stroke
+
+200
+00:15:20,040 --> 00:15:24,300
+incidence الإمكانية إن يحصل فيه stroke incidence
+
+201
+00:15:24,300 --> 00:15:29,340
+مش بس الـ MI الـ Myocardial Infection الجلطة
+
+202
+00:15:29,340 --> 00:15:35,740
+القلبيةتنزل إمكانية إنها تحدث عند المريض طبع الضغط
+
+203
+00:15:35,740 --> 00:15:40,540
+لو عالجته بطريقة سليمة من 20% لـ 25% رقم رهائن
+
+204
+00:15:40,540 --> 00:15:43,480
+وطبعا هذه الأرقام تتحسن مع الوجد يا جماعة يعني
+
+205
+00:15:43,480 --> 00:15:49,180
+بمعنى كل ما تطور الطب وكل ما العيان أو المريض أو
+
+206
+00:15:49,180 --> 00:15:54,550
+الإنسان الإجباليعاوز معايا أكتر كل ما اتحسنت
+
+207
+00:15:54,550 --> 00:15:59,210
+الأمور أكتر مش بس حتى الـ heart failure heart
+
+208
+00:15:59,210 --> 00:16:05,590
+failure فشل القلب بجل ليش لأنه مش مضطر يضخ ضد ضغط
+
+209
+00:16:05,590 --> 00:16:09,910
+عالي أكتر من 50% هذه ال benefits of treatment يعني
+
+210
+00:16:09,910 --> 00:16:14,550
+لأ احنا مش بس يعني مش بس يعني و هذا طبعا مش لازم
+
+211
+00:16:14,550 --> 00:16:17,110
+يكون بس بالدواء يا جماعة لأنه أنا في عندي عدة طرق
+
+212
+00:16:17,110 --> 00:16:21,880
+to manage a hypertensionهنحكي عنها بعد الآن بس
+
+213
+00:16:21,880 --> 00:16:27,140
+يعني جزدي إذا أدينا علاج بتحسن المريض بنسب معينة
+
+214
+00:16:27,140 --> 00:16:31,600
+ومش بس بخفف من إمكانية الـ Complications الـ
+
+215
+00:16:31,600 --> 00:16:35,480
+Complications اللي هي Stroke Myocardial Infection
+
+216
+00:16:35,480 --> 00:16:38,660
+Heart Failure ويمكن لو نطلع على باقى الأعضاء
+
+217
+00:16:38,660 --> 00:16:45,600
+هنلاقيها أنها بتتحسن كمان بنسب أو بأخرى طيب يمكن
+
+218
+00:16:45,600 --> 00:16:50,240
+كتير هتسمعوا مني خلال الفصلموضوع الـ lifestyle
+
+219
+00:16:50,240 --> 00:16:58,620
+modification لسبب أنه هو جزء في العلاج الطبيعي من
+
+220
+00:17:00,620 --> 00:17:05,840
+الـ Medical System بمعنى، ماقدرش واحد ييجي يقول
+
+221
+00:17:05,840 --> 00:17:10,600
+أنا بس علاج طبيعي، أنا باجي و بعمله و بدلكله و
+
+222
+00:17:10,600 --> 00:17:14,500
+بمشيه و بظبطله العضلات و بس، لأ هو جزء من الـ
+
+223
+00:17:14,500 --> 00:17:17,700
+Education للناس، بيكون فيها أخصائي العلاج الطبيعي،
+
+224
+00:17:17,700 --> 00:17:20,000
+بيكون فيها الممرض، بيكون فيها الدكتور، يعني زي ما
+
+225
+00:17:20,000 --> 00:17:23,560
+يقولوها Teamwork، و في الآخر المريض يا جماعة بقعد
+
+226
+00:17:23,560 --> 00:17:28,520
+عندكفترة أكتر من أكتر المهن اللي بقعد المريض فيها
+
+227
+00:17:28,520 --> 00:17:34,420
+مع المعالج العلاج الطبيعي أكتر المهن الطبية أقصد
+
+228
+00:17:34,420 --> 00:17:40,800
+صحيًا يعني الطبيب خمس دقايق عشر دقايق الممرض يعني
+
+229
+00:17:40,800 --> 00:17:43,540
+زي ما بقولوها زي النحلة بروح بعمل الشغل و بطلع
+
+230
+00:17:43,540 --> 00:17:46,120
+مقعدش عند العيال ولا عيال موجود عنده لكن العلاج
+
+231
+00:17:46,120 --> 00:17:51,810
+الطبيعي لأ في مما في في في فترة فالفترة هايبتكون
+
+232
+00:17:51,810 --> 00:17:54,770
+غنية في الغالب بيظل هم مش ساكتين هما اتنين تبع
+
+233
+00:17:54,770 --> 00:17:58,390
+العلاج الطبيعي و المريض بيصير المريض يبدأ يسأل
+
+234
+00:17:58,390 --> 00:18:01,530
+ماينفعش أخصائي العلاج الطبيعي يقول والله لأ مش
+
+235
+00:18:01,530 --> 00:18:05,610
+عارف مامرش عليا ماينفعش وانا مطلوب انه يجاوب غلط
+
+236
+00:18:05,610 --> 00:18:11,730
+عشان ما تفهمونيش غلط بس مطلوب انه يجاوبه صح ويدله
+
+237
+00:18:11,730 --> 00:18:16,510
+صح فرصة انه انا أستاذ القجارية أحاول أفهم الناس و
+
+238
+00:18:16,510 --> 00:18:22,450
+أشرح لهم شرح صح طبعا زي ما قلنافبنضطر نشرح، فمهم
+
+239
+00:18:22,450 --> 00:18:26,250
+تعرفوا الحاجات هاي انتوا، انه لازم أشرح للعيان عن
+
+240
+00:18:26,250 --> 00:18:29,790
+الـ Life Style Modification، هذا في كل الأمراض
+
+241
+00:18:29,790 --> 00:18:34,650
+غالبا حاليها، على رأسها الـ Weight Reduction، طبعا
+
+242
+00:18:34,650 --> 00:18:37,590
+الناس بتكره الكلمة هاي، بدك تنزل وزنك، مابتدوش
+
+243
+00:18:37,590 --> 00:18:38,210
+ينزلوا وزنك
+
+244
+00:18:41,190 --> 00:18:47,090
+أو العمر كبير، 60 أو 65 سنة ولم تكن قادرة على
+
+245
+00:18:47,090 --> 00:18:50,410
+المشي ولم تكن قادرة على الذهاب للجيم وليس هناك
+
+246
+00:18:50,410 --> 00:18:55,470
+جيم، يعني أمور كثيرة تمنع الموضوع هذا، لكن يا
+
+247
+00:18:55,470 --> 00:18:59,900
+جماعةإذا وصلت لـ Normal Body Weight Body Mass
+
+248
+00:18:59,900 --> 00:19:07,640
+Index من 18.5 إلى 24.9 كيلو جرام لكل متر مربع ممكن
+
+249
+00:19:07,640 --> 00:19:13,340
+ينزل الضغط من 5 إلى 20 كيلو
+
+250
+00:19:13,340 --> 00:19:22,420
+جرام لكل 10 كيلو جرام هذا
+
+251
+00:19:22,420 --> 00:19:29,610
+رقم كبيريعني أنا لو كان الضغط 145 على 90 ونزلها
+
+252
+00:19:29,610 --> 00:19:34,130
+لجد زي ما بنقول معناته احنا في نعمة كبيرة حينزل
+
+253
+00:19:34,130 --> 00:19:38,450
+حينزل الضغط على الطبيعي عملياً بس بال body weight
+
+254
+00:19:38,450 --> 00:19:43,610
+reduction توي حكينا عن واحدة اسمها body mass index
+
+255
+00:19:43,610 --> 00:19:48,870
+هذه طبعا هي ال homework الوحيد اللي هتعمله what is
+
+256
+00:19:48,870 --> 00:19:59,500
+body mass indexأيش بيمثل كيب من حسب والدرجات تبعته
+
+257
+00:19:59,500 --> 00:20:08,000
+درجات تبعته فيه درجات وهذا بدياه بس ايش بديش copy
+
+258
+00:20:08,000 --> 00:20:13,940
+paste بدياه مكتوب بخط اليد ومصور و تبعتولياه لما
+
+259
+00:20:13,940 --> 00:20:18,500
+ابعتلكوا الفيديو هحددلكوا على ال model لو أكتش
+
+260
+00:20:18,500 --> 00:20:25,500
+الإجابةيبقى بخط اليد جدول body mass index الشوة
+
+261
+00:20:25,500 --> 00:20:32,540
+ماذا يعني كيبن حسب والدرجات تبعته okay الحاجة
+
+262
+00:20:32,540 --> 00:20:35,460
+التانية اللي أنا عشان من ال lifestyle modification
+
+263
+00:20:35,460 --> 00:20:41,740
+بعملها هو حاجة اسمها dash eating plan هنحكي عنها
+
+264
+00:20:41,740 --> 00:20:47,090
+بعد شوية وهذه لو أنا مشيت فيهابالمناسبة هي الـ
+
+265
+00:20:47,090 --> 00:20:52,110
+«Dash High» تتناسب جداً جداً مع حياة الإنسان الصحي
+
+266
+00:20:52,110 --> 00:20:58,190
+هو اللي بنعمله الـ «Accidentally Lifestyle» اللي
+
+267
+00:20:58,190 --> 00:21:02,290
+بنشتغله اللابتوب والجوال وطول النهار عن الـ «نت»
+
+268
+00:21:02,290 --> 00:21:08,490
+والكولة جانان كبيرة ومجات قهوة وماعرفش الشاورما
+
+269
+00:21:08,490 --> 00:21:13,190
+هذه كلها أصلا مش الوضع الطبيعي هذه مش هي الوضع
+
+270
+00:21:13,190 --> 00:21:17,850
+الطبيعي الوضع الطبيعي أنه أنابأكل Fruit كفاية،
+
+271
+00:21:17,850 --> 00:21:22,270
+Vegetable كفاية اللي هي Low-fat في الغالب، بأخد
+
+272
+00:21:22,270 --> 00:21:28,070
+منتجات لبنية Low-fat كفاية، الأكل تبعي بيبقى مش
+
+273
+00:21:28,070 --> 00:21:37,990
+Full of saturated fat، هذا الأصلوبيتزا واندومي
+
+274
+00:21:37,990 --> 00:21:40,790
+وماعرفش إيش، لأ مش هو هذا الأصل، الأصل أنه أنا
+
+275
+00:21:40,790 --> 00:21:43,890
+بأكل طبيعي زي البني أدمين عادي، fruits
+
+276
+00:21:43,890 --> 00:21:48,450
+وvegetables، جطعة لحمة وفواكه وخضار وهي الأكل
+
+277
+00:21:48,450 --> 00:21:52,050
+تبعنا، هذا الأصل، هذا الموضوع لو أنا مشيت عليه
+
+278
+00:21:52,050 --> 00:21:57,610
+اللي هو ال «Dash Eating Plan» بنزل الضغط من 8 إلى
+
+279
+00:21:57,610 --> 00:21:59,590
+14 systolic
+
+280
+00:22:01,730 --> 00:22:09,310
+طبعاً كثير بنسمع عن خفف الملح في أكلك Reduce
+
+281
+00:22:09,310 --> 00:22:16,310
+Dietary Sodium Intake To No More Than حوالي احنا
+
+282
+00:22:16,310 --> 00:22:23,930
+اللي بنقوله 2.5 جرام في النهار وهذا بنزلي 2 إلى 8
+
+283
+00:22:23,930 --> 00:22:30,850
+في الضغط السيستالينيالحاجة الأخيرة اللي بنقول عنها
+
+284
+00:22:30,850 --> 00:22:33,270
+لموضوع الـ Lifestyle Modification اللي هي الـ
+
+285
+00:22:33,270 --> 00:22:38,810
+Physical Activity بنلاقي إنه كل ما مشينا بريسك
+
+286
+00:22:38,810 --> 00:22:44,370
+ووكينج يعني بهمة لتلاتين دقيقة Most days of the
+
+287
+00:22:44,370 --> 00:22:49,520
+week هذه ممكن تنزلي من أربعة لتسعة في الضغططبعاً
+
+288
+00:22:49,520 --> 00:22:52,120
+لما أنا بعمل كل ال activities هايم أنا تضغطي هينزل
+
+289
+00:22:52,120 --> 00:22:55,880
+بصورة واضحة وأنا لسه في lifestyle modification أنا
+
+290
+00:22:55,880 --> 00:23:00,780
+مادخلتش لسه على drugs في
+
+291
+00:23:00,780 --> 00:23:03,620
+ناس ميشي بيقولك طب وانت قاعد بتقولي جلل أكلك كيف
+
+292
+00:23:03,620 --> 00:23:06,280
+بدأ أجلل الأكل في هذا الوضع يا جماعة بدنا نحاول
+
+293
+00:23:06,280 --> 00:23:09,940
+جدر الإمكان نعمل dietary changes نقول عنها كيف
+
+294
+00:23:09,940 --> 00:23:15,460
+يعني؟ يعني أجلل ال portion size تبع الأكلمش بس طب
+
+295
+00:23:15,460 --> 00:23:18,900
+كيف بدأ اجلله دايما انا بنصحه لما تيجي تاكله يا
+
+296
+00:23:18,900 --> 00:23:24,180
+جماعة اكله في صحن صغير صحن صغير حطه في صحن صغير كل
+
+297
+00:23:24,180 --> 00:23:27,340
+ما كبرت الصحن عين .. تعرفوا عين البني آدم فارغة
+
+298
+00:23:27,340 --> 00:23:31,160
+بده يعبيه أكتر لأ حط في صحن صغير جد ما تحط فيه
+
+299
+00:23:31,160 --> 00:23:35,600
+بيبين كتير يعني اضحك على حالك هيك بمعنى أخر الأمر
+
+300
+00:23:35,600 --> 00:23:38,620
+التاني لما تستعمله يا بتستعمل معلجة صغيرة المعلجة
+
+301
+00:23:38,620 --> 00:23:44,020
+بتاعة الشاي في الأكل عشان تطوره وانت بتاكليا إما
+
+302
+00:23:44,020 --> 00:23:48,140
+بتستخدم الشوكة الشوكة بيضلش عليها رز و حاجات تاك
+
+303
+00:23:48,140 --> 00:23:51,600
+كتير خاصة لما يكون مقارنة و رز و كذا حاول الشوكة
+
+304
+00:23:51,600 --> 00:23:57,980
+الشوكة بتاخدش كتير فبتطول في الأكل هذه من الحاجات
+
+305
+00:23:57,980 --> 00:24:03,380
+اللي احنا ممكن نستخدمها عشان أجلل مثلا الخبز على
+
+306
+00:24:03,380 --> 00:24:07,260
+سبيل المثال دايما ماأكلش الطازة على الآخر الطريق
+
+307
+00:24:07,260 --> 00:24:10,780
+لأن هذا سهل المضغ و سهل البلع و سهل كل حاجة خد
+
+308
+00:24:10,780 --> 00:24:16,230
+دايما الناشف أو المحمرمش عشان أقل قيمة غذائية من
+
+309
+00:24:16,230 --> 00:24:19,210
+ناحية الـ Calories لأ لأنه بطول و أنا بمضغ فيه و
+
+310
+00:24:19,210 --> 00:24:22,750
+أنا باكل فيه فهذه بتخليه الأمر التالت و المهم
+
+311
+00:24:22,750 --> 00:24:27,110
+حاولوا جدر الإمكان تبعدوا عن السفرة عن جاعة السفرة
+
+312
+00:24:27,110 --> 00:24:32,610
+هذه لللمّة اللي فيها كل ما لذ و طاب حاولوا تبعدوا
+
+313
+00:24:32,610 --> 00:24:40,640
+عنها هذه بتضمنلك أن انت تاخد كمياتبسيطة من الأكل،
+
+314
+00:24:40,640 --> 00:24:43,260
+ساندوشات أنت عارف إيش اللي بدك إياه أو بتاخد على
+
+315
+00:24:43,260 --> 00:24:46,460
+الطريقة الأمريكية كله لنفسه صحن صغير على جدك
+
+316
+00:24:46,460 --> 00:24:50,140
+وبتقعد فيه وبتاكل وخلصنا أما السفرة هات إيدك وهات
+
+317
+00:24:50,140 --> 00:24:53,660
+رجلك وعبي وعبي وعبي وعبي، هذه مشكلة كبيرة، صحيح؟
+
+318
+00:24:53,660 --> 00:24:57,140
+وبتخلي الواحد يظل ياكل بلا حدود، بلا حدود، خاصة لو
+
+319
+00:24:57,140 --> 00:25:01,160
+في تسلية جباله في منافسة وهذا بياكل وهذا بتاكل
+
+320
+00:25:01,160 --> 00:25:06,510
+وهذا بياكل وهذا بتاكلهذه مشكلة طيب الأمر التاني
+
+321
+00:25:06,510 --> 00:25:12,170
+نحاول نجلّل .. نجلّل نحن لا نحرم حالنا يا جماعة
+
+322
+00:25:12,170 --> 00:25:15,750
+أنا مش طالب من حد يحرم حاله لكن reduce portion
+
+323
+00:25:15,750 --> 00:25:19,950
+size or frequency of consumption of calorie
+
+324
+00:25:19,950 --> 00:25:24,550
+containing beverages مش ممنوع ولا حرام ولا عيب
+
+325
+00:25:24,550 --> 00:25:29,170
+أشرب كولة بس ممنوع أشرب جنيتين وتلاتة أو جنية
+
+326
+00:25:29,170 --> 00:25:33,920
+كبيرة أو علبتين وتلاتة أو أربعة في النهارأو جنيه
+
+327
+00:25:33,920 --> 00:25:37,620
+لتر أو جنيتيه لتر في النهار لكن مسموح لي أشرب شوية
+
+328
+00:25:37,620 --> 00:25:42,480
+كولا مش قضية كبيرة الأمر التاني اللي هو decrease
+
+329
+00:25:42,480 --> 00:25:45,400
+time in sedentary behavior طول النهار قاعد على
+
+330
+00:25:45,400 --> 00:25:50,460
+التليفزيون وعلى ال video games وشغال online وطبعا
+
+331
+00:25:50,460 --> 00:25:56,320
+أكل أكل أكل أكل وخاصة الأمر الآخر اللي هي regular
+
+332
+00:25:56,320 --> 00:26:01,240
+physical activity avoidance of tobacco use طبعا و
+
+333
+00:26:01,240 --> 00:26:06,920
+stress managementيبقى احنا هنا مش طالبين معجزات
+
+334
+00:26:06,920 --> 00:26:15,080
+عمليًا المفروض انه الأكل الصحي هيك زي ما قلنا كمان
+
+335
+00:26:15,080 --> 00:26:20,940
+مرة اجلل ال food اشتغل exercise manage weight stop
+
+336
+00:26:20,940 --> 00:26:26,620
+smoking cut salt limit alcohol manage stress كلام
+
+337
+00:26:26,620 --> 00:26:31,630
+معقول مافيش عليه كلام كتيرطب ايش الـ Dash-Diety؟
+
+338
+00:26:31,630 --> 00:26:34,930
+الـ Dash-Diety عبارة اختصار طبعاً كلمة الـ Dash
+
+339
+00:26:34,930 --> 00:26:41,430
+الـ D عن Dietary Approach أو Approaches الـ A Stop
+
+340
+00:26:41,430 --> 00:26:47,690
+S Hypertension الـ H Diet Nutritional Therapy
+
+341
+00:26:47,690 --> 00:26:51,930
+اسمها وطبعاً اجروا عنها كتير مكتوب عليها الأساس
+
+342
+00:26:51,930 --> 00:26:59,690
+تبعها Sodium Restriction مش فسيخSodium Restriction
+
+343
+00:26:59,690 --> 00:27:04,970
+حاول نجلل الملح جذراً إمكاناً من الأساسيات هي Diet
+
+344
+00:27:04,970 --> 00:27:09,310
+Rich in Vegetable Fruit and Non-Fat Dietary
+
+345
+00:27:09,310 --> 00:27:15,850
+Products دايماً حتى لو أخدت منتجات لبنية منتجات
+
+346
+00:27:15,850 --> 00:27:20,490
+حليب لابد أن يكون كمية الفات وهنا سنتحدث بعد ذلك
+
+347
+00:27:20,490 --> 00:27:24,990
+في محاضرة كاملة ممكن أنتم تتحدثوا فيها عن أنواع
+
+348
+00:27:24,990 --> 00:27:33,030
+الفاتأو calorie restriction if overweight وقدر
+
+349
+00:27:33,030 --> 00:27:38,030
+الإمكان one medic يعني إذا اشتغلت عليها هاي زي ما
+
+350
+00:27:38,030 --> 00:27:41,750
+حكينا على الحاجات اللي جامل هيك تقريبا بتأثر تأثير
+
+351
+00:27:42,730 --> 00:27:47,950
+الدوة تلقى واحد one medication هذا بالنسبة للـ
+
+352
+00:27:47,950 --> 00:27:51,190
+Dash ياريت يكون فيه قراءة حرة عن الـ Dash يعني كل
+
+353
+00:27:51,190 --> 00:27:54,610
+واحد يحاول يشوفله حاجة يجراها عن الـ Dash لأنه
+
+354
+00:27:54,610 --> 00:27:57,370
+لابد أنه نعرف عنها أنا مش هحكي عنها تفاصيل زي ما
+
+355
+00:27:57,370 --> 00:28:04,050
+قلتلكوا لكن هي أساسياتها طبعاً فينا الآن موضوع بعد
+
+356
+00:28:04,050 --> 00:28:06,670
+ما خلصنا اللي هو العلاج
+
+357
+00:28:09,880 --> 00:28:17,540
+زي عندما خلصنا العلاج بالـ Lifestyle Modification
+
+358
+00:28:17,540 --> 00:28:23,260
+نحكي عن العلاج بالأدوية واللي دا داير طبعا الآن هو
+
+359
+00:28:23,260 --> 00:28:26,480
+الشغال زي ما أحنا عارفين أن الـBlood Pressure هو
+
+360
+00:28:26,480 --> 00:28:31,080
+عبارة عن ناتج الـCardiac Output، ماجدش الجلب بدخ،
+
+361
+00:28:31,080 --> 00:28:36,340
+ماجدش فيه مقاومة في الشرايين مقابل الدخ اللي بيصير
+
+362
+00:28:36,340 --> 00:28:40,260
+فيهاللي هو Blood Pressure يساوي Cardiac Output ضرب
+
+363
+00:28:40,260 --> 00:28:46,520
+Systemic Vascular Resistance وبنستخدم أدوية كلها
+
+364
+00:28:46,520 --> 00:28:50,840
+تخفف إما من الـ Cardiac Output أو بتخفف من الـ
+
+365
+00:28:50,840 --> 00:28:54,660
+Systemic Vascular Resistance بتجلل منه وبتزيد
+
+366
+00:28:54,660 --> 00:28:59,120
+Cardiac Output فبتجلل
+
+367
+00:28:59,120 --> 00:29:06,470
+Blood Pressure من الأدوية اللي بنستخدمهاالـ
+
+368
+00:29:06,470 --> 00:29:13,770
+Diuretics هي الأدوية التي تزيد كمية البُقى لأنها
+
+369
+00:29:13,770 --> 00:29:18,850
+بتخفف من الـ Systemic Vascular Resistance وبتخفف
+
+370
+00:29:18,850 --> 00:29:23,610
+من كمية الـ Cardiac Output الـ Beta Blockers الـ
+
+371
+00:29:23,610 --> 00:29:28,170
+Direct Vasodilators لأنها بتوسع الشرايين فالضغط
+
+372
+00:29:28,170 --> 00:29:28,670
+تخف
+
+373
+00:29:31,760 --> 00:29:36,420
+ونفس الفكرة وفيها الـ calcium channel blockers
+
+374
+00:29:38,420 --> 00:29:42,580
+أيضًا من الأدوية أو المجموعات الأدوية اللي
+
+375
+00:29:42,580 --> 00:29:48,680
+بيستخدمها ك blood pressure reducer طبعًا الآن أنا
+
+376
+00:29:48,680 --> 00:29:51,540
+مش هشرح عنهم هدول ولا احنا لازمنا حقيقة بس ده يعني
+
+377
+00:29:51,540 --> 00:29:55,180
+كويس لو نعرف المجموعات كفاية ال drugs انتوا مش
+
+378
+00:29:55,180 --> 00:29:58,980
+مطلوبة منكوا لكن يعرف احنا وين اتجاهنا في الأدوية
+
+379
+00:29:58,980 --> 00:30:05,780
+ايه رايحة هذا بالنسبة لمحاضرة ال hypertensionوزي
+
+380
+00:30:05,780 --> 00:30:08,580
+ما اتحدثنا يا جماعة موضوع الـ Hypertension موضوع
+
+381
+00:30:08,580 --> 00:30:14,040
+أضخم من هيك بكتير في عليه كتب أصلا، لكن احنا أخدنا
+
+382
+00:30:14,040 --> 00:30:18,540
+من زاوية خاصة بينا في العلاج الطبيعي بحيث أنه يكفي
+
+383
+00:30:18,540 --> 00:30:22,800
+عمليا
+
+384
+00:30:22,800 --> 00:30:27,160
+لإلنا ويكون عندنا فكرة عن الموضوع بس بدأ أذكركم في
+
+385
+00:30:27,160 --> 00:30:33,100
+سؤال أوجهناه الـ Body Mass Index ما هو؟ كيف يتم
+
+386
+00:30:33,100 --> 00:30:39,810
+حسابه؟وجدول عن الدرجات تبعته وهذا كله مكتوب بخط
+
+387
+00:30:39,810 --> 00:30:43,850
+اليد وتبعتوليها على ال model تحديد الوجب تبع ال
+
+388
+00:30:43,850 --> 00:30:49,030
+model على ال model هتلاقي الكلام هذا
+
+389
+00:30:52,220 --> 00:30:55,960
+يعني إن شاء الله تكونوا أنا عارف إنه التعليم
+
+390
+00:30:55,960 --> 00:31:00,580
+بالطريقة هذه مش هو لكن أنا هكون سعيد بكل ملاحظة
+
+391
+00:31:00,580 --> 00:31:05,760
+ممكن ملاحظة إيجابية طبعاً إنه أقدر أحسن في وصول
+
+392
+00:31:05,760 --> 00:31:13,120
+المعلومة إلكوا بأي صورة لأن الفرصة هذه مش موجودة
+
+393
+00:31:13,120 --> 00:31:19,410
+في تعليم الـالإلكتروني لكن لو كان وجاهي بقدر واحد
+
+394
+00:31:19,410 --> 00:31:24,570
+يشرح ويزيد ويعيد وكذا لكن الآن حقيقة محصور لأ أي
+
+395
+00:31:24,570 --> 00:31:28,190
+حد بده أي حاجة على الرقم اللي عندكم على الواتس أب
+
+396
+00:31:28,190 --> 00:31:32,670
+يرسلني مباشرة ماعندي مشكلة حتى يعني بدون .. بدون
+
+397
+00:31:32,670 --> 00:31:37,310
+ما اتبايلوا على فريزة ابعتولي لو في أي حاجة إيش
+
+398
+00:31:37,310 --> 00:31:42,450
+بقدر أعمل حاول .. حاول نخفف بدنا نعدي هذه الفترة
+
+399
+00:31:42,450 --> 00:31:45,570
+على خير ونشوف وشكوا على خير إن شاء الله السلام
+
+400
+00:31:45,570 --> 00:31:45,870
+عليكم
+
diff --git a/PL9fwy3NUQKwaf4aPlmtIQXiASpDxVLEnO/4FK6MUObdtw_raw.json b/PL9fwy3NUQKwaf4aPlmtIQXiASpDxVLEnO/4FK6MUObdtw_raw.json
new file mode 100644
index 0000000000000000000000000000000000000000..b0b16889a9b2da39b5f6bbcc48f3cef49627df8f
--- /dev/null
+++ b/PL9fwy3NUQKwaf4aPlmtIQXiASpDxVLEnO/4FK6MUObdtw_raw.json
@@ -0,0 +1 @@
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0.667236328125}, {"start": 1407.9, "end": 1408.18, "word": " بتبقى", "probability": 0.847900390625}, {"start": 1408.18, "end": 1408.68, "word": " مقدمة", "probability": 0.9358723958333334}, {"start": 1408.68, "end": 1409.22, "word": " لإيش", "probability": 0.77294921875}, {"start": 1409.22, "end": 1409.86, "word": " للـ", "probability": 0.6163330078125}, {"start": 1409.86, "end": 1410.1, "word": " gall", "probability": 0.238525390625}, {"start": 1410.1, "end": 1410.56, "word": " stones", "probability": 0.440185546875}, {"start": 1410.56, "end": 1411.8, "word": " formation", "probability": 0.54541015625}], "temperature": 1.0}, {"id": 59, "seek": 143781, "start": 1415.11, "end": 1437.81, "text": "بالفحص، medical history، ultrasound، وممكن بعض الأحيان نعمل X-ray Complication of cholecystitis وهنا طبعا احنا بيصير الواحد شوية مشاكل ليش؟ لأنه بطلت بتصير المشكلة مش restricted على جول الـ bladder itself إنما ممكن توصلي تعملي sepsis يعني bacterial super infection with cholangitis", "tokens": 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"end": 1417.23, "word": " history،", "probability": 0.8046875}, {"start": 1417.23, "end": 1418.37, "word": " ultrasound،", "probability": 0.61865234375}, {"start": 1418.37, "end": 1418.57, "word": " وممكن", "probability": 0.744140625}, {"start": 1418.57, "end": 1418.85, "word": " بعض", "probability": 0.895751953125}, {"start": 1418.85, "end": 1419.35, "word": " الأحيان", "probability": 0.9903564453125}, {"start": 1419.35, "end": 1419.75, "word": " نعمل", "probability": 0.9794921875}, {"start": 1419.75, "end": 1419.99, "word": " X", "probability": 0.78369140625}, {"start": 1419.99, "end": 1420.23, "word": "-ray", "probability": 0.83447265625}, {"start": 1420.23, "end": 1420.97, "word": " Complication", "probability": 0.6588236490885416}, {"start": 1420.97, "end": 1421.11, "word": " of", "probability": 0.97607421875}, {"start": 1421.11, "end": 1421.93, "word": " cholecystitis", "probability": 0.81806640625}, {"start": 1421.93, "end": 1422.55, "word": " وهنا", "probability": 0.816162109375}, {"start": 1422.55, "end": 1422.85, "word": " طبعا", "probability": 0.86175537109375}, {"start": 1422.85, "end": 1423.11, "word": " احنا", "probability": 0.7568359375}, {"start": 1423.11, "end": 1423.67, "word": " بيصير", "probability": 0.9151611328125}, {"start": 1423.67, "end": 1424.23, "word": " الواحد", "probability": 0.9075520833333334}, {"start": 1424.23, "end": 1425.17, "word": " شوية", "probability": 0.86474609375}, {"start": 1425.17, "end": 1425.67, "word": " مشاكل", "probability": 0.98388671875}, {"start": 1425.67, "end": 1425.93, "word": " ليش؟", "probability": 0.700927734375}, {"start": 1425.93, "end": 1426.11, "word": " لأنه", "probability": 0.7415364583333334}, {"start": 1426.11, "end": 1426.51, "word": " بطلت", "probability": 0.82421875}, {"start": 1426.51, "end": 1427.21, "word": " بتصير", "probability": 0.7705891927083334}, {"start": 1427.21, "end": 1427.61, "word": " المشكلة", "probability": 0.9896240234375}, {"start": 1427.61, "end": 1427.81, "word": " مش", "probability": 0.98828125}, {"start": 1427.81, "end": 1428.61, "word": " restricted", "probability": 0.59912109375}, {"start": 1428.61, "end": 1428.85, "word": " على", "probability": 0.6494140625}, {"start": 1428.85, "end": 1429.11, "word": " جول", "probability": 0.4166259765625}, {"start": 1429.11, "end": 1429.37, "word": " الـ", "probability": 0.36328125}, {"start": 1429.37, "end": 1429.67, "word": " bladder", "probability": 0.4736328125}, {"start": 1429.67, "end": 1430.51, "word": " itself", "probability": 0.513671875}, {"start": 1430.51, "end": 1431.59, "word": " إنما", "probability": 0.73046875}, {"start": 1431.59, "end": 1431.99, "word": " ممكن", "probability": 0.959228515625}, {"start": 1431.99, "end": 1432.67, "word": " توصلي", "probability": 0.6678466796875}, {"start": 1432.67, "end": 1433.33, "word": " تعملي", "probability": 0.8258463541666666}, {"start": 1433.33, "end": 1434.09, "word": " sepsis", "probability": 0.9013671875}, {"start": 1434.09, "end": 1434.53, "word": " يعني", "probability": 0.883544921875}, {"start": 1434.53, "end": 1435.13, "word": " bacterial", "probability": 0.68603515625}, {"start": 1435.13, "end": 1435.53, "word": " super", "probability": 0.89404296875}, {"start": 1435.53, "end": 1436.03, "word": " infection", "probability": 0.6298828125}, {"start": 1436.03, "end": 1436.77, "word": " with", "probability": 0.91162109375}, {"start": 1436.77, "end": 1437.81, "word": " cholangitis", "probability": 0.692138671875}], "temperature": 1.0}, {"id": 60, "seek": 145320, "start": 1439.28, "end": 1453.2, "text": "أو يدّيني سبسيس Gold Bladder Perforation ممكن الـ Gold Bladder هي ملتهبة، إيش تعملي؟ حواليها تعملي زي أبسس مدة وتعملي زي .. و ترتبط", "tokens": [10721, 2407, 7251, 3215, 11703, 9957, 1829, 8608, 3555, 3794, 1829, 3794, 6731, 2177, 29901, 3026, 2994, 399, 3714, 43020, 2423, 39184, 6731, 2177, 29901, 39896, 3714, 1211, 47395, 49401, 12399, 11933, 1829, 8592, 6055, 25957, 20292, 22807, 11331, 2407, 6027, 1829, 11296, 6055, 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1876.57, "word": " أو", "probability": 0.75634765625}, {"start": 1876.57, "end": 1876.95, "word": " goal", "probability": 0.293212890625}, {"start": 1876.95, "end": 1877.33, "word": " bladder", "probability": 0.61181640625}, {"start": 1877.33, "end": 1877.79, "word": " draining", "probability": 0.89599609375}, {"start": 1877.79, "end": 1878.51, "word": " لو", "probability": 0.69140625}, {"start": 1878.51, "end": 1878.89, "word": " حطينا", "probability": 0.93701171875}, {"start": 1878.89, "end": 1879.17, "word": " كمان", "probability": 0.7979329427083334}, {"start": 1879.17, "end": 1879.51, "word": " فيديو", "probability": 0.9322916666666666}, {"start": 1879.51, "end": 1879.93, "word": " بيبقى", "probability": 0.8119140625}, {"start": 1879.93, "end": 1880.37, "word": " رائع", "probability": 0.9934895833333334}, {"start": 1880.37, "end": 1880.63, "word": " كتير", "probability": 0.978515625}, {"start": 1880.63, "end": 1880.89, "word": " كتير", "probability": 0.9471028645833334}, {"start": 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diff --git a/PL9fwy3NUQKwaf4aPlmtIQXiASpDxVLEnO/7Dh_YwaAVoc.srt b/PL9fwy3NUQKwaf4aPlmtIQXiASpDxVLEnO/7Dh_YwaAVoc.srt
new file mode 100644
index 0000000000000000000000000000000000000000..061ef25a5e1ddea1c2fb964541828c0ad477de97
--- /dev/null
+++ b/PL9fwy3NUQKwaf4aPlmtIQXiASpDxVLEnO/7Dh_YwaAVoc.srt
@@ -0,0 +1,1511 @@
+1
+00:00:02,010 --> 00:00:04,290
+بسم الله الرحمن الرحيم الحمد لله رب العالمين و
+
+2
+00:00:04,290 --> 00:00:08,250
+الصلاة والسلام علي سيدنا محمد وسيد المرسلين إذا ما
+
+3
+00:00:08,250 --> 00:00:11,430
+إن شاء الله تعالى اليوم هنتحدث عن infective 
+
+4
+00:00:11,430 --> 00:00:18,070
+endocarditis من القسم طبعا بلاحظين نحن هنحكي عن
+
+5
+00:00:18,070 --> 00:00:23,270
+infections of the heart infections of the heart
+
+6
+00:00:25,710 --> 00:00:31,370
+وبعدها هنكمل على الـ myocardiopathies اللي هي بعض 
+
+7
+00:00:31,370 --> 00:00:35,990
+أمراض عضلات القلب اللي بتأثر على أداء و بتؤدي
+
+8
+00:00:35,990 --> 00:00:41,390
+للـ heart failure الـinfective endocarditis هو
+
+9
+00:00:41,390 --> 00:00:47,870
+تقريبا أشهر أنواع ال infection في القلب و طبعا زي
+
+10
+00:00:47,870 --> 00:00:51,900
+ما نحن شايفين أو زي ما نحن سامعين إنه
+
+11
+00:00:51,900 --> 00:00:56,540
+endocarditis، endocardium بيصير له التهاب اللي هو 
+
+12
+00:00:56,540 --> 00:01:00,920
+infection أو inflammation والسبب تبعه infection
+
+13
+00:01:04,800 --> 00:01:13,840
+الـ Infective Endocarditis اللي بيصير فيه أنه مش
+
+14
+00:01:13,840 --> 00:01:18,700
+زي أي التهاب في أي مكان في الجسم ممكن لكن هو بأثر
+
+15
+00:01:18,700 --> 00:01:25,280
+تأثير مباشر على بما أنه الصمامات هي امتداد للـ
+
+16
+00:01:25,280 --> 00:01:29,650
+Endocardium فهو بأثر تأثير مباشرة على الصمامات
+
+17
+00:01:29,650 --> 00:01:34,630
+تصوروا أنتم تخيلوا قدامكم كيف أن الصمامات في القلب
+
+18
+00:01:34,630 --> 00:01:38,190
+و هي من الـ endocardium و صار فيها التهاب يعني فيه
+
+19
+00:01:38,190 --> 00:01:41,750
+هنا أكثر من حاجة هتصير في الغالب أنها مش هتقدر
+
+20
+00:01:41,750 --> 00:01:45,850
+تشتغل ما يتوجب عليها أنها تشتغله و بيصير عندنا
+
+21
+00:01:45,850 --> 00:01:49,490
+حاجة اسمها severe valvular insufficiency فشل في
+
+22
+00:01:49,490 --> 00:01:55,850
+عمل الصمامات بتصير تهرب وهذا طبعا تصور الصمامات
+
+23
+00:01:55,850 --> 00:02:01,870
+بتهرب و بتدخل دم أكثر لل ventricles إيش بيصير؟ آه 
+
+24
+00:02:01,870 --> 00:02:05,250
+صحيح بيصير احتقان أو congestive heart failure
+
+25
+00:02:05,250 --> 00:02:08,690
+احتقان في عضلة القلب لأنه مش قادر تبقى بالشغل
+
+26
+00:02:08,690 --> 00:02:13,030
+بتعب دم لكن مش قادر تبقى بالشغل ومش بس دايما 
+
+27
+00:02:13,030 --> 00:02:16,330
+الالتهابات ال infective endocarditis ممكن تؤدي إلى
+
+28
+00:02:16,330 --> 00:02:22,650
+نشوء abscesses دمامل جوا القلب
+
+29
+00:02:25,410 --> 00:02:29,470
+الـ Infective Endocarditis produces a wide variety
+
+30
+00:02:29,470 --> 00:02:33,910
+of systemic signs and symptoms، مش بس أجد الجلب،
+
+31
+00:02:33,910 --> 00:02:38,570
+إنما ممكن في كل الجسم، عن طريق آليات كثيرة، في منهم 
+
+32
+00:02:38,570 --> 00:02:42,630
+ثلاث أربع آليات، احنا هنتحدث عن آليتين، كيف بأثر
+
+33
+00:02:42,630 --> 00:02:47,810
+الـ Infective Endocarditis على الجسم
+
+34
+00:02:50,320 --> 00:02:55,880
+بس قبل هيك بدنا نحكي عن الـ Etiology ومقصود فيها ال
+
+35
+00:02:55,880 --> 00:02:58,940
+organisms causing major clinical forms of
+
+36
+00:02:58,940 --> 00:03:03,500
+endocarditis أننا نوعين مشهورين كبار أو نوعين من
+
+37
+00:03:03,500 --> 00:03:08,200
+البكتيريا اللي بتؤدي إلى endocarditis أو infective
+
+38
+00:03:08,200 --> 00:03:12,240
+endocarditis وتقريبا كل نوع من البكتيريا هذه 
+
+39
+00:03:12,810 --> 00:03:17,450
+ووصفولها نوع من الـEndocarditis اللي هو 
+
+40
+00:03:17,450 --> 00:03:20,690
+الـChronicity of Endocarditis يعني هنا بنحدد احنا
+
+41
+00:03:20,690 --> 00:03:25,050
+عن Staphylococcus aureus وهذا الـBacterium أو 
+
+42
+00:03:25,050 --> 00:03:29,430
+الـMicroorganism the most common cause of
+
+43
+00:03:29,430 --> 00:03:33,050
+infective endocarditis أكثر من نصف الحالات are not
+
+44
+00:03:33,050 --> 00:03:36,290
+associated with underlying valvular diseases يعني 
+
+45
+00:03:36,290 --> 00:03:43,620
+لما Staph aureus بدرب Endocardium أو بدرب القلب،
+
+46
+00:03:43,620 --> 00:03:49,100
+بروحش للقلب العليل والتعبان واللي عنده مشاكل، لا 
+
+47
+00:03:49,100 --> 00:03:53,260
+هو بروح على القلب السليم الـ Mortality Rate of
+
+48
+00:03:53,260 --> 00:03:58,260
+Staph Aureus Infective Endocarditis حوالي 40 ل 50%
+
+49
+00:03:58,260 --> 00:04:01,360
+بموتوا لما يصير عنده الواحد Infective Endocarditis
+
+50
+00:04:01,360 --> 00:04:07,200
+نتيجة Staph Aureus Staph Aureus عبارة عن نوع من
+
+51
+00:04:07,200 --> 00:04:10,120
+أنواع البكتيريا أخدتها في الـ Microbiology غالباً
+
+52
+00:04:11,000 --> 00:04:16,060
+وفيها النوع الثاني اللي أقل حدة واللي هو 
+
+53
+00:04:16,060 --> 00:04:22,680
+Streptococcus viridans وهذا طبعاً بيأثر لي أو
+
+54
+00:04:22,680 --> 00:04:26,560
+بيديني 50 ل 60% of cases of sub acute diseases
+
+55
+00:04:26,560 --> 00:04:29,820
+احنا هنحكي الآن ال acute diseases و ال sub acute
+
+56
+00:04:29,820 --> 00:04:33,200
+diseases ال acute endocarditis و ال sub acute
+
+57
+00:04:33,200 --> 00:04:37,120
+endocarditis ال Staph aureus هيدينا acute
+
+58
+00:04:37,120 --> 00:04:41,960
+endocarditisولكن الـ Streptococcus viridans هيدينا
+
+59
+00:04:41,960 --> 00:04:47,500
+Subacute Endocarditis
+
+60
+00:04:47,500 --> 00:04:51,480
+وفي طبعاً Other Microorganisms لكن هدول أهم
+
+61
+00:04:51,480 --> 00:04:54,400
+اثنين، الـ Streptococcus viridans أو Streptococcus Coccus
+
+62
+00:04:54,400 --> 00:04:59,360
+Viridans بصيب القلب اللي عنده مشكلة من الأول، هذه
+
+63
+00:04:59,360 --> 00:05:03,040
+الفرق بين التانين كمان، فرق من الفروق الـ Acute
+
+64
+00:05:03,040 --> 00:05:05,940
+Endocarditis usually occurs when heart valves are
+
+65
+00:05:05,940 --> 00:05:10,170
+colonized عن طريق البكتيريا الـ virulent in the
+
+66
+00:05:10,170 --> 00:05:14,490
+course of microbial يعني لما بيصير في عندي انتشار
+
+67
+00:05:14,490 --> 00:05:17,170
+للبكتيريا سواء الـ Staph aureus أو الـ viridans
+
+68
+00:05:17,170 --> 00:05:21,990
+بيصير عندي acute endocarditis اللي هو التهاب
+
+69
+00:05:21,990 --> 00:05:27,470
+الصمامات عندما 
+
+70
+00:05:27,470 --> 00:05:33,260
+الصمامات تعشش فيها البكتيريا بتدينا كمان acute
+
+71
+00:05:33,260 --> 00:05:41,780
+endocarditis الـ Subacute
+
+72
+00:05:41,780 --> 00:05:44,840
+endocarditis الpatient with subacute endocarditis
+
+73
+00:05:44,840 --> 00:05:48,400
+usually have underlying valvular heart disease
+
+74
+00:05:48,400 --> 00:05:53,540
+يعني subacute أو الـ Streptococcus viridans بصيب
+
+75
+00:05:53,540 --> 00:05:56,680
+للقلب اللي عنده مشاكل من الأول مش لازم لكن
+
+76
+00:05:56,680 --> 00:06:06,960
+الغالبية وهي أقل حدة من البكتيريا عن الـ Staph Aureus
+
+77
+00:06:06,960 --> 00:06:11,720
+كمان مرة الـ classification فيها acute
+
+78
+00:06:11,720 --> 00:06:16,260
+endocarditis و sub acute endocarditis الـ acute
+
+79
+00:06:16,260 --> 00:06:22,340
+بيصيب normal heart valves و
+
+80
+00:06:22,340 --> 00:06:28,000
+rapidly destructive مدمر، عشان هي كانت بموتوا 40 إلى 
+
+81
+00:06:28,000 --> 00:06:33,460
+50% مازال الموضوع هذا، ومش بس بيبعت لي metastatic
+
+82
+00:06:33,460 --> 00:06:38,520
+foci يعني بيبعت لي بكتيريا من المناطق هذه، من
+
+83
+00:06:38,520 --> 00:06:43,240
+القلب على باقي الجسم كمان، يعني ممكن نلاقي عاملة
+
+84
+00:06:43,240 --> 00:06:45,160
+في ال brain abscess، أسمي المثال
+
+85
+00:06:47,940 --> 00:06:51,220
+و commonly الـ acute بيبقى عن طريق ال staff زي ما
+
+86
+00:06:51,220 --> 00:06:54,520
+حكينا قبل شوية ال staff ال staff له coccus aureus و
+
+87
+00:06:54,520 --> 00:06:59,000
+if not treated usually fatal within six weeks يعني
+
+88
+00:06:59,000 --> 00:07:03,920
+إذا ما ديناهوش علاج ممكن يموت خلال ست أسابيع ال
+
+89
+00:07:03,920 --> 00:07:09,520
+sub acute غالبا
+
+90
+00:07:09,520 --> 00:07:16,500
+بصيب damaged heart valves اللي هي الصمامات اللي
+
+91
+00:07:16,500 --> 00:07:22,620
+فيها مشكلة وده not treated بياخد واجهة أكثر fatal
+
+92
+00:07:22,620 --> 00:07:27,740
+by one عادة
+
+93
+00:07:27,740 --> 00:07:34,710
+صحة الأسنان هي أهم عنصر من عناصرالـ Prevention of
+
+94
+00:07:34,710 --> 00:07:39,830
+Endocarditis لأن الـ Primary Portals هي منطقة 
+
+95
+00:07:39,830 --> 00:07:49,830
+الأسنان اللي ممكن تصدرنا بكتيريا
+
+96
+00:07:49,830 --> 00:07:53,510
+على القلب عشان هيك في حاجة اسمها Endocarditis
+
+97
+00:07:53,510 --> 00:07:54,370
+Prophylaxis
+
+98
+00:07:58,100 --> 00:08:02,140
+عندما أجي أخلع سن أو أظبط في سنان عند الناس اللي
+
+99
+00:08:02,140 --> 00:08:06,760
+عندها مشاكل في القلب طيب،
+
+100
+00:08:06,760 --> 00:08:11,940
+الشيء ال clinical قلنا احنا قبل شوي إنه الـ 
+
+101
+00:08:11,940 --> 00:08:16,040
+endocarditis بيضلش على جد المشكلة اللي عاملها في
+
+102
+00:08:16,040 --> 00:08:21,000
+القلب، في عنده كمان امتدادات لبرا، يعني مش بس
+
+103
+00:08:21,000 --> 00:08:25,180
+بأثر لي على القلب، إنما ممكن يأثر لي بميكانيزم معينة
+
+104
+00:08:26,570 --> 00:08:33,870
+بُرّة القلب زي destruction of the intra-cardiac
+
+105
+00:08:33,870 --> 00:08:39,070
+valve
+
+106
+00:08:39,070 --> 00:08:46,150
+leaflets الكرديتنديني ممكن يعمل في الآخر برضه 
+
+107
+00:08:46,150 --> 00:08:50,170
+valvular stenosis وفي الآخر يؤدي لـ congestive
+
+108
+00:08:50,170 --> 00:08:54,170
+heart failure يبقى أنا مشكلة الـ endocarditis مش
+
+109
+00:08:54,170 --> 00:08:58,670
+بس locally في القلب على ال valves اللي هي طبعا
+
+110
+00:08:58,670 --> 00:09:01,650
+مشكلة ممكن يعملي valve insufficient ممكن يعملي 
+
+111
+00:09:01,650 --> 00:09:05,790
+valve perforation ممكن يعملي valve rupture
+
+112
+00:09:07,660 --> 00:09:12,220
+وفي الآخر يدّيني heart failure أو congestive heart
+
+113
+00:09:12,220 --> 00:09:17,660
+failure لكن في كمان حاجة ثانية أنه نفس ال .. أنتو 
+
+114
+00:09:17,660 --> 00:09:20,520
+.. احنا يعني يمكن ما اتكلمناش أكثر أنه إيش اللي
+
+115
+00:09:20,520 --> 00:09:24,000
+بيصير على ال valves؟ يعني هي صحيح صار فيها 
+
+116
+00:09:24,000 --> 00:09:26,400
+infection، تعرفوا لما يصير infection في أي مكان،
+
+117
+00:09:26,400 --> 00:09:30,920
+إيش بيصير عندنا؟ نوع من أنواع الـ Debris، مدة، طبعا
+
+118
+00:09:30,920 --> 00:09:34,880
+المدة هي مع شوية calcium، مع كذا بتدينا حاجة اسمها 
+
+119
+00:09:34,880 --> 00:09:39,610
+vegetations«Vegetations» مثل «Vegetatio»
+
+120
+00:09:39,610 --> 00:09:42,770
+«Vegetations» يعني حاجة نباتية حاجة زي شكل
+
+121
+00:09:42,770 --> 00:09:48,790
+القرنبيط بيعملي على الصمامات حاجة هيك مدة وتكوينات 
+
+122
+00:09:48,790 --> 00:09:55,590
+بكتيرية هذه الـ «Vegetations» ممكن إنها يصير لها
+
+123
+00:09:55,590 --> 00:10:00,220
+«embolization» تفك عن الـ «valve» تفك وتروح مع
+
+124
+00:10:00,220 --> 00:10:04,460
+تيار الدم وتعملي infection أو Infection في
+
+125
+00:10:04,460 --> 00:10:07,640
+الـSpleen، في الـKidney، في الـMeninges، في
+
+126
+00:10:07,640 --> 00:10:10,440
+الـBrain، في الـBone، في الـPericardium، في
+
+127
+00:10:10,440 --> 00:10:15,800
+الـSynovium، فهي ممكن تعملي مشاكل كثيرة، هذه من 
+
+128
+00:10:15,800 --> 00:10:19,740
+جملة الأمور اللي ممكن يعملها الـEndocarditis مش
+
+129
+00:10:19,740 --> 00:10:24,920
+على جد القلب الـ Clinical Features of Infective
+
+130
+00:10:24,920 --> 00:10:28,800
+Endocarditis طبعاً fever, chills and sweats,
+
+131
+00:10:29,840 --> 00:10:34,160
+anorexia, weight loss, malaise, myalgia,
+
+132
+00:10:34,620 --> 00:10:40,000
+arthralgia, back pain, heart murmur، heart murmur
+
+133
+00:10:40,000 --> 00:10:46,000
+طبعاً ممكن يكون شيء جديد أو  Mitral Regurgitation
+
+134
+00:10:46,000 --> 00:10:50,200
+heart murmur أو murmur يعني بدأ يبين في لغط في القلب
+
+135
+00:10:50,200 --> 00:10:55,520
+نتيجة إصابة الصمامات أصوات في القلب غير طبيعية
+
+136
+00:10:55,520 --> 00:11:02,110
+وجديدة و Petechiae ال Petechiae هي عبارة عن نقاط حمراء أو
+
+137
+00:11:02,110 --> 00:11:07,350
+تكوينات تجميعات حمراء دموية في الجلد وذلك
+
+138
+00:11:07,350 --> 00:11:11,010
+نتيجة زي ما قلنا Embolization of Vegetations of
+
+139
+00:11:11,010 --> 00:11:16,450
+Infection to the other parts of the body حلو لو
+
+140
+00:11:16,450 --> 00:11:25,210
+طلعته على شكل ال Petechiae في صور على ال Google أو 
+
+141
+00:11:25,210 --> 00:11:32,460
+حاجة بتصير في عنده تصور واحد إيش ممكن يصير من الـ
+
+142
+00:11:32,460 --> 00:11:37,120
+Clinical Features أثار الـ arterial embolism بيصير 
+
+143
+00:11:37,120 --> 00:11:42,560
+عندنا Splenomegaly إيش يعني؟ Spleen؟ طحال؟ Megaly؟
+
+144
+00:11:42,560 --> 00:11:46,960
+بيكبر الطحال؟ بيصير عندنا Clapping إيش يعني
+
+145
+00:11:46,960 --> 00:11:50,880
+Clapping؟
+
+146
+00:11:50,880 --> 00:11:55,380
+Clapping صح، زي ما قلتي، آه صح، زي ما قلتي
+
+147
+00:11:55,380 --> 00:11:57,620
+Clapping
+
+148
+00:11:59,350 --> 00:12:02,230
+أنا مش هحكي عن الـ Clapping كتير، ولكن لأن هذا الـ
+
+149
+00:12:02,230 --> 00:12:05,710
+Next Assignment تبعنا، الـ Clapping، What is
+
+150
+00:12:05,710 --> 00:12:12,290
+Clapping، درجات الـ Clapping، و الـ
+
+151
+00:12:12,290 --> 00:12:15,190
+Differential Diagnosis of Clapping، يعني خمس، ست
+
+152
+00:12:15,190 --> 00:12:18,790
+حاجات بدون ما تشرحوا عنهم إيش ممكن يعملي Clapping
+
+153
+00:12:18,790 --> 00:12:23,650
+في الجسم بعض الأحيان بيصير عنها a neurological
+
+154
+00:12:23,650 --> 00:12:27,210
+manifestation زي الـ stroke على سبيل المثال أو
+
+155
+00:12:27,210 --> 00:12:33,110
+brain abscess، a peripheral manifestation زي حاجة
+
+156
+00:12:33,110 --> 00:12:37,670
+اسمها Osler's nodes و subungual hemorrhages
+
+157
+00:12:41,480 --> 00:12:45,940
+طبعاً فينا Lab Features of Infective Endocarditis
+
+158
+00:12:45,940 --> 00:12:50,140
+زي الأنيميا، الـ Leukocytosis، إيش عن الـ
+
+159
+00:12:50,140 --> 00:12:55,140
+Leukocytosis؟ آه، صح، زيادة في عدد الكريات البيضاء
+
+160
+00:12:55,140 --> 00:13:00,840
+و Leucopenia، نقص في عدد الكريات البيضاء عدد
+
+161
+00:13:00,840 --> 00:13:04,080
+الكريات البيضاء عادة بيبقى من كم لکم، من أربع
+
+162
+00:13:04,080 --> 00:13:10,740
+تلاف لعشرة تلاف الآن لويكولد سيتوز هلاجيها أكتر من
+
+163
+00:13:10,740 --> 00:13:15,460
+عشرة تلاف بيصير عندنا كمان microscopic hematuria
+
+164
+00:13:15,460 --> 00:13:19,780
+بالمايكروسكوب تنشف عبارة عن ما يسمى hematuria
+
+165
+00:13:19,780 --> 00:13:24,820
+عرفنا يومها إيش الـ hematuria بيصير عندنا elevated
+
+166
+00:13:24,820 --> 00:13:29,560
+erythrocyte sedimentation rate أكتر من 90% أنا بس
+
+167
+00:13:29,560 --> 00:13:32,280
+أوقف شوي بس عشان الأداء
+
+168
+00:13:37,040 --> 00:13:41,420
+طيب، إيش الـ «Elevated Erythrocyte Sedimentation 
+
+169
+00:13:41,420 --> 00:13:46,640
+Rate»؟ بنقول عنه الـ ESR لما يكون عالي طبعاً هذا
+
+170
+00:13:46,640 --> 00:13:51,820
+الجزء التاني من ال assignment «What is Erythrocyte
+
+171
+00:13:51,820 --> 00:13:59,440
+Sedimentation Rate؟» على إيش بدلنا؟ وإيش يعني
+
+172
+00:13:59,440 --> 00:14:00,360
+elevated؟
+
+173
+00:14:02,790 --> 00:14:06,030
+مهم بالنسبة لنا هذا اللي بيقوله عنه سرعة الترسيب
+
+174
+00:14:06,030 --> 00:14:10,830
+عندك عالي اللي هي الأهمية احنا بدنا نحاول قدر
+
+175
+00:14:10,830 --> 00:14:17,830
+الإمكان أنه نحيط إيش إيش يعني ESR إيش معناتها إيش
+
+176
+00:14:17,830 --> 00:14:22,060
+اللي بيصير بس short sentences بحاولوا ما تكتبوش
+
+177
+00:14:22,060 --> 00:14:25,920
+كتير أنا بدي عليها هاي ال-retro site أو ال-
+
+178
+00:14:25,920 --> 00:14:29,160
+elevated retro site sedimentation rate اللي هو
+
+179
+00:14:29,160 --> 00:14:33,360
+سرعة الترسيب توع الكريات الحمر بس على جد زي ما
+
+180
+00:14:33,360 --> 00:14:37,500
+قلنا إيش هو وإيش ال- differential diagnosis تبعه
+
+181
+00:14:37,500 --> 00:14:42,000
+وقت إيش بيبقى عالي على أساس إنه نصير فاهمين عنه إن
+
+182
+00:14:42,000 --> 00:14:46,830
+شاء الله من الحاجات اللي بنشوفها، طب وأنا حكيتلكوا
+
+183
+00:14:46,830 --> 00:14:52,490
+عن أن البتيتشيا، نحاول نشوفها على ال google، لكن
+
+184
+00:14:52,490 --> 00:14:56,330
+لا، احنا فيه إن حاجة هي زي هيك ال بتيشيا بتبقى
+
+185
+00:14:56,330 --> 00:15:00,450
+موجودة، non-cardiac manifestation، احنا قلنا مشكلة
+
+186
+00:15:00,450 --> 00:15:03,030
+ال endocarditis أنه بدينا cardiac manifestation و
+
+187
+00:15:03,030 --> 00:15:05,230
+بدينا non-cardiac manifestation، هذه إنها ال
+
+188
+00:15:05,230 --> 00:15:10,320
+بتيتشيا اللي ممكن يعملها أو الـ vasculitis البتيتشيا
+
+189
+00:15:10,320 --> 00:15:13,300
+هي نوع من أنواع ال vasculitis أو presentation من
+
+190
+00:15:13,300 --> 00:15:16,320
+presentation ال vasculitis ال clapping
+
+191
+00:15:21,900 --> 00:15:25,780
+من الحاجات اللي بنشوفها حقيقة اللي هي أنا حكيت
+
+192
+00:15:25,780 --> 00:15:30,300
+عنها قبل شوية وقولت إنها شكلها زي شكل جطة القرنبيط
+
+193
+00:15:30,300 --> 00:15:36,040
+بتكوينات اللي هي vegetation تكوينات من ال fibrin و
+
+194
+00:15:36,040 --> 00:15:39,260
+calcium وبعض الحاجات بتبقى موجودة على ال valves وهي
+
+195
+00:15:39,260 --> 00:15:44,410
+اللي في الآخربتعملنا الـ Stenosis، بتعملنا الـ
+
+196
+00:15:44,410 --> 00:15:48,570
+Regurgitation of the valve، بعض الأحيان بيصير لها
+
+197
+00:15:48,570 --> 00:15:53,130
+Embolization، بتنفصل، و بتهاجر من القلب وبتروح على
+
+198
+00:15:53,130 --> 00:15:56,150
+أماكن ثانية، بتعملنا مشاكل. ال picture هاي اللي
+
+199
+00:15:56,150 --> 00:15:59,610
+احنا شايفينها هي عبارة عن Echocardiogram و ال
+
+200
+00:15:59,610 --> 00:16:04,400
+Vegetations وين الـ Mitral Valve الـ Mitral Valve بين
+
+201
+00:16:04,400 --> 00:16:08,040
+مين ومين، أه صح، بين الـ Left Atrium والـ Left
+
+202
+00:16:08,040 --> 00:16:14,300
+Ventricle، وزي ما احنا شايفين وين السهم هو هذه
+
+203
+00:16:14,300 --> 00:16:20,440
+الـ Vegetation Skip، بنشوفها بالـ Eco-cardiography،
+
+204
+00:16:20,440 --> 00:16:25,220
+نفس الكلام الـ Vegetation اللي هي زي ما شايفينها
+
+205
+00:16:25,220 --> 00:16:30,680
+القطع اللي بيبين السهم شايفين السهم الأسود هذه
+
+206
+00:16:30,680 --> 00:16:36,220
+عبارة عن vegetation على الصمامات على الـ mitral
+
+207
+00:16:36,220 --> 00:16:40,440
+valve نتيجة نتيجة endocarditis
+
+208
+00:16:42,730 --> 00:16:45,350
+العلاج مش مطلوب منكوا يا جماعة، لكن لازم تبقوا
+
+209
+00:16:45,350 --> 00:16:47,850
+عارفين أنه Antibiotic Treatment for Infective
+
+210
+00:16:47,850 --> 00:16:51,750
+Endocarditis مطلوب، لازم أدّي Antibiotic، و لازم
+
+211
+00:16:51,750 --> 00:16:55,250
+أدّي Antibiotic زي ما أنتو شايفين، كل هي تقف four
+
+212
+00:16:55,250 --> 00:16:58,250
+weeks، four weeks، two weeks، four weeks، يعني
+
+213
+00:16:58,250 --> 00:17:03,050
+القضية أنه بدّي Antibiotic بصورة كبيرة و strong و
+
+214
+00:17:03,050 --> 00:17:07,650
+لفترة طويلة، يعني الغالب، الغالب في العالم بيعطوا
+
+215
+00:17:07,650 --> 00:17:08,290
+الـ Benzylpenicillin
+
+216
+00:17:12,410 --> 00:17:18,150
+How to prevent الـ Endocarditis، Endocarditis يمكن أن
+
+217
+00:17:18,150 --> 00:17:21,150
+يتم حفظه بما أن البكتيريا تأتي من الخارج
+
+218
+00:17:21,150 --> 00:17:26,310
+للـ Endocardium، يعني أن أي عملية أعملها في الجسم
+
+219
+00:17:26,310 --> 00:17:30,710
+خاصة في منطقة الأسنان، لابد أن أدّي مضاد حيوي،
+
+220
+00:17:30,710 --> 00:17:36,190
+بيقول عنه Pre-procedure Antibiotic، وعلى أساس أنه أحمي
+
+221
+00:17:36,190 --> 00:17:39,850
+المريض اللي عنده مشكلة بالذات في القلب، من أن
+
+222
+00:17:39,850 --> 00:17:42,430
+يأخذ مضاد حيوي، بالذات الناس اللي عندها مشكلة في
+
+223
+00:17:42,430 --> 00:17:49,410
+القلب دايمًا أحافظ على جود oral hygiene وده الـ most
+
+224
+00:17:49,410 --> 00:17:54,690
+أو more effective in the overall prevention of
+
+225
+00:17:54,690 --> 00:18:00,670
+valvular infection، ليش؟ لأن الجينجيفيتيس
+
+226
+00:18:00,670 --> 00:18:07,790
+التهاب اللثة هو أكثر source للبكتيريا طبعاً ممكن
+
+227
+00:18:07,790 --> 00:18:10,910
+ما يؤثرش على أعضاء ثانية لكن هو بيأثرلي على مين؟
+
+228
+00:18:10,910 --> 00:18:15,370
+بيأثرلي على الـ Endocardium بتديني Endocarditis
+
+229
+00:18:15,370 --> 00:18:19,950
+وكل فترة وفترة الـ American Heart Association
+
+230
+00:18:19,950 --> 00:18:24,990
+بتطلع Recommendation للـ Endocarditis Prophylaxis
+
+231
+00:18:24,990 --> 00:18:30,730
+Infective Endocarditis Prophylaxis هذا بتدخلنا على
+
+232
+00:18:30,730 --> 00:18:38,120
+موضوع أو على جزئية ثانية اسمها أمراض التوسع القلب و
+
+233
+00:18:38,120 --> 00:18:40,540
+الـ myocarditis و الـ pericardial disease احنا
+
+234
+00:18:40,540 --> 00:18:44,080
+هنحكي بس كمان slide عن الموضوع هذا ما هنحكي شيء
+
+235
+00:18:44,080 --> 00:18:50,160
+كثير cardiomyopathy هذا يدل على أنه في عندي في عضلة
+
+236
+00:18:50,160 --> 00:18:53,900
+القلب في مرض مرض في عضلة القلب Myocarditis التهاب
+
+237
+00:18:53,900 --> 00:18:58,520
+عضلة القلب و pericardial disease اللي هو ما يصيب
+
+238
+00:18:58,520 --> 00:19:02,140
+الـ pericardium الـ myocarditis هو عبارة عن
+
+239
+00:19:02,140 --> 00:19:05,160
+inflammation of the myocardium احنا بس قاعدين
+
+240
+00:19:05,160 --> 00:19:08,560
+بنعرف الأمراض تعريفها بنحكي عنها جمل بسيطة يعني
+
+241
+00:19:08,560 --> 00:19:13,480
+احنا ما هذا ما موضوع موضوع بياخد أكثر من هيك في كتب
+
+242
+00:19:13,480 --> 00:19:17,680
+البطنة لكن احنا بنحاول نعرج ونقول يعني بحيث لما
+
+243
+00:19:17,680 --> 00:19:20,180
+أسمع والله myocarditis أبقى فاهم عن إيش الموضوع
+
+244
+00:19:20,180 --> 00:19:23,740
+بيصير لما أسمع endocarditis أبقى فاهم عن إيش بنحكي
+
+245
+00:19:23,740 --> 00:19:28,180
+inflammation of the myocardium can be a result of
+
+246
+00:19:28,180 --> 00:19:31,300
+systemic disorder أو infectious agent
+
+247
+00:19:35,470 --> 00:19:38,570
+الأسباب هنا بتتراوح طبعاً قبل شوية في الـ
+
+248
+00:19:38,570 --> 00:19:42,430
+Endocarditis كنا الكلام تبعنا كله عن بكتيريا، لكن
+
+249
+00:19:42,430 --> 00:19:45,430
+الآن بدأنا نحكي عن viruses الـ Myocarditis ممكن
+
+250
+00:19:45,430 --> 00:19:49,490
+يكون سببها viruses زي الـ Viral Coxsackie B، الـ
+
+251
+00:19:49,490 --> 00:19:51,790
+Echovirus، الـ Influenza، البار Influenza، الـ
+
+252
+00:19:51,790 --> 00:19:56,310
+Epstein-Barr Virus و الـ HIV الـ HIV، إيش بيفكركوا
+
+253
+00:19:56,310 --> 00:20:02,800
+هذا الـ HIV؟ أه صح، هذا الـ AIDS الـ AIDS كما ممكن
+
+254
+00:20:02,800 --> 00:20:07,100
+أنواع من بعض أنواع البكتيريا ممكن تعملنا
+
+255
+00:20:07,100 --> 00:20:19,740
+Myocarditis زي الـ Diphtheria زي الـ Mycoplasma
+
+256
+00:20:19,740 --> 00:20:24,720
+Pneumonia زي الـ Beta-Hemolytic Streptococci وفي
+
+257
+00:20:24,720 --> 00:20:27,780
+الغالب لما يكون في عندنا Myocarditis يبقى في عندنا
+
+258
+00:20:27,780 --> 00:20:31,340
+أيضاً الـ Pericarditis منطقة البريكارديوم
+
+259
+00:20:31,340 --> 00:20:36,690
+الـ Myocarditis clinical feature تدينافيرور،
+
+260
+00:20:36,690 --> 00:20:40,170
+تاكيكارديا، ميالجيا، headache، Kragers، chest pain
+
+261
+00:20:40,170 --> 00:20:43,910
+احنا حد تحدثنا مرة بتهيألي أو قولنا واحد من ال
+
+262
+00:20:43,910 --> 00:20:49,970
+assignments كان chest pain chest pain assignment
+
+263
+00:20:49,970 --> 00:20:54,190
+differential diagnosis severe cases may have
+
+264
+00:20:54,190 --> 00:21:00,090
+congestive heart failure symptoms كيف بشخص ال
+
+265
+00:21:00,090 --> 00:21:07,560
+myocarditis ب ECG؟ ممكن يديني FA Block، ممكن يديني
+
+266
+00:21:07,560 --> 00:21:12,620
+Prolonged QRS Duration، ممكن يديني ST Elevation،
+
+267
+00:21:12,620 --> 00:21:16,400
+الـ Chest X-Ray بيلقى طبيعي، الـ Enzymes ممكن
+
+268
+00:21:16,400 --> 00:21:20,060
+تلاقيها عالية، الـ Enzymes، وطبعاً في
+
+269
+00:21:20,060 --> 00:21:22,320
+الـ Myocarditis، مش هو لما واحد يصير عنده Chest
+
+270
+00:21:22,320 --> 00:21:25,140
+Pain، مش أول حاجة بفكر فيها Myocarditis، لأ، احنا
+
+271
+00:21:25,140 --> 00:21:28,610
+عندنا مشاكل بتصير في القلب، زي الإنفكشن والـ
+
+272
+00:21:28,610 --> 00:21:32,930
+valvular diseases بتبقى more common than
+
+273
+00:21:32,930 --> 00:21:41,430
+myocarditis في إن الـ acute pericarditis اللي بيصير
+
+274
+00:21:41,430 --> 00:21:47,190
+أنه بيصير أنه lose visceral pericardium and parietal
+
+275
+00:21:47,190 --> 00:21:52,530
+pericardium surround the heart
+
+276
+00:21:54,810 --> 00:21:58,290
+الـ Pericardial space may contain لحد 50 ملّي
+
+277
+00:21:58,290 --> 00:22:02,490
+normally اللي هو Pericardial effusion الـ
+
+278
+00:22:02,490 --> 00:22:06,110
+Etiologies of subacute pericarditis فيه viral،
+
+279
+00:22:06,110 --> 00:22:09,150
+bacterial، fungal، أي حاجة ممكن، بس أنا لو كان في
+
+280
+00:22:09,150 --> 00:22:14,250
+عندي pericarditis يا جماعة لابد أن دايمًا أبحث عن 
+
+281
+00:22:14,250 --> 00:22:18,760
+السبب ما ينفعش أقول والله فلان عنده، ليش؟ لأنه زي ما
+
+282
+00:22:18,760 --> 00:22:23,440
+أنتم شايفين، احنا عندنا مليون سبب وسبب عشان، وعلى 
+
+283
+00:22:23,440 --> 00:22:26,600
+رأسها طبعا الأمور الـinfection والـmalignancy
+
+284
+00:22:26,600 --> 00:22:30,780
+والـdrugs، كلها بتدّي pericardial effusion، فلابد 
+
+285
+00:22:30,780 --> 00:22:34,660
+أنه في الـpericardial effusion، صحيح هو ممكن يكون 
+
+286
+00:22:34,660 --> 00:22:37,700
+pericarditis، لكن في أسباب تانية بتعملنا 
+
+287
+00:22:37,700 --> 00:22:39,400
+pericardial effusion،
+
+288
+00:22:41,740 --> 00:22:46,680
+الـ Clinical Features Of Acute Pericarditis Most
+
+289
+00:22:46,680 --> 00:22:51,780
+Common Sudden Or Gradual Onset Of Sharp Or
+
+290
+00:22:51,780 --> 00:22:55,200
+Stabbing، Stabbing زي ما تحط سكينة في صدر واحد
+
+291
+00:22:55,200 --> 00:22:58,740
+With Radiation To Back, Neck And Left Shoulder Or
+
+292
+00:22:58,740 --> 00:23:05,880
+Arm، Pain، بيدينا pain وطبعا ممكن يوصل ال pain هذا
+
+293
+00:23:05,880 --> 00:23:11,030
+للـ Left Trapezial Ridge الصعوبة أكثر عالية عندما 
+
+294
+00:23:11,030 --> 00:23:16,710
+تلقى صحابة مقفلة ومساعدة عند المقفلكل ما جاعد
+
+295
+00:23:16,710 --> 00:23:20,390
+الواحد كل ما كان أحسن لما يتمدد بيبقى ال pain أسوأ
+
+296
+00:23:20,390 --> 00:23:23,410
+و بيبقى عندنا low grade fever dyspnea ممكن 
+
+297
+00:23:23,410 --> 00:23:28,870
+dysphagia و ممكن يصير عندنا transient friction rub
+
+298
+00:23:28,870 --> 00:23:31,630
+عشان ال friction rub زي ما احنا عارفين أن ال
+
+299
+00:23:31,630 --> 00:23:35,250
+pericardium عبارة عن جتائتين عن جزئين فيه في جلبه
+
+300
+00:23:35,250 --> 00:23:40,050
+في واحد فيه التهاب و فيه fluid بينهم بيصيروا يحكوا 
+
+301
+00:23:40,050 --> 00:23:45,350
+بعض فهذا دينا صوت بالسمعة اسمه friction rub بندخل
+
+302
+00:23:45,350 --> 00:23:50,250
+الآن على الـ Cardiomyopathies الـ
+
+303
+00:23:50,250 --> 00:23:54,210
+cardiac dysfunction caused by myocardial disease
+
+304
+00:23:54,210 --> 00:24:00,150
+بنقول عنه Cardiomyopathy ويعتبر 
+
+305
+00:24:00,150 --> 00:24:04,770
+هو الـ third most common form of heart disease in
+
+306
+00:24:04,770 --> 00:24:10,690
+US لكن second most common cause of adolescent
+
+307
+00:24:10,690 --> 00:24:12,050
+sudden death
+
+308
+00:24:15,620 --> 00:24:18,860
+و Directly affect cardiac structure Cardiomyopathy
+
+309
+00:24:18,860 --> 00:24:23,020
+يعني خلاص هو الجسد جسد القلب بيصير مصاب و impairs
+
+310
+00:24:23,020 --> 00:24:28,780
+myocardial function فينا three types منه ال DCM و
+
+311
+00:24:28,780 --> 00:24:33,560
+ال HCM و restrictive cardiomyopathy dilated
+
+312
+00:24:33,560 --> 00:24:39,500
+cardiomyopathy dilated يعني متوسع يعني القلب عضلة
+
+313
+00:24:39,500 --> 00:24:46,690
+القلب متوسعة زيادة عن اللازم الـ HCM هايباتروفايد
+
+314
+00:24:46,690 --> 00:24:55,070
+كارديوبتي بالعكس متضخم القلب و restrictive زي بدنا 
+
+315
+00:24:55,070 --> 00:25:01,230
+نتصور كيف يعني restrictive شفته الجلد
+
+316
+00:25:02,060 --> 00:25:07,900
+لما أعمل منه وعاء مثلا هممكن يساله خمسة لتر مثلا
+
+317
+00:25:07,900 --> 00:25:12,400
+لو عملت أنا من جلد جلد جلد وعاء الجلد غير متمدد
+
+318
+00:25:12,400 --> 00:25:16,500
+بمعنى ما أقدرش أحط فيه عشرة مش زي ال nylon أو زي 
+
+319
+00:25:16,500 --> 00:25:21,400
+البلون أو كذا ممكن أحط فيه أكتر الجلد لا بتجاوب في
+
+320
+00:25:21,400 --> 00:25:24,820
+أحط فيه أكتر ولا أحط فيه أقل، يعني حتى ما بتصغرش،
+
+321
+00:25:24,820 --> 00:25:28,040
+مش زي البلون، بصغر، فهو restrictive، معناته لا 
+
+322
+00:25:28,040 --> 00:25:31,960
+يتحرك، بيضل زي ما هو restrictive، restrictive
+
+323
+00:25:31,960 --> 00:25:33,600
+cardiomyopathy
+
+324
+00:25:37,240 --> 00:25:40,320
+بيصير فيه عندنا dilatation and compensatory
+
+325
+00:25:40,320 --> 00:25:42,860
+hypertrophy of myocardium بيصير عندنا توسع في 
+
+326
+00:25:42,860 --> 00:25:45,580
+الـmyocardium بيصير عندنا hypertrophy لكن 
+
+327
+00:25:45,580 --> 00:25:50,060
+compensatory depressed systolic function and pump
+
+328
+00:25:50,060 --> 00:25:56,320
+failure with low cardiac output طبيعي 80% من ال
+
+329
+00:25:56,320 --> 00:26:00,120
+dilated cardiomyopathy are idiopathic
+
+330
+00:26:03,210 --> 00:26:05,830
+غير معروفة السبب «idiopathic» يعني غير معروف 
+
+331
+00:26:05,830 --> 00:26:10,030
+السبب، مش أنه مافيش سبب، غير معروف السبب، ولاجوا
+
+332
+00:26:10,030 --> 00:26:19,890
+أن  الـ «أفريقان أمريكان وملابسهم» لديهم 2.5% fold 
+
+333
+00:26:19,890 --> 00:26:23,710
+increased risk وmost common age of diagnosis من 20
+
+334
+00:26:23,710 --> 00:26:28,350
+إلى 50، وهذا مين؟ هذا dilated cardiomyopathy، الـ
+
+335
+00:26:28,350 --> 00:26:31,350
+hypertrophic cardiomyopathy الـ Asymmetric left
+
+336
+00:26:31,350 --> 00:26:37,490
+heart hypertrophy وممكن يكون معاها الـ right heart
+
+337
+00:26:37,490 --> 00:26:42,250
+وفي الغالب بيبقى معاهم الـ septum كمان without أي
+
+338
+00:26:42,250 --> 00:26:47,130
+نتيجة، بدون تواسع في القلب، يعني تضخم في الـ left
+
+339
+00:26:47,130 --> 00:26:50,490
+ممكن تاخد معاها ال right و ال septum بيبقى متضخم
+
+340
+00:26:50,490 --> 00:26:55,470
+أتصور في صورة بتبريكيها 50% are hereditary بنخلق 
+
+341
+00:26:55,470 --> 00:26:59,610
+فيه prevalence واحد من كل خمسمية mortality واحد في
+
+342
+00:26:59,610 --> 00:27:04,710
+المية ال mortality بتزيد في الأطفال وال adolescent
+
+343
+00:27:07,230 --> 00:27:11,070
+الـ Hypertrophic Cardiomyopathy سمتوم severity
+
+344
+00:27:11,070 --> 00:27:16,510
+progress with age ديسنيا كل ما بيكبر واحد بتعب 
+
+345
+00:27:16,510 --> 00:27:19,890
+أكتر ديسنيا on exertion most common initial or
+
+346
+00:27:19,890 --> 00:27:25,310
+presenting symptoms بتعب لما سدروا بتعب، بيصير
+
+347
+00:27:25,310 --> 00:27:31,410
+عندهم ضيق نفس لما بتعب وممكن نتيجة تضخم ال cardiac
+
+348
+00:27:31,410 --> 00:27:37,740
+muscle يبطل يصل كفاية دم للـ muscles فيصير أنها
+
+349
+00:27:37,740 --> 00:27:41,300
+angina-like دايماً chest pain ممكن يصير أنها 
+
+350
+00:27:41,300 --> 00:27:45,840
+palpitation لأنه برضه ال propagation of the
+
+351
+00:27:45,840 --> 00:27:52,080
+discharge of the heart بيصير
+
+352
+00:27:52,080 --> 00:27:55,800
+مش سليم فبيصير أنها palpitation وممكن يصير أنها 
+
+353
+00:27:55,800 --> 00:28:01,260
+كمان Psychosis لأن الجلب ما يقدرش يضخ كفاية دم 
+
+354
+00:28:01,260 --> 00:28:04,870
+للدمار فبيصير أنها Psychosis الـ Restrictive
+
+355
+00:28:04,870 --> 00:28:07,730
+Cardiomyopathy one of least common
+
+356
+00:28:07,730 --> 00:28:11,650
+Cardiomyopathies يعني جليل Ventricular volume and
+
+357
+00:28:11,650 --> 00:28:17,010
+septum are normal لكن زي ما قلنا ما بتجاوبش مع 
+
+358
+00:28:17,010 --> 00:28:24,130
+دخول الانبساط والستوليكوم والدياستوليكوم تبقى 
+
+359
+00:28:24,130 --> 00:28:28,320
+القلب mostly idiopathic ممكن بعض الأحيان يكون 
+
+360
+00:28:28,320 --> 00:28:33,240
+Familiar Systemic Disorder غالبا بيبقى وراه زي الـ 
+
+361
+00:28:33,240 --> 00:28:35,860
+Amyloidosis زي الـ Zarcoidosis زي الـ
+
+362
+00:28:35,860 --> 00:28:39,680
+Hemochromatosis زي الـ Scleroderma والـ Carcinoid
+
+363
+00:28:39,680 --> 00:28:46,700
+هذه عبارة عن زي ما بنقول عنها connective tissue
+
+364
+00:28:46,700 --> 00:28:52,080
+diseases وغالبا هي في جزء كبير من الأسباب
+
+365
+00:28:52,080 --> 00:28:56,740
+لـrestrictive cardiomyopathy يعني احنا هنا بس
+
+366
+00:28:56,740 --> 00:29:06,360
+حاولنا جدر الإمكان نلقي الضوء على بعض أمراض القلب
+
+367
+00:29:06,360 --> 00:29:10,240
+زي الـ Cardiomyopathy زي الـ Myocarditis و الـ
+
+368
+00:29:10,240 --> 00:29:18,700
+Pericardial diseases وتحدثنا شوية بزيادة عن الـ 
+
+369
+00:29:18,700 --> 00:29:23,100
+Infective Endocarditis على أساس هو الـ most common
+
+370
+00:29:24,750 --> 00:29:29,670
+نوعين منهم اللي هو الـ acute و ال sub acute و
+
+371
+00:29:29,670 --> 00:29:32,950
+ربطنا ال acute و ال sub acute ربط مش نهائي طبعا مش
+
+372
+00:29:32,950 --> 00:29:37,150
+100% مع ال staff aureus و ال streptococcus
+
+373
+00:29:37,150 --> 00:29:42,870
+viridans يعني شوية المادة احتمال تكون ناشفة شوية
+
+374
+00:29:42,870 --> 00:29:47,450
+لكن لابد أنهم يكونوا عادين عليها بتمنى لكوا انهار
+
+375
+00:29:47,450 --> 00:29:53,250
+سعيد زي ما قلنا ال assignment كان جزئين الـ
+
+376
+00:29:53,250 --> 00:29:59,850
+Clapping و الـ Differential Diagnosis و الجزء
+
+377
+00:29:59,850 --> 00:30:05,930
+التاني كان الـ ESR تعريفه و على ماذا يدل و الـ
+
+378
+00:30:05,930 --> 00:30:07,290
+Differential Diagnosis 
diff --git a/PL9fwy3NUQKwaf4aPlmtIQXiASpDxVLEnO/EDmMWZTraVE_postprocess.srt b/PL9fwy3NUQKwaf4aPlmtIQXiASpDxVLEnO/EDmMWZTraVE_postprocess.srt
new file mode 100644
index 0000000000000000000000000000000000000000..de126713d950e42a26d5cd518f716c50a0480f77
--- /dev/null
+++ b/PL9fwy3NUQKwaf4aPlmtIQXiASpDxVLEnO/EDmMWZTraVE_postprocess.srt
@@ -0,0 +1,1684 @@
+1
+00:00:01,500 --> 00:00:03,900
+بسم الله الرحمن الرحيم الحمد لله ورحمة الله وسلام
+
+2
+00:00:03,900 --> 00:00:07,940
+علي سيدنا محمد وسيد المسيحين اليوم هنتحدث إن شاء
+
+3
+00:00:07,940 --> 00:00:11,720
+الله في عضو آخر من أعضاء الجهاز التنفسي اللي هو
+
+4
+00:00:11,720 --> 00:00:16,080
+liver و هنحكي عن ال hepatic disorders بقصد هنا
+
+5
+00:00:16,080 --> 00:00:20,020
+طبعا أنا ال hepatitis طبعا في liver cirrhosis اللي
+
+6
+00:00:20,020 --> 00:00:28,120
+هو عملية التشمع بتاعة الكبد مش هنحكي عنها لكن كتير
+
+7
+00:00:28,120 --> 00:00:31,280
+من الحاجات اللي هنحكيها بتروح على liver cirrhosis
+
+8
+00:00:32,050 --> 00:00:34,930
+الـ Hepatic Disorders اللي هنحكي عنه الآن اللي هو
+
+9
+00:00:34,930 --> 00:00:39,210
+الـ Hepatitis بنا ناخد بالنا جماعة لما نحكي
+
+10
+00:00:39,210 --> 00:00:43,590
+Hepatitis Hepat اللي هو كبد إتس معناته
+
+11
+00:00:43,590 --> 00:00:47,730
+inflammation و ال Hepatitis إيه له أسباب كتير
+
+12
+00:00:48,530 --> 00:00:53,890
+أكترها أو خلّيني أقول أشهرها في البلاد غير أوروبا
+
+13
+00:00:53,890 --> 00:00:57,890
+وأمريكا اللي هو ال infection لكن في أوروبا وأمريكا
+
+14
+00:00:57,890 --> 00:01:04,330
+الأسباب أكتر الألكحول هو السبب الأساسي يبقى كمان
+
+15
+00:01:04,330 --> 00:01:07,310
+مرة ال hepatic disorders hepatitis هنحكي عن
+
+16
+00:01:07,310 --> 00:01:13,130
+hepatitis liver صورة عامة بس هيكي نتحدث حاجة خفيفة
+
+17
+00:01:14,150 --> 00:01:17,590
+is located in the upper right quadrant of the
+
+18
+00:01:17,590 --> 00:01:20,730
+abdomen وطبعا هذا المكان اتعرفنا عليه قبل هيك
+
+19
+00:01:20,730 --> 00:01:25,510
+وقتاش لما حكينا عن ال goal bladder صح؟ في ال right
+
+20
+00:01:25,510 --> 00:01:30,050
+في ال upper right quadrant او ال right upper
+
+21
+00:01:30,050 --> 00:01:34,890
+quadrant الكبد شغلته ايش؟ الكبد تخدوا بالكم من اهم
+
+22
+00:01:34,890 --> 00:01:41,940
+الاعضاء في الجسمهو بيعملنا cleaning للـ blood،
+
+23
+00:01:41,940 --> 00:01:45,640
+regulate hormones، helps with blood clotting، اللي
+
+24
+00:01:45,640 --> 00:01:49,640
+عملية التخفة والتجلط، من عنده بتبدأ، هو اللي بيفرز
+
+25
+00:01:49,640 --> 00:01:57,000
+الـ bile، هو اللي بيصنع الـ bile، وهو طبعا أهم
+
+26
+00:01:57,000 --> 00:02:02,200
+صانع لبروتينز، maintaining blood sugar level،
+
+27
+00:02:02,200 --> 00:02:07,860
+وكتير حاجات تانية، مافيش داعي لأذكرها الآن،الـ
+
+28
+00:02:07,860 --> 00:02:12,300
+liver is essential for life بدون liver ما بنقدر
+
+29
+00:02:12,300 --> 00:02:18,660
+نعيش ال anatomy بتاعة ال liver زي ما قلنا هو في ال
+
+30
+00:02:18,660 --> 00:02:23,840
+right upper quadrant في الاتجاه الآخر للمعدة يعني
+
+31
+00:02:23,840 --> 00:02:27,020
+المعدة على جهة هو على جهة وفي تحت منه ال
+
+32
+00:02:27,020 --> 00:02:31,940
+gallbladder بيطلع من ال liver
+
+33
+00:02:31,940 --> 00:02:36,580
+زي ما قلنا ال common bile ductوبدخل لإله الـ
+
+34
+00:02:36,580 --> 00:02:43,520
+portal vein و الـ hepatic artery what is
+
+35
+00:02:43,520 --> 00:02:48,100
+hepatitis؟ hepatitis means inflammation ماحدش
+
+36
+00:02:48,100 --> 00:02:52,160
+يقوللي infection ال infection معناته في عندي micro
+
+37
+00:02:52,160 --> 00:02:56,640
+organism هو العملية عمللي inflammation لكن ال
+
+38
+00:02:56,640 --> 00:02:58,920
+inflammation ممكن يكون هو
+
+39
+00:03:02,320 --> 00:03:06,700
+Inflammation بشيء تاني غير الفيروس أو البكتيريا أو
+
+40
+00:03:06,700 --> 00:03:10,740
+ال micro organism ممكن يكون زي ما قلنا نتيجة مشاكل
+
+41
+00:03:10,740 --> 00:03:15,980
+مشاكل هنحكي عن ال causes بعد شوية زي الألكوهول عسب
+
+42
+00:03:15,980 --> 00:03:21,430
+المثاليبقى viral hepatitis means there is a
+
+43
+00:03:21,430 --> 00:03:24,390
+specific virus that is causing the liver to
+
+44
+00:03:24,390 --> 00:03:27,950
+inflame طبعا inflammation بياخد كل صفات ال
+
+45
+00:03:27,950 --> 00:03:33,150
+inflammation swelling or become larger than normal
+
+46
+00:03:33,150 --> 00:03:40,530
+ال causes of hepatitis reactions
+
+47
+00:03:40,530 --> 00:03:47,760
+to chemical agents drugs and toxinsحلو لو انا في
+
+48
+00:03:47,760 --> 00:03:54,080
+الواحد ايش chemicals agent او chemical agents ايش
+
+49
+00:03:54,080 --> 00:03:58,380
+drugs ايش toxins ممكن تديني تعملي hepatitis او
+
+50
+00:03:58,380 --> 00:04:04,480
+autoimmune disease و viruses هذه هي الأسباب ال
+
+51
+00:04:04,480 --> 00:04:11,350
+virusesViral Hepatitis دينا Hepatitis A وB وC وD
+
+52
+00:04:11,350 --> 00:04:15,890
+وE على أيامنا بالمناسبة لما أنا درست كان عندنا A
+
+53
+00:04:15,890 --> 00:04:21,970
+وB وفي الآخر في واحد اللي جوا واكتشفوا اللي هو C
+
+54
+00:04:21,970 --> 00:04:26,750
+بس ماسموهوش C كانوا يقولوا none A none Bلا A ولا B
+
+55
+00:04:26,750 --> 00:04:31,890
+يعني حاجة تانية C ولكن هي مع الوجه الآن صار إنها C
+
+56
+00:04:31,890 --> 00:04:36,870
+وD وE واحتمال في السنوات الجاية يكون في اكتشاف لا
+
+57
+00:04:36,870 --> 00:04:43,610
+فيروسات تانية الـ Hepatitis A and E الـ A and E
+
+58
+00:04:43,610 --> 00:04:46,590
+بدأ نبدأ يا جماعة من الآن نفرج بين الخمس أنواع
+
+59
+00:04:46,590 --> 00:04:49,610
+لأنه
+
+60
+00:04:49,610 --> 00:04:55,650
+فعلا بختلفوا عن بعض الـ A and Eالذين يتسببون
+
+61
+00:04:55,650 --> 00:04:59,690
+بالإنجاش من الطعام المخطط أو الماء
+
+62
+00:05:02,840 --> 00:05:08,400
+الطريق تبقى الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+63
+00:05:08,400 --> 00:05:08,860
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+64
+00:05:08,860 --> 00:05:09,500
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+65
+00:05:09,500 --> 00:05:09,700
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+66
+00:05:09,700 --> 00:05:11,040
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+67
+00:05:11,040 --> 00:05:12,900
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+68
+00:05:12,900 --> 00:05:17,220
+الـ الـ الـ الـ الـ الـ
+
+69
+00:05:17,220 --> 00:05:20,480
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+70
+00:05:20,480 --> 00:05:20,940
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+71
+00:05:20,940 --> 00:05:32,780
+الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+72
+00:05:32,910 --> 00:05:39,350
+الـ B والـ C والـ D عادةً
+
+73
+00:05:39,350 --> 00:05:42,970
+caused أو typically caused by contact with
+
+74
+00:05:42,970 --> 00:05:48,390
+contaminated blood or blood fluids blood or blood
+
+75
+00:05:48,390 --> 00:05:51,770
+fluids نقصد فيها طبعا أي fluid بيبقى فيها الـ
+
+76
+00:05:51,770 --> 00:05:54,190
+semen أسباب مثال بيبقى فيها ال blood
+
+77
+00:06:01,840 --> 00:06:05,420
+يبقى هذه عملية الانتقال تبعتهم A وB وC وD وE
+
+78
+00:06:05,420 --> 00:06:08,700
+بالخصوص
+
+79
+00:06:08,700 --> 00:06:14,180
+الـ Hepatitis type B وC بوصلوني إلى chronic
+
+80
+00:06:14,180 --> 00:06:21,060
+disease في مئات الملايين من الناس وهم the most
+
+81
+00:06:21,060 --> 00:06:24,840
+common cause of liver cirrhosis and cancer مع بعض
+
+82
+00:06:24,840 --> 00:06:26,160
+التانين مع بعض الـB وC
+
+83
+00:06:36,270 --> 00:06:40,410
+يبقى الـ POC كمان مرة، lead to chronic disease in
+
+84
+00:06:40,410 --> 00:06:44,910
+hundreds of millions of people and together are
+
+85
+00:06:44,910 --> 00:06:48,250
+the most common cause of liver cirrhosis and
+
+86
+00:06:48,250 --> 00:06:54,670
+cancer واحد مليون die each year from causes
+
+87
+00:06:54,670 --> 00:07:00,430
+related to viral hepatitisالأن كمان مرة الـ type
+
+88
+00:07:00,430 --> 00:07:05,310
+of hepatitis A و B و C و D الان ال source of virus
+
+89
+00:07:05,310 --> 00:07:10,510
+ال A قلنا و ال E ال faces بوراز فيكو oral بينجل من
+
+90
+00:07:10,510 --> 00:07:15,350
+ال faces ال بوراز إلى ال oral و B و C و D عن طريق
+
+91
+00:07:15,350 --> 00:07:22,540
+ال blood و ال blood derived و ال body fluidsالـ
+
+92
+00:07:22,540 --> 00:07:27,640
+Root of Transmission A فيكو أورال مش Orophical
+
+93
+00:07:27,640 --> 00:07:32,620
+فيكو أورال والـ A فيكو أورال لكن الـ B و الـ C و
+
+94
+00:07:32,620 --> 00:07:38,700
+الـ D عادة عن طريق الـ Mucosa Percutaneous ميوكوزا
+
+95
+00:07:43,720 --> 00:07:48,180
+chronic infection الـ E والـ E الـ A والـ E فيش
+
+96
+00:07:48,180 --> 00:07:52,060
+chronic infection لكن الـ B والـ C والـ D نعم ال
+
+97
+00:07:52,060 --> 00:07:55,200
+prevention وهذا طبعا مهم بالنسبة لنا ال prevention
+
+98
+00:07:55,200 --> 00:08:00,620
+أنه ال A حاجة منقول عنها pre and post exposure
+
+99
+00:08:00,620 --> 00:08:08,140
+immunization فيه قلوطات طيب ال B برضه فيه قلوطات
+
+100
+00:08:08,140 --> 00:08:11,560
+طيب C
+
+101
+00:08:12,980 --> 00:08:18,280
+مافيش إله تطعيم الـD إله جثة تانية، هنحكيها بعد
+
+102
+00:08:18,280 --> 00:08:23,520
+شوية هو إله تطعيم بس من خلال تطعيم الـB والـE
+
+103
+00:08:23,520 --> 00:08:26,360
+مالوش تطعيم، يبقى اللي مالهم مش أكيد تطعيم الـC
+
+104
+00:08:26,360 --> 00:08:33,490
+والـE الـA والـB والـD اللي هم تطعيملكن الـ D مش
+
+105
+00:08:33,490 --> 00:08:37,450
+تطعيم خصوصي بيه، إنما أنا من خلال تطعيمي للـ B
+
+106
+00:08:37,450 --> 00:08:43,110
+بأثر على الـ D بالصورة هذه، هنحكي الآن عن كل واحد،
+
+107
+00:08:43,110 --> 00:08:46,270
+إيش
+
+108
+00:08:46,270 --> 00:08:48,330
+ال signs و ال symptoms اللي بنشوفها في الناس اللي
+
+109
+00:08:48,330 --> 00:08:56,490
+عندهم Hepatitis، بيبقوا tired، nausea، muscle or
+
+110
+00:08:56,490 --> 00:08:59,810
+joint pain، ملاحظين أن هذه الحاجات اللي بنسمعها في
+
+111
+00:08:59,810 --> 00:09:03,910
+كتير من ال viral infectionTrouble sleeping Loss of
+
+112
+00:09:03,910 --> 00:09:10,990
+appetite Light stools Weight loss Abdominal pain
+
+113
+00:09:10,990 --> 00:09:16,530
+Itchiness من الصفار بيصير في حكة Depression Dark
+
+114
+00:09:16,530 --> 00:09:22,250
+urine والبول بيبقى غامق والبراص بيبقى فاتح
+
+115
+00:09:26,760 --> 00:09:31,340
+من أهم ال symptoms اللي احنا بنشوفها اللي هو
+
+116
+00:09:31,340 --> 00:09:35,600
+joinedess ال joinedess يعني سفار يولوينج of skin
+
+117
+00:09:35,600 --> 00:09:43,370
+or eyes طبعا or and eyes ممكنSkin or Eyes
+
+118
+00:09:43,370 --> 00:09:50,270
+Yellowing بيسمي إيش Jointess وإيش كمعاملة الـ
+
+119
+00:09:50,270 --> 00:09:52,730
+Symptoms؟ بتلاقي في عنده Pale Stool اللي قلنا قبل
+
+120
+00:09:52,730 --> 00:09:56,390
+شوية عنه Light Stool بتلاقي إنه البراز إذا فكرين
+
+121
+00:09:56,390 --> 00:09:59,910
+المرة اللي فات تحدتنا بيبقى شكله زي طحانية بيضة
+
+122
+00:09:59,910 --> 00:10:07,420
+Pale Pale ال Hepatitis Aمن أهم الأنواع الـ
+
+123
+00:10:07,420 --> 00:10:12,080
+Hepatitis واللي هو معظمنا عملها، أخدها، وإحنا
+
+124
+00:10:12,080 --> 00:10:15,560
+صغار، لأن هو الـ Hepatitis اللي بنقول عنه اللي هو
+
+125
+00:10:15,560 --> 00:10:20,740
+التهاب الكبد الوبائي، الـA بالذات، التهاب الكبد
+
+126
+00:10:20,740 --> 00:10:24,740
+الوبائي هو طبعا الكلمة كبيرة، لكن هو الأمر أسهل
+
+127
+00:10:24,740 --> 00:10:27,460
+وكافي، بيصير في الأطفال الـHepatitis A، ممكن يكون
+
+128
+00:10:27,460 --> 00:10:32,040
+في الأكبر، لكن هو is a pediatric is a child
+
+129
+00:10:32,040 --> 00:10:38,610
+diseaseمعظمنا أخدوا وعدّى عليه يوم أو يومين وكأنه
+
+130
+00:10:38,610 --> 00:10:42,590
+انفلونزا وكأنه رشح وكأنه كذا شوية صفار وانتهت
+
+131
+00:10:42,590 --> 00:10:48,790
+الأمور هنحكي في الفيروسات الآن أنه لكل فيروس فيه
+
+132
+00:10:48,790 --> 00:10:49,770
+إيقاع
+
+133
+00:10:52,300 --> 00:10:57,680
+الـ Incubation Period بنتفق أنه أنه أنا هحكي مش
+
+134
+00:10:57,680 --> 00:11:01,800
+Range هحكي Average عشان أسهل لإلكوا وانتوا خدوا
+
+135
+00:11:01,800 --> 00:11:06,160
+بالكم الـ Average الـ Incubation اللي هو الفترة
+
+136
+00:11:06,160 --> 00:11:11,740
+التي تمر بينما أنه الطفل ياخد العدوى وبينما تظهر
+
+137
+00:11:11,740 --> 00:11:20,790
+الأعراض حوالي 30 يوم ال Age Groupاللي أقل من 6
+
+138
+00:11:20,790 --> 00:11:28,490
+سنين بتصير معاهم أقل من 10% 6 إلى 14 سنة ربعيه لـ
+
+139
+00:11:28,490 --> 00:11:33,370
+50% أكتر من 14 سنة بيصير عندهم Jaundice يعني الآن
+
+140
+00:11:33,370 --> 00:11:41,430
+هنلاقي أنه اللي أقل من 6 سنين يعني عملياً أقل من
+
+141
+00:11:41,430 --> 00:11:44,690
+10% من الأطفال أقل من 6 سنين بيصير معاهم Jaundice
+
+142
+00:11:44,690 --> 00:11:49,580
+يعني كتير منهم بمر عليه بياخد الفيروسبدون ما يصير
+
+143
+00:11:49,580 --> 00:11:55,540
+عنده jaundice كل ما كبر الواحد كل ما الصفار بيّن
+
+144
+00:11:55,540 --> 00:11:58,600
+عنده يعني الناس اللي بين ست سنين واربعتاشر سنة في
+
+145
+00:11:58,600 --> 00:12:02,160
+العمر لأ اربعين لخمسين في المية منهم يعني من كل
+
+146
+00:12:02,160 --> 00:12:05,700
+اتنين واحد بيبين عنده صفار قولنا الأقل من ست سنين
+
+147
+00:12:05,700 --> 00:12:08,580
+تقريبا معظمهم بيبينش يعني أقل من عشر في المية
+
+148
+00:12:08,580 --> 00:12:11,980
+بيبين عندهم صفار لأن فجر اربعتاشر في المية
+
+149
+00:12:11,980 --> 00:12:19,030
+اربعتاشر سنة الأكبار لو صار عنده hepatitis A70 ل
+
+150
+00:12:19,030 --> 00:12:26,450
+80% بيبين عنده صفار أمراض الأطفال يا جماعة اللي هي
+
+151
+00:12:26,450 --> 00:12:30,590
+specific للأطفال لو صارت عند الكبار تتعبهم جدا في
+
+152
+00:12:30,590 --> 00:12:35,130
+حين أن الطفل مابتعبش منها كتير لكن الكبار بتعبوا
+
+153
+00:12:35,130 --> 00:12:37,510
+منها جدا وهذا اللي بنلاحظه الآن على سبيل المثال في
+
+154
+00:12:37,510 --> 00:12:41,230
+الكورونا أن الأطفال مابتعب من فيروس الكورونا كتير
+
+155
+00:12:41,230 --> 00:12:45,900
+okay في مجموعة لكن الغالبالـ Hepatitis نفس الكلام
+
+156
+00:12:45,900 --> 00:12:49,740
+فينا لـ Chickenpox اللي هو الجدير الماء برضه إذا
+
+157
+00:12:49,740 --> 00:12:53,840
+أخذوا طفوله الواحد و هو طفل ولا في مشاكل لكن إذا
+
+158
+00:12:53,840 --> 00:12:57,520
+هو كبير أخذه بيتعب ممكن يحتاج مستشفيات كمان، نوع
+
+159
+00:12:57,520 --> 00:13:02,380
+مستشفي أهم complication للـ Hepatitis A اللي هو
+
+160
+00:13:02,380 --> 00:13:05,400
+الـ Fulminating Hepatitis إيش يعني Fulminating؟ ده
+
+161
+00:13:05,400 --> 00:13:10,840
+اللي يصير إن التهاب حاد شديد للكبد و طبعا ممكن
+
+162
+00:13:10,840 --> 00:13:17,490
+يروح فيها الواحدولكن هذا Complication أو Relapsing
+
+163
+00:13:17,490 --> 00:13:22,670
+Hepatitis يعني ممكن اللي أجاله Hepatitis A يجيله
+
+164
+00:13:22,670 --> 00:13:28,890
+كمان مرة نعم لكن بالنسبة جليلة بالنسبة جليلة و ال
+
+165
+00:13:28,890 --> 00:13:32,370
+Hepatitis A عمره بيصير chronic عمره بيصير chronic
+
+166
+00:13:32,370 --> 00:13:36,550
+يبقى احنا الآن حكينا مجموعة حقائق عن ال Hepatitis
+
+167
+00:13:36,550 --> 00:13:44,220
+A بداية من طريقة العدوى قلنا فيكوا Oralقلنا إن الـ
+
+168
+00:13:44,220 --> 00:13:48,240
+Incubation تبعه حوالي تلاتين يوم الناس كل ما كبر
+
+169
+00:13:48,240 --> 00:13:51,660
+الواحد الـ jaundice بيبين عنده أكتر وقلنا النسب ال
+
+170
+00:13:51,660 --> 00:13:54,960
+complications of fulminant hepatitis وrelapsing
+
+171
+00:13:54,960 --> 00:13:58,180
+hepatitis الـ relapsing يعني بروحه بيجي و الـ no
+
+172
+00:13:58,180 --> 00:14:05,960
+chronic sequelae طيب
+
+173
+00:14:05,960 --> 00:14:09,770
+هو ممكن يندجل عن طريق الدم؟نعم لكن نادر كتير كتير
+
+174
+00:14:09,770 --> 00:14:14,700
+كتير كتير هو بنتجل عادة close personal contactزي
+
+175
+00:14:14,700 --> 00:14:20,640
+الرودات والحضانات وقلنا كمان أنه بنتجل عن طريق الـ
+
+176
+00:14:20,640 --> 00:14:24,860
+contaminated food و ال water الناس البائعين
+
+177
+00:14:24,860 --> 00:14:29,240
+الجوالين الناس اللي بتبيع و بتحافظش بتحافظش على
+
+178
+00:14:29,240 --> 00:14:32,880
+الناس اللي بيأكلوا السمك نهي الناس البائعين
+
+179
+00:14:32,880 --> 00:14:40,360
+الجوالين اللي على الكارات اللي يعني عمليا نسبة
+
+180
+00:14:40,360 --> 00:14:45,660
+النضافة بتبقى عندهم zeroطبعاً هدول واحد بيخاف منهم
+
+181
+00:14:45,660 --> 00:14:48,480
+أنهم يعملوا و blood exposure in very rare
+
+182
+00:14:48,480 --> 00:14:52,680
+Transmission
+
+183
+00:14:52,680 --> 00:14:59,060
+الـ Hepatitis B هو طبعاً مختلف لأنه احنا قولنا الـ
+
+184
+00:14:59,060 --> 00:15:02,940
+Hepatitis B بنجل عن طريق blood و blood products و
+
+185
+00:15:02,940 --> 00:15:07,340
+عن طريق fluids of the body ال incubation تبعه
+
+186
+00:15:07,340 --> 00:15:12,980
+حوالي من شهرية لتلتة شهر ال averageنحكي عن الـ
+
+187
+00:15:12,980 --> 00:15:19,440
+Average الـ Clinical Illness بالنسبة
+
+188
+00:15:19,440 --> 00:15:25,180
+للصفار أقل من 5 سنين أقل من 10% لكن 5 سنين وفوق من
+
+189
+00:15:25,180 --> 00:15:30,960
+30 ل 50% يعني النص يبين عندهم صفار يعني ممكن واحد
+
+190
+00:15:30,960 --> 00:15:35,520
+ياخد عدوة Hepatitis وما يبينش عنده أه ممكن طبعا
+
+191
+00:15:36,190 --> 00:15:39,870
+الخطر هو أن أجالي من خمس سنين تلاتي لتسعين في
+
+192
+00:15:39,870 --> 00:15:44,830
+المائية بيصير عندهم chronicity لكن خمس سنين وفوق
+
+193
+00:15:44,830 --> 00:15:47,970
+من اتنين لعشرة في المائية بيصير عندهم chronicity
+
+194
+00:15:47,970 --> 00:15:52,130
+وممكن طبعا الواحد لأنه احنا قلنا بيصير chronic
+
+195
+00:15:52,130 --> 00:15:55,790
+disease من خمستاشر لخمسة وعشرين في المائية ممكن
+
+196
+00:15:55,790 --> 00:15:59,010
+يموتوا premature mortality من chronic liver
+
+197
+00:15:59,010 --> 00:16:05,770
+diseaseالـ Hepatitis B إيه بعض الحاجات اللي لازم
+
+198
+00:16:05,770 --> 00:16:09,250
+ننتبه عليها اللي هو الـ Transmission Mode of
+
+199
+00:16:09,250 --> 00:16:18,910
+Hepatitis B الأول حاجة اللي هو الـ Perinatal لأن
+
+200
+00:16:18,910 --> 00:16:23,410
+في عندنا الـ Perinatal Transmissionبينجل الـ
+
+201
+00:16:23,410 --> 00:16:30,870
+«فايروس» من الأم الحامل للطفل 90% من الأطفال
+
+202
+00:16:30,870 --> 00:16:34,870
+المصابين بيصبحوا مصابين بشكل كرونيكي
+
+203
+00:16:34,870 --> 00:16:37,890
+والمامي
+
+204
+00:16:37,890 --> 00:16:41,790
+اللي مصابة بشكل كرونيكي هي أكتر ممكن تتعامل مع
+
+205
+00:16:41,790 --> 00:16:47,870
+أطفالها للطفل ليش؟ لأن هو بياخد الدم يعني عن طريق
+
+206
+00:16:47,870 --> 00:16:48,210
+الـ
+
+207
+00:16:50,900 --> 00:16:54,580
+بياخد الدم، بياخد عن طريق الدم، طبيعي، وفيه طبعا
+
+208
+00:16:54,580 --> 00:16:59,880
+الـ Sexual كمان غير طبعا عن طريق الدم، هذه طريقة
+
+209
+00:16:59,880 --> 00:17:02,940
+modes of transmission of Hepatitis B الـ Sex
+
+210
+00:17:02,940 --> 00:17:07,900
+worker and homosexuals are particular at risk زي
+
+211
+00:17:07,900 --> 00:17:14,840
+الـ HIV، نفس الكلام بيندجل اللي هو الـ Hepatitis
+
+212
+00:17:14,840 --> 00:17:16,000
+عن طريق
+
+213
+00:17:20,010 --> 00:17:26,370
+عن طريق اللي هو الـ sexual contacts طبعا سواء كانت
+
+214
+00:17:26,370 --> 00:17:30,150
+في إطار القانون أو في إطار خارج القانون ال health
+
+215
+00:17:30,150 --> 00:17:32,810
+care worker ممكن ينتجلهم child to child ممكن
+
+216
+00:17:32,810 --> 00:17:36,710
+ينتجلهم زي ما قلنا ال contaminated needles الناس
+
+217
+00:17:36,710 --> 00:17:42,070
+اللي بتاخد مخدرات وبياخدوا needles من بعض هدول
+
+218
+00:17:42,070 --> 00:17:46,130
+كمان ممكن ينتجلهم ال blood transfusion عشان هيكهذا
+
+219
+00:17:46,130 --> 00:17:48,650
+بنقول عنه «Horizontal Transmission» اللي هو الـ
+
+220
+00:17:48,650 --> 00:17:52,370
+«host to other» «Recipient» لكن من الـ «mother»
+
+221
+00:17:52,370 --> 00:17:57,170
+للـ «infant» بنقول عنه «perinatal» أو «vertical»
+
+222
+00:17:57,170 --> 00:18:01,730
+Transmission عشان أشخص أنا الـ Hepatitis B في
+
+223
+00:18:01,730 --> 00:18:06,610
+مجموعة من الـ «serological tests» بتنعمل مش لعيلة
+
+224
+00:18:06,610 --> 00:18:11,550
+ذكرة أو متابعتها لأن هذه حاجة تقصصية للـ «acute»
+
+225
+00:18:11,550 --> 00:18:14,070
+والـ «chronic» Hepatitis B طب كيف بنعالج الـ
+
+226
+00:18:14,070 --> 00:18:14,750
+Hepatitis؟
+
+227
+00:18:18,540 --> 00:18:22,720
+بالـ interferon من 30 ل 40% الـ response rate يعني
+
+228
+00:18:22,720 --> 00:18:25,720
+ضعيف عامليا مش كتير successful response to
+
+229
+00:18:25,720 --> 00:18:29,280
+treatment و result in the disappearance of
+
+230
+00:18:29,280 --> 00:18:34,200
+hepatitis B antigen المهم حقيقة في موضوعنا في ال
+
+231
+00:18:34,200 --> 00:18:37,400
+hepatitis B هو ال prevention و ركزوا في موضوع ال
+
+232
+00:18:37,400 --> 00:18:39,920
+hepatitis مش على ال treatment على ال prevention
+
+233
+00:18:39,920 --> 00:18:47,340
+vaccinationوطبعاً هذا الكل بياخده في برنامج
+
+234
+00:18:47,340 --> 00:18:51,800
+التطعيمات العالمي ودي موجود عندنا في القطاع من
+
+235
+00:18:51,800 --> 00:18:56,340
+واحد هو طفل بياخده Highly effective recombinant
+
+236
+00:18:56,340 --> 00:19:00,780
+vaccine are now available Vaccine can be given to
+
+237
+00:19:00,780 --> 00:19:04,100
+those who are at increased risk of Hepatitis B
+
+238
+00:19:04,100 --> 00:19:07,460
+virus infection زي الـHealth Care Worker، يعني لو
+
+239
+00:19:07,460 --> 00:19:11,030
+واحد بده يروح يتوظففي الحكومة بيعملوله بيشوف ال
+
+240
+00:19:11,030 --> 00:19:13,110
+antibody عنده لو لقوا عنده عنده مشكلة في ال
+
+241
+00:19:13,110 --> 00:19:16,450
+antibodies بيطعموه كمان بيعملوله vaccination it is
+
+242
+00:19:16,450 --> 00:19:20,270
+also given routinely to neonates as a universal
+
+243
+00:19:20,270 --> 00:19:24,710
+vaccination in many countries Hepatitis B طيب واحد
+
+244
+00:19:24,710 --> 00:19:28,710
+الآن ولد وامه عنده Hepatitis B بينعطى
+
+245
+00:19:32,690 --> 00:19:39,090
+اللي هو Hemoglobin بـ Hepatitis B immunoglobulin
+
+246
+00:19:39,480 --> 00:19:42,760
+ممكن يتم استخدامها لحماية الشخصين اللي اتفاقوا بـ
+
+247
+00:19:42,760 --> 00:19:50,360
+Hepatitis B وزي ما قلنا انها تحدث في 48
+
+248
+00:19:50,360 --> 00:19:54,280
+ساعة من الحادثة هنا طبعا بنحكي احنا الفرق بين
+
+249
+00:19:54,280 --> 00:19:58,080
+اللقاح و الـ muscle الvaccination هو اللقاح ال
+
+250
+00:19:58,080 --> 00:20:01,500
+muscle اللي هو ال immunoglobulin اللي بنعطيه الآن
+
+251
+00:20:01,500 --> 00:20:05,300
+على أساس انه يدينا effect ال vaccination بياخد وقت
+
+252
+00:20:05,300 --> 00:20:09,950
+عبارة من ما بيجيبولكن الـ muscle بيساعد على طول في
+
+253
+00:20:09,950 --> 00:20:16,170
+إنه يديني أكون efficient it may also be given to
+
+254
+00:20:16,170 --> 00:20:20,650
+neonate who are at increased risk of contracting
+
+255
+00:20:20,650 --> 00:20:24,970
+hepatitis B ممكن يكون أخر مجارات أخرى Screening of
+
+256
+00:20:24,970 --> 00:20:29,490
+blood donors blood and blood fluid precautions
+
+257
+00:20:29,490 --> 00:20:30,850
+طبعاً
+
+258
+00:20:33,380 --> 00:20:37,940
+في العلاقات على سبيل المثال سواء الزوجية أو غير
+
+259
+00:20:37,940 --> 00:20:40,620
+الزوجية على أساس إنه بنتجيل كمان sexual من واحد
+
+260
+00:20:40,620 --> 00:20:46,240
+positive لواحد مش positive أو للطرف الآخر مش لابد
+
+261
+00:20:46,240 --> 00:20:51,320
+من أخد سبل الوقاية في هذا الموضوع إنقل الدم في
+
+262
+00:20:51,320 --> 00:20:54,640
+المستشفيات يتم الانتباه لهذا الموضوع بنفحص الدم
+
+263
+00:20:54,640 --> 00:21:01,320
+المتبرع على hepatitis B مش بس على hepatitis C كمان
+
+264
+00:21:01,320 --> 00:21:07,450
+و بنفحص علىالـ HIV اللي هو الـ AIDS وهذا بنجيلنا
+
+265
+00:21:07,450 --> 00:21:11,490
+لموضوع الـ Hepatitis C هو طبعا الـ Hepatitis C بدأ
+
+266
+00:21:11,490 --> 00:21:15,250
+ينتشر بصورة مهولة خاصة في دول زي جيراننا في مصر
+
+267
+00:21:15,250 --> 00:21:19,030
+الـ Incubation period الـ average ستة لسبع أسابيع
+
+268
+00:21:20,360 --> 00:21:24,080
+أحفظوا الـ Average يا جماعة الـ Jaundice 30% لـ 40
+
+269
+00:21:24,080 --> 00:21:28,160
+% ممكن تشوفوا عند الناس اللي بيصير معهم يعني مش كل
+
+270
+00:21:28,160 --> 00:21:30,660
+واحد بيصير عنده Hepatitis بيصير عنده Jaundice
+
+271
+00:21:30,660 --> 00:21:35,600
+Chronicity كبيرة 70% Persistent infection بيظل
+
+272
+00:21:35,600 --> 00:21:40,760
+الفيروس موجود في الكبد وال Immunity مافيش أنه
+
+273
+00:21:40,760 --> 00:21:46,420
+Protective Antibody Response Identified لحد تلقاه
+
+274
+00:21:46,420 --> 00:21:51,480
+مافيش حد تطيب لإلهالـ Hepatitis C تأثير على كل شخص
+
+275
+00:21:51,480 --> 00:21:54,920
+بشكل مختلف في الآخر ما صار المرض عند كل واحد مختلف
+
+276
+00:21:54,920 --> 00:21:57,620
+لا يوجد مصادر مصادر مصادر مصادر مصادر مصادر مصادر
+
+277
+00:21:57,620 --> 00:21:57,680
+مصادر مصادر مصادر مصادر مصادر مصادر مصادر مصادر
+
+278
+00:21:57,680 --> 00:22:00,440
+مصادر مصادر مصادر مصادر مصادر مصادر مصادر مصادر
+
+279
+00:22:00,440 --> 00:22:02,340
+مصادر مصادر مصادر مصادر مصادر مصادر مصادر مصادر
+
+280
+00:22:02,340 --> 00:22:03,900
+مصادر مصادر مصادر مصادر مصادر مصادر مصادر مصادر
+
+281
+00:22:03,900 --> 00:22:18,080
+مصادر مصادر مصادر مصادر مصادر مصادر مصاد
+
+282
+00:22:19,720 --> 00:22:23,440
+إنه لما يصير عندى acute infection بالـ Hepatitis
+
+283
+00:22:23,440 --> 00:22:26,720
+C، only 20 will show symptoms initially، زي ما
+
+284
+00:22:26,720 --> 00:22:30,740
+اتفرجنا، يعني مش بالضرورة كله يبينلي أي symptoms،
+
+285
+00:22:30,740 --> 00:22:33,860
+وبدخل في إيش؟ في chronicity، شايفين احنا ال curve؟
+
+286
+00:22:33,860 --> 00:22:37,060
+بدخل في chronic infection، بعد ال chronic الجسم
+
+287
+00:22:37,060 --> 00:22:42,880
+بتغلب في 20% على الفيروس، لكن في عندنا 80% بيظل هم
+
+288
+00:22:42,880 --> 00:22:47,990
+موجودين،في الكبد وبيعملولنا liver damage وبعدين
+
+289
+00:22:47,990 --> 00:22:51,950
+بدخلوا في chronicity زي ما حكينا chronic hematitis
+
+290
+00:22:51,950 --> 00:22:56,430
+وبعدين 20 ل 30% منهم بدخلوا على liver cirrhosis
+
+291
+00:22:56,430 --> 00:23:02,830
+وفي منهم بيصير عنده liver cancer من 1 ل 4% لكل سنة
+
+292
+00:23:02,830 --> 00:23:05,750
+most symptoms begin to show only when liver is
+
+293
+00:23:05,750 --> 00:23:08,850
+more severely damaged يعني لما تبدأ تبين ال
+
+294
+00:23:08,850 --> 00:23:14,180
+symptoms عادة بيبقى في عنا مشكلة كبيرة فيهنا مشكلة
+
+295
+00:23:14,180 --> 00:23:21,280
+كبيرة في الكبد Risk
+
+296
+00:23:21,280 --> 00:23:24,720
+factors associated with transmission of hepatitis
+
+297
+00:23:24,720 --> 00:23:31,270
+C virus زي ما قلنا Transfusionأو transplant من
+
+298
+00:23:31,270 --> 00:23:36,010
+واحد infected donor أو استخدام نفس الـ lineات نفس
+
+299
+00:23:36,010 --> 00:23:40,350
+الكانيولات من واحد معدي عنده hepatitis C لواحد ما
+
+300
+00:23:40,350 --> 00:23:44,230
+عندهوش injecting drug a use الناس اللي بيعملوا
+
+301
+00:23:44,230 --> 00:23:49,160
+غسيل كدهaccidental injury with needle or sharps
+
+302
+00:23:49,160 --> 00:23:53,540
+الإبار وكذا في المستشفيات sexual أو household
+
+303
+00:23:53,540 --> 00:23:57,360
+exposure to antihepatitis C virus positive
+
+304
+00:23:57,360 --> 00:24:03,860
+contacts multiple sexual partners فاهمين birth to
+
+305
+00:24:03,860 --> 00:24:08,480
+hepatitis C virus infected mothers كمان وطبعا
+
+306
+00:24:08,480 --> 00:24:13,760
+التشخيص يحصل من خلال surgical tests treatment
+
+307
+00:24:16,820 --> 00:24:24,580
+متافرون وrebutherينيعني ممكن لوحده الـ «انتافيرون»
+
+308
+00:24:24,580 --> 00:24:30,760
+يدّينا نسبة 50% response مع الـ«ريبافيرين» تتحسن
+
+309
+00:24:30,760 --> 00:24:34,880
+النسبة المهم كمان مرة مش العلاج، okay بنعرف إن
+
+310
+00:24:34,880 --> 00:24:40,840
+الـ«ريبافيرين» و«انتافيرون» كويسين لكن أهم حاجة
+
+311
+00:24:40,840 --> 00:24:44,620
+لـprevention screening of blood organ and tissue
+
+312
+00:24:44,620 --> 00:24:48,340
+donors high risk behavior modification blood and
+
+313
+00:24:48,340 --> 00:24:52,060
+body fluid precautionsطبعاً لازم نبقى فاهمين بعض
+
+314
+00:24:52,060 --> 00:24:55,700
+الحاجات أنه الـ Hepatitis C لا تتم تفريغها
+
+315
+00:24:55,700 --> 00:25:02,580
+بـHugging، Kissing اللي هو الـCasual Contacts،
+
+316
+00:25:02,580 --> 00:25:07,020
+الـHugging، يعني الحضن Sharing, Eating, Utensils,
+
+317
+00:25:07,580 --> 00:25:13,520
+Drinking, Glasses، العطس، Scuffing، Shaking Hands،
+
+318
+00:25:13,520 --> 00:25:17,140
+Sitting on a Toilet Seat، يعني هذه ما بتعملي عملية
+
+319
+00:25:17,140 --> 00:25:24,080
+نقل للفيروسالأن زي ما عرفنا الـ Current Vaccine
+
+320
+00:25:24,080 --> 00:25:27,440
+حكيناها احنا قبل شوي قولنا الـ Hepatitis A والـ B
+
+321
+00:25:27,440 --> 00:25:35,540
+والـ C في اللهم تطعيم عفواً الـ A والـ B والـ D
+
+322
+00:25:35,540 --> 00:25:38,780
+اللهم تطعيم الـ D indirect حنحكي بعد شوي والـ C
+
+323
+00:25:38,780 --> 00:25:43,720
+مالوش طبعا للأسفالـ Hepatitis D بنقول عنه Delta
+
+324
+00:25:43,720 --> 00:25:49,780
+Virus هو عبارة عن بيجي إما كـ Co-infection، severe
+
+325
+00:25:49,780 --> 00:25:53,560
+acute disease، low risk of chronic infection أو
+
+326
+00:25:53,560 --> 00:25:58,120
+super infection، يعني إما بيجي مع الـB مع بعض أو
+
+327
+00:25:58,990 --> 00:26:02,370
+بيجي usually develop chronic hepatitis D virus
+
+328
+00:26:02,370 --> 00:26:05,690
+infection high risk to severe chronic liver
+
+329
+00:26:05,690 --> 00:26:09,870
+disease may present as an acute hepatitis بيجي بعد
+
+330
+00:26:09,870 --> 00:26:13,750
+الـB ما ييجي يعني مش بيجي مع الـB بيجي بعد الـB
+
+331
+00:26:17,420 --> 00:26:21,460
+وطبعاً ممكن ييجي عن طريق الـPercutaneous Exposure
+
+332
+00:26:21,460 --> 00:26:24,880
+زي الـInjecting Drug Used أو عن طريق الـPer
+
+333
+00:26:24,880 --> 00:26:29,380
+-Mucosal Exposure اللي هو الـSexual Contacts
+
+334
+00:26:29,380 --> 00:26:36,910
+Prevention الـHepatitis B Virusمع الـ Hepatitis D
+
+335
+00:26:36,910 --> 00:26:40,650
+Virus Co-infection لما يجوا مع بعض بينفعنا مين لـ
+
+336
+00:26:40,650 --> 00:26:44,730
+Pre- أو Post-Exposure Prophylaxis لتمنع الـ
+
+337
+00:26:44,730 --> 00:26:53,230
+Hepatitis B Virus لكن لما يكون Super Infection
+
+338
+00:26:53,230 --> 00:27:01,240
+بصرا بفتش كتير التطعيم، ليش؟ لأنه هو أجابعد الـB،
+
+339
+00:27:01,240 --> 00:27:05,080
+يعني صار في عندنا اتهاب B فبينفجر الـEducation to
+
+340
+00:27:05,080 --> 00:27:08,500
+reduce risk behavior among persons with chronic
+
+341
+00:27:08,500 --> 00:27:12,980
+hepatitis B virus ما هيفيد التطعيم في الأحوال
+
+342
+00:27:12,980 --> 00:27:16,880
+زيكي، بقى التطعيم بيجي في الـHepatitis D لما
+
+343
+00:27:16,880 --> 00:27:20,840
+بيستفيد منه مريض الـHepatitis D لما يكون Co
+
+344
+00:27:20,840 --> 00:27:25,640
+-infection، يعني Co معاه جاي، مش Super، Super يعني
+
+345
+00:27:25,640 --> 00:27:30,170
+موجود الـHepatitis Bوأجعله الـD فوق منه
+
+346
+00:27:30,170 --> 00:27:33,710
+الـHepatitis E قولنا هو عبارة عن الـAverage تبعه
+
+347
+00:27:33,710 --> 00:27:37,870
+حوالي 40 Days والـCase
+
+348
+00:27:37,870 --> 00:27:42,250
+Fatality Rate حوالي 1% لـ3% طبعاً pregnant woman
+
+349
+00:27:42,250 --> 00:27:47,710
+بالذات إن السيفيريتي increased with age وإنه مافيش
+
+350
+00:27:47,710 --> 00:27:53,310
+Chronic Sequel في العادة وهذا بنلاقي عادة في
+
+351
+00:27:53,310 --> 00:27:57,940
+الـFacially Contaminated DrinkingWater المجاري
+
+352
+00:27:57,940 --> 00:28:03,960
+بتطف على المية العادية Prevention تبع الـ E هو زي
+
+353
+00:28:03,960 --> 00:28:09,280
+ما قلنا Avoid drinking water Uncooked
+
+354
+00:28:09,280 --> 00:28:12,720
+shell-filled fish Uncooked fruits and vegetables
+
+355
+00:28:12,720 --> 00:28:19,760
+Not peeled or prepared by travelersوممكن ياخد
+
+356
+00:28:19,760 --> 00:28:25,260
+الواحد immunoglobulin بس
+
+357
+00:28:25,260 --> 00:28:31,030
+يعني في الآخر مفيدش كتير للتطبيقالـ Diagnostic
+
+358
+00:28:31,030 --> 00:28:33,950
+طبعاً كل الـ Hepatitis أنا بعمل حاجة اسمها
+
+359
+00:28:33,950 --> 00:28:40,310
+بالفحوصات «Liver Function Test» «Alkaline
+
+360
+00:28:40,310 --> 00:28:43,430
+Phosphatase» «Serum Bilirubin» «Liver Biopsy»
+
+361
+00:28:43,430 --> 00:28:47,210
+«Antigen Specific» «Specific» حقيقة يعني ملزمكوش
+
+362
+00:28:47,210 --> 00:28:50,590
+كتير الموضوع هذا لكن اللي ملزمنا أكتر الـ
+
+363
+00:28:50,590 --> 00:28:55,410
+Prevention وTreatment الـ «Treatment Diet» «High
+
+364
+00:28:55,410 --> 00:28:58,590
+Calories Protein Low Fat» على أساس أريح الكبد
+
+365
+00:29:00,270 --> 00:29:06,530
+الفيتامين زي الـ B و الـ K الـ Stop Alcohol Drugs
+
+366
+00:29:06,530 --> 00:29:11,770
+الفيتامين B و K يا جماعة هما بنقول عنهم الـ Edica
+
+367
+00:29:11,770 --> 00:29:16,810
+الـ E و الـ D و الـ K و الـ A هدول الفيتامينات
+
+368
+00:29:16,810 --> 00:29:22,470
+اللي هي بنقول عنها تذوب في الفات تذوب في الفات و
+
+369
+00:29:22,470 --> 00:29:26,150
+طبعا احنا هنا بيبقى انه انا عندي مشكلةفي الكبد
+
+370
+00:29:26,150 --> 00:29:31,390
+فهدولة الذوبان تبعهم بيبقى جالين فأنا بديهم خارجي
+
+371
+00:29:31,390 --> 00:29:35,330
+في العادة ليش؟ لأن الكبد لا يستطيع أنه يتعامل
+
+372
+00:29:35,330 --> 00:29:38,310
+معاهم stop alcohol and drugs fluid management bed
+
+373
+00:29:38,310 --> 00:29:41,350
+rest وطبعا drug therapy زي ال interferon
+
+374
+00:29:41,350 --> 00:29:44,930
+والميفودين والrimavirin والحاجات هاي و prevention
+
+375
+00:29:44,930 --> 00:29:51,390
+للهيباطايتس A والهيباطايتس B الآن في ال prevention
+
+376
+00:29:53,710 --> 00:29:59,310
+Never Share Drug Equipment ممنوع النيدل، السرنجات،
+
+377
+00:29:59,310 --> 00:30:02,450
+الـwater، الـfilter، الـcooker، الـpipes، هذا كله
+
+378
+00:30:02,450 --> 00:30:05,230
+الـstraws، الـpills، كله ممنوع نستخدمه مع بعض
+
+379
+00:30:05,230 --> 00:30:10,030
+Never Share Toothbrushes، فراش السنان، الماكنات
+
+380
+00:30:10,030 --> 00:30:13,630
+الإحلاقية اللي هي الشوفر، ليش؟ لأن هذا ممكن في
+
+381
+00:30:13,630 --> 00:30:18,590
+لحظة من لحظة تنجل عدوات Prevention بصورة عامة،
+
+382
+00:30:18,590 --> 00:30:23,310
+خدوا بالكم يا جماعةalways make sure a new and
+
+383
+00:30:23,310 --> 00:30:27,210
+sterilized equipment is being used for tattooing
+
+384
+00:30:27,210 --> 00:30:29,850
+and piercing يمكن ماعناش احنا كتير هيك لكن
+
+385
+00:30:29,850 --> 00:30:32,830
+tattooing اللي هو الوشم و ال piercing لما بيحطوا
+
+386
+00:30:32,830 --> 00:30:36,450
+جطع في المناخير و في الدان هيك الحلوج و كذا فهذه
+
+387
+00:30:36,450 --> 00:30:42,210
+طبعا بتنجل عدوات مش بس حتى الحبر اللي بيعمله منه
+
+388
+00:30:42,210 --> 00:30:48,250
+ال tattooing اللي هو عملية الوشمكمان ممكن يكون فيه
+
+389
+00:30:48,250 --> 00:30:53,190
+Hepatitis نوع من الأنواع اللي بتنتقل عن طريق الجلد
+
+390
+00:30:53,190 --> 00:30:56,750
+«Don't touch dirty needles» الإبر في المستشفيات
+
+391
+00:30:56,750 --> 00:30:59,630
+«without proper equipment» أو «following proper
+
+392
+00:30:59,630 --> 00:31:02,830
+procedures» «Main prevention measures for
+
+393
+00:31:02,830 --> 00:31:05,730
+Hepatitis B and C» وهذا من المناطق المهمة جدا
+
+394
+00:31:05,730 --> 00:31:08,930
+عندنا في الامتحانات «Promote vaccination against
+
+395
+00:31:08,930 --> 00:31:12,490
+Hepatitis B» «Ensure needles and syringe program
+
+396
+00:31:12,490 --> 00:31:17,580
+to avoid sharing injectingالتجارب التجاربية تحفظ
+
+397
+00:31:17,580 --> 00:31:21,100
+الهواتف تحفظ الـ Safe Sex تحفظ التجارب التجاربية
+
+398
+00:31:21,100 --> 00:31:23,940
+التحفظ الهواتف التحفظ الهواتف التحفظ الهواتف
+
+399
+00:31:23,940 --> 00:31:24,460
+التحفظ الهواتف التحفظ الهواتف التحفظ الهواتف
+
+400
+00:31:24,460 --> 00:31:24,940
+التحفظ الهواتف التحفظ الهواتف التحفظ الهواتف
+
+401
+00:31:24,940 --> 00:31:26,600
+التحفظ الهواتف التحفظ الهواتف التحفظ الهواتف
+
+402
+00:31:26,600 --> 00:31:26,760
+التحفظ الهواتف التحفظ الهواتف التحفظ الهواتف
+
+403
+00:31:26,760 --> 00:31:29,940
+التحفظ الهواتف التحفظ الهواتف التحفظ الهواتف
+
+404
+00:31:29,940 --> 00:31:34,630
+التحفظ الهواتف التحفظ الهواتف التحفظكـ population.
+
+405
+00:31:34,930 --> 00:31:38,830
+Dirty needles، إيش عامل فيها؟ Handle only if you
+
+406
+00:31:38,830 --> 00:31:42,930
+have proper equipment. Place needles in puncture
+
+407
+00:31:42,930 --> 00:31:46,630
+proof containers. في containers بنحط فيه ال
+
+408
+00:31:46,630 --> 00:31:52,710
+needles هاي. Can dispose containers in garbage but
+
+409
+00:31:52,710 --> 00:31:57,010
+better if it is taken to health clinic or needle
+
+410
+00:31:57,010 --> 00:32:01,510
+exchange زي المحارج وكذا. إذا إذا إذا أصلا عندك
+
+411
+00:32:01,510 --> 00:32:02,150
+needle break
+
+412
+00:32:05,460 --> 00:32:08,980
+انغزيت بإبرة، بغض النظر فيها أو ما فيها، don't
+
+413
+00:32:08,980 --> 00:32:13,140
+milk brick side، ما تقعدش تعصص عليها عشان بدك تطلع
+
+414
+00:32:13,140 --> 00:32:16,680
+دم منها، wash the area with soap and water، go to
+
+415
+00:32:16,680 --> 00:32:19,600
+nearest emergency department for assessment and
+
+416
+00:32:19,830 --> 00:32:23,930
+Treatment، بهيك احنا بنبقى عملنا محاضرة الـ
+
+417
+00:32:23,930 --> 00:32:27,090
+Hepatitis، أنا كنت أتوقع حقيقة تاخد وقت أكتر من
+
+418
+00:32:27,090 --> 00:32:31,390
+هيك، بس ربنا يسر ومشيته الأمور، بتمنى لكوا انهار
+
+419
+00:32:31,390 --> 00:32:35,830
+سعيد، واهتموا في موضوع الـ Hepatitis، مهم كتير
+
+420
+00:32:35,830 --> 00:32:39,590
+بالنسبالنا، خاصة في الامتحانات، مكان مهم للأسئلة
+
+421
+00:32:39,590 --> 00:32:40,490
+لكتير
+
diff --git a/PL9fwy3NUQKwaf4aPlmtIQXiASpDxVLEnO/FR-vBag9siY_raw.json b/PL9fwy3NUQKwaf4aPlmtIQXiASpDxVLEnO/FR-vBag9siY_raw.json
new file mode 100644
index 0000000000000000000000000000000000000000..80e3d8d1eeacfd70b5d40c5ce29f56aa5276cf4b
--- /dev/null
+++ b/PL9fwy3NUQKwaf4aPlmtIQXiASpDxVLEnO/FR-vBag9siY_raw.json
@@ -0,0 +1 @@
+{"segments": [{"id": 1, "seek": 1565, "start": 1.35, "end": 15.65, "text": "بسم الله الرحمن الرحيم الحمد لله رب العالمين والصلاة والسلام علي سيدنا محمد سيد المرسلين أجمعين يعني اليوم ان شاء الله نحاول نتحدث عن الاخر محاضرة هي في الصدرية عمليا", "tokens": [3555, 38251, 21984, 34892, 5016, 27842, 34892, 5016, 32640, 21542, 2304, 3215, 24976, 3224, 12602, 3555, 18863, 45340, 9957, 16070, 9381, 15040, 3660, 16070, 3794, 37440, 25894, 8608, 25708, 8315, 3714, 35571, 3215, 8608, 25708, 9673, 2288, 3794, 1211, 9957, 5551, 7435, 2304, 3615, 9957, 37495, 22653, 45595, 20498, 16472, 13412, 16606, 21984, 8717, 5016, 995, 12610, 8717, 2655, 24401, 12984, 18871, 2423, 47283, 2288, 3714, 5016, 46958, 25720, 39896, 8978, 31767, 3215, 2288, 10632, 6225, 2304, 20292, 995], "avg_logprob": -0.14326171781867741, "compression_ratio": 1.8703703703703705, "no_speech_prob": 3.212690353393555e-05, "words": [{"start": 1.35, "end": 1.63, "word": "بسم", "probability": 0.76806640625}, {"start": 1.63, "end": 1.81, "word": " الله", "probability": 0.9609375}, {"start": 1.81, "end": 2.17, "word": " الرحمن", "probability": 0.9765625}, {"start": 2.17, "end": 2.51, "word": " الرحيم", "probability": 0.9952799479166666}, {"start": 2.51, "end": 2.81, "word": " الحمد", "probability": 0.8154296875}, {"start": 2.81, "end": 2.95, "word": " لله", "probability": 0.94580078125}, {"start": 2.95, "end": 3.09, "word": " رب", "probability": 0.7587890625}, {"start": 3.09, "end": 3.47, "word": " العالمين", "probability": 0.8590494791666666}, {"start": 3.47, "end": 3.85, "word": " والصلاة", "probability": 0.8753662109375}, {"start": 3.85, "end": 4.21, "word": " والسلام", "probability": 0.9435221354166666}, {"start": 4.21, "end": 4.43, "word": " علي", "probability": 0.521484375}, {"start": 4.43, "end": 4.79, "word": " سيدنا", "probability": 0.9900716145833334}, {"start": 4.79, "end": 5.37, "word": " محمد", "probability": 0.9832356770833334}, {"start": 5.37, "end": 6.43, "word": " سيد", "probability": 0.7132568359375}, {"start": 6.43, "end": 6.93, "word": " المرسلين", "probability": 0.9888671875}, {"start": 6.93, "end": 7.53, "word": " أجمعين", "probability": 0.9111328125}, {"start": 7.53, "end": 8.27, "word": " يعني", "probability": 0.90185546875}, {"start": 8.27, "end": 8.71, "word": " اليوم", "probability": 0.904052734375}, {"start": 8.71, "end": 8.87, "word": " ان", "probability": 0.52880859375}, {"start": 8.87, "end": 9.03, "word": " شاء", "probability": 0.9775390625}, {"start": 9.03, "end": 9.17, "word": " الله", "probability": 0.94921875}, {"start": 9.17, "end": 9.75, "word": " نحاول", "probability": 0.9134521484375}, {"start": 9.75, "end": 11.53, "word": " نتحدث", "probability": 0.987548828125}, {"start": 11.53, "end": 11.89, "word": " عن", "probability": 0.9912109375}, {"start": 11.89, "end": 13.53, "word": " الاخر", "probability": 0.6064046223958334}, {"start": 13.53, "end": 13.95, "word": " محاضرة", "probability": 0.96484375}, {"start": 13.95, "end": 14.09, "word": " هي", "probability": 0.68896484375}, {"start": 14.09, "end": 14.23, "word": " في", "probability": 0.97705078125}, {"start": 14.23, "end": 14.79, "word": " الصدرية", "probability": 0.9759521484375}, {"start": 14.79, "end": 15.65, "word": " عمليا", "probability": 0.77215576171875}], "temperature": 1.0}, {"id": 2, "seek": 3545, "start": 17.35, "end": 35.45, "text": "طبعا ممكن في الآخر يطلع معنا كمان محاضرة اسمها الـ Core Pulmonale لكن الآن COPD Chronic Obstructive Pulmonary Disease حسب كيف بيكفل وجد بالنسبة لموضوع الـ Core Pulmonale ممكن نعملها محاضرة ممكن ننسيبها حسب إيش بيصير معنا لكن الـCOPD", "tokens": [9566, 3555, 3615, 995, 3714, 43020, 8978, 6024, 95, 34740, 7251, 9566, 1211, 3615, 20449, 8315, 9122, 2304, 7649, 3714, 5016, 46958, 25720, 24525, 2304, 11296, 2423, 39184, 14798, 35568, 3317, 1220, 44381, 6024, 48506, 383, 12059, 35, 1721, 11630, 4075, 372, 21673, 35568, 46386, 35360, 11331, 35457, 9122, 33911, 4724, 1829, 4117, 5172, 1211, 49610, 3215, 20666, 1863, 35457, 3660, 32767, 2407, 11242, 45367, 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"probability": 0.980224609375}, {"start": 20.19, "end": 20.49, "word": " اسمها", "probability": 0.92138671875}, {"start": 20.49, "end": 20.61, "word": " الـ", "probability": 0.576904296875}, {"start": 20.61, "end": 20.81, "word": " Core", "probability": 0.493408203125}, {"start": 20.81, "end": 21.51, "word": " Pulmonale", "probability": 0.6736653645833334}, {"start": 21.51, "end": 22.45, "word": " لكن", "probability": 0.71142578125}, {"start": 22.45, "end": 23.11, "word": " الآن", "probability": 0.67724609375}, {"start": 23.11, "end": 23.91, "word": " COPD", "probability": 0.6910400390625}, {"start": 23.91, "end": 24.67, "word": " Chronic", "probability": 0.595703125}, {"start": 24.67, "end": 25.25, "word": " Obstructive", "probability": 0.9484049479166666}, {"start": 25.25, "end": 25.79, "word": " Pulmonary", "probability": 0.925537109375}, {"start": 25.79, "end": 26.55, "word": " Disease", "probability": 0.90478515625}, {"start": 26.55, "end": 27.19, "word": " حسب", "probability": 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"end": 930.96, "text": "بيبقى ماعندوش symptoms لكن هو عنده COPD. 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new file mode 100644
index 0000000000000000000000000000000000000000..d5bcc3fa9351e7e200418d26362d0d41fea3594e
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+++ b/PL9fwy3NUQKwaf4aPlmtIQXiASpDxVLEnO/G--MWAi7z5k_raw.json
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diff --git a/PL9fwy3NUQKwaf4aPlmtIQXiASpDxVLEnO/G--MWAi7z5k_raw.srt b/PL9fwy3NUQKwaf4aPlmtIQXiASpDxVLEnO/G--MWAi7z5k_raw.srt
new file mode 100644
index 0000000000000000000000000000000000000000..a09acdb8f13b44a8ca47840b8095fe1ae7b4ad42
--- /dev/null
+++ b/PL9fwy3NUQKwaf4aPlmtIQXiASpDxVLEnO/G--MWAi7z5k_raw.srt
@@ -0,0 +1,1484 @@
+1
+00:00:01,430 --> 00:00:03,110
+بسم الله الرحمن الرحيم الحمد لله و الرحمن الرحيم
+
+2
+00:00:03,110 --> 00:00:05,210
+الصلاة والسلام علي سيدنا محمد وسيد المرسلين
+
+3
+00:00:05,210 --> 00:00:09,050
+الأجمعين اليوم ان شاء الله بنحكي في الجزء التالت
+
+4
+00:00:09,050 --> 00:00:14,470
+من محاضرة ال infective and obstructive and
+
+5
+00:00:14,470 --> 00:00:18,590
+restrictive pulmonary disorders اللي الآن هنحكي
+
+6
+00:00:18,590 --> 00:00:21,690
+عنها restrictive pulmonary disorders كنا حكينا
+
+7
+00:00:21,690 --> 00:00:25,410
+طبعا احنا عن infections جزئيا حكينا عن ال
+
+8
+00:00:25,410 --> 00:00:29,480
+obstructive حكينا عن الاللي هو الـ Enphysema أو
+
+9
+00:00:29,480 --> 00:00:32,960
+الـ Chronic Bronchitis الـ Acute Bronchitis حكينا
+
+10
+00:00:32,960 --> 00:00:35,460
+عن الـ Bronchial Asthma الآن بنحكي عن Restrictive
+
+11
+00:00:35,460 --> 00:00:39,860
+Pulmonary Disorders وعلى رأسها حاجة بنسمعها كتير
+
+12
+00:00:39,860 --> 00:00:43,940
+موجودة في المستشفيات اللي هي الـ Pneumothorax
+
+13
+00:00:43,940 --> 00:00:49,620
+Pneumothorax يعني صدر Pneumo معناته زي هوى يعني
+
+14
+00:00:49,620 --> 00:00:56,100
+Pneumothorax Pneumothorax
+
+15
+00:00:56,100 --> 00:01:00,100
+هو عبارة عنEntry of air into the pleural cavity
+
+16
+00:01:00,100 --> 00:01:03,380
+احنا فاكرين في الـ Anatomy لما حكينا أنه فينا
+
+17
+00:01:03,380 --> 00:01:09,080
+pleural cavity حوالين الـ lung وكل pleural cavity
+
+18
+00:01:09,080 --> 00:01:14,580
+لكل lung لوحدة بتسكر على ال lung لوحدة in which
+
+19
+00:01:14,580 --> 00:01:17,820
+the lung resides يبقى بنوع Motorax هو عبارة عن
+
+20
+00:01:17,820 --> 00:01:23,330
+دخول هواالى الـ plural cavity in which the lung
+
+21
+00:01:23,330 --> 00:01:26,470
+resides طيب ليش هو يعني إيش فيه في ال plural
+
+22
+00:01:26,470 --> 00:01:29,310
+cavity؟ ال plural cavity عبارة عن cavity فيها
+
+23
+00:01:29,310 --> 00:01:33,350
+fluid عادة و فيه negative pressure كيف يعني
+
+24
+00:01:33,350 --> 00:01:36,310
+negative pressure؟ negative pressure يعني عندنا ال
+
+25
+00:01:36,310 --> 00:01:42,970
+atmosphere برا اللي هو الجو في ال plural cavity
+
+26
+00:01:42,970 --> 00:01:45,230
+في عندنا negative pressure و في ال lung في عندنا
+
+27
+00:01:45,230 --> 00:01:48,530
+positive pressure الآن لو دخل هوا بأي طريقة من
+
+28
+00:01:48,530 --> 00:01:53,500
+الطرق هنحكيللـ pleural cavity إيش اللي بيصير؟
+
+29
+00:01:53,500 --> 00:01:57,720
+بيعلى ال pressure في ال pleural cavity فال lung
+
+30
+00:01:57,720 --> 00:02:01,940
+بتعملنا collapsing خلاص ليش؟ لأنه بيزيد الضغط جوا
+
+31
+00:02:01,940 --> 00:02:07,600
+ال pleural cavity عن الضغط في جوا ال lung فبتنسحب
+
+32
+00:02:07,600 --> 00:02:13,220
+ال lung و بتعملنا بنوع Motorax يبقىأحنا بصورة عامة
+
+33
+00:02:13,220 --> 00:02:17,020
+عشان الـ lung تقدر تعمل expansion في التنفس، رايح
+
+34
+00:02:17,020 --> 00:02:19,700
+جاي، لازم يكون negative pressure كما قلنا في الـ
+
+35
+00:02:19,700 --> 00:02:24,060
+pleural cavity مقارنة بالإيش؟ بالأتمسفير، بالجو
+
+36
+00:02:24,060 --> 00:02:28,360
+البراني الـ inside of the pleural cavity is
+
+37
+00:02:28,360 --> 00:02:32,580
+essentially a vacuum عادة، vacuum يعني فراغ، مش
+
+38
+00:02:32,580 --> 00:02:37,910
+إشي، هو fluid، لكن فراغ، ضغط، قلنا negativeand
+
+39
+00:02:37,910 --> 00:02:40,830
+when air enters the pleural cavity the negative
+
+40
+00:02:40,830 --> 00:02:44,030
+pressure is lost بروح ال negative pressure and
+
+41
+00:02:44,030 --> 00:02:47,850
+lungs collapse طبيعي زي البلون يعني بيبقى كأنك
+
+42
+00:02:47,850 --> 00:02:53,700
+نفست البلون بعد ما نفختهوزي ما قلنا كلها جاعدة
+
+43
+00:02:53,700 --> 00:02:56,460
+لوحدها في cavity فلما بيصير عند بنوية موتوراكس في
+
+44
+00:02:56,460 --> 00:03:01,080
+العادة إذا ماكنش ال cause على الجهتين فهو بيعملنا
+
+45
+00:03:01,080 --> 00:03:04,640
+على جهة واحدة في الغالب البنوية موتوراكس فيه تلاتة
+
+46
+00:03:04,640 --> 00:03:07,200
+types متعرفة تلاتة من ال types
+
+47
+00:03:10,150 --> 00:03:15,330
+الأول الـ Open or Communicating بنوع من Torax Open
+
+48
+00:03:15,330 --> 00:03:17,630
+أو Communicating Communicating يعني متواصل مع
+
+49
+00:03:17,630 --> 00:03:21,190
+الخارج هيك معناته متواصل مع الخارج usually
+
+50
+00:03:21,190 --> 00:03:25,270
+involves a traumatic chest wound سكينة في صدر واحد
+
+51
+00:03:25,870 --> 00:03:29,510
+الهواء بدخل إلى الـblurred cavity من برة من
+
+52
+00:03:29,510 --> 00:03:36,030
+الاتмосفير الـlung بتقول للـcavity يا ستة يا هيني
+
+53
+00:03:36,030 --> 00:03:39,790
+تركتك لوحدك وانا هيني ساحبة حالي أقوم بسحبة وعملة
+
+54
+00:03:39,790 --> 00:03:47,190
+lung collapse ليش؟ لأنه صار الـblurred cavity
+
+55
+00:03:47,190 --> 00:03:51,220
+pressure عاليبتصارت الـ language مش قادرة تقاتل ضد
+
+56
+00:03:51,220 --> 00:03:55,300
+الأخر فبتنسحب في أن الحاجة التانية يبقى ال open و
+
+57
+00:03:55,300 --> 00:03:58,220
+ال communicating بيبقى ال opening اللي صارت في ال
+
+58
+00:03:58,220 --> 00:04:04,680
+chest مع الخارج مع الخارج okay هذا ال open بنوع
+
+59
+00:04:04,680 --> 00:04:08,140
+motorax أو communicating بنوع motorax تشوف الصورة
+
+60
+00:04:08,140 --> 00:04:13,730
+اللي موجودةطبعًا الهواء بيروح وبيجي في الأحوال زي
+
+61
+00:04:13,730 --> 00:04:19,090
+هيك يعني مش بيثبتش زي ما هنشوف في نوع من الأنواع
+
+62
+00:04:19,090 --> 00:04:22,650
+في الـ tension بنوع Motorax الـ Closed أو
+
+63
+00:04:22,650 --> 00:04:27,830
+Spontaneous بنوع Motorax بيصير لما الـ Lung يصير
+
+64
+00:04:27,830 --> 00:04:32,470
+فيها leaks يصير فيها leakage يصير فيها خرم لسبب من
+
+65
+00:04:32,470 --> 00:04:37,750
+الأسباب ويصير الهواء ييجي من الـ Lung عوين على الـ
+
+66
+00:04:37,750 --> 00:04:44,660
+Cavityيبقى occurs when air هو leaks from the lung
+
+67
+00:04:44,660 --> 00:04:48,500
+into the pleural cavity من أيش طيب هذا ممكن يصير؟
+
+68
+00:04:48,500 --> 00:04:52,120
+اه هذا حاجة داخلية معناته lung cancer rupture
+
+69
+00:04:52,120 --> 00:04:57,760
+بلمونري disease بلمونري
+
+70
+00:04:57,760 --> 00:04:58,240
+disease
+
+71
+00:05:05,540 --> 00:05:10,180
+يبقى كمان مرة لـ Closed بيبقى occurs when air
+
+72
+00:05:10,180 --> 00:05:13,340
+leaks from the lungs into the pleural cavity ممكن
+
+73
+00:05:13,340 --> 00:05:18,420
+يكون داخل السبب لأن المشكلة في ال lung lung cancer
+
+74
+00:05:18,420 --> 00:05:21,780
+ربشة pulmonary disease where the increased pleural
+
+75
+00:05:21,780 --> 00:05:25,800
+pressure prevents lung expansion طبيعي كل ما زاد
+
+76
+00:05:25,800 --> 00:05:30,460
+الضغط جوا ال cavity بمنع أن ال lung تحاول تعمل
+
+77
+00:05:30,460 --> 00:05:37,740
+expansion تعمل تمددفي أثناء أخذ الشحيق فبتظلها
+
+78
+00:05:37,740 --> 00:05:43,620
+collapsed النوع التالت واللي هو أخطرهم طبعاً اللي
+
+79
+00:05:43,620 --> 00:05:46,860
+هو ال tension بنوي متركز طب وين الخطر تبعه ال
+
+80
+00:05:46,860 --> 00:05:50,560
+tension يا جماعة زي ال open بس اللامة in ال open
+
+81
+00:05:50,560 --> 00:05:55,840
+بدخله بطلع هوا لكن في ال tension بنوي متركز ال
+
+82
+00:05:55,840 --> 00:06:01,490
+condition if there is a one way movement of airbut
+
+83
+00:06:01,490 --> 00:06:05,130
+not out of the pleural cavity يعني بدخل هوا على
+
+84
+00:06:05,130 --> 00:06:09,570
+الـ pleural cavity وما بطلعش هوا one-valve
+
+85
+00:06:09,570 --> 00:06:12,150
+mechanism يعني زي الـ valve mechanism يعني في
+
+86
+00:06:12,150 --> 00:06:15,990
+اتجاه واحد طب وده إيش بيعمل؟ أه طبعا خدوا بالكوا
+
+87
+00:06:15,990 --> 00:06:19,590
+كل ما أخدنا نفس بدخل هوا ومابطلعش بدخل هوا
+
+88
+00:06:19,590 --> 00:06:23,930
+ومابطلعش بدخل هوا ومابطلعش في الآخر بيصير a severe
+
+89
+00:06:23,930 --> 00:06:27,610
+life
+
+90
+00:06:27,610 --> 00:06:32,200
+threatening conditionليش؟ لأن الـ pressure في الـ
+
+91
+00:06:32,200 --> 00:06:36,520
+pleural cavity continuous to increase may result
+
+92
+00:06:36,520 --> 00:06:41,940
+in further lung compression، compression of the
+
+93
+00:06:41,940 --> 00:06:46,780
+large blood vessels و الـ heart، كل إشي بنضغط،
+
+94
+00:06:46,780 --> 00:06:49,660
+القضية مش قضية، بس الآن إنه ينضغط، شوف الصورة اللي
+
+95
+00:06:49,660 --> 00:06:55,460
+عندنا، إنه بس ينضغط، بس ينضغط ال lung، لأ، بتنضغط
+
+96
+00:06:55,460 --> 00:06:59,570
+كل chest organs، هذا المشكلةيبقى احنا فيه عندنا
+
+97
+00:06:59,570 --> 00:07:02,270
+open or communicating بنوع Motorax، عندنا closed
+
+98
+00:07:02,270 --> 00:07:05,130
+or spontaneous بنوع Motorax، وفيه عندنا tension
+
+99
+00:07:05,130 --> 00:07:08,210
+headache، tension عفوا headache، tension بنوع
+
+100
+00:07:08,210 --> 00:07:14,510
+Motorax، اللي هو أخطرهم وmost severe، كيف طب
+
+101
+00:07:14,510 --> 00:07:19,950
+بنعرفه؟الـPneumothorax، تكيبنية، دسنية، chest pain
+
+102
+00:07:19,950 --> 00:07:24,210
+طبعاً لو أنا في عندى واحد محطوط صدر في صدره سكينة
+
+103
+00:07:24,210 --> 00:07:27,650
+باعرف أنه غالباً هيصير معاه Pneumothorax لكن نايم
+
+104
+00:07:27,650 --> 00:07:30,530
+عندى مثلا واحد في العينية المركزة عسمى المثال ومرة
+
+105
+00:07:30,530 --> 00:07:34,110
+واحدة صار عنده لسبب من الأسباب دسنية، تكيبنية،
+
+106
+00:07:34,110 --> 00:07:38,170
+chest pain طبعا ممكن على طول أفكر في Pneumothorax
+
+107
+00:07:38,170 --> 00:07:41,270
+كيف بعرفها؟ بعرفها طبعا من الـChest X-Ray، واضح
+
+108
+00:07:41,270 --> 00:07:46,320
+الـTreatment of Pneumothoraxكيف نعالج بنويمو
+
+109
+00:07:46,320 --> 00:07:50,560
+توراكس وهذه سأتركها كلها في الجزء الأول من
+
+110
+00:07:50,560 --> 00:07:54,640
+الassignment اللي هو how to treat بنويمو توراكس
+
+111
+00:07:54,640 --> 00:07:59,760
+وبرضه ما تزيدوش يا جماعة عن 6-7 تسطور في الكتابة و
+
+112
+00:07:59,760 --> 00:08:02,920
+how to treat بنويمو توراكس
+
+113
+00:08:05,260 --> 00:08:09,640
+وطبعا زي ما قلنا يا جما مرة تانية أي homework أي
+
+114
+00:08:09,640 --> 00:08:14,360
+assignment بطلبه هو يعتبر عمليا منهج بسأل فيه في
+
+115
+00:08:14,360 --> 00:08:17,940
+ال course ذاته في الامتحانات وفي كل مكان حتى لو في
+
+116
+00:08:17,940 --> 00:08:22,360
+أسئلة مقالة هيسأل عنها كمان فاخدوا بالكوا هذا مش
+
+117
+00:08:22,360 --> 00:08:27,420
+خارج المنهجصحيح إحنا بدنا نوسع المدارك بالمطالعة
+
+118
+00:08:27,420 --> 00:08:33,080
+لكن هذا هو عبارة عن منهج طيب من ال restrictive
+
+119
+00:08:33,080 --> 00:08:37,600
+pulmonary disorders لـ atelictasis الـ atelictasis
+
+120
+00:08:37,600 --> 00:08:42,800
+is a condition in which there is incomplete
+
+121
+00:08:42,800 --> 00:08:47,920
+expansion of lung tissues incomplete expansion
+
+122
+00:08:47,920 --> 00:08:50,060
+يعني ال lung
+
+123
+00:08:53,870 --> 00:08:57,430
+بتفتحش زي ما بدنا، ليش؟ لأنه فيه Blockage of the
+
+124
+00:08:57,430 --> 00:09:02,240
+Airways أو Compression على الـ Alveolar Sacsمنها
+
+125
+00:09:02,240 --> 00:09:05,380
+الـ Types زي Absorption of Lactases تصير عندما
+
+126
+00:09:05,380 --> 00:09:10,040
+يكون شخص عنده على سبيل المثال برونكيل بساجز تكون
+
+127
+00:09:10,040 --> 00:09:15,460
+Blocked with a mucous tumor أو اديمة وطبعاً هذه
+
+128
+00:09:15,460 --> 00:09:17,680
+بنشوفها كتير في chronic bronchitis الناس اللي
+
+129
+00:09:17,680 --> 00:09:20,780
+عندهم بلغم كتير chronic bronchitis فاكرين؟ أو
+
+130
+00:09:20,780 --> 00:09:25,540
+Cystic Fibrosis إيش ال cystic fibrosis؟ تعرفوه
+
+131
+00:09:25,540 --> 00:09:28,960
+سمعتوا عنه؟طبعاً الـ Cystic Fibrosis من الحاجات يا
+
+132
+00:09:28,960 --> 00:09:32,080
+جماعة اللي هيشتغلوا في الأطفال هيشتغلوا معاهم
+
+133
+00:09:32,080 --> 00:09:37,180
+بالضرورة لأن هدول يزد أساسي من علاجهم اللي هو الـ
+
+134
+00:09:37,180 --> 00:09:41,100
+Mucus removal
+
+135
+00:09:41,100 --> 00:09:45,900
+عن طريق الـ Chest Physotherapy وSevere حقيقة فبرضه
+
+136
+00:09:45,900 --> 00:09:49,860
+الـ Cystic Fibrosis هتكتبولي عن ما هو Cystic
+
+137
+00:09:49,860 --> 00:09:55,260
+Fibrosis إيش الأسباب التبعته؟‏how to manage فيه
+
+138
+00:09:55,260 --> 00:10:01,360
+على قدر صفحة يعني بس على جد صفحة واحدة مش ورقة
+
+139
+00:10:01,360 --> 00:10:05,060
+صفحة واحدة تكتبولي عن الـ cystic fibrosis الحاجات
+
+140
+00:10:05,060 --> 00:10:09,620
+المهمة الـcystic fibrosis الحاجة التانية اللي هي
+
+141
+00:10:09,620 --> 00:10:11,560
+في الاتليكتازيس النوع التاني اللي هو compression
+
+142
+00:10:11,560 --> 00:10:17,520
+هو يكون انه ال lung tissue compressed externally
+
+143
+00:10:17,520 --> 00:10:22,900
+by air blood fluid أو ال tumorكيف الـ
+
+144
+00:10:22,900 --> 00:10:26,360
+«manifestation»؟ «Dyspnea» كاف نفس الكلام طبيعي
+
+145
+00:10:26,360 --> 00:10:30,900
+لما أنا يكون عندي جزء من الـLung Expand ومش قادر
+
+146
+00:10:30,900 --> 00:10:34,020
+يعمل
+
+147
+00:10:34,020 --> 00:10:38,840
+عملية تبادل للغازات
+
+148
+00:10:38,840 --> 00:10:42,280
+أنه يكون عندي Reduce Gas Exchange طبعاً بيصير
+
+149
+00:10:42,280 --> 00:10:44,400
+عندنا حاجة اسمها إذا فكريني لما حكينا عن
+
+150
+00:10:44,400 --> 00:10:49,960
+الـVentilation Perfusion Couplingبيصير طبيعي أن
+
+151
+00:10:49,960 --> 00:10:55,920
+الدم اللي كان رايح المفروض يروح لمنطقة اللي فيها
+
+152
+00:10:55,920 --> 00:10:58,760
+أتلك تازس بيبطل يروح عليها و بيصير يروح لمناطق
+
+153
+00:10:58,760 --> 00:11:03,300
+جادرة تعمل gas exchange العلاج تبعها الباب اللي
+
+154
+00:11:03,300 --> 00:11:07,500
+بيجيك منه الريح سد و استريح removal of airway
+
+155
+00:11:07,500 --> 00:11:13,020
+blockage اللي عامل السبب لازم أشيله طبعا سواء كان
+
+156
+00:11:13,360 --> 00:11:19,120
+Blood, Fluids, Tumor زي ما بيقولوها ميوكاس لازم
+
+157
+00:11:19,120 --> 00:11:26,500
+ينشال كمان مرة عندما أقول إن في حد عنده أتلكتازيس
+
+158
+00:11:26,500 --> 00:11:32,480
+معناته lung collapse with loss of internal air فش
+
+159
+00:11:32,480 --> 00:11:36,740
+هوا جواته وهذا ممكن يصيب لأ sub unit of the loop
+
+160
+00:11:36,740 --> 00:11:40,800
+بنقول انه sub segmental أتلكتازيس ممكن للوب كله
+
+161
+00:11:41,830 --> 00:11:44,430
+فاكرين انتوا كام لوب احنا قلنا فيه في ال right
+
+162
+00:11:44,430 --> 00:11:49,730
+lung و left lung فاكرين؟ لوب كامل أو يكون ال whole
+
+163
+00:11:49,730 --> 00:11:52,990
+lung ات الاكتزاز لكن ال common كتير كتير اللي هو
+
+164
+00:11:52,990 --> 00:11:59,190
+sub-segmental الاسباب حكينا عنها قبل شوية مش
+
+165
+00:11:59,190 --> 00:12:02,430
+محتاجين ان نحكي عنها مرة تانية طيب انا فيه في ال X
+
+166
+00:12:02,430 --> 00:12:05,350
+-ray انتوا برضه في العلاج الطبيعي بتمس كل X-ray و
+
+167
+00:12:05,350 --> 00:12:09,070
+لازم تشوفوا خاصة بتكون تعرفوا اين بتعملوا علاج
+
+168
+00:12:09,070 --> 00:12:14,940
+طبيعي على منطقة الصدرأو whole lung atlectasis
+
+169
+00:12:14,940 --> 00:12:20,260
+بلاقي إبسي لاترال يعني
+
+170
+00:12:20,260 --> 00:12:26,620
+في نفس الجهة بلاقي أن التركيا انزحت لنفس الجهة زي
+
+171
+00:12:26,620 --> 00:12:30,440
+ما أحنا شايفين في الصورة اللي تحت أنه في عند ال
+
+172
+00:12:30,440 --> 00:12:37,600
+lung collapse على ال left side والتركياوين الأسهم
+
+173
+00:12:37,600 --> 00:12:41,180
+الحمراء اللي فوق ميلة أو زيها عندها Epsilateral
+
+174
+00:12:41,180 --> 00:12:46,120
+Tracheal displacement بيصير عندي increased density
+
+175
+00:12:46,120 --> 00:12:51,660
+of collapsed lung tissue بيصير عندي Epsilateral
+
+176
+00:12:51,660 --> 00:12:57,400
+mediastinal shift شايفين كل الأعضاء، التراكية وقلب
+
+177
+00:12:57,400 --> 00:13:04,460
+كله رايح على الجهة المصابة elevation of the hymen
+
+178
+00:13:05,890 --> 00:13:13,670
+وهي ارتفاع الديافرام بيصير عالي وهرنياشن وطبعا ال
+
+179
+00:13:13,670 --> 00:13:17,730
+lung التانية لاجت واسع فبتروح هي التانية زايحة
+
+180
+00:13:17,730 --> 00:13:23,150
+وداخلة عبر ال midline تبع ال lung الأخرى بتعتدي
+
+181
+00:13:23,150 --> 00:13:27,710
+على حرومات الآخرين ليش؟ لاجتها ضعيفة فهي ايش؟
+
+182
+00:13:27,710 --> 00:13:29,650
+بتستجو ال lung التانية فبتفرد
+
+183
+00:13:33,050 --> 00:13:40,350
+زي ما حكينا الـ Collapse بيبين في الـ X-Ray
+
+184
+00:13:40,350 --> 00:13:43,370
+الرسمات
+
+185
+00:13:43,370 --> 00:13:46,950
+هي بس للخبرة بعد هيك كيف الـ Collapse بيبين هنا من
+
+186
+00:13:46,950 --> 00:13:48,930
+أهم الصور اللي بنشوفها في الـ Collapse على سبيل
+
+187
+00:13:48,930 --> 00:13:52,010
+المثال في الـ Foreign Body Inhalation بلاقي على
+
+188
+00:13:52,010 --> 00:13:57,810
+الجهة اليمن Right Upper Lobe Collapse في الـ
+
+189
+00:13:57,810 --> 00:14:00,330
+Foreign Body Inhalation الأطفال اللي بيتشردقوا
+
+190
+00:14:00,880 --> 00:14:04,520
+البذرة بكذا بنلاقي أنه بتروح في المنطقة هذه اللي
+
+191
+00:14:04,520 --> 00:14:10,240
+هي البذرة وتعمل نبس في المنطقة هذه في عندنا حاجة
+
+192
+00:14:10,240 --> 00:14:13,680
+اسمها Bronchectasis Bronchectasis يعني احنا بس
+
+193
+00:14:13,680 --> 00:14:16,780
+هنحكي slide واحد عنها عشان نكون سمعانين فيها لأنه
+
+194
+00:14:16,780 --> 00:14:19,590
+برضه هي بتعملنا accumulation of mucusis a
+
+195
+00:14:19,590 --> 00:14:22,810
+condition that results from prolonged injury or
+
+196
+00:14:22,810 --> 00:14:25,810
+inflammation of respiratory airways and
+
+197
+00:14:25,810 --> 00:14:29,330
+bronchioles يعني بعد فترة من إصابة ال respiratory
+
+198
+00:14:29,330 --> 00:14:31,890
+airways أو ال bronchioles بinflammation أو
+
+199
+00:14:31,890 --> 00:14:35,750
+infection أو أي injury بيصير عندنا إيش؟ condition
+
+200
+00:14:35,750 --> 00:14:41,110
+بيقول عنها bronchectasis إيش هي؟ dilatation of the
+
+201
+00:14:41,110 --> 00:14:43,830
+bronchus and or bronchi بتتوسع
+
+202
+00:15:10,290 --> 00:15:24,120
+وهذا بدخلنا على موضوعبهم بنسمع فيه دايماً الآن
+
+203
+00:15:24,120 --> 00:15:28,040
+هنحكي عن حاجة اسمها الـ ARDS الحقيقة الآن الـ ARDS
+
+204
+00:15:28,040 --> 00:15:31,440
+بنشوفها في كتير من الأحوال
+
+205
+00:15:35,870 --> 00:15:38,570
+اللي هي الـ Adult Respiratory Distress Syndrome
+
+206
+00:15:38,570 --> 00:15:42,190
+بتشوفها في العينايات الأوروبية الناس اللي بيصير
+
+207
+00:15:42,190 --> 00:15:44,890
+عندها نمونيا نتيجة الـ Corona بيصير عندهم حاجة
+
+208
+00:15:44,890 --> 00:15:49,250
+اسمها ARDS Adult Respiratory Distress Syndrome
+
+209
+00:15:49,250 --> 00:15:54,260
+احفظوا الافتصار يا جماعةARDS في الخارج، لن تجد أحد
+
+210
+00:15:54,260 --> 00:15:56,880
+في مستشفى الأوروبي أو في الشفاء يقول لك «أدلت
+
+211
+00:15:56,880 --> 00:15:59,740
+راسبيراتوري ديسترست سندروم» مع طول بيقولوها «ARDS»
+
+212
+00:15:59,740 --> 00:16:02,840
+«ARDS» يعني عبر عن سندروم associated with
+
+213
+00:16:02,840 --> 00:16:08,120
+destruction of alveolar membranes ومش بس والـ
+
+214
+00:16:08,120 --> 00:16:10,780
+«كابيلاريز» اللي related لإلها يعني الـ
+
+215
+00:16:10,780 --> 00:16:13,440
+«كابيلاريز» والـ «كابيلاريز» يعني تدمير عمليًا
+
+216
+00:16:13,440 --> 00:16:17,540
+تدمير للـ tissue of the lung it may occur as a
+
+217
+00:16:17,540 --> 00:16:22,950
+result of direct injury to the lungs مثلانمونيا أو
+
+218
+00:16:22,950 --> 00:16:27,970
+كورونا نمونيا أو as a result of dramatic decrease
+
+219
+00:16:27,970 --> 00:16:30,450
+in blood flow of the lungs دائما نقول عنها shock
+
+220
+00:16:30,450 --> 00:16:39,630
+lung يعني صار مافيش دم يروح على ال lungs لسبب من
+
+221
+00:16:39,630 --> 00:16:43,910
+الأسباب زي ما مافيش دم يروح على الكلة أو مافيش دم
+
+222
+00:16:43,910 --> 00:16:46,870
+يروح على الجلب إيش بيصير؟ إنه shock kidney أو
+
+223
+00:16:46,870 --> 00:16:52,320
+shock heart shock lung نقول عنهاوهذه بتعملنا
+
+224
+00:16:52,320 --> 00:16:58,720
+Damage في الـ Alveolar membranes و الـ related
+
+225
+00:16:58,720 --> 00:17:01,680
+capillaries طبعا إذا ال membranes راحت ال alveolar
+
+226
+00:17:01,680 --> 00:17:05,600
+membranes طب مين بده يعمل blood exchange، gas
+
+227
+00:17:05,600 --> 00:17:10,840
+exchange؟ مين؟ الأسباب اللي بتعملي ARDS مش كلها
+
+228
+00:17:10,840 --> 00:17:16,200
+lung، sepsisيوريميا ارتفاع اليوريا يعني الكلة ما
+
+229
+00:17:16,200 --> 00:17:20,960
+تشتغلش فاليوريا بترتفع ولمّا ترتفع اليوريا بتعمل
+
+230
+00:17:20,960 --> 00:17:25,040
+مشاكل في الرئتين هيك كمان اه طبعا trauma near
+
+231
+00:17:25,040 --> 00:17:30,220
+drowning اشيا near drowning غرق بس ماتش drowning
+
+232
+00:17:30,220 --> 00:17:36,420
+معناته غرق ومات near drowning يعني
+
+233
+00:17:36,660 --> 00:17:40,340
+جرّى بيموت من الغرق لكن ما ماتش Inhalation of
+
+234
+00:17:40,340 --> 00:17:44,300
+toxic gases or agents الناس اللي بتتنفس غازات
+
+235
+00:17:44,300 --> 00:17:49,140
+حارقة و irritative يعني حاجات بتعمل irritation
+
+236
+00:17:49,140 --> 00:17:54,840
+Aspiration of gastric continents إيش هاي؟ أه الناس
+
+237
+00:17:54,840 --> 00:17:57,520
+اللي بتتشردك بالحمض زي الأطفال على سبيل المثال
+
+238
+00:17:57,520 --> 00:18:01,060
+ممكن widespread pneumonia pneumonia نفسها Drug
+
+239
+00:18:01,060 --> 00:18:05,240
+overdose systemicشوك أي شيء بيعمللي شوك في الجسم
+
+240
+00:18:05,240 --> 00:18:10,680
+ممكن يديني ARDS المنفستيشن تبقيتها ديسنيا تاكبنيا
+
+241
+00:18:10,680 --> 00:18:15,080
+هيبوكسيميا طبعا CO2 بيصير عالي في الدم الأكسجين
+
+242
+00:18:15,080 --> 00:18:18,850
+بيقل في الدمInfiltration of lung tissue with
+
+243
+00:18:18,850 --> 00:18:21,730
+immune cells that release inflammatory mediators
+
+244
+00:18:21,730 --> 00:18:24,290
+بيصير بعد ذلك accumulation of fluids في
+
+245
+00:18:24,290 --> 00:18:27,950
+الـAlveoli، around الـAlveoli وبيصير change في
+
+246
+00:18:27,950 --> 00:18:32,630
+الـBlood pH اللي هو حموضة الدم وكمية الـCO2 في
+
+247
+00:18:32,630 --> 00:18:36,150
+الـPulmonary Fibrosis في الآخر، حتى الجدر تبع
+
+248
+00:18:36,150 --> 00:18:40,810
+الـLung بيصير عملية Fibrosis وبعدها طبعا بدخل في
+
+249
+00:18:40,810 --> 00:18:43,650
+Respiratory Failure إيش يعني Respiratory Failure؟
+
+250
+00:18:44,130 --> 00:18:48,490
+يعني الـ Lungs لا تستطيع أن تقوم بشغلها اللي هو
+
+251
+00:18:48,490 --> 00:18:52,130
+اسمه Gas Exchange، Respiratory Failure، هنحكي عنها
+
+252
+00:18:52,130 --> 00:18:56,250
+بعد شوية أكمل الـ slide كيف أعالج الـ ARDS؟ طبعا،
+
+253
+00:18:56,250 --> 00:19:00,070
+بدأ أشيل السبب، و بدي أكسجين، Anti-inflammatory
+
+254
+00:19:00,070 --> 00:19:04,670
+drugs، Diuretics، عشان لو فيه أديمة أخففها، و أصلح
+
+255
+00:19:04,670 --> 00:19:07,630
+الـ Acid Base Balance، و أصلح طبعا الـ Condition
+
+256
+00:19:07,630 --> 00:19:13,890
+الأساسي الـ Respiratory Failureهي مصالح تنتج عندما
+
+257
+00:19:13,890 --> 00:19:16,990
+لا يستطيع الحصان الحصان الحصان الحصان الحصان
+
+258
+00:19:16,990 --> 00:19:17,650
+الحصان الحصان الحصان الحصان الحصان الحصان الحصان
+
+259
+00:19:17,650 --> 00:19:20,890
+الحصان الحصان الحصان الحصان الحصان الحصان الحصان
+
+260
+00:19:20,890 --> 00:19:25,930
+الحصان الحصان الحصان الحصان الحصان الحصان الحصان
+
+261
+00:19:25,930 --> 00:19:29,630
+الحصان الحصان الحصان الحصان الحصان الحصان الحصان
+
+262
+00:19:29,630 --> 00:19:30,290
+الحصان الحصان الحصان الحصان الحصان الحصان الحصان
+
+263
+00:19:30,290 --> 00:19:30,810
+الحصان الحصان الحصان الحصان الحصان الحصان الحصان
+
+264
+00:19:30,810 --> 00:19:30,930
+الحصان الحصان الحصان الحصان الحصان الحصان الحصان
+
+265
+00:19:30,930 --> 00:19:30,930
+الحصان الحصان الحصان الحصان الحصان الحصان الحصان
+
+266
+00:19:30,930 --> 00:19:32,170
+الحصان الحصان الحصان الحصان الحصان الحصان الحصان
+
+267
+00:19:32,170 --> 00:19:37,070
+الحصان
+
+268
+00:19:37,070 --> 00:19:37,590
+الح
+
+269
+00:19:42,810 --> 00:19:46,390
+والعادة يدخل فيه الـ Respiratory Failure في أخرها
+
+270
+00:19:46,390 --> 00:19:52,490
+كنهاية أو قد يكون هذا حدث مضحك يتسبب بالفكتورات
+
+271
+00:19:52,490 --> 00:19:57,190
+مثل بنويموتوراكس أو Drug Overdose من الناس اللي
+
+272
+00:19:57,190 --> 00:20:00,270
+بتاخد المخدرات هدول أو صار عنده بنويموتوراكس ممكن
+
+273
+00:20:00,270 --> 00:20:03,130
+يعمل Respiratory Failure يبقى Respiratory Failure
+
+274
+00:20:03,130 --> 00:20:07,930
+ممكن يكونdue to chronic condition وممكن يكون due
+
+275
+00:20:07,930 --> 00:20:12,170
+to acute condition مبناش يصعب الدنيا لكن الـ
+
+276
+00:20:12,170 --> 00:20:14,970
+Respiratory failure is a condition that results
+
+277
+00:20:14,970 --> 00:20:18,590
+when the lungs are no longer able to oxygenate the
+
+278
+00:20:18,590 --> 00:20:22,850
+blood sufficiently or to remove CO2 from it مهم
+
+279
+00:20:22,850 --> 00:20:29,370
+بالنسبة لي يا جماعة هذا التعريف بالذات وإيش ال
+
+280
+00:20:29,370 --> 00:20:33,580
+manifestation اللي بنشوفه كيف أنا بعرفأن فلان دخل
+
+281
+00:20:33,580 --> 00:20:36,160
+في الـ Respiratory Failure فشل في التنفس،
+
+282
+00:20:36,160 --> 00:20:41,220
+هيبوكسيميا، هيبوكسيميا، طب كيف أنا بعرف الـ
+
+283
+00:20:41,220 --> 00:20:46,900
+هيبوكسيميا؟ بحط الشخص
+
+284
+00:20:46,900 --> 00:20:51,960
+أو المريض على بلس أكسوميتر أو Monitor وبيجيسلي
+
+285
+00:20:51,960 --> 00:20:57,890
+كمية الأكسجين في الدمهيبركابنيا نفس الكلام بجيسلي
+
+286
+00:20:57,890 --> 00:21:01,710
+كمية ثانية أكسيد الكربون في الدم أو باخد حاجة
+
+287
+00:21:01,710 --> 00:21:05,090
+اسمها blood gases arterial blood gases إيش هذا ال
+
+288
+00:21:05,090 --> 00:21:08,490
+arterial blood gases؟ وعبارة عن فحص باخده من ال
+
+289
+00:21:08,490 --> 00:21:15,130
+capillaries أو حتى من ال artery وبافحص اللي هي
+
+290
+00:21:15,130 --> 00:21:22,550
+كمياتالأكسجين partial pressure of oxygen في الفحص
+
+291
+00:21:22,550 --> 00:21:27,140
+بافحص ال partial pressure of CO2بنقول عنه arterial
+
+292
+00:21:27,140 --> 00:21:32,800
+blood gases وبشوف فيه كمية الحموضة الـ pH جداش
+
+293
+00:21:32,800 --> 00:21:39,120
+وجداش فيه زيادة في ال base excess في القواعد في ال
+
+294
+00:21:39,120 --> 00:21:42,360
+blood gases هذا بس باختصار عشان تكونوا سمعتوا إيش
+
+295
+00:21:42,360 --> 00:21:45,220
+ال arterial blood gases في نفس الوقت بقدر من ال
+
+296
+00:21:45,220 --> 00:21:52,170
+monitor أحطه بس أحط على الأصبع اللي هوزي Gas هو
+
+297
+00:21:52,170 --> 00:21:55,050
+بتطلعلي على الـ Monitor بس بتطلعلي كمية الـ
+
+298
+00:21:55,050 --> 00:21:59,990
+Partial Pressure of Oxygen و اللي هي طبعاً لا يصح
+
+299
+00:21:59,990 --> 00:22:06,910
+أنها تجيل عن رقم معين 95% بدون أكسجين إيش كمان
+
+300
+00:22:06,910 --> 00:22:11,520
+ممكن Manifestation of Respiratory Disorder؟لكن ليس
+
+301
+00:22:11,520 --> 00:22:15,020
+بضرورة يجب أن يكون في ذراعك بعد ذلك الـ
+
+302
+00:22:15,020 --> 00:22:16,840
+Respiratory Failure يعني أنه تبطّر، يروح فيه
+
+303
+00:22:16,840 --> 00:22:20,240
+أكسيجين وين؟ على الدماغ، على المخ، فيصير يعني CNS
+
+304
+00:22:20,240 --> 00:22:26,390
+Symptoms CNS Symptoms زي Confusionالـ slurred
+
+305
+00:22:26,390 --> 00:22:29,470
+speech بيصير عندنا impaired motor function بيصير
+
+306
+00:22:29,470 --> 00:22:31,590
+عندنا زي ما قلنا توفر ال blood لما شرحنا عن ال
+
+307
+00:22:31,590 --> 00:22:36,870
+blood gases Altered blood pH بتغير ال pH Initially
+
+308
+00:22:36,870 --> 00:22:40,650
+في الأول في الأول الجسم بيحاول يكيّف مع الموضوع
+
+309
+00:22:40,650 --> 00:22:44,350
+هذا بيعمل تكيكارديا بيعمل increase cardiac output
+
+310
+00:22:44,350 --> 00:22:49,660
+لكن بعد هيكبقدرش يكمل الجسم أو بقدرش يكمل الـ
+
+311
+00:22:49,660 --> 00:22:53,120
+heart بيصير عندها bodycardia and decreased cardiac
+
+312
+00:22:53,120 --> 00:22:57,980
+output وهنا إيش الخطورة، نعمل بس مقارنة بين
+
+313
+00:22:57,980 --> 00:23:01,700
+الأسباب بتاعة ال respiratory failure أسباب acute
+
+314
+00:23:01,700 --> 00:23:04,840
+وأسباب chronic الأكيو زي pneumotoxic، drug
+
+315
+00:23:04,840 --> 00:23:09,440
+overdose، pleural effusion، airway obstruction،
+
+316
+00:23:09,440 --> 00:23:13,480
+status asthmaticus، inhalation of toxinsأو نكسيات
+
+317
+00:23:13,480 --> 00:23:18,100
+غازية كرونية زي الانفزيامة Intestinal lung disease
+
+318
+00:23:18,100 --> 00:23:20,720
+Cystic fibrosis Spinal cord أو brain injury
+
+319
+00:23:20,720 --> 00:23:23,260
+Congestive heart failure Neuromuscular disorder
+
+320
+00:23:23,260 --> 00:23:27,980
+بلمونري إمبولاي Diffused pneumonia بلمونري إديمة
+
+321
+00:23:27,980 --> 00:23:33,000
+يا جماعة إذا احنا في جامعة الآن يعني مش مدرسة كمان
+
+322
+00:23:33,000 --> 00:23:38,380
+مرة اللي بيلاقي بعض الحاجات مش فاهمها مش بالضرورة
+
+323
+00:23:38,380 --> 00:23:44,210
+دايما يشرحها المدرسيروح يكشف عنها يجرى أكتر يشوف
+
+324
+00:23:44,210 --> 00:23:49,180
+الان كل كلمة من هدولكلنا مسؤولين عنها وانتوا
+
+325
+00:23:49,180 --> 00:23:52,560
+مسؤولين عنها يعني في الآخر ممكن أنا أسأل عن هذه
+
+326
+00:23:52,560 --> 00:23:56,040
+إيش هي إيش هي مش لازم أكون شرحتها انتوا لازم
+
+327
+00:23:56,040 --> 00:23:59,540
+ترجعوا في عندكم كتب و في مراجع و في ال internet
+
+328
+00:23:59,540 --> 00:24:05,640
+بالصلاة و عن النبي فيعني بسهولة جدا ممكن أجي أقول
+
+329
+00:24:05,640 --> 00:24:09,820
+والله plural effusion إيش هي طب هات أشوف فيديو طيب
+
+330
+00:24:09,820 --> 00:24:14,440
+كيف بيشيلوا ال effusion؟مثلًا، حلو لوحد يعرف
+
+331
+00:24:14,440 --> 00:24:19,540
+الحاجات هاي طيب الـ Airways Obstruction، إيش يعني؟
+
+332
+00:24:19,540 --> 00:24:23,960
+إيش ممكن اسمها Status Asthmaticus؟ طب Inhalation
+
+333
+00:24:23,960 --> 00:24:28,320
+of Toxins or Noxious Gases؟ إيش الـ Toxins هاي؟
+
+334
+00:24:28,320 --> 00:24:31,360
+إيش الـ Noxious Gases هاي؟ هذا إنتوا لازم لوحدكم
+
+335
+00:24:31,360 --> 00:24:35,600
+تحاولوا تكشفوا عنها طبعًا ماحدش يقول كتير عنها
+
+336
+00:24:35,600 --> 00:24:40,260
+وعنا شغل كتير وعنا دراسة كتير وعنا يعني هيك حال
+
+337
+00:24:40,260 --> 00:24:43,690
+الدنيا يعني في الآخرالـ Respiratory Failure
+
+338
+00:24:43,690 --> 00:24:50,930
+Treatment تبعه كيف بيعالجه؟ طبعاً مرة تانية لازم
+
+339
+00:24:50,930 --> 00:24:52,550
+أصلح السبب في الموضوع
+
+340
+00:24:56,360 --> 00:24:59,260
+طبعا مش لازم أصلحه وماعملش حاجة لأ أنا بدأ أصلح
+
+341
+00:24:59,260 --> 00:25:01,640
+السبب لكن في عندي حاجات في الأول بدأ أعملها
+
+342
+00:25:01,640 --> 00:25:05,360
+Prongodilators موسيعات الشعب الهوائية Corruption
+
+343
+00:25:05,360 --> 00:25:10,380
+of blood gas الـ pH لازم أصلح الغازات الدم لازم
+
+344
+00:25:10,380 --> 00:25:12,920
+أديه أكسيجين وممكن في الآخر الأكسيجين العادي
+
+345
+00:25:12,920 --> 00:25:15,840
+مايجيبش أضطر أعمله حاجة اسمها mechanical
+
+346
+00:25:15,840 --> 00:25:22,160
+ventilation mechanical ventilation بحطله
+
+347
+00:25:22,160 --> 00:25:23,660
+تيوب عمليت زي
+
+348
+00:25:26,650 --> 00:25:31,470
+من الانف أو من الفم وبحطه على جهاز وفيه مجسات
+
+349
+00:25:31,470 --> 00:25:35,970
+معينة بيصير الجهاز يصلي
+
+350
+00:25:35,970 --> 00:25:40,890
+يديني أكسجين وياخد ثاني أكسيد الكربون عمليًا طبعًا
+
+351
+00:25:40,890 --> 00:25:43,570
+قضية الـ mechanical ventilation هي في حالتها تجسسة
+
+352
+00:25:43,990 --> 00:25:46,330
+بنقدرش انا في محاضرة اقعد احكي عن الـ Mechanical
+
+353
+00:25:46,330 --> 00:25:49,270
+Ventilation انتوا لكن ممكن يروح يجرى عنها ممكن
+
+354
+00:25:49,270 --> 00:25:53,050
+يشوفله فيديو كيف بيصير عملية الـ Mechanical
+
+355
+00:25:53,050 --> 00:25:55,370
+Ventilation Functions of Mechanical Ventilation
+
+356
+00:25:55,370 --> 00:25:58,750
+يعني المفروض انه زي ما قلتلكوا تحاولوا جدًا لإمكان
+
+357
+00:25:58,750 --> 00:26:03,550
+فرصة انه نتعرف على الطب عن قرب عندكوا لما تتخرجوا
+
+358
+00:26:03,550 --> 00:26:06,730
+ان شاء الله تعالى هتلاقوا حالكوا في الآخر منغمسين
+
+359
+00:26:06,730 --> 00:26:10,370
+بس في ايش؟ في المادة تبعاتكوا في التخصص تبعكوا
+
+360
+00:26:11,310 --> 00:26:15,370
+وهتبعدوا شويه عن مواد الطب اللي هي .. يعني احنا ..
+
+361
+00:26:15,370 --> 00:26:18,470
+يعني إيش بقولوها team واحد في الآخر، يعني لازم
+
+362
+00:26:18,470 --> 00:26:22,110
+الممرض، لازم أخصائي العلاج الطبيعي، أخصائي الـ
+
+363
+00:26:22,110 --> 00:26:24,390
+occupational therapy والطبيب يكونوا تقريبا
+
+364
+00:26:24,390 --> 00:26:27,590
+مستوعبين كتير بالحاجات دي، ممكن مش بنفس القدر، لكن
+
+365
+00:26:27,590 --> 00:26:32,610
+في الآخر لازم يكون في لغة تفاهم واضحة بينهم اللي
+
+366
+00:26:32,610 --> 00:26:39,200
+هي لغة الطب قبل كل شيءبأتمنى لكوا انهار سعيد و ..
+
+367
+00:26:39,200 --> 00:26:44,980
+و .. يعني إن شاء الله من هلأ نهاية الأسبوع بنعمل
+
+368
+00:26:44,980 --> 00:26:49,800
+محاضرة التانية اللي هي COPD آخر محاضرة فيه اللي هو
+
+369
+00:26:49,800 --> 00:26:55,020
+موضوع الـ Respiratory System بذكركوا بالـ
+
+370
+00:26:55,020 --> 00:26:59,520
+Assignment كان عبارة عن جزئين بتلاقوهم جوا
+
+371
+00:26:59,520 --> 00:27:00,880
+المحاضرة، يعطيكوا العافية
+
diff --git a/PL9fwy3NUQKwaf4aPlmtIQXiASpDxVLEnO/_Q5kQ1JQWrE.srt b/PL9fwy3NUQKwaf4aPlmtIQXiASpDxVLEnO/_Q5kQ1JQWrE.srt
new file mode 100644
index 0000000000000000000000000000000000000000..46b1c63a37ecccab68b1b3814284ca6c5973e555
--- /dev/null
+++ b/PL9fwy3NUQKwaf4aPlmtIQXiASpDxVLEnO/_Q5kQ1JQWrE.srt
@@ -0,0 +1,2020 @@
+1
+00:00:01,040 --> 00:00:03,020
+بسم الله الرحمن الرحيم الحمد لله رب العالمين
+
+2
+00:00:03,020 --> 00:00:06,800
+والصلاة والسلام علي سيدنا محمد سيد المرسلين أجمعين
+
+3
+00:00:06,800 --> 00:00:09,760
+يعني اللي نلتقي اليوم إن شاء الله تعالى في محاضرة
+
+4
+00:00:09,760 --> 00:00:13,400
+جديدة وحنتحدث عن الـ congenital heart disease المرة
+
+5
+00:00:13,400 --> 00:00:17,360
+اللي فاتة حكينا عن الـ valvular heart disease طبعاً
+
+6
+00:00:17,360 --> 00:00:20,740
+هي بتتقاطع مع الـ congenital لأن بعض أنواع الـ
+
+7
+00:00:20,740 --> 00:00:24,770
+valvular كانت إيش؟ كانت أو الـ congenital إذا 
+
+8
+00:00:24,770 --> 00:00:28,650
+فاكرين، زي الـ pulmonary artery stenosis على سبيل
+
+9
+00:00:28,650 --> 00:00:32,970
+المثال أو الـ aortic stenosis، هذا كلها valvular
+
+10
+00:00:32,970 --> 00:00:37,290
+عملياً، لكن هي أيضاً congenital heart disease في بعض
+
+11
+00:00:37,290 --> 00:00:40,130
+الأحيان في الأطفال، الحقيقة موضوع الـ Congenital
+
+12
+00:00:40,130 --> 00:00:45,070
+Heart Disease مهم بالنسبة للعلاج الطبيعي للناس
+
+13
+00:00:45,070 --> 00:00:47,930
+اللي بتشتغل مع الأطفال لأن كثير من الأطفال هدول
+
+14
+00:00:47,930 --> 00:00:51,270
+اللي بيبقى عندهم Complex Heart Disease أو Complex
+
+15
+00:00:51,270 --> 00:00:53,890
+Congenital Heart Disease لأن فينا simple و فينا
+
+16
+00:00:53,890 --> 00:00:59,310
+complex، هدول الناس بيصير عندهم مشاكل في الموتور
+
+17
+00:00:59,310 --> 00:01:03,850
+على سبيل المثال، و هم ممكن يحتاجوا كمان علاج طبيعي
+
+18
+00:01:04,120 --> 00:01:08,420
+الـ congenital heart disease common pediatric
+
+19
+00:01:08,420 --> 00:01:13,840
+problem و الـ general incidence كل ألف مولود ستة
+
+20
+00:01:13,840 --> 00:01:16,720
+بيبقى عندهم، يعني احنا قلنا مرة إيه ممكن أو ما عرفش
+
+21
+00:01:16,720 --> 00:01:21,560
+إذا ما قلناش إنه قطاع غزة فيه كام مولود في السنة
+
+22
+00:01:21,560 --> 00:01:28,180
+في القطاع، صح؟ خمس آلاف في الشهر تقريباً، ستين ألف في
+
+23
+00:01:28,180 --> 00:01:32,420
+السنة، ضافيهم بركة، مش هيك؟ آه، ستين ألف في السنة،
+
+24
+00:01:32,420 --> 00:01:37,080
+فأنا لما أجي بحكي عن ستة ستة في الألف، من أننا
+
+25
+00:01:37,080 --> 00:01:39,860
+بنحكي عن حوالي أربعمية حالة أو تلاتمية أو ستين
+
+26
+00:01:39,860 --> 00:01:43,820
+حالة، طبعاً كل أنواع الـ congenital heart disease في
+
+27
+00:01:43,820 --> 00:01:47,150
+السنة، و دا أحس بهيك بسيطة، ولا جوا إنه في الـ
+
+28
+00:01:47,150 --> 00:01:50,890
+statistics إن الـ males and females are equally
+
+29
+00:01:50,890 --> 00:01:56,110
+affected، و في كثير بالأحيان بيبقى في جزء من الأسباب
+
+30
+00:01:56,110 --> 00:01:59,630
+genetic لكن ممكن يكون multifactorial inheritance
+
+31
+00:01:59,630 --> 00:02:03,670
+زي على سبيل المثال ممارسات جديدة اللي بتصير عند
+
+32
+00:02:03,670 --> 00:02:07,910
+بعض الأمهات إلى جانب طبعاً زي التدخين على سبيل
+
+33
+00:02:07,910 --> 00:02:11,770
+المثال أو السمنة أو حاجة زي هي، 30% of cases
+
+34
+00:02:11,770 --> 00:02:18,270
+associated with syndromes، يعني مشكلة
+
+35
+00:02:18,270 --> 00:02:20,710
+القلب الـ congenital heart disease جزء من
+
+36
+00:02:20,710 --> 00:02:24,790
+المتلازمة، فيه حاجات تانية زي cleft palate زي
+
+37
+00:02:24,790 --> 00:02:28,570
+hydrocephalus، حاجات تانية لكن هم الـ congenital
+
+38
+00:02:28,570 --> 00:02:34,350
+heart disease بيبقى جزء من السيندروم في 30% و 10%
+
+39
+00:02:34,350 --> 00:02:38,430
+with extra cardiac malformation، Commonly divided
+
+40
+00:02:38,430 --> 00:02:42,850
+into non-cyanotic and cyanotic، يعني إحنا لما نقول
+
+41
+00:02:42,850 --> 00:02:48,030
+عن واحد congenital heart بدي أقول هل هو cyanotic
+
+42
+00:02:48,030 --> 00:02:52,410
+ولا non-cyanotic، يعني إيش يعني cyanotic؟
+
+43
+00:02:52,410 --> 00:02:55,270
+والـ cyanosis اللي بنقصدها hence central cyanosis
+
+44
+00:02:56,460 --> 00:02:58,620
+فيه فرق بين الـ central؟ في حاجة تانية غير الـ
+
+45
+00:02:58,620 --> 00:03:01,760
+central؟ آه فيه peripheral لكن احنا احنا بنحكي هنا
+
+46
+00:03:01,760 --> 00:03:05,960
+عن central cyanosis واحنا بنحكي عن slide مهم الآن
+
+47
+00:03:05,960 --> 00:03:08,820
+is the bluish discoloration of the lips, tongue
+
+48
+00:03:08,820 --> 00:03:11,540
+and mucous membrane of the oral cavity and
+
+49
+00:03:11,540 --> 00:03:16,700
+fingernails، المواضح للتعريف الـ degree of cyanosis
+
+50
+00:03:16,700 --> 00:03:20,820
+في أنها mild or early cases و بيبقى mainly
+
+51
+00:03:20,820 --> 00:03:25,060
+exertional بس لما بعيط الطفل أو لما بيرضع بسرعة أو
+
+52
+00:03:25,060 --> 00:03:28,060
+كذا، and appears only during crying, sucking or
+
+53
+00:03:28,060 --> 00:03:32,700
+exertion، في أن established cases أو الدرجة التانية
+
+54
+00:03:32,700 --> 00:03:40,340
+دائماً عنده زراج، دائماً ولكن بيرجع أكثر وأكثر بيغمج
+
+55
+00:03:40,340 --> 00:03:44,660
+أكثر وأكثر on accession، وفي long standing cases
+
+56
+00:03:44,660 --> 00:03:48,740
+هدول الناس اللي بيقعدوا usually above ست شهر growth
+
+57
+00:03:48,740 --> 00:03:52,700
+failure، clapping of the fingers occurs، clapping of
+
+58
+00:03:52,700 --> 00:03:56,600
+the fingers، هذا بيذكرنا في حاجة، وين مرت علينا
+
+59
+00:03:56,600 --> 00:03:59,520
+clapping of the fingers ولا ما مرتش علينا، احنا مع
+
+60
+00:03:59,520 --> 00:04:06,540
+بعض لسه، طيب، بدناش نعمل منها الآن موضوع «clapping
+
+61
+00:04:06,540 --> 00:04:09,080
+of the finger»، كل واحد يكشف لوحده، لإله، يعني في
+
+62
+00:04:09,080 --> 00:04:10,860
+الموضوع «clapping of the finger»، «differential
+
+63
+00:04:10,860 --> 00:04:14,020
+diagnosis»، «clapping of the finger»، زي ما أجيها
+
+64
+00:04:14,020 --> 00:04:17,540
+عند الأطفال، إيش يعني؟ ماذا تعني؟ وين لازم أفكر؟
+
+65
+00:04:17,540 --> 00:04:21,780
+هل بس في «congenital heart disease»؟ ولا في «else
+
+66
+00:04:21,780 --> 00:04:26,220
+other diagnosis»؟ آه، طبعاً، في حاجات تانية، كويسة
+
+67
+00:04:26,220 --> 00:04:29,200
+أقول معلومة هذه الـ Clapping of the fingers
+
+68
+00:04:29,200 --> 00:04:33,120
+differential diagnosis مش همور كده يلقوا أنتو
+
+69
+00:04:33,120 --> 00:04:38,240
+اللي بديوا يطلع يطلع الـ
+
+70
+00:04:38,240 --> 00:04:42,200
+etiology is a small percentage of congenital heart
+
+71
+00:04:42,200 --> 00:04:44,620
+disease are related زي ما قلنا قبل شوي
+
+72
+00:04:44,620 --> 00:04:48,100
+لـ chromosomal abnormalities زي الـ «Trisomy» أو
+
+73
+00:04:48,100 --> 00:04:53,280
+«Down Syndrome»، «Trisomy 21»، حوالي 50% قائمة أكبر
+
+74
+00:04:53,280 --> 00:04:55,720
+من المصابين بالقلق من قلق قلق قلق قلق قلق قلق قلق
+
+75
+00:04:55,720 --> 00:04:58,340
+قلق قلق قلق قلق قلق قلق قلق قلق قلق قلق قلق قلق
+
+76
+00:04:58,340 --> 00:05:04,040
+قلق قلق قلق قلق قلق قلق قلق قلق قلق قلق قلق قلق
+
+77
+00:05:04,040 --> 00:05:06,240
+قلق قلق قلق قلق قلق قلق قلق قلق قلق قلق قلق قلق
+
+78
+00:05:06,240 --> 00:05:06,300
+قلق قلق قلق قلق قلق قلق قلق قلق قلق قلق قلق قلق
+
+79
+00:05:06,300 --> 00:05:09,040
+قلق قلق قلق قلق قلق قلق قلق قلق قلق قلق قلق قلق
+
+80
+00:05:09,040 --> 00:05:16,820
+قلق قلق ق
+
+81
+00:05:20,100 --> 00:05:24,760
+Infection اللي بنقول عنه إيش؟ Torch Infection
+
+82
+00:05:24,760 --> 00:05:28,820
+ما بعرفش إذا عمرو كنت سمعته في الـ Torch Infection
+
+83
+00:05:28,820 --> 00:05:35,700
+الـ T معناته دي عبارة عن بدايات أحرف لكلمات، أحرف
+
+84
+00:05:35,700 --> 00:05:42,100
+بدايات لكلمة، الـ Torch، T تكسب لزموزز، O Others هنحكي
+
+85
+00:05:42,100 --> 00:05:48,920
+إيش يعني؟ R Rubella اللي هي الحصبة الألمانية، C
+
+86
+00:05:48,920 --> 00:05:55,300
+Cytomegalovirus، H.Herpesvirus
+
+87
+00:05:55,300 --> 00:06:00,400
+O.أي حاجة بتعملي Infection Intrauterine Other
+
+88
+00:06:00,400 --> 00:06:06,980
+viruses، حرف الـ O في other، أي فيروس ممكن يعمل أو
+
+89
+00:06:06,980 --> 00:06:09,500
+كثير من الفيروسات ممكن تعمل Intrauterine infection
+
+90
+00:06:09,500 --> 00:06:13,300
+وبالتالي تعمل مشاكل بالإضافة لأمور تانية في الجسم
+
+91
+00:06:13,300 --> 00:06:17,140
+اللي هي المشاكل القلبية، ويلاحظوا إنه No racial
+
+92
+00:06:17,140 --> 00:06:20,220
+differences in the occurrence of a congenital
+
+93
+00:06:20,220 --> 00:06:25,400
+heart disease as a whole has been noted، طيب هو
+
+94
+00:06:25,400 --> 00:06:28,860
+خدوا بالكوا إحنا قبل يمكن نتحدث عن
+
+95
+00:06:31,770 --> 00:06:38,050
+عن الـ Genital Heart Disease، لابد إنه نعرف اللي هي
+
+96
+00:06:38,050 --> 00:06:44,930
+الـ Fetal Circulation، لأنه مهم بالنسبة لنا الـ
+
+97
+00:06:44,930 --> 00:06:51,610
+Fetal Circulation، أنا هحاول الآن أمشي فيديو تشوفوه
+
+98
+00:06:51,610 --> 00:06:55,170
+عن الـ Fetal Circulation حوالي عشر دقائق أو أقل
+
+99
+00:07:03,920 --> 00:07:05,940
+Hello، in this video we're going to talk about
+
+100
+00:07:05,940 --> 00:07:12,000
+fetal circulation، The fetus is you when you were
+
+101
+00:07:12,000 --> 00:07:16,500
+inside your mother's womb, the uterus، Now, the
+
+102
+00:07:16,500 --> 00:07:20,780
+fetal circulation is actually very complicated and
+
+103
+00:07:20,780 --> 00:07:24,020
+different to the circulation of a baby which is
+
+104
+00:07:24,020 --> 00:07:27,590
+essentially the circulation of an adult، في هذا
+
+105
+00:07:27,590 --> 00:07:32,890
+الفيديو، يعتمد على أن المشاهد، أنت، سيحصل على فهم
+
+106
+00:07:32,890 --> 00:07:35,850
+عام لشخص طبيعي بشكل عام لشخص طبيعي بشكل عام لشخص
+
+107
+00:07:35,850 --> 00:07:37,050
+طبيعي بشكل عام لشخص طبيعي بشكل عام لشخص طبيعي بشخص
+
+108
+00:07:37,050 --> 00:07:38,230
+طبيعي بشخص طبيعي بشخص طبيعي بشخص طبيعي بشخص طبيعي
+
+109
+00:07:38,230 --> 00:07:39,150
+بشخص طبيعي بشخص طبيعي بشخص طبيعي بشخص طبيعي بشخص
+
+110
+00:07:39,150 --> 00:07:42,330
+طبيعي بشخص طبيعي بشخص طبيعي بشخص طبيعي بشخص طبيعي
+
+111
+00:07:42,330 --> 00:07:49,310
+بشخص طبيعي بشخص طبيعي
+
+112
+00:07:49,310 --> 00:07:54,590
+بشخص
+
+113
+00:07:54,590 --> 00:07:59,540
+طبيعي، الجزء الآخر من القلب يسئول أن يدفع الدم إلى
+
+114
+00:07:59,540 --> 00:08:05,760
+الحلقات الدم يعود للقلب، إلى الجزء الآخر، من خلال
+
+115
+00:08:05,760 --> 00:08:09,380
+الحلقات الموجودة هنا في اللون الأزرق ويدخل الهيئة
+
+116
+00:08:09,380 --> 00:08:10,960
+الأيمنية الأيمنية الأيمنية الأيمنية الأيمنية
+
+117
+00:08:10,960 --> 00:08:11,520
+الأيمنية الأيمنية الأيمنية الأيمنية الأيمنية
+
+118
+00:08:11,520 --> 00:08:13,960
+الأيمنية الأيمنية الأيمنية الأيمنية الأيمنية
+
+119
+00:08:13,960 --> 00:08:15,000
+الأيمنية الأيمنية الأيمنية الأيمنية الأيمنية
+
+120
+00:08:15,000 --> 00:08:15,660
+الأيمنية الأيمنية الأيمنية الأيمنية الأيمنية
+
+121
+00:08:15,660 --> 00:08:16,100
+الأيمنية الأيمنية الأيمنية الأيمنية الأيمنية
+
+122
+00:08:16,100 --> 00:08:21,170
+الولكن في المتجر الحيواني، هناك فتحة عملية عملية
+
+123
+00:08:21,170 --> 00:08:22,390
+عملية عملية عملية عملية عملية عملية عملية عملية
+
+124
+00:08:22,390 --> 00:08:24,470
+عملية عملية عملية عملية عملية عملية عملية عملية
+
+125
+00:08:24,470 --> 00:08:28,710
+عملية عملية عملية عملية عملية عملية عملية عملية
+
+126
+00:08:28,710 --> 00:08:29,410
+عملية عملية عملية عملية عملية عملية عملية عملية
+
+127
+00:08:29,410 --> 00:08:34,270
+عملية عملية عملية عملية عملية عملية عملية عملية
+
+128
+00:08:34,270 --> 00:08:40,350
+عملية 
+
+129
+00:08:42,750 --> 00:08:46,450
+الجهة اليسرى من القلب تحتوي على الـ left atrium و
+
+130
+00:08:46,450 --> 00:08:50,310
+left ventricle، الجهة اليسرى من القلب مسؤولة عن
+
+131
+00:08:50,310 --> 00:08:52,830
+مدفع الدم للجميع من جسم الفيتول
+
+132
+00:08:57,350 --> 00:09:00,970
+بالرغم من الهول التشريحي، الـ patent foramen
+
+133
+00:09:00,970 --> 00:09:07,190
+ovale، هناك اتصال مهم جداً أو هول اسمه الـ patent
+
+134
+00:09:07,190 --> 00:09:12,510
+ductus arteriosus، الـ patent ductus arteriosus هو
+
+135
+00:09:12,510 --> 00:09:19,130
+الاتصال بين الارتريا الرئوية والأورطا وذلك في
+
+136
+00:09:19,130 --> 00:09:23,450
+المتطورة الفيطالية، الدم يتم تحميله إلى بقية الدم
+
+137
+00:09:23,450 --> 00:09:30,010
+الفيطالي، الدم يمكن أن يتجاوز الجلد لأن هناك الكثير
+
+138
+00:09:30,010 --> 00:09:34,570
+من الضرر في الجلد الفيطالي، لذلك الدم لا يمكن أن
+
+139
+00:09:34,570 --> 00:09:41,330
+يتمسك بسهولة في الـ Aorta، الـPatent Foramen Ovale
+
+140
+00:09:41,330 --> 00:09:45,830
+وPatent Ductus Arteriosus يسمحوا بـRight to Left
+
+141
+00:09:45,830 --> 00:09:50,830
+وهذا يعني أن الدم يدخل في الجانحة اليسرى من
+
+142
+00:09:50,830 --> 00:09:54,110
+القلب بسهولة، أن يكون شنطا إلى الجانحة اليسرى من
+
+143
+00:09:54,110 --> 00:09:58,610
+القلب، ويغيّر الحلوانات في تجارب الحيوانات، لا 
+
+144
+00:09:58,610 --> 00:10:02,090
+يوجد حاجة لدم يدخل في الحيوانات لأن الحيوانات لا
+
+145
+00:10:02,090 --> 00:10:05,830
+تتنفس في الهواء على أية حالة، بل الحيوانات تعتمد
+
+146
+00:10:05,830 --> 00:10:09,710
+على الأكسجين الذي تم إعطاؤه تجريبياً في 
+
+147
+00:10:09,710 --> 00:10:15,520
+الـPlacenta. رأيك، الـPlacenta هي الاتصال المهم بين
+
+148
+00:10:15,520 --> 00:10:19,320
+الأم والـFetus. الـPlacenta تمنح الحيوانات والعيون 
+
+149
+00:10:19,320 --> 00:10:21,960
+للـGrowing Fetus من خلال الـCord السري
+
+150
+00:10:21,960 --> 00:10:26,800
+الـCord السري يوصل الـFetus إلى الـPlacenta
+
+151
+00:10:26,800 --> 00:10:31,020
+الـCord السري ليس فقط قطة، إنه يحتوي على
+
+152
+00:10:31,020 --> 00:10:35,540
+أوعية دموية وعضلات وعضلات وعضلات سريّة الـVein الموجودة هنا
+
+153
+00:10:35,540 --> 00:10:40,770
+في الأرجل يتجه من الـPlacenta إلى الـLiver هنا،
+
+154
+00:10:40,770 --> 00:10:44,570
+هناك واحدة من الأوعية الدموية، وهي مرسومة هنا في
+
+155
+00:10:44,570 --> 00:10:48,490
+اللون الأحمر، لأنها محفوظة، وهي محتوية من الدم
+
+156
+00:10:48,490 --> 00:10:53,130
+المحفوظ الذي تم إعطاؤه من الأمهات، اسمها وعاء دموي
+
+157
+00:10:53,130 --> 00:11:00,050
+دموي لأن الوعاء يعود إلى قلب. مرة أخرى، الـVein 
+
+158
+00:11:00,050 --> 00:11:00,930
+الوحيد 
+
+159
+00:11:00,930 --> 00:11:01,090
+ 
+160
+00:11:01,090 --> 00:11:02,830
+ 
+161
+00:11:02,830 --> 00:11:19,830
+ 
+162
+00:11:21,120 --> 00:11:24,780
+الـ Inferior Vena Cava التي ستدخل جزءًا من جزء من
+
+163
+00:11:24,780 --> 00:11:30,580
+جزء من جزء من جزء من جزء من جزء من جزء من جزء من
+
+164
+00:11:30,580 --> 00:11:31,020
+جزء من جزء من جزء من جزء من جزء من جزء من جزء من
+
+165
+00:11:31,020 --> 00:11:31,100
+جزء من جزء من جزء من جزء من جزء من جزء من جزء من
+
+166
+00:11:31,100 --> 00:11:34,020
+جزء من جزء من جزء من جزء من جزء من جزء من جزء من
+
+167
+00:11:34,020 --> 00:11:44,680
+جزء من
+
+168
+00:11:44,680 --> 00:11:50,170
+ومن ثم يدخل في الجانحة اليسرى، ومن ثم سيكون الدم
+
+169
+00:11:50,170 --> 00:11:56,470
+مدفوعًا إلى الـAorta وإلى المتحركات الدم
+
+170
+00:11:56,470 --> 00:12:00,890
+المدفوع من الجانحة اليسرى، يمكن أيضًا أن يدخل في
+
+171
+00:12:00,890 --> 00:12:03,090
+الجانحة اليسرى، ومن ثم يدخل في الجانحة اليسرى، ومن ثم
+
+172
+00:12:03,090 --> 00:12:06,350
+يدخل في الجانحة اليسرى، ومن ثم يدخل في الجانحة اليسرى،
+
+173
+00:12:06,350 --> 00:12:08,550
+ومن ثم يدخل في الجانحة اليسرى، ومن ثم يدخل في الجانحة
+
+174
+00:12:08,550 --> 00:12:11,510
+اليسرى، ومن ثم يدخل في الجانحة اليسرى، ومن ثم يدخل
+
+175
+00:12:11,510 --> 00:12:15,730
+ثم يدخل في الج Okay, so now you have partly
+
+176
+00:12:15,730 --> 00:12:20,330
+oxygenated blood in the aorta where it will now be
+
+177
+00:12:20,330 --> 00:12:24,450
+delivered to the fetal tissue. The fetus needs
+
+178
+00:12:24,450 --> 00:12:28,570
+oxygenated blood. Once the oxygenated blood is
+
+179
+00:12:28,570 --> 00:12:31,510
+used by the tissue, blood will then normally
+
+180
+00:12:31,510 --> 00:12:35,350
+return back to the heart via the veins. So here
+
+181
+00:12:35,350 --> 00:12:38,170
+the veins are in blue which indicates deoxygenated
+
+182
+00:12:38,170 --> 00:12:42,740
+blood. ولكن في تجارب الحيوانات، يوجد بالفعل أوعية دموية
+
+183
+00:12:42,740 --> 00:12:44,160
+أوعية دموية أوعية دموية أوعية دموية أوعية دموية أوعية دموية أوعية دموية أوعية دموية أوعية دموية أوعية دموية
+
+184
+00:12:44,160 --> 00:12:47,280
+أوعية دموية أوعية دموية أوعية دموية أوعية دموية أوعية دموية أوعية دموية أوعية دموية أوعية دموية أوعية دموية
+
+185
+00:12:47,280 --> 00:12:48,640
+أوعية دموية أوعية دموية أوعية دموية أوعية دموية أوعية دموية أوعية دموية أوعية دموية أوعية دموية أوعية دموية
+
+186
+00:12:48,640 --> 00:13:04,300
+أوعية دموية أوعية دموية أوعية دموية أوعية دموية
+
+187
+00:13:16,180 --> 00:13:19,160
+نرجع للمحاضرة، إحنا شفنا كيف الـ «فيتال
+
+188
+00:13:19,160 --> 00:13:22,520
+تسيركوليشن» بيصير. كان مهمًا أن نتعرف على الـ «فيتال
+
+189
+00:13:22,520 --> 00:13:25,680
+تسيركوليشن». طبعًا هو الأصل أن أنا هذا بشرحه في 
+
+190
+00:13:25,680 --> 00:13:29,520
+العادة على اللوح بالألوان حتى نعرف كيف الـ
+
+191
+00:13:29,520 --> 00:13:32,800
+«أكسجينيتد» و «دي أكسجينيتد» blood delivered to
+
+192
+00:13:32,800 --> 00:13:35,720
+the fetus وكيف يخرج من الـ «فيتس» وكيف الـ
+
+193
+00:13:35,720 --> 00:13:38,860
+«سيركوليشن» جوا القلب زي ما لاحظنا أنه في ثلاثة
+
+194
+00:13:38,860 --> 00:13:42,800
+changes موجودة في الـ «فيتال تسيركوليشن» مش موجودة
+
+195
+00:13:42,800 --> 00:13:46,330
+عند الـ Adult. ولو ضلت موجودة بتعمل لي مشكلة. يعني لما
+
+196
+00:13:46,330 --> 00:13:49,810
+بينولد الطفل إذا ضلت الأماكن الثلاثة اللي اتحدثت
+
+197
+00:13:49,810 --> 00:13:52,870
+عنهم في الفيديو إذا ضلت هذه بتعمل لي مشكلة زي ال
+
+198
+00:13:52,870 --> 00:13:57,150
+Patent أو ال foramen ovale Patent. معناته ضل Patent
+
+199
+00:13:57,150 --> 00:14:00,770
+foramen ovale. إذا ضلّ فاتحًا بعد الولادة بيصير إن
+
+200
+00:14:00,770 --> 00:14:05,190
+دي مشكلة. الأمر الثاني اللي هو ductus arteriosus أو
+
+201
+00:14:05,190 --> 00:14:11,650
+Botalli. هذا اللي بربط عبارة عن فتحة أو معبر بين
+
+202
+00:14:11,650 --> 00:14:17,420
+الأورطة وبين ال pulmonary artery. لو ضال بعد
+
+203
+00:14:17,420 --> 00:14:23,640
+الولادة هذا بيعمل دكتوس أرتريوزوس بوتالي أو بيتان
+
+204
+00:14:23,640 --> 00:14:27,340
+دكتوس أرتريوزوس. معناته مفتوح وهذا بده شغل وفي
+
+205
+00:14:27,340 --> 00:14:33,900
+عندنا الأقل أهمية دكتوس فينوزوس. عادة ما بيعمل لنا أي
+
+206
+00:14:34,310 --> 00:14:38,630
+مشاكل. لكن برضه بينتهي دوره بمجرد إنه ينولد الطفل. 
+
+207
+00:14:38,630 --> 00:14:41,770
+طيب هو إيش اللي بأثر لما أنا بطلع الطفل وبيصير 
+
+208
+00:14:41,770 --> 00:14:47,670
+inflating لل lungs. فبزيد الضغوط، بتزيد في ال left
+
+209
+00:14:47,670 --> 00:14:53,690
+في ال left side of the heart. بيزيد الضغط فتتسكر
+
+210
+00:14:53,690 --> 00:15:00,920
+معظم الفتحات هذه اللي موجودة. طيب احنا الآن المنظر
+
+211
+00:15:00,920 --> 00:15:04,960
+القلب هذا. طبعًا تعودنا عليه من الأمام وإيش المشاكل 
+
+212
+00:15:04,960 --> 00:15:09,940
+اللي ممكن تصير الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+213
+00:15:09,940 --> 00:15:10,740
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+214
+00:15:10,740 --> 00:15:13,020
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ 
+
+215
+00:15:13,020 --> 00:15:13,440
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+216
+00:15:13,440 --> 00:15:14,980
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+217
+00:15:14,980 --> 00:15:21,120
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+218
+00:15:21,120 --> 00:15:23,160
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ 
+
+219
+00:15:23,160 --> 00:15:26,060
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+220
+00:15:26,060 --> 00:15:29,800
+الـ الـ الـ الـ الـ الـ 
+
+221
+00:15:29,800 --> 00:15:34,520
+الـ 
+
+222
+00:15:34,720 --> 00:15:39,100
+الـ two ventricles. صار عندي defect في فتحة بنقول
+
+223
+00:15:39,100 --> 00:15:45,740
+عنها VSD. Ventricular Septum Defect بتمثل 25 إلى
+
+224
+00:15:45,740 --> 00:15:49,940
+30% من حالات القلب اللي بتصير. يعني أنا لو عندي
+
+225
+00:15:49,940 --> 00:15:53,020
+congenital heart disease بيتحالة هذه بتمثل من 25
+
+226
+00:15:53,020 --> 00:15:58,760
+إلى 30%. يعني ال most common defect أو Legion في ال
+
+227
+00:15:58,760 --> 00:16:01,780
+congenital heart disease هي الـventricular septum
+
+228
+00:16:01,780 --> 00:16:05,250
+defect. في الحاجات الثانية اللي بتيجي بعدها اللي هي
+
+229
+00:16:05,250 --> 00:16:10,790
+«atrial septum defect». بتمثل من 6 لـ8% وبيبقى
+
+230
+00:16:10,790 --> 00:16:14,370
+الـforamen ovale اللي إحنا شفناه قبل شوية أو 
+
+231
+00:16:14,370 --> 00:16:19,260
+الثقب اللي بين الأذينين ضالّ فاتحًا. بعدين في عندنا الـ
+
+232
+00:16:19,260 --> 00:16:22,540
+patent ductus arteriosus. الـ ductus arteriosus هذا
+
+233
+00:16:22,540 --> 00:16:27,700
+الربطة أو الوصلة الشريانية بيقول عنها بين الأورطة
+
+234
+00:16:27,700 --> 00:16:31,220
+وبين البلمونري. أرض البلمونري. أرض ضلت فاتحة فقلنا
+
+235
+00:16:31,220 --> 00:16:35,360
+عنها patent ductus arteriosus. في عندنا حاجة ثانية
+
+236
+00:16:35,360 --> 00:16:38,100
+الـ Coarctation of أورطة. وين مرت علينا هي يا جماعة
+
+237
+00:16:38,100 --> 00:16:41,340
+فاكرين أنتم؟ في ال secondary causes of 
+
+238
+00:16:41,340 --> 00:16:47,940
+hypertension. في عندنا كمان التترالوجي of Fallot.
+
+239
+00:16:47,940 --> 00:16:52,280
+Fallot. ده اسم عالم. تترالوجي. رباعية. رباعية Fallot
+
+240
+00:16:52,280 --> 00:16:56,800
+خمسة لسبعة فالمية. في عندنا بولموناري valve stenosis
+
+241
+00:16:56,800 --> 00:17:00,860
+مرت علينا في الـvalvular heart diseases. خمسة لسبعة 
+
+242
+00:17:00,860 --> 00:17:04,200
+فالمية. aortic valve stenosis أربع لسبعة فالمية. وفي
+
+243
+00:17:04,200 --> 00:17:09,160
+حاجة بعدها أنا هوقف بعدها لأن Transposition of
+
+244
+00:17:09,160 --> 00:17:14,180
+Great Artery. تصوروا أن الأورطة ينتج المحل 
+
+245
+00:17:14,180 --> 00:17:17,320
+البلمونري. R3 والبولمونري. R3 ينتج المحل 
+
+246
+00:17:20,000 --> 00:17:24,720
+الأورطة. أورطة كبيرة ومشكلة كبيرة. وفي مجموعة أقل 
+
+247
+00:17:24,720 --> 00:17:31,060
+بتصير لكن هم هدول الـ most common legions بتصير في
+
+248
+00:17:31,060 --> 00:17:34,860
+الـ heart. طبعًا التقسيمة الأهم اللي هي cyanotic and
+
+249
+00:17:34,860 --> 00:17:37,940
+non-cyanotic. Non-cyanotic congenital heart disease
+
+250
+00:17:37,940 --> 00:17:43,020
+حوالي 80% من ال cases يعني رحمة من ربنا هي non
+
+251
+00:17:43,020 --> 00:17:47,370
+-cyanotic. فشيه cyanosis بكشفها بس بالسبعة ومش 
+
+252
+00:17:47,370 --> 00:17:51,290
+عاملالي مشاكل كبيرة، لكن بيدها متابعة. منسيا نوتك
+
+253
+00:17:51,290 --> 00:17:53,910
+congenital heart disease حوالي 80% of cases.
+
+254
+00:17:53,910 --> 00:17:57,510
+ventricular septum defect 25% is by far the most
+
+255
+00:17:57,510 --> 00:18:02,990
+common lesion followed by ASD, BDA، الآن بدنا نقول
+
+256
+00:18:02,990 --> 00:18:05,890
+اختصارات، يا جماعة، كده بقالكوا، ventricular
+
+257
+00:18:05,890 --> 00:18:09,610
+septum defect VSD, atrial septum defect ASD
+
+258
+00:18:11,240 --> 00:18:16,920
+بوتالدكتاس أرتريوزوس BDA والموراريستينوزيس BS أو 
+
+259
+00:18:16,920 --> 00:18:22,380
+أورتكستينوزيس AS Transposition
+
+260
+00:18:22,380 --> 00:18:28,880
+of Great Arteries TGA. فلوت أو تترالوجي of فلوت TOF
+
+261
+00:18:28,880 --> 00:18:37,260
+هذه اختصارات مشهورة. المجموعة هذه اللي هي الـ Non
+
+262
+00:18:37,260 --> 00:18:40,520
+-cyanotic. الـ Clinical diagnosis can be made
+
+263
+00:18:40,520 --> 00:18:44,660
+confirmed by simple investigations. إيكو مثلاً، إيكو
+
+264
+00:18:44,660 --> 00:18:49,420
+كارديوغرافي، Prognosis في العادة good حتى لو
+
+265
+00:18:49,420 --> 00:18:52,380
+محتاجة. يعني في كثير من الأحيان ما يحتاجش corrective
+
+266
+00:18:52,380 --> 00:18:55,660
+surgery، ما يحتاجش جراحات، إنما ممكن بعض التدخلات 
+
+267
+00:18:55,660 --> 00:19:00,400
+أو just waiting مع المتابعة. الـ Non-cyanotic زي ما
+
+268
+00:19:00,400 --> 00:19:07,940
+قلنا ASD، VSD، BDA والحاجات اللي بيصير فيها اللي هو
+
+269
+00:19:07,940 --> 00:19:11,480
+BS و AS أو أورتكستينوزي أمبربونرستينوزي. هذا
+
+270
+00:19:11,480 --> 00:19:14,720
+بنسميها obstruction of blood to blood flow. يعني
+
+271
+00:19:14,720 --> 00:19:20,900
+عندما يجيء يطلع الدم من القلب هذا اسمه flow وفيه
+
+272
+00:19:20,900 --> 00:19:26,260
+بيبقى obstruction هناك. الآن حقيقة يعني بدي أدق
+
+273
+00:19:26,260 --> 00:19:30,540
+كثير كثير. لكن بدي أشرح بس عن الـ ASD بعض الحاجات
+
+274
+00:19:30,540 --> 00:19:33,780
+يعني هذه من الحاجات common هتسمعوها أنه في والله
+
+275
+00:19:33,780 --> 00:19:38,080
+فتحة بين الأذينين. الـ ASD هي abnormal opening
+
+276
+00:19:38,080 --> 00:19:42,720
+between the two atria allowing blood to shift from
+
+277
+00:19:42,720 --> 00:19:48,300
+left to right atrium. هذه هي مشكلتنا أن الدم بدل ما
+
+278
+00:19:48,300 --> 00:19:52,100
+ينزل على الـRight ventricle كالوضع الطبيعي و
+
+279
+00:19:52,100 --> 00:19:54,240
+الـRight ventricle يودع على الـPulmonary. لا بيصير
+
+280
+00:19:54,240 --> 00:20:00,080
+الدم يدخل من الـLeft atrium to 
+
+281
+00:20:00,080 --> 00:20:05,170
+Right atrium causing the atrium to hypertrophy لأن
+
+282
+00:20:05,170 --> 00:20:09,550
+الضغط في الـ Left Atrium أعلى من الـ Right Atrium
+
+283
+00:20:09,550 --> 00:20:15,950
+فبيصير الدم يدخل للـ Right Atrium وإذا دخل للـ
+
+284
+00:20:15,950 --> 00:20:18,930
+Right Atrium بيصير عنده Hypertrophy للـ Right 
+
+285
+00:20:18,930 --> 00:20:23,050
+Atrium ومش بس ينزل الآن الدم وين؟ على الـ Right
+
+286
+00:20:23,050 --> 00:20:25,270
+Ventricle الـ Right Ventricle بيصير له Hypertrophy
+
+287
+00:20:25,270 --> 00:20:30,890
+كمان وطبعا في الآخر بيعمللي Right Ventricle 
+
+288
+00:20:30,890 --> 00:20:31,910
+Failure
+
+289
+00:20:34,320 --> 00:20:40,740
+الوضع الطبيعي ما فيه مشاكل لكن لو صار فيه منه مشاكل
+
+290
+00:20:40,740 --> 00:20:45,280
+يعني الوضع الطبيعي الـ  symptomatic لكن لو صار فيه
+
+291
+00:20:45,280 --> 00:20:50,350
+مشاكل بيعملنا اللي حكيناها قبل شوية عادة ما فيه
+
+292
+00:20:50,350 --> 00:20:54,510
+symptoms كتير إله لأنه asymptomatic اللهم إن
+
+293
+00:20:54,510 --> 00:20:57,870
+إحنا في السماعة بنسمع حاجة اسمها systolic murmur
+
+294
+00:20:57,870 --> 00:21:03,250
+وممكن في الـ X-ray يبين لي في عندي في الـ X-ray
+
+295
+00:21:03,250 --> 00:21:07,330
+enlargement of the cardiac shadow ليش؟ لأن الـ
+
+296
+00:21:07,330 --> 00:21:09,110
+right ventricle اتضخم
+
+297
+00:21:11,610 --> 00:21:15,810
+والتشخيص عادة بنعمل إما بالـ echocardiography أو 
+
+298
+00:21:15,810 --> 00:21:23,070
+حتى بال cardiac catheter بالقسطرة القلبية ال
+
+299
+00:21:23,070 --> 00:21:25,930
+treatment أحنا زي ما قلت لكم مش هنحكي كتير في 
+
+300
+00:21:25,930 --> 00:21:29,110
+الأمراض لكن كمثل هنحكي عن الـ ASD
+
+301
+00:21:32,120 --> 00:21:35,720
+كانوا زمان يدخلوا فيه open heart وكذا لكن الآن 
+
+302
+00:21:35,720 --> 00:21:38,880
+طبعا ما حدش بدخل الكل بدخل في حاجة اسمها cardiac
+
+303
+00:21:38,880 --> 00:21:41,960
+catheter هذا لو كان interventional cardiac
+
+304
+00:21:41,960 --> 00:21:46,440
+catheter هذا لو كان الـ defect كبير مش كل defect
+
+305
+00:21:46,440 --> 00:21:49,520
+بيندخله لو كان الـdefect كبير هنحكي الآن على
+
+306
+00:21:49,520 --> 00:21:53,680
+الـVSD ونشوف جداش حتى في المساحة يعني في مجسات
+
+307
+00:21:53,680 --> 00:21:58,860
+بنحكي عنها to repair the defect وعن طريق القسطرة 
+
+308
+00:21:58,860 --> 00:22:02,640
+القلبية بدخله زي حاجة هيك device شايفينه أنتم في 
+
+309
+00:22:02,640 --> 00:22:07,420
+الصورة وبدخل بين الـ two atrium وبدخلها هناك
+
+310
+00:22:07,420 --> 00:22:13,520
+وبيعمل عملية زي بلون لكن إيش بيسكر لي الفتحة
+
+311
+00:22:13,520 --> 00:22:18,120
+وبتنتهي المشكلة لهذا الـ Atrial Septal Defect الـ
+
+312
+00:22:18,120 --> 00:22:21,480
+ASD defect in term infant may close spontaneously
+
+313
+00:22:21,480 --> 00:22:26,750
+في الغالب الـ ASD بيسكر في الأطفال بالذات اللي خلص
+
+314
+00:22:26,750 --> 00:22:30,170
+أربعين أسبوع وانولد كامل The results after
+
+315
+00:22:30,170 --> 00:22:33,150
+surgical or device closure in children with
+
+316
+00:22:33,150 --> 00:22:36,770
+moderate to large shunt are excellent Early and
+
+317
+00:22:36,770 --> 00:22:40,170
+midterm results with device closure are excellent
+
+318
+00:22:40,170 --> 00:22:43,950
+يعني حتى اللي بنضطر نحط له الـ device هذا في الـ
+
+319
+00:22:43,950 --> 00:22:48,410
+septum defect أصلا النتائج تبعته بتبقى رائعة وزي
+
+320
+00:22:48,410 --> 00:22:50,970
+ما قلنا جزء كبير من الناس دول بتنتهي مشكلتهم
+
+321
+00:22:50,970 --> 00:22:59,520
+spontaneously بتسكر خلال تقريبا سنة بيسكر الـ ASD 
+
+322
+00:22:59,520 --> 00:23:02,400
+بالنسبة للـ VSD قلنا إحنا قبل شوية أنه الـ most
+
+323
+00:23:02,400 --> 00:23:09,000
+common congenital heart disease وبيمثل حوالي 35%
+
+324
+00:23:09,000 --> 00:23:14,920
+of all congenital heart disease الـ VSD الـ three
+
+325
+00:23:14,920 --> 00:23:18,960
+major types فيه الـ small و هذا بيمثل 85% أقل من 3
+
+326
+00:23:18,960 --> 00:23:23,890
+ملي تصوروا 3 ملي في الغالب الـ Small هذا
+
+327
+00:23:23,890 --> 00:23:27,850
+Hemodynamically Insignificant يعني مش عامل مشاكل All
+
+328
+00:23:27,850 --> 00:23:33,090
+Close Spontaneously ممكن لحد School Years يكون
+
+329
+00:23:33,090 --> 00:23:38,030
+سكر لوحده ال Small هذا اللي أقل من تلاتة مليون
+
+330
+00:23:38,030 --> 00:23:40,750
+عشان هيك المتابعات بتصير في المستشفيات عشان يشوفوا
+
+331
+00:23:40,750 --> 00:23:45,370
+حجم الـ VSD بيصغر ولا بيكبر ولا بيظل زي ما هو 
+
+332
+00:23:45,370 --> 00:23:48,070
+بده تدخل ولا بدهوش تدخل في أن الـ Moderate VSD
+
+333
+00:23:48,070 --> 00:23:51,830
+من 3 إلى 5 ملي في الـ Diameter و هو الـ least
+
+334
+00:23:51,830 --> 00:23:55,850
+common group of children 3 إلى 5% without evidence
+
+335
+00:23:55,850 --> 00:23:58,890
+of congenital of congestive heart بدون احتقان في 
+
+336
+00:23:58,890 --> 00:24:01,190
+القلب أو برمونية hypertension may be followed
+
+337
+00:24:01,190 --> 00:24:04,650
+until spontaneous closure حتى هذا أه حتى هذا في أن
+
+338
+00:24:04,650 --> 00:24:07,150
+الـ Large هم اللي بيبقوا عاملين لنا مشكلة من 6 إلى
+
+339
+00:24:07,150 --> 00:24:11,530
+10 ملي في الـ Diameter usually محتاج تدخل جراحي
+
+340
+00:24:11,530 --> 00:24:15,250
+وإذا ما تدخلش الواحد جراحيا بيعمل لنا مشاكل
+
+341
+00:24:15,250 --> 00:24:18,690
+congestive heart failure وfailure to thrive FTT
+
+342
+00:24:18,690 --> 00:24:26,110
+يعني FTT failure to thrive يعني فشل في النمو CHF 
+
+343
+00:24:26,110 --> 00:24:32,510
+congestive heart failure يعني فشل قلبي احتقاني هيك
+
+344
+00:24:32,510 --> 00:24:38,160
+يعني الـ VSD عادة بنشوفه إحنا في ال .. في ال ..في
+
+345
+00:24:38,160 --> 00:24:42,100
+الـ ECO وزي ما قلنا ممكن يعمل لنا fatigue،
+
+346
+00:24:42,100 --> 00:24:45,980
+diaphoresis، feeding and poor growth عشان هيك يدخل
+
+347
+00:24:45,980 --> 00:24:49,920
+عيّن في FTT والعلاج تبعه بيبقى a significant
+
+348
+00:24:49,920 --> 00:24:53,200
+number، 30 ل 50% of small defects close
+
+349
+00:24:53,200 --> 00:24:58,140
+spontaneously before the age of أربع سنوات، 
+
+350
+00:24:58,140 --> 00:25:02,380
+ملاحظين أنتم؟ يعني جزء كبير بتحصل لوحدها some long
+
+351
+00:25:02,380 --> 00:25:07,040
+term studies of adult with unoperated small VSD
+
+352
+00:25:07,040 --> 00:25:11,870
+shock والتحقيق في الـ Arrhythmia والتعامل والتعامل
+
+353
+00:25:11,870 --> 00:25:15,450
+والتعامل والتعامل والتعامل والتعامل والتعامل
+
+354
+00:25:15,450 --> 00:25:21,770
+والتعامل والتعامل والتعامل والتعامل والتعامل
+
+355
+00:25:21,770 --> 00:25:21,870
+والتعامل والتعامل والتعامل والتعامل والتعامل
+
+356
+00:25:21,870 --> 00:25:23,190
+والتعامل والتعامل والتعامل والتعامل والتعامل
+
+357
+00:25:23,190 --> 00:25:31,110
+والتعامل والتعامل والتعامل
+
+358
+00:25:31,110 --> 00:25:36,150
+و
+
+359
+00:25:38,230 --> 00:25:41,270
+indicated للـ VSD associated with congestive heart
+
+360
+00:25:41,270 --> 00:25:44,090
+failure FTT or Pulmonary hypertension ارتفاع في 
+
+361
+00:25:44,090 --> 00:25:48,290
+الضغط الرئوي والناس اللي عندها poor growth الناس
+
+362
+00:25:48,290 --> 00:25:50,630
+اللي عندها poor exercise tolerance الناس اللي
+
+363
+00:25:50,630 --> 00:25:54,010
+بيصير عندها cardiomyopathy تحتاج أنه نعملها repair
+
+364
+00:25:54,010 --> 00:26:00,490
+surgical repair والمفروض أنه ال results of primary 
+
+365
+00:26:00,490 --> 00:26:04,090
+surgical repair excellent وما فيه مشاكل leading to
+
+366
+00:26:04,090 --> 00:26:11,370
+long term problem أو الـ Rare Problems دكتس
+
+367
+00:26:11,370 --> 00:26:15,650
+أرتريوزوس أو الـ PDA Patent Ductus Arteriosus أو دكتس 
+
+368
+00:26:15,650 --> 00:26:19,430
+بوتالي اسمه في بعض الأحيان هو بـ resistance of
+
+369
+00:26:19,430 --> 00:26:23,270
+normal fetal vessel joining the pulmonary artery
+
+370
+00:26:23,270 --> 00:26:26,970
+and the aorta أو to the aorta وكلها spontaneously
+
+371
+00:26:26,970 --> 00:26:31,570
+ال term الموجود 40 أسبوع على اليوم التالت لليوم
+
+372
+00:26:31,570 --> 00:26:36,500
+الخامس بيسكر حوالي 10% بيمثل من الـ Congenital heart
+
+373
+00:26:36,500 --> 00:26:40,080
+disease و Most common in female، ليش؟ ما حدش عارف،
+
+374
+00:26:40,080 --> 00:26:43,880
+في الصورة اللي بعدها شايفين إنه ال treatment بيصير
+
+375
+00:26:43,880 --> 00:26:51,020
+إنه لو ضال وعمل لي مشاكل، بنحتاج إله، خاصة لما يكون
+
+376
+00:26:51,020 --> 00:26:54,880
+البدأ large، بنحتاج a surgical intervention يعني
+
+377
+00:26:54,880 --> 00:26:58,020
+عملية رابط إله، عن طريق ال catheter أو trans
+
+378
+00:26:58,020 --> 00:27:02,720
+catheter closure of small defects، وتنتهي القضية
+
+379
+00:27:05,500 --> 00:27:11,630
+وهكذا ندخل على الـ cyanotic heart disease اللي هي
+
+380
+00:27:11,630 --> 00:27:18,990
+for 20% of cases على رأسها طبعا الـ most common في 
+
+381
+00:27:18,990 --> 00:27:22,570
+الـ cyanotic الـ most common في الـ cyanotic اللي
+
+382
+00:27:22,570 --> 00:27:27,990
+هي الـ Tetralogy بتمثل 10% and transposition of great
+
+383
+00:27:27,990 --> 00:27:31,530
+artery بتمثل 5% are the most common من الـ 
+
+384
+00:27:31,530 --> 00:27:34,550
+congenital heart disease لكن هي الـ cyanotic بتمثل
+
+385
+00:27:34,550 --> 00:27:37,610
+فقط 20% والـ cyanotic يا جماعة مش بس هدول التانين
+
+386
+00:27:37,610 --> 00:27:42,040
+في حاجات تانية لكن هدول ال most common الـ Cardiac
+
+387
+00:27:42,040 --> 00:27:48,180
+catheter ضروري ينعمل للتشخيص واحد اتنين لل
+
+388
+00:27:48,180 --> 00:27:50,720
+corrective surgery إذا بدى عمله corrective surgery
+
+389
+00:27:50,720 --> 00:27:55,520
+لابد من قسطرة قلبية الـ Prognosis is generally bad
+
+390
+00:27:55,520 --> 00:27:58,780
+unless a corrective surgery is performed early
+
+391
+00:27:58,780 --> 00:28:03,220
+يعني الأمور مش بالسهولة اللي بنتصورها الـ Main
+
+392
+00:28:03,220 --> 00:28:07,720
+presentation في الناس هدول اللي هي إيش؟ التي هي الـ
+
+393
+00:28:07,720 --> 00:28:12,300
+Central Cyanosis طبعا في الـ TGA الـ
+
+394
+00:28:12,300 --> 00:28:15,860
+Transposition of Great Arteries بنلاقيها later
+
+395
+00:28:15,860 --> 00:28:19,300
+بتيجي الـ Cyanosis لكن في الـ TOF تيجي early
+
+396
+00:28:19,300 --> 00:28:22,640
+Clinical evaluation is useful to put some
+
+397
+00:28:22,640 --> 00:28:25,840
+possibilities forward but it can never provide an
+
+398
+00:28:25,840 --> 00:28:30,010
+accurate diagnosis يعني ما ينفعش أقول بالسمعة هذا
+
+399
+00:28:30,010 --> 00:28:35,930
+Fallot وهذا TGA، لأ، إنما لازم أعمله Echo، وممكن 
+
+400
+00:28:35,930 --> 00:28:38,570
+زي ما قلنا نحتاج إله Echo-cardiography، نحتاج إله 
+
+401
+00:28:38,570 --> 00:28:41,450
+اللي هو نحتاج إله
+
+402
+00:29:00,980 --> 00:29:06,840
+التترالوجي والتترالوجي والتترالوجي والتترالوجي
+
+403
+00:29:06,840 --> 00:29:07,740
+والتترالوجي والتترالوجي والتترالوجي والتترالوجي
+
+404
+00:29:07,740 --> 00:29:08,120
+والتترالوجي والتترالوجي والتترالوجي والتترالوجي
+
+405
+00:29:08,320 --> 00:29:18,420
+Tetralogy of Fallot أربعية، Tetralogy of Fallot أربعية،
+
+406
+00:29:18,420 --> 00:29:22,540
+Tetralogy of Fallot أربعية، Tetralogy of Fallot أربعية، Tetralogy of
+
+407
+00:29:22,540 --> 00:29:26,240
+Fallot أربعية، Tetralogy of Fallot أربعية، Tetralogy of Fallot أربعية، 
+
+408
+00:29:26,240 --> 00:29:26,340
+Fallot أربعية، Tetralogy of Fallot أربعية، Tetralogy of
+
+409
+00:29:26,340 --> 00:29:34,460
+Fallot أربعية، Tetralogy of Fallot أربعية، Tetralogy of Fallot أربعية، ف
+
+410
+00:29:34,510 --> 00:29:37,790
+وفي عندنا كمان بيصير pulmonary stenosis وهو هذا 
+
+411
+00:29:37,790 --> 00:29:41,950
+اللي عن جد بعقد لنا الأمور كتير، ليش؟ لأن الـ right
+
+412
+00:29:41,950 --> 00:29:45,270
+ventricle على عكس المنتظر لاجينا فيه إيش؟
+
+413
+00:29:45,270 --> 00:29:47,470
+hypertrophy، فيه thickness، يبقى أربع حاجات
+
+414
+00:29:47,470 --> 00:29:51,070
+overriding أوورطة، pulmonary stenosis، ventricular
+
+415
+00:29:51,070 --> 00:29:54,460
+septal defect، right ventricular hypertrophy كل هذا
+
+416
+00:29:54,460 --> 00:29:58,000
+طب إحنا كنا نحكي عن حاجة واحدة VSD و ASD و Doctors
+
+417
+00:29:58,000 --> 00:30:03,340
+Arteriosis ومش خالصين، الآن أربع حاجات في قلب
+
+418
+00:30:03,340 --> 00:30:10,140
+الطفل وهذا اللي بيسوّق الـ Prognosis أنه أنا بحكيش
+
+419
+00:30:10,140 --> 00:30:14,700
+عن يعني بمعنى آخر، الجلب معجّج يعني، ما فيه إشي راكم
+
+420
+00:30:14,700 --> 00:30:18,320
+على بعضه زي الناس وهذه مشكلة «TOF» is a common 
+
+421
+00:30:18,320 --> 00:30:23,450
+حوالي 10% من كل  the cyanotic congenital heart disease
+
+422
+00:30:23,450 --> 00:30:25,890
+الـ Congenital Heart Disease لكن من الـ cyanotic
+
+423
+00:30:25,890 --> 00:30:30,650
+هو بمثل 50% خدوا بالكوا للنسبة و التنقص 10% من الـ
+
+424
+00:30:30,650 --> 00:30:34,330
+Congenital Heart Disease كلها لكن 50% من مين؟ من
+
+425
+00:30:34,330 --> 00:30:39,590
+الـ cyanotic group معظم المريضين هم مريضين عشان شهر
+
+426
+00:30:39,590 --> 00:30:44,830
+أو شهرين من عمرهم، وهي عادية تتزيد، وبتدخل في حاجة
+
+427
+00:30:44,830 --> 00:30:49,910
+اسمها Hypoxemic Spills أو Tet Spills، واحدة من
+
+428
+00:30:49,910 --> 00:30:55,030
+أعراض التترالوجية السيفيرة الـ typical features
+
+429
+00:30:55,030 --> 00:30:59,130
+بنلاقي بعض الأحيان عندهم cyanosis after the
+
+430
+00:30:59,130 --> 00:31:02,830
+neonatal period، hypoxemic spells ممكن يحصل أثناء
+
+431
+00:31:02,830 --> 00:31:06,310
+الـ infancy، إيش يعني hypoxemic spells؟ بتلاقي
+
+432
+00:31:06,310 --> 00:31:12,510
+بدخل في نوبات ذرجان ونقص أكسيجين، right-sided
+
+433
+00:31:12,510 --> 00:31:17,070
+aortic arch في 25% of all patients، مش إنه بروح
+
+434
+00:31:17,070 --> 00:31:19,130
+على الـ left زي ما احنا بنعرف، لأ بروح على ال
+
+435
+00:31:19,130 --> 00:31:22,430
+right اللي هو ال .. ال .. ال .. ال .. ال .. ال ..
+
+436
+00:31:22,430 --> 00:31:22,690
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+437
+00:31:22,690 --> 00:31:22,850
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+438
+00:31:22,850 --> 00:31:27,020
+.. ال .. ال .. ال ..الـ «جنة تو» اللي هي موضوع الـ
+
+439
+00:31:27,020 --> 00:31:32,880
+«Tet Spells» أن الطفل بيزرج فإيش بروح بيعمل الطفل
+
+440
+00:31:32,880 --> 00:31:37,280
+بيقعد الجادة هاية القرفصاء أو لو نايم على بطنه
+
+441
+00:31:37,280 --> 00:31:41,500
+بنعمله أهله بيعملوله الجادة هاية وكأنه هو نايم على
+
+442
+00:31:41,500 --> 00:31:48,040
+بطنه لكن بنضغط الرجلتين على البطن، ليش؟لأن الموضوع
+
+443
+00:31:48,040 --> 00:31:54,080
+هذا بزيد لـ Systemic Ventricular Resistance ‏in
+
+444
+00:31:54,080 --> 00:31:57,260
+the lower extremities فبتحسن الـCyanosis ‏بنقول
+
+445
+00:31:57,260 --> 00:32:02,200
+عنه، هي عبارة عن الـHypercyanotic Spills أو الـTet
+
+446
+00:32:02,200 --> 00:32:06,960
+Spills بدخل المريض في knee chest position عشان يحسن
+
+447
+00:32:06,960 --> 00:32:11,920
+وضعه بطريقة لا إرادية بعملها يا جماعة بطريقة لا
+
+448
+00:32:11,920 --> 00:32:16,320
+إرادية to relieve cyanotic spells بروح بقعد
+
+449
+00:32:16,320 --> 00:32:21,070
+الجعدهية الـ Tetralogy of Fallot بنعمل ع Stage
+
+450
+00:32:21,070 --> 00:32:25,290
+العملية الجراحية طبعته طبعا جراحة Staged or
+
+451
+00:32:25,290 --> 00:32:27,430
+Complete في ناس بتعملها مرة واحدة في ناس بتعملها ع
+
+452
+00:32:27,430 --> 00:32:33,070
+Staging Total repair is done by 6 months if 
+
+453
+00:32:33,070 --> 00:32:35,370
+cyanotic spills Surgery is not necessarily
+
+454
+00:32:35,370 --> 00:32:39,890
+curative يعني مش بيظل الجلب فيه مشكلة But most
+
+455
+00:32:39,890 --> 00:32:42,970
+have improved quality of life and improved
+
+456
+00:32:42,970 --> 00:32:44,370
+longevity
+
+457
+00:32:47,240 --> 00:32:50,540
+الـ Transposition of Great Arteries – TGA – is the
+
+458
+00:32:50,540 --> 00:32:52,800
+second most common cyanotic congenital heart
+
+459
+00:32:52,800 --> 00:32:58,660
+disease وهو بيمثل 5% من كل الـ cyanotic heart
+
+460
+00:32:58,660 --> 00:33:02,820
+disease اللي بيصير إن الأورطة arises from the
+
+461
+00:33:02,820 --> 00:33:05,500
+right ventricle والبلمونري أرتري بيطلع من الـ left
+
+462
+00:33:05,500 --> 00:33:07,980
+ventricle طبعاً في نوعيات إلو يا جماعة، بس أنا بدي
+
+463
+00:33:07,980 --> 00:33:12,040
+أديكوا hint بس يعني إنه في هنا حاجة اسمها TGA مش
+
+464
+00:33:12,040 --> 00:33:14,600
+شرط النوعيات تبعته، طبعا النوعيات تبعته لما يصير
+
+465
+00:33:14,600 --> 00:33:18,390
+له subclasses بصير أسوأ الـ Results بيصير عندى Two
+
+466
+00:33:18,390 --> 00:33:24,070
+Independent Circuits Which Cannot Maintain Life مش
+
+467
+00:33:24,070 --> 00:33:27,070
+هيظبط لما يصير عندى أنا دورة دموية بتروح ال lungs
+
+468
+00:33:27,070 --> 00:33:31,150
+لحالها ودورة دموية للجسم لوحده طيب وين العملية ال
+
+469
+00:33:31,150 --> 00:33:36,330
+Oxygenation و ال Deoxygenation؟ مش هيظبط كتير فهذه
+
+470
+00:33:36,330 --> 00:33:41,170
+مش ماشية مع الحياة عشان هيك سبحان الله سبحان الله
+
+471
+00:33:41,170 --> 00:33:48,250
+العظيم أن ربنا بترك أمور عشان الدورتين يخلطواهنا
+
+472
+00:33:48,250 --> 00:33:54,290
+بيجي دور من الفرام القفالي اللي كنا بنطالب يسكر و
+
+473
+00:33:54,290 --> 00:33:59,550
+PDA اللي كنا بنطالب يسكر و لو لاجينا اي منهم لو
+
+474
+00:33:59,550 --> 00:34:02,670
+واحد منهم بيكفي لكن طب لو لاجينا هم سكرين التانين
+
+475
+00:34:02,670 --> 00:34:07,390
+لازم نعمل اجراءات عشان بمجرد انه يولد الطفل لأنه
+
+476
+00:34:07,390 --> 00:34:10,990
+هيموت لازم نفتح فتح اما الفرام القفالي بنعمله
+
+477
+00:34:10,990 --> 00:34:17,400
+عملية تفجير فتحلو سكر و الـ PDA لو سكر بدي أدوية
+
+478
+00:34:17,400 --> 00:34:24,300
+عشان يظل maintained عشان يظل فاتح الـ Central
+
+479
+00:34:24,300 --> 00:34:28,220
+cyanosis and tachypnea appear at birth or during
+
+480
+00:34:28,220 --> 00:34:31,400
+the first few days cardiac examination usually
+
+481
+00:34:31,400 --> 00:34:35,720
+reveal no murmur chest x-ray reveal حاجة اسمها egg
+
+482
+00:34:35,720 --> 00:34:39,870
+shape الـ Heart شكله بيبقى على صورة الأشعة زي صورة
+
+483
+00:34:39,870 --> 00:34:44,530
+البيضة urgent diagnosis لابد انه يتشخص بصورة طارقة
+
+484
+00:34:44,530 --> 00:34:49,870
+essential urgent management include اللي بنعمله
+
+485
+00:34:49,870 --> 00:34:55,010
+عشان نخلط الدم تبع الدورة الدموية الرئوية والدورة
+
+486
+00:34:55,010 --> 00:34:58,670
+الدموية تحت الجسم فبندى حاجة اسمها Prostaglandin E
+
+487
+00:34:58,670 --> 00:35:08,350
+Infusion وهذا والدكتاز البيتانتي يجعله فاتح على
+
+488
+00:35:08,350 --> 00:35:13,050
+أساس يخلط دم وبمنع إغلاق الدكتاز وممكن نعمل حاجة
+
+489
+00:35:13,050 --> 00:35:17,040
+تانية الـBallon، واحد منهم يعني الـBallon Atrial
+
+490
+00:35:17,040 --> 00:35:20,880
+Septostomy تفجير سبتستومي، فتح الـSeptum بين
+
+491
+00:35:20,880 --> 00:35:23,980
+الـTwo Atrium اللي هو الفرامين أوفالي، لو كان مسكر
+
+492
+00:35:23,980 --> 00:35:28,340
+وهذا بيسمح للـMixture of Blood وطبعاً بعدين بنعمل
+
+493
+00:35:28,340 --> 00:35:31,920
+حاجة اسمها Switch Operation إيش يعني Switch؟ بجلب
+
+494
+00:35:31,920 --> 00:35:36,920
+الأورطة مرة تانية على مكانه والـPulmonary على
+
+495
+00:35:36,920 --> 00:35:42,160
+مكانه وهذا يفضل إنه ينعمل بسرعة يعني بسرعة، بسرعة،
+
+496
+00:35:42,160 --> 00:35:46,270
+بسرعة، هيك بنبقى تحدثنا عن الـ .. الـ .. الـ ..
+
+497
+00:35:46,270 --> 00:35:50,950
+طبعاً بيظل ال quotation of .. of the أوورتا و اللي
+
+498
+00:35:50,950 --> 00:35:54,470
+عملنا secondary hypertension رجاء إلكوا يمكن تجري
+
+499
+00:35:54,470 --> 00:35:59,550
+.. تجروها أنا مش .. مش هتحدث عنها هان و مش هطلبها
+
+500
+00:35:59,550 --> 00:36:02,790
+في assignment المرة هاي يعني أنا ما طلبتش
+
+501
+00:36:02,790 --> 00:36:06,670
+assignment المرة هاي لكن اجروها و اجروا النقطة
+
+502
+00:36:06,670 --> 00:36:09,930
+اللي قلتلكوا عنها قبل هيك واعتبروهم من المادة يعني
+
+503
+00:36:09,930 --> 00:36:14,410
+اجروهم واعتبروهم من المادة اللي هي quotation of the
+
+504
+00:36:14,410 --> 00:36:18,790
+aorta وش التانية كانت فاكرين اللي تبقوا معانا
+
+505
+00:36:18,790 --> 00:36:24,730
+فاكرين اكيد ماشي ان شاء الله وانهاركوا سعيد ان شاء
+
+506
+00:36:24,730 --> 00:36:26,730
+الله يعطيكوا العافية السلام عليكم
diff --git a/PL9fwy3NUQKwaf4aPlmtIQXiASpDxVLEnO/bBU74brkUFs_raw.srt b/PL9fwy3NUQKwaf4aPlmtIQXiASpDxVLEnO/bBU74brkUFs_raw.srt
new file mode 100644
index 0000000000000000000000000000000000000000..0867cd501b5c7580dc285ee216170e7e74b1ebbd
--- /dev/null
+++ b/PL9fwy3NUQKwaf4aPlmtIQXiASpDxVLEnO/bBU74brkUFs_raw.srt
@@ -0,0 +1,1536 @@
+1
+00:00:02,160 --> 00:00:04,300
+بسم الله الرحمن الرحيم الحمد لله رب العالمين
+
+2
+00:00:04,300 --> 00:00:06,700
+والصلاة والسلام علي سيدنا محمد وسيد المرسلين
+
+3
+00:00:06,700 --> 00:00:10,980
+الأجمعين يعني اليوم إن شاء الله تعالى بنكمل لكن
+
+4
+00:00:10,980 --> 00:00:16,720
+بنكمل محاضرات البطن والقلب والصدرية لكن هنبدأ
+
+5
+00:00:16,720 --> 00:00:21,520
+بأولى خليني أقول chapter محاضرات chapter اللي هو
+
+6
+00:00:21,520 --> 00:00:26,760
+respiratory systemهنحاول زي ما عملنا في القلب حتى
+
+7
+00:00:26,760 --> 00:00:29,820
+في الـ Anatomy و الـ Physiology نحاول ناخد من كل
+
+8
+00:00:29,820 --> 00:00:33,900
+بستان زهرة يعني ما هنقول والله منهج متكامل لأنه مش
+
+9
+00:00:33,900 --> 00:00:38,080
+حقيقي في الموضوع لكن هنحكي الأمور المهمة التي قد
+
+10
+00:00:38,080 --> 00:00:42,280
+في لحظة من اللحظات اتساعدنا في فهم طبيعة الأمراض
+
+11
+00:00:42,280 --> 00:00:47,700
+للـ Respiratory System والـ Respiratory System زي
+
+12
+00:00:47,700 --> 00:00:53,600
+ما احنا عارفين يعني واضح أنه أهم حاجة عندهالعملية
+
+13
+00:00:53,600 --> 00:01:01,100
+الرئيسية هي الهواء أو الهواء وطبعا معروف أن عملية
+
+14
+00:01:01,100 --> 00:01:06,680
+الهواء أو الهواء بتتم على أساس أنه تمنح أكسجين
+
+15
+00:01:06,680 --> 00:01:12,940
+لبطانيا وأنه يزيل كربون ديوكسيد CO2 من بطانيا عن
+
+16
+00:01:12,940 --> 00:01:18,800
+طريق طبعا اللغةExternal Respiration Exchange of
+
+17
+00:01:18,800 --> 00:01:22,240
+Oxygen and Carbon Dioxide هذا الـ External
+
+18
+00:01:22,240 --> 00:01:25,860
+Respiration إذا في External Respiration معناته في
+
+19
+00:01:25,860 --> 00:01:31,260
+Internal Respiration وهذه فرصة لوحدكوا لإلكوا
+
+20
+00:01:31,260 --> 00:01:35,120
+تتطلعوا ما هو ال Internal Respiration بما أن ال
+
+21
+00:01:35,120 --> 00:01:38,740
+External Respiration هو عبارة عن Exchange of
+
+22
+00:01:38,740 --> 00:01:41,680
+Oxygen and Carbon Dioxide ما هو ال Internal
+
+23
+00:01:41,680 --> 00:01:47,640
+Respirationأقضاء اللي طبعا اللي حكينا بالنسبة
+
+24
+00:01:47,640 --> 00:01:51,860
+للـinternal respiration مش assignment لكل واحد
+
+25
+00:01:51,860 --> 00:01:57,460
+الحرية يكشف كما يريدالـ organs of the respiratory
+
+26
+00:01:57,460 --> 00:02:01,240
+system زي ما احنا عارفين بتبدأ بالـ nose، بالـ
+
+27
+00:02:01,240 --> 00:02:03,240
+pharynx، الـ pharynx اللي هو البلعوم، الـ larynx
+
+28
+00:02:03,240 --> 00:02:06,520
+الحنجرة، تراكية القصبة الهوائية، bronchial tree
+
+29
+00:02:06,520 --> 00:02:13,140
+شجرة القصبة الهوائية، و الرئتين ال lungs ال organs
+
+30
+00:02:13,140 --> 00:02:17,580
+هي بس باختصار جدا إيه وظيفة كل عضو
+
+31
+00:02:20,360 --> 00:02:26,940
+النازل كافيتى في الأول نبدأ من الناس من الناس نازل
+
+32
+00:02:26,940 --> 00:02:31,560
+سبتم دفع النازل كافيتى إلى الجزء اليسار والجزء
+
+33
+00:02:31,560 --> 00:02:39,620
+اليسار نازل كونشي يستمع النازل نازل كافيتىالعملية
+
+34
+00:02:39,620 --> 00:02:43,920
+الـ moistening والـ warmness of the air بصورة أحسن
+
+35
+00:02:43,920 --> 00:02:49,100
+ميوكاس منبرين warms and moistens the air and سيليا
+
+36
+00:02:49,100 --> 00:02:51,740
+helps eliminate particles في أن الـ Paranasal
+
+37
+00:02:51,740 --> 00:02:56,340
+sinus واللي احنا بنسمع دايما عنها افلان في عنده
+
+38
+00:02:56,340 --> 00:03:01,140
+جيوب هي ال Paranasal sinuses هي الجيوب اللي احنا
+
+39
+00:03:01,140 --> 00:03:08,600
+بنقصدها دايما وهذه عبارة عنأماكن أو spaces within
+
+40
+00:03:08,600 --> 00:03:14,620
+the skull bone في جلب الـ bone«أيرفيلد» فيها هوا
+
+41
+00:03:14,620 --> 00:03:22,300
+وهي طبعاً الـ «أسكال» لو نتصورها عضم صبة جامد كان
+
+42
+00:03:22,300 --> 00:03:25,600
+على جهها أدقل من هي كتير وهذا تقل على أيش؟ على
+
+43
+00:03:25,600 --> 00:03:31,920
+الجسم وحتى لما ينخبط الواحد في رأسه أو يجعب لو
+
+44
+00:03:31,920 --> 00:03:37,000
+كانت عضم بحت ممكن تأثر تأثيرات كبيرة لكن لما تكون
+
+45
+00:03:37,000 --> 00:03:38,180
+«أيرفيلد»
+
+46
+00:03:40,370 --> 00:03:47,510
+بتخلّي حجم أو وزن الـ skull لجل وبتدّي كمان مش بس
+
+47
+00:03:47,510 --> 00:03:52,010
+بما أن في عندنا غرف هوائية بتخلّي الـ voice له حدة
+
+48
+00:03:52,010 --> 00:03:59,270
+معينة أيضًا طول معين الـ skull bones withالصينيات
+
+49
+00:03:59,270 --> 00:04:01,570
+تحتوي على الصينيات الصحيحة الصحيحة الصحيحة الصحيحة
+
+50
+00:04:01,570 --> 00:04:02,070
+الصحيحة الصحيحة الصحيحة الصحيحة الصحيحة الصحيحة
+
+51
+00:04:02,070 --> 00:04:04,910
+الصحيحة الصحيحة الصحيحة الصحيحة الصحيحة الصحيحة
+
+52
+00:04:04,910 --> 00:04:07,910
+الصحيحة الصحيحة الصحيحة الصحيحة الصحيحة الصحيحة
+
+53
+00:04:07,910 --> 00:04:09,030
+الصحيحة الصحيحة الصحيحة الصحيحة الصحيحة الصحيحة
+
+54
+00:04:09,030 --> 00:04:13,750
+الصحيحة الصحيحة الصحيحة الصحيحة الصحيحة الصحيحة
+
+55
+00:04:13,750 --> 00:04:18,610
+الصحيحة الصحيحة الصحيحة الصحيحة الصحيحة الصحيحة
+
+56
+00:04:18,610 --> 00:04:24,510
+الصحيحة الصحيحة
+
+57
+00:04:24,510 --> 00:04:31,970
+الصعنا طبعا مدخل الأنف بالـInferior Turbinate
+
+58
+00:04:31,970 --> 00:04:38,150
+والـMiddle وSuperior Turbinate والأن بنشوفه في
+
+59
+00:04:38,150 --> 00:04:41,330
+صورة تانية الـSinus Cavities أحنا قولنا كام كام
+
+60
+00:04:41,330 --> 00:04:47,330
+Sinus فيه عندنا Frontal Sinus أثمويدل Sinus
+
+61
+00:04:47,330 --> 00:04:51,510
+frontal فوق العين على طول فوق الحاجب أثمويدل بين
+
+62
+00:04:51,510 --> 00:04:56,890
+الإنتين منطقة بين الإنتينفي عندنا الـ maxillary
+
+63
+00:04:56,890 --> 00:05:01,590
+sinus وفي الـ maxillary sinus طبعا جنب الأنف وفي
+
+64
+00:05:01,590 --> 00:05:05,110
+الـ sphenoidal اللي هو من الصورة من الأمام بنشوفوش
+
+65
+00:05:05,110 --> 00:05:09,010
+لكن ممكن من ال to the sides sphenoidal sinus نشوف
+
+66
+00:05:09,010 --> 00:05:16,950
+هذه صورة X-rayللـ sinuses دي اللي ماببعت انا اي حد
+
+67
+00:05:16,950 --> 00:05:20,530
+يعمل صورة sinus بيعمله الصورة هاي زي ما احنا
+
+68
+00:05:20,530 --> 00:05:24,210
+شايفين رقم واحد ال nasal septum رقم اتنين frontal
+
+69
+00:05:24,210 --> 00:05:29,830
+sinus فوق رقم تلاتة ال maxillary sinus ethmoidal
+
+70
+00:05:29,830 --> 00:05:35,170
+sinus رقم اربعة اللي هو بين ال ال ال two eyes
+
+71
+00:05:36,590 --> 00:05:40,970
+الـ Nose وفي عندنا خمسة اللي هو الـ Inferior
+
+72
+00:05:40,970 --> 00:05:45,750
+Turbinate اللي بيبين أنها لفوق منه في اللي اللي هو
+
+73
+00:05:45,750 --> 00:05:51,790
+ال middle وستة في عندنا ال Odontoid Process اللي
+
+74
+00:05:51,790 --> 00:05:57,110
+هو منطقة الأسدان هنا صورة جديدة الصورة هذه بتقولي
+
+75
+00:05:57,110 --> 00:06:01,110
+على الجهة الخضراء على الجهة اليمين Healthy Sinuses
+
+76
+00:06:01,110 --> 00:06:05,510
+شايفين أن الـ Sinuses Free عاملياً برضه كمان مرة
+
+77
+00:06:05,510 --> 00:06:11,270
+frontalSphenoidal, ethmoidal and maxillary sinus
+
+78
+00:06:11,270 --> 00:06:16,450
+لكن على الجهة التانية Inflamed sinus lining وسبب
+
+79
+00:06:16,450 --> 00:06:19,070
+ال inflammation في الغالب في الغالب بيبقى
+
+80
+00:06:19,070 --> 00:06:22,350
+infection لكن برضه بعض الأحيان ممكن يكون allergy
+
+81
+00:06:22,350 --> 00:06:29,990
+أو fluid filled أو ميوكس يبقى موجود وهذه بتعملنا
+
+82
+00:06:29,990 --> 00:06:34,110
+التجول والتهابات في الجيوب الأنفية خاصة بعد
+
+83
+00:06:34,110 --> 00:06:38,430
+الإصابة بال influenzaمن الـ organs of the
+
+84
+00:06:38,430 --> 00:06:41,390
+respiratory system اللي هو الـ pharynx البلعوم
+
+85
+00:06:41,390 --> 00:06:47,290
+وطبعا البلعوم هو منطقة مشتركة بين ال digestive
+
+86
+00:06:47,290 --> 00:06:54,480
+system and respiratory systemLarynx Voice Box هو
+
+87
+00:06:54,480 --> 00:06:59,240
+صندوق الصوت زي ما بيقولوها حبة الصوت زي ما بيقول
+
+88
+00:06:59,240 --> 00:07:04,520
+في الحاجات التانية Larynx Voice Box moves air in
+
+89
+00:07:04,520 --> 00:07:11,160
+and out of the trachea هو المدخل و المخرج نتيجة
+
+90
+00:07:11,160 --> 00:07:16,500
+دخول و فروج الهواءبيصير عينها vice وبتكون ال
+
+91
+00:07:16,500 --> 00:07:21,760
+larynx من تلت أمور من تلت cartilage اللي هي ال
+
+92
+00:07:21,760 --> 00:07:26,660
+cartilage الغضروف من تلت غضاريف thyroid cartilage
+
+93
+00:07:26,660 --> 00:07:33,120
+ال epiglottic cartilage و ال cricoid cartilage زي
+
+94
+00:07:33,120 --> 00:07:35,820
+ما احنا شايفين في الصورة
+
+95
+00:07:38,020 --> 00:07:43,180
+من مساوية هذا التعليم إن الواحد ما بيقدرش يأشر على
+
+96
+00:07:43,180 --> 00:07:48,280
+الحاجات لكن أجرهم و أتبعوا بالظبط ال vocal cord
+
+97
+00:07:48,280 --> 00:07:52,160
+ننزل على الأحبال الصوتية هي عبارة عن cords من
+
+98
+00:07:52,160 --> 00:07:57,380
+اسمها أحبال stretched between ممدودة بين thyroid
+
+99
+00:07:57,380 --> 00:08:01,960
+cartilage and the cracoid cartilage بين الثيرويد
+
+100
+00:08:01,960 --> 00:08:06,940
+والكريكويدالـ Upper vocal cords are false cords
+
+101
+00:08:06,940 --> 00:08:10,520
+and lower are true vocal cords اللي هي الأصلية
+
+102
+00:08:10,520 --> 00:08:11,880
+اللي احنا شايفينها في الصورة
+
+103
+00:08:15,200 --> 00:08:18,820
+الـ Glottis is the opening between the vocal cords
+
+104
+00:08:18,820 --> 00:08:22,140
+لما يفتح ال vocal cords عن بعض، بيبين قدامنا أش
+
+105
+00:08:22,140 --> 00:08:26,160
+للـ Glottis The major vocal cords are stretched
+
+106
+00:08:26,160 --> 00:08:32,920
+the more عفواً كل ما شدت ال cords كل ما كان الصوت
+
+107
+00:08:32,920 --> 00:08:40,990
+high pitchedأو higher the voice pitch بيصير حاد
+
+108
+00:08:40,990 --> 00:08:47,390
+بالصوت هذا بيصير بعدين
+
+109
+00:08:47,390 --> 00:08:50,690
+بيجينا لتركيا و لبرونكاي و لبرونكيولز زي ما احنا
+
+110
+00:08:50,690 --> 00:08:53,470
+شايفين الترتيب تركيا هو القصارى الهوائية
+
+111
+00:08:53,470 --> 00:08:57,670
+البرونكيولز هي عبارة عن القصايبات و البرونكيولز
+
+112
+00:08:57,670 --> 00:09:01,350
+البرونكاي هي عبارة عن القصايبات و البرونكيولز
+
+113
+00:09:01,350 --> 00:09:06,100
+عبارة عن الشعيبات الهوائيةوالتركية زي ما احنا
+
+114
+00:09:06,100 --> 00:09:13,920
+بنعرفه إله اسم باللغة الإنجليزية اسمه windpipe وهو
+
+115
+00:09:13,920 --> 00:09:18,320
+عبارة عن tubular organ made of rings عبارة عن
+
+116
+00:09:18,320 --> 00:09:24,380
+حلقاتمن الـ cartilage و smooth muscles عبارة عن
+
+117
+00:09:24,380 --> 00:09:29,040
+عضلات و cartilage و جواته في حاجة جوة التركية
+
+118
+00:09:29,040 --> 00:09:38,960
+السيلية اللي هي الهديبات التنفسية و التركية
+
+119
+00:09:38,960 --> 00:09:43,700
+بيمتد من الـ larynx حتى الـ bronchi السيلية أو
+
+120
+00:09:43,700 --> 00:09:49,780
+الهديبات هاي بتحركنا البلغموتطلعوا من عن طريق
+
+121
+00:09:49,780 --> 00:09:55,360
+الحلق وطبعا هذا اللي بعد هيك احنا اما بيصير يعني
+
+122
+00:09:55,360 --> 00:10:00,680
+بيكحوا الواحد أو بتفو أو كذا ما هان أول مشاكل
+
+123
+00:10:00,680 --> 00:10:06,650
+الأولى مشاكلالـ Smoking اللي هو Destroy السيليا
+
+124
+00:10:06,650 --> 00:10:11,730
+عشان هيك بنلاقي إن المدخنين الصبح دايما بيطلعوا
+
+125
+00:10:11,730 --> 00:10:15,870
+كميات مهولة من البلغم اللي اتجمعت نتيجة التدخين
+
+126
+00:10:15,870 --> 00:10:19,830
+ونتيجة ال environment ونتيجة ولم تخرج في ساعتها
+
+127
+00:10:19,830 --> 00:10:23,610
+يعني ما قامتش السيلياس بإخراجها تتجمع في السدر
+
+128
+00:10:23,610 --> 00:10:28,830
+وتاني يوم الصبح بيحاول يطلعها اللي هو المدخن
+
+129
+00:10:30,550 --> 00:10:33,090
+الـ lungs طبعا زي ما احنا كلنا عارفين هي عبارة عن
+
+130
+00:10:33,090 --> 00:10:36,670
+cone-shaped organs right lung has three lobes
+
+131
+00:10:36,670 --> 00:10:42,530
+loops and left lung has two loops ال left عنده
+
+132
+00:10:42,530 --> 00:10:45,470
+تنتيل و ال right عنده تلاتة the membrane
+
+133
+00:10:45,470 --> 00:10:53,170
+surrounding the lungs is called pleura ال
+
+134
+00:10:53,170 --> 00:10:56,070
+lung contains connective tissue, bronchial tree,
+
+135
+00:10:56,250 --> 00:10:59,430
+nerves, lymphatic vessels and blood vessels
+
+136
+00:10:59,430 --> 00:11:03,680
+باختصارطيب هو إيش اللي بيصير بالظبط الـ mechanism
+
+137
+00:11:03,680 --> 00:11:07,080
+of breathing؟ بيصير عندنا inspiration and
+
+138
+00:11:07,080 --> 00:11:14,360
+expiration inspiration هو عبارة عن عملية الشهيق
+
+139
+00:11:14,360 --> 00:11:22,740
+and auto rich air هو محمل بالأكسجين يدخل الناس من
+
+140
+00:11:22,740 --> 00:11:26,700
+الجو الديافرام إيش بيصير في الديافرام اللي هو
+
+141
+00:11:26,700 --> 00:11:36,350
+الحجاب الحاجز؟ contractsor flattens يعني هو يتسطح
+
+142
+00:11:36,350 --> 00:11:42,410
+هو contracts ويتسطح the intercostal muscle raise
+
+143
+00:11:42,410 --> 00:11:50,150
+the ribs اللي هو القفص الصدري بالعضلات يكبر ويرتفع
+
+144
+00:11:50,980 --> 00:11:54,060
+الـ Breathing أو الـ Oral Pulmonary Ventilation
+
+145
+00:11:54,060 --> 00:11:57,220
+Consists of Inspiration, Inhalation and Expiration
+
+146
+00:11:57,220 --> 00:12:01,140
+اللي هو الـ Exhalation و اللي بتحكم فيه زي ما قلنا
+
+147
+00:12:01,140 --> 00:12:07,580
+طبعا الـ Expiration هو عبارة عن خروج CO2
+
+148
+00:12:07,580 --> 00:12:12,680
+Rich Air من الـ Lung الديافرام بالعكس طبعا زي ال
+
+149
+00:12:12,680 --> 00:12:15,640
+Inspiration بيصيروا عملية Relaxation والـ
+
+150
+00:12:15,640 --> 00:12:20,030
+Intercostal طبعا Relaxation معناته هيرتفعوالـ
+
+151
+00:12:20,030 --> 00:12:27,890
+Intercostal muscles lower the ribs اللي بتحكم في
+
+152
+00:12:27,890 --> 00:12:31,690
+عملية الـ Ventilation الـ Breathing هو الميدالة
+
+153
+00:12:31,690 --> 00:12:35,290
+Oblongate هنحكي عنها بزيادة شوي بعد شوي Control
+
+154
+00:12:35,290 --> 00:12:39,730
+the rhythm and depth of the breathing الـ Pons
+
+155
+00:12:39,730 --> 00:12:46,500
+اللي هو تحته للميدالة Oblongateالجزء من الدماغ
+
+156
+00:12:46,500 --> 00:12:52,060
+يتحكم عفوا فوقه للمدلة أبلونجيت يتحكم على رتب
+
+157
+00:12:52,060 --> 00:12:57,400
+الهواء يبقى الـ Depth وRhythm يتحكم عليها المدلة
+
+158
+00:12:57,400 --> 00:13:01,380
+أبلونجيت والـ Bones يتحكم عليها رتب الهواء، سنشوف
+
+159
+00:13:01,380 --> 00:13:01,720
+كيف
+
+160
+00:13:10,630 --> 00:13:16,570
+واضح إنه لابد إنه يرتبط الأكسجين بالدم ويدّينا
+
+161
+00:13:16,570 --> 00:13:22,510
+أكسو-هوميقلوبين في حين إن الـCarbon Dioxide بيدخل
+
+162
+00:13:22,510 --> 00:13:28,250
+Bloodstream كـBicarbonate وبعد
+
+163
+00:13:28,250 --> 00:13:29,570
+ما يدخل بالصورة هذه
+
+164
+00:13:32,380 --> 00:13:38,140
+بيتحلل مرة تانية وجزء منه اللي هو سانيوكسيد
+
+165
+00:13:38,140 --> 00:13:42,260
+الكربون بيرتبط على الـ hemoglobin وجزء منه بيتحلل
+
+166
+00:13:42,260 --> 00:13:45,380
+إلى carbon dioxide and water بيصير له عملية
+
+167
+00:13:45,380 --> 00:13:51,330
+expiration أو exhalation والwater كبخار مالفينا
+
+168
+00:13:51,330 --> 00:13:55,230
+حاجة اسمها كيف أنا أجيس اللي هو الـ Pulmonary
+
+169
+00:13:55,230 --> 00:13:58,490
+Function Test يعني أحنا فحوصات الـ Lung مش كتيرة
+
+170
+00:13:58,490 --> 00:14:03,810
+حقيقة، اللي هي بعض الفحوصات، لكن أهمها ومشهور عنها
+
+171
+00:14:03,810 --> 00:14:10,790
+اللي هو Pulmonary Function Test، Lung Function
+
+172
+00:14:10,790 --> 00:14:15,990
+Test اللي هو بنقول عنه باسم آخر Spirometer، جهاز
+
+173
+00:14:15,990 --> 00:14:20,680
+Spirometer زي هيكهدف الجهاز هذا بيحاول إن الـ
+
+174
+00:14:20,680 --> 00:14:25,000
+Instrument consisting of a hollow inverted over
+
+175
+00:14:25,000 --> 00:14:29,860
+water used to evaluate respiratory function للـ
+
+176
+00:14:29,860 --> 00:14:34,120
+functions، Spirometer إيش بيعمل؟ بيفرج لبين، هل
+
+177
+00:14:34,120 --> 00:14:36,440
+اللي أنا بحكي عنه، المرض اللي بحكي عنه
+
+178
+00:14:36,440 --> 00:14:43,290
+obstructive، و هنعرف الفرق، pulmonary diseaseكل ما
+
+179
+00:14:43,290 --> 00:14:46,030
+كان فيه عنوان Obstruction كل ما كان الـ Resistance
+
+180
+00:14:46,030 --> 00:14:52,770
+أعلى تبع الـ Airway و لا Restrictive الـ
+
+181
+00:14:52,770 --> 00:14:56,830
+Restrictive بيعمل Restriction للـ Lung Volume الـ
+
+182
+00:14:56,830 --> 00:15:01,290
+Capacity of the lung بتقل نتيجة مشاكل في الـ
+
+183
+00:15:01,290 --> 00:15:06,040
+Structure أو الـ Functional Lung Changesهذا شغلت
+
+184
+00:15:06,040 --> 00:15:08,360
+الـ Spirometer اللي يبقى Spirometer أو الـ
+
+185
+00:15:08,360 --> 00:15:12,460
+Pulmonary lung function أو test function test lung
+
+186
+00:15:12,460 --> 00:15:18,900
+test بيفرقلي بين الـ Obstructive Pulmonary Disease
+
+187
+00:15:18,900 --> 00:15:23,160
+والـ Restrictive Pulmonary Disease وقلنا الفرق بين
+
+188
+00:15:23,160 --> 00:15:25,620
+الأتنين الـ Obstructive Pulmonary Disease بيبقى
+
+189
+00:15:25,620 --> 00:15:28,580
+عندنا increased airway resistance يعني الهواء
+
+190
+00:15:28,580 --> 00:15:32,290
+نتيجة الـ Obstructionما بتخلش بالصورة اللي بدنا
+
+191
+00:15:32,290 --> 00:15:35,270
+إيها restrictive معناته فيها reduction في الـ
+
+192
+00:15:35,270 --> 00:15:40,710
+total lung capacity نتيجة مشكلة في الـ structure
+
+193
+00:15:40,710 --> 00:15:47,390
+أو الـ function of the lung ال
+
+194
+00:15:47,390 --> 00:15:51,810
+respiratory volumes اللي أنا بشوفها مسجلة عندى أو
+
+195
+00:15:51,810 --> 00:15:58,400
+عن طريق رسمة في الصورة هايبتديني بعض الـ hints
+
+196
+00:15:58,400 --> 00:16:03,500
+يمكن نتعرف عليها من باب اللي بروح lung
+
+197
+00:16:03,500 --> 00:16:07,300
+rehabilitation على سبيل المثال أو حاجة أو
+
+198
+00:16:07,300 --> 00:16:10,300
+respiratory rehabilitation مستقبلاً لابد أن يكون
+
+199
+00:16:10,300 --> 00:16:11,400
+سامع بالموضوع
+
+200
+00:16:14,330 --> 00:16:17,610
+ان يكون الواحد عارف عن الموضوع Respirator volumes
+
+201
+00:16:17,610 --> 00:16:23,810
+اللي بيجيس الجهاز حاجة اسمها tidal volume وإيش هذا
+
+202
+00:16:23,810 --> 00:16:28,390
+tidal volume؟ هذا عبارة عن amount of air كمية
+
+203
+00:16:28,390 --> 00:16:34,410
+الهواء that moves in or out the lung during a
+
+204
+00:16:34,410 --> 00:16:38,870
+normal breath حوالي 500 ملي يعني أنا و أنا بتنفس
+
+205
+00:16:38,870 --> 00:16:42,310
+عادي، نفس عادي، بطلع و بنزل هذا بقول عنه tidal
+
+206
+00:16:42,310 --> 00:16:47,020
+volume حوالي 500 مليreflects the amount of air
+
+207
+00:16:47,020 --> 00:16:51,300
+that moves in or out of the lungs during a normal
+
+208
+00:16:51,300 --> 00:16:56,200
+breath في عندنا بعد ذلك حاجة تانية اللي هي الـ
+
+209
+00:16:56,200 --> 00:17:00,400
+inspiratory reserve volume والـ expiratory reserve
+
+210
+00:17:00,400 --> 00:17:04,760
+volume من اسمها reserve، reserve معناته طبقةهو
+
+211
+00:17:04,760 --> 00:17:10,020
+عبارة عن كمية الهواء اللي can be forcefully
+
+212
+00:17:10,020 --> 00:17:13,500
+inhaled بعد normal inhalation يعني صار واحد عنده
+
+213
+00:17:13,500 --> 00:17:22,540
+normal inhalation عادي جداً جداً جداً لكن وأطلع
+
+214
+00:17:22,540 --> 00:17:27,540
+كمية الهواء جد ما يستطيع أكتر حاجة مقدر يطلعها هذه
+
+215
+00:17:27,540 --> 00:17:32,400
+بقول أنا inspiratory reserve volumeهي حوالي 2 لتر
+
+216
+00:17:32,400 --> 00:17:38,360
+لـ 3 لتر، 2100 لـ 3200 ملي، كمية كبيرة طب الـ
+
+217
+00:17:38,360 --> 00:17:42,340
+expiratory reserve volume هي عبارة عن كمية الـ air
+
+218
+00:17:42,340 --> 00:17:46,180
+that can be forcefully exhaled following a normal
+
+219
+00:17:46,180 --> 00:17:51,340
+exhalation يعني أنا عملت normal exhalation، تنفس،
+
+220
+00:17:51,340 --> 00:17:59,110
+زفير طبيعي وروحت طلعت كام هوة كمان بالقوةفوق اللي
+
+221
+00:17:59,110 --> 00:18:05,470
+أخرجته طبيعي بأطلع حوالي ألف لألف و متين مليون دول
+
+222
+00:18:05,470 --> 00:18:08,170
+بتلاقيهم في ال curve اللي موجودة تحت بالمناسبة
+
+223
+00:18:08,170 --> 00:18:13,790
+بتقدر تطلع على long function test على اليوتيوب في
+
+224
+00:18:13,790 --> 00:18:17,830
+ناس بتشرحها بالورجة أو الجلم بصورة رائعة وأنصح
+
+225
+00:18:17,830 --> 00:18:21,510
+بالموضوعالـ Reserve Volume هي عبارة عن كمية الـ
+
+226
+00:18:21,510 --> 00:18:24,690
+Volume of Air That Always Remains اللي بيضل دايماً
+
+227
+00:18:24,690 --> 00:18:30,050
+في الـ Lung حتى بعد ما أطلق كمية الهواء الكاملة
+
+228
+00:18:30,050 --> 00:18:34,810
+بيضل كمية صغيرة 1200 يعني مافيش Lung بيصيرلها Zero
+
+229
+00:18:34,810 --> 00:18:38,870
+يعني مافيش Lung بتطلع كل الهواء اللي فيها وخلاص
+
+230
+00:18:38,870 --> 00:18:41,870
+وتصير Zero فالكمية اللي بتضل في ال Lung بعد
+
+231
+00:18:41,870 --> 00:18:48,980
+Forcefully Exhalationيعني إذا فير بآخر ما أستطيع
+
+232
+00:18:48,980 --> 00:18:53,060
+اللي بيضال في ال lung بقول عنه residual volume فيه
+
+233
+00:18:53,060 --> 00:18:57,220
+إن ال vital capacity وهو عبارة عن حسبة هي كمية
+
+234
+00:18:57,220 --> 00:19:01,720
+الدواء الهواء اللي can be forcefully exhaled after
+
+235
+00:19:01,720 --> 00:19:08,360
+the deepest inhalation يعني أنا بعد ما أخد نفس
+
+236
+00:19:08,360 --> 00:19:14,320
+deep على الآخر كمية الهواء اللي بطلع بطلعها اللي
+
+237
+00:19:14,320 --> 00:19:19,960
+هي عبارة عن total volumeعبارة عن Inspiratory
+
+238
+00:19:19,960 --> 00:19:23,320
+Reserve Volume وExpiratory Reserve Volume هي عبارة
+
+239
+00:19:23,320 --> 00:19:26,480
+عن الـ Vital Capacity وفيه عنا الـ Total Lung
+
+240
+00:19:26,480 --> 00:19:33,100
+Capacity وهو عبارة عن حسبة
+
+241
+00:19:33,100 --> 00:19:38,700
+الـ Total Amount Of Air Of The Lung Can Be Hold هو
+
+242
+00:19:38,700 --> 00:19:42,740
+أكتر كمية من الهواء ممكن تدخل الـ Lungs بأي صورة
+
+243
+00:19:42,740 --> 00:19:46,740
+من الصورة وهي عبارة عن 6 لترالـ Pulmonary Function
+
+244
+00:19:46,740 --> 00:19:51,640
+Test في له حسبات تانية حقيقية غير هدول اللي إحنا
+
+245
+00:19:51,640 --> 00:19:56,060
+بنعتبرهم وفي بعض الأمراض بيعتبروهم تشخصيين يعني هم
+
+246
+00:19:56,060 --> 00:20:00,680
+بشخصهم عملياً زي الـ Total Ventilation وهي حسبة
+
+247
+00:20:00,680 --> 00:20:04,940
+عبارة عن Total Amount of Gas Flow in to or out of
+
+248
+00:20:04,940 --> 00:20:08,260
+the respiratory tract in one minute يعني كمية
+
+249
+00:20:08,260 --> 00:20:14,200
+الهواء بتروح وبتيجي في الrespiratory tract في one
+
+250
+00:20:14,200 --> 00:20:19,670
+minuteأو في إن الـForced Vital Capacity عبارة عن
+
+251
+00:20:19,670 --> 00:20:24,550
+جاز Forcibly expelled after taking a deep
+
+252
+00:20:24,550 --> 00:20:30,310
+breathing وهذا اللي قلنا عنه قبل شوية هو الـVital
+
+253
+00:20:30,310 --> 00:20:33,930
+Capacity أو في إن الـForced Expiratory Volume هو
+
+254
+00:20:33,930 --> 00:20:37,430
+عبارة عن كمية الجاز expelled during specific time
+
+255
+00:20:37,430 --> 00:20:45,270
+intervals of the Forced Vital Capacityهلال الرسم
+
+256
+00:20:45,270 --> 00:20:50,810
+عنا موجودة؟ بنلاحظ إنه الـTidal Volume، بعدين غير
+
+257
+00:20:50,810 --> 00:20:53,650
+الـTidal Volume، في عنا الـInspiratory Reserve
+
+258
+00:20:53,650 --> 00:20:56,590
+Volume، عنا الـExpiratory Reserve Volume، وفي عنا
+
+259
+00:20:56,590 --> 00:20:58,690
+الـResidual، لأن لما نجمع الـInspiratory Reserve
+
+260
+00:20:58,690 --> 00:21:01,230
+Volume والـTidal Volume والـExpiratory Reserve
+
+261
+00:21:01,230 --> 00:21:04,730
+Volume والـResidual Volume، إيش بدينا؟ صحيح، بدينا
+
+262
+00:21:04,730 --> 00:21:09,800
+Total Land Capacityالـ Vital Capacity بتجمع بين
+
+263
+00:21:09,800 --> 00:21:12,320
+الـ Inspiratory Reserve Volume مع الـ Tidal Volume
+
+264
+00:21:12,320 --> 00:21:16,200
+والـ Inspiratory Reserve Volume والـ Total Lung
+
+265
+00:21:16,200 --> 00:21:21,260
+Capacity هي عبارة عن كل 6 لتر كل الهواء اللي
+
+266
+00:21:21,260 --> 00:21:25,860
+يستطيع أن تحتويه الـ Lung كمان مرة بنصح بإنه على
+
+267
+00:21:25,860 --> 00:21:30,490
+اليوتيوبوهي حاجة بيشرحها بالإيد، بالرسم، بيبقى
+
+268
+00:21:30,490 --> 00:21:36,130
+كتير رائع، وبيفهمها الواحد بصورة أكبر، وهي simple،
+
+269
+00:21:36,130 --> 00:21:40,090
+مافيش فيها، بس يفهم إيش الواحد ال background تبعه
+
+270
+00:21:43,270 --> 00:21:45,970
+respiratory air moves احنا كنا نحكي inspiration و
+
+271
+00:21:45,970 --> 00:21:50,510
+expiration لكن ال movements للهوى فيه حاجة non
+
+272
+00:21:50,510 --> 00:21:53,950
+-respiratory حاجات أنا بدخل هوى و بطلع هوى لكن مش
+
+273
+00:21:53,950 --> 00:21:58,870
+عشان التنفس هي عبارة عن reflex action زي على سبيل
+
+274
+00:21:58,870 --> 00:22:03,480
+المثال الكحة لما بكح بطلع هوىوباخد هوايا،
+
+275
+00:22:03,480 --> 00:22:07,340
+sneezing، العطس، crying، laughing، hiccuping، and
+
+276
+00:22:07,340 --> 00:22:14,360
+yawning هذه الحاجات كلها
+
+277
+00:22:14,360 --> 00:22:19,930
+non-respiratory air movementفي إن حاجة برضه زي ما
+
+278
+00:22:19,930 --> 00:22:23,810
+قلنا كمان مرة أحنا بنحاول جدر الإمكان نلقي نظرة
+
+279
+00:22:23,810 --> 00:22:28,150
+على كل حاجة ممكن تفيدنا في المحاضرات بعد هيك
+
+280
+00:22:28,150 --> 00:22:31,350
+نشرحها في الأول وممكن بعد هيك نسيب فيها في
+
+281
+00:22:31,350 --> 00:22:35,210
+المحاضرات زي الـ Ventilation-Perfusion Coupling
+
+282
+00:22:35,210 --> 00:22:40,650
+الـ Ventilation is the amount of gas reaching the
+
+283
+00:22:40,650 --> 00:22:43,270
+alveoli والـ Perfusion is the blood flow reaching
+
+284
+00:22:43,270 --> 00:22:46,230
+the alveoli، بس للتعريفالـ Ventilation هو كمية
+
+285
+00:22:46,230 --> 00:22:50,790
+الهواء اللي بدخل على الـ Alveoli الـ Perfusion
+
+286
+00:22:50,790 --> 00:22:55,610
+كمية الدم اللي بتروح على الـ Alveoli لازم يكون فيه
+
+287
+00:22:55,610 --> 00:23:01,390
+تناغم بين الـ Ventilation مع الـ Perfusion لأن لو
+
+288
+00:23:01,390 --> 00:23:05,210
+أنا عندي تسكير في الشرايين والـ Veins يعني الـ
+
+289
+00:23:05,210 --> 00:23:11,350
+Perfusion حيصير عندي مشكلة فيه، صح؟طيب لما يكون
+
+290
+00:23:11,350 --> 00:23:14,430
+عنده الـ Perfusion فيه مشكلة لكن الهواء رايح جاي
+
+291
+00:23:14,430 --> 00:23:18,330
+بفيد؟ مفيدش كتير فهو لازم يكون فيه تناغم
+
+292
+00:23:18,330 --> 00:23:21,730
+ventilation and perfusion must be tightly
+
+293
+00:23:21,730 --> 00:23:26,550
+regulated for efficient gas exchange مش عشان
+
+294
+00:23:26,550 --> 00:23:32,030
+براعوا خواطر بعض لأ لأنه واحد بدون التاني ال
+
+295
+00:23:32,030 --> 00:23:33,930
+ventilation بدون ال perfusion بقدرش يشتغل ال
+
+296
+00:23:33,930 --> 00:23:36,130
+perfusion بدون ال ventilation بقدرش يشتغل حتى لو
+
+297
+00:23:36,130 --> 00:23:38,770
+كان في عندي perfusion كويس لكن مافيش ventilation
+
+298
+00:23:38,770 --> 00:23:43,540
+واتجبتفمهم يكونوا التانيين ventilation and
+
+299
+00:23:43,540 --> 00:23:50,280
+perfusion coupled أو متزامنين مين بتحكم أكتر واحد
+
+300
+00:23:50,280 --> 00:23:55,500
+في الموضوع هذا؟ الـCO2، changes in PCO2 اللي هو
+
+301
+00:23:55,500 --> 00:23:58,500
+الـpartial pressure P، هذه الـpartial pressure
+
+302
+00:23:58,500 --> 00:24:03,100
+عبارة عن الضغط، ضغط الـCO2 in the alveolicus،
+
+303
+00:24:03,100 --> 00:24:08,890
+changes in the diameter of bronchiolesوبنلاقي الـ
+
+304
+00:24:08,890 --> 00:24:13,970
+Passageways Serving Areas Where Alveolar Carbon
+
+305
+00:24:13,970 --> 00:24:18,330
+Dioxide Is High Dilate يعني كل ما زاد كمية ثاني
+
+306
+00:24:18,330 --> 00:24:21,490
+أكسيد الكربون في مكان بيصير عندى dilatation في وين
+
+307
+00:24:21,490 --> 00:24:26,590
+في الـ alveolar في ال .. في ال .. كل ما زاد ثاني
+
+308
+00:24:26,590 --> 00:24:29,930
+أكسيد الكربون كل ما زاد اللي هو في الارتريويل ال
+
+309
+00:24:29,930 --> 00:24:35,010
+dilatation وكل ما جال ال carbon dioxide في ال
+
+310
+00:24:35,010 --> 00:24:43,280
+alveolsكلما جال، بيصير عندي تضيق في الـ… في الـ…
+
+311
+00:24:43,280 --> 00:24:45,560
+في الـ… في الـ… في الـ… في الـ… في الـ… في الـ… في
+
+312
+00:24:45,560 --> 00:24:47,420
+الـ… في الـ… في الـ… في الـ… في الـ… في الـ… في
+
+313
+00:24:47,420 --> 00:24:47,560
+الـ… في الـ… في الـ… في الـ… في الـ… في الـ… في
+
+314
+00:24:47,560 --> 00:24:47,560
+الـ… في الـ… في الـ… في الـ… في الـ… في الـ… في
+
+315
+00:24:47,560 --> 00:24:52,200
+الـ… في الـ… في الـ… في الـ… في الـ… في الـ… في
+
+316
+00:24:52,200 --> 00:24:52,860
+الـ… في الـ… في الـ… في الـ… في الـ… في الـ… في
+
+317
+00:24:52,860 --> 00:24:52,860
+الـ… في الـ… في الـ… في الـ… في الـ… في الـ… في
+
+318
+00:24:52,860 --> 00:24:55,560
+الـ… في الـ… في الـ… في الـ… في الـ… في الـ… في
+
+319
+00:24:55,560 --> 00:25:02,580
+الـ… في الـ… في
+
+320
+00:25:04,730 --> 00:25:09,490
+لما كان عندى reduce alveolar ventilation فاش هوا
+
+321
+00:25:09,490 --> 00:25:14,210
+كتير شايفين انتوا الفقاعة اللى هى الصفرة الفقاعة
+
+322
+00:25:14,210 --> 00:25:18,770
+جال ال alveolar ventilation طبعا لما يجي ال
+
+323
+00:25:18,770 --> 00:25:21,710
+alveolar ventilation ايش بيصير؟ بيزيد ال CO2 طبعا
+
+324
+00:25:21,710 --> 00:25:24,530
+لأنه مافيش عملية تبادل oxygen و ثانو oxygen
+
+325
+00:25:24,530 --> 00:25:27,130
+الكربون فبيزيد ثانو oxygen الكربون و بيبقى فيه
+
+326
+00:25:27,130 --> 00:25:30,190
+عندنا excessive perfusion ايش بروح بيصير؟ لما يزيد
+
+327
+00:25:30,190 --> 00:25:33,290
+ثانو oxygen الكربون على طول بتضيق البولبنري
+
+328
+00:25:33,290 --> 00:25:40,190
+artulesأرتيويل بيصيروا تضيق والأرتيويل اللي بروح
+
+329
+00:25:40,190 --> 00:25:43,010
+على الالفيول هاي بيصيروا تضيق ولمّا يصيروا تضيق
+
+330
+00:25:43,010 --> 00:25:48,730
+على طول بيصير عندنا reduced perfusion بمعنى أنا
+
+331
+00:25:48,730 --> 00:25:52,230
+لما جيت لما يزيد سانو اكسيد الكربون في الالفيول
+
+332
+00:25:52,230 --> 00:25:59,080
+والبرفيوجين لسه شغال زي الناسبيأخد Action الجسم
+
+333
+00:25:59,080 --> 00:26:05,120
+وبيخلّي الـ «أرتيويل» يـ Constrict نتيجة ارتفاع
+
+334
+00:26:05,120 --> 00:26:08,820
+ثاني أكسيد الكربون فبيجيل كمية الدم اللي بتروح
+
+335
+00:26:08,820 --> 00:26:14,240
+هناك بالصورة هاي يعني مش عملية عكسية عملية بيرتبوا
+
+336
+00:26:14,240 --> 00:26:19,000
+وشغلوا مع بعض والله جال الهواء اللي بيروح وبيجي
+
+337
+00:26:19,000 --> 00:26:22,700
+فبيزيد ثاني أكسيد الكربون وبيضيق الـ «أرتيويل»
+
+338
+00:26:22,700 --> 00:26:29,740
+الآن بالعكسزاد الـ Alveolar Ventilation فجال الـ
+
+339
+00:26:29,740 --> 00:26:32,180
+Alveolar Ventilation زاد معناته سينيوكسيد الكربون
+
+340
+00:26:32,180 --> 00:26:35,180
+بيصير ينطرد كتير لما ينطرد كتير معناته بيجيل في
+
+341
+00:26:35,180 --> 00:26:41,820
+الـ Alveolar Ventilation فأنا إيش بعمل؟ لما يجيل
+
+342
+00:26:41,820 --> 00:26:45,260
+سينيوكسيد الكربون بيتوسع الارتريول لما يتوسع
+
+343
+00:26:45,260 --> 00:26:52,230
+الارتريول بيزيد اللي هو الـ Perfusionبشتغل مع
+
+344
+00:26:52,230 --> 00:26:55,990
+التاني هذه اسمها ventilation-perfusion coupling
+
+345
+00:26:55,990 --> 00:27:01,090
+ممكن نسمعها في المحاضرات بعدها ال service area and
+
+346
+00:27:01,090 --> 00:27:04,410
+thickness برضه وردة جديدة يعني زي ما قلنا مواضيع
+
+347
+00:27:04,410 --> 00:27:08,310
+مختلفة بندخل من حاجة بندخل في حاجة لكن أنا للتوضيح
+
+348
+00:27:08,310 --> 00:27:15,480
+يعني وهذه آخر slide للمحاضرة اليوماللي هو سبحانك
+
+349
+00:27:15,480 --> 00:27:19,900
+ربي يعني عملية الـ Gas Exchange اللي بتسير في الـ
+
+350
+00:27:19,900 --> 00:27:29,120
+Lungs بتسير فيه جدر جدر كتير fine فبنلاقي إنه الـ
+
+351
+00:27:29,120 --> 00:27:33,780
+Respiratory Membranesحوالي نص لواحد Micro-Thick
+
+352
+00:27:33,780 --> 00:27:37,160
+Allowing for Efficient Gas Exchange عشان يصير
+
+353
+00:27:37,160 --> 00:27:42,140
+عملية يعني إيش أكتر حاجة بتتصوروا إنها أرجيجة
+
+354
+00:27:42,140 --> 00:27:46,780
+ونعمة هادي أجل منها يدوب مبينة عمليًا يدوب مبينة
+
+355
+00:27:46,780 --> 00:27:52,280
+عمليًا من 0.5 نص لواحد Micro-Meter Thick Allowing
+
+356
+00:27:52,280 --> 00:27:55,720
+Efficient Gas Exchange لو تيجي تجمع في الـLungs
+
+357
+00:27:55,720 --> 00:27:58,620
+عند الرجال ع سبيل المثال بتلاقي إنه حوالي 60 متر
+
+358
+00:27:58,620 --> 00:28:05,640
+مربعهذه جد أربعين مرة That one's skin جد الجلد لما
+
+359
+00:28:05,640 --> 00:28:09,360
+تجمع Total Surface Area في الـMills ويقال إنها
+
+360
+00:28:09,360 --> 00:28:19,340
+تساوي مساحة ملعب كرة التنس لأن لو صار في عندي
+
+361
+00:28:19,340 --> 00:28:25,170
+أديمةبالموناري إديمة إيش بيصير في الـ .. في الـ
+
+362
+00:28:25,170 --> 00:28:30,370
+membranes هاي بيصير thick أكتر وأكتر طبعاً لما كل
+
+363
+00:28:30,370 --> 00:28:34,270
+ما زاد ال thickness تبعها بيصير ال gas exchange
+
+364
+00:28:34,270 --> 00:28:38,230
+inadequate و بيصير عندنا نقص أكسجين بيصير عندنا
+
+365
+00:28:38,230 --> 00:28:42,470
+gas deprivation ليش؟ لأن ال lung هذه اللي فيها
+
+366
+00:28:42,470 --> 00:28:47,200
+edematous walls جواتهامش قادرة تعمل gas exchange
+
+367
+00:28:47,200 --> 00:28:51,400
+زي الناس فبصير عندى hypo ventilation أو بيصير عندى
+
+368
+00:28:51,400 --> 00:28:56,080
+oxygen أو asphyxia معنى آخر لأن كل ما جل ال
+
+369
+00:28:56,080 --> 00:29:02,890
+surfaceزي في الـ Enphysema ‏when walls of alveoli
+
+370
+00:29:02,890 --> 00:29:07,350
+break through يعني بتقل الـ service وبرضه بتقل
+
+371
+00:29:07,350 --> 00:29:13,450
+عملية الـ oxygen exchange نتيجة وجود Enphysema،
+
+372
+00:29:13,450 --> 00:29:18,390
+ليش؟ لأن الـ service أصلا اتدهورفبالصورة هذه بيصير
+
+373
+00:29:18,390 --> 00:29:23,910
+عندنا نفس الـ Effect إنه بدخل عند العيّان في
+
+374
+00:29:23,910 --> 00:29:28,990
+أسفكسيا بطريقة أو بأخرى وهاكذا معظم الـ Mechanisms
+
+375
+00:29:28,990 --> 00:29:35,890
+يعني أنا عارف إنه الممكن فيه شويه ناشفة المعلومات
+
+376
+00:29:35,890 --> 00:29:40,550
+لأنه مش practical كتير لكن هي زي ما قلنا عبارة عن
+
+377
+00:29:40,550 --> 00:29:43,410
+بعض الإضاءات
+
+378
+00:29:44,020 --> 00:29:47,120
+في الـ Anatomy and Physiology of Respiratory
+
+379
+00:29:47,120 --> 00:29:52,780
+System وبنحاول في المحاضرات الأمراض أنه نرجع لل
+
+380
+00:29:52,780 --> 00:29:56,720
+basics هاي هي مهمة كتير حقيقة وحتى يعني احنا
+
+381
+00:29:56,720 --> 00:29:59,840
+بنستهدفها في الامتحانات كتير على أساس أنه ال
+
+382
+00:29:59,840 --> 00:30:03,320
+basics لازم الواحد يكون فاهمه قدر الممكن أنا
+
+383
+00:30:03,320 --> 00:30:09,920
+بشكرلكم حسن الاستماع ويعطيكم الف عافية السلام
+
+384
+00:30:09,920 --> 00:30:10,300
+عليكم
+
diff --git a/PL9fwy3NUQKwaf4aPlmtIQXiASpDxVLEnO/bWJL4CfrdXk_raw.srt b/PL9fwy3NUQKwaf4aPlmtIQXiASpDxVLEnO/bWJL4CfrdXk_raw.srt
new file mode 100644
index 0000000000000000000000000000000000000000..84333ca2f2529a299dc06b31c36ead5952628706
--- /dev/null
+++ b/PL9fwy3NUQKwaf4aPlmtIQXiASpDxVLEnO/bWJL4CfrdXk_raw.srt
@@ -0,0 +1,1492 @@
+1
+00:00:01,670 --> 00:00:04,930
+بسم الله الرحمن الرحيم الحمد لله رب العالمين بدنا
+
+2
+00:00:04,930 --> 00:00:09,450
+يا جماعة نقترب من النهاية اليوم هنتحدث .. ممكن
+
+3
+00:00:09,450 --> 00:00:13,550
+ماحدش حتى الآن خلال دراستكوا تحدث عن ال cancer
+
+4
+00:00:13,550 --> 00:00:18,150
+طبعا احنا مش هروح بتوسع لكن بدنا نكون عارفين بعض
+
+5
+00:00:18,150 --> 00:00:22,810
+ال terminology بعض المبادئ الأساسية في ال cancer
+
+6
+00:00:22,810 --> 00:00:26,710
+هو ان انتوا في يوم من الأيام في لحظة من اللحظات
+
+7
+00:00:26,710 --> 00:00:34,740
+هتكون عندكواشغل مع مرضى الـ Cancer من خلال فرق الـ
+
+8
+00:00:34,740 --> 00:00:36,940
+Palliative أو من خلال الناس اللي بتحتاج إيداش
+
+9
+00:00:36,940 --> 00:00:39,820
+طبيعي أو الناس اللي عندها breast cancer و اللي
+
+10
+00:00:39,820 --> 00:00:48,020
+بنعملها طبعاً lymph drainage فـ what is cancer؟
+
+11
+00:00:48,020 --> 00:00:52,280
+cancer begins when cells in a part of the body
+
+12
+00:00:52,280 --> 00:00:57,140
+become abnormal لما تتحول خلايا لخلايا غير طبيعية
+
+13
+00:00:59,430 --> 00:01:02,710
+مش اللي بيصير more abnormal cells forming a mass
+
+14
+00:01:02,710 --> 00:01:11,090
+of tissue called growth or cancerous tumor يعني ال
+
+15
+00:01:11,090 --> 00:01:15,390
+tumor هو عبارة عن ورم، مش شرط يكون ورم، ورم خبير،
+
+16
+00:01:15,390 --> 00:01:19,150
+ورم ممكن يكون الدم اللي بعمل ورم، فهذا ما نقول عنه
+
+17
+00:01:19,150 --> 00:01:22,150
+space-occupying lesion
+
+18
+00:01:29,250 --> 00:01:34,210
+ليجن مش شرط يكون ماليجنان ممكن يكون tumor أبسس
+
+19
+00:01:34,210 --> 00:01:40,550
+نسيج نسيج lipoma ممكن يكون أكتر من حاجة لكن هو
+
+20
+00:01:40,550 --> 00:01:46,350
+بيظل ورم فالورم مش اللي احنا هنقص ضهان مش اللي
+
+21
+00:01:46,350 --> 00:01:50,230
+دارج على الإسان العام الناس اللي بتحكي ان ورم
+
+22
+00:01:50,230 --> 00:01:57,560
+معناته خلاص cancer و هيموت وكذا لأ ورمهو عبارة عن
+
+23
+00:01:57,560 --> 00:02:03,360
+mass of tissue بنقول عنها growth أو tumor وهذا
+
+24
+00:02:03,360 --> 00:02:07,860
+ممكن يكون cancerous يعني حاجة سرطانية malignant
+
+25
+00:02:07,860 --> 00:02:15,060
+cancer cells can break away and spread وهذه
+
+26
+00:02:15,060 --> 00:02:21,010
+خطورتها أن بعض ال cancersبتعمل «Mets» أو
+
+27
+00:02:21,010 --> 00:02:24,810
+«Metastases» «Mets» المختصر تبعها «Metastatic
+
+28
+00:02:24,810 --> 00:02:30,410
+Cancer» بتنتجل من مكان إلى مكان في الجسم مش كل
+
+29
+00:02:30,410 --> 00:02:34,730
+خدوا بالكم الجملة هاي «Tumors are cancerous» مش كل
+
+30
+00:02:34,730 --> 00:02:42,070
+ورم معناته cancer «Benign tumors don't spread to
+
+31
+00:02:42,070 --> 00:02:44,990
+other parts of the body» وهذه القاعدة وهذه من
+
+32
+00:02:44,990 --> 00:02:49,500
+الحاجات اللي بتفرج بين الـ «Benign»وبين الـ
+
+33
+00:02:49,500 --> 00:02:57,100
+Malignant عكس Benign حميد Malignant خبيث Cancer
+
+34
+00:02:57,100 --> 00:03:01,080
+usually start in one part of the bodyولكن إذا أصبح
+
+35
+00:03:01,080 --> 00:03:04,740
+عضو الحرارة أكبر، فهو يمكن أن يؤذي، وليس فقط
+
+36
+00:03:04,740 --> 00:03:07,800
+بندقل، لكن هو ممكن يضغط على حاجات حواليه زي
+
+37
+00:03:07,800 --> 00:03:11,140
+الـcancer of brain عسم المثال، ممكن يسكرلي أي
+
+38
+00:03:11,140 --> 00:03:15,560
+فرامن ويعمللي hydrocephalus مثلا أو يخرّبني
+
+39
+00:03:15,560 --> 00:03:18,960
+النسيجي لجنبه، وإذا وجدت عضو الحرارة قريبًا،
+
+40
+00:03:18,960 --> 00:03:23,650
+فالتعامل هو أكثر ممكن أن يعملاللي هو بيعتمد عليه
+
+41
+00:03:23,650 --> 00:03:28,750
+الموضوع الـ Screening إنه كل ما اكتشفت الـ Cancer
+
+42
+00:03:28,750 --> 00:03:36,330
+بدري كل ما كانت إمكانية أنه أعالجه كل ما كانت أفضل
+
+43
+00:03:36,330 --> 00:03:39,690
+وهذا طبعاً بتمش إلا بالـ Screening اللي هي عملية
+
+44
+00:03:39,690 --> 00:03:42,970
+الافتشاف
+
+45
+00:03:42,970 --> 00:03:48,870
+مبكر Early Treatmentوكل ما علجناه بدري، هذا can
+
+46
+00:03:48,870 --> 00:03:52,510
+shrink or destroy the cancer tumor and stop it
+
+47
+00:03:52,510 --> 00:03:56,310
+from growing and spreading كل ما علجناه بدري، كل
+
+48
+00:03:56,310 --> 00:03:59,770
+ما سيطرنا عليه بدري، كل ما جدرش الـ cancer هذا إيش
+
+49
+00:03:59,770 --> 00:04:01,670
+ينجل إلى أماكن أخرى
+
+50
+00:04:04,090 --> 00:04:11,590
+عندما تحصل على شكب عادى وكيفية معرفة شكب عادى وشكب
+
+51
+00:04:11,590 --> 00:04:14,630
+عادى
+
+52
+00:04:14,630 --> 00:04:16,390
+وشكب عادى وشكب عادى وشكب عادى وشكب عادى وشكب عادى
+
+53
+00:04:16,390 --> 00:04:17,750
+وشكب عادى وشكب عادى وشكب عادى وشكب عادى وشكب عادى
+
+54
+00:04:17,750 --> 00:04:19,310
+وشكب عادى وشكب عادى وشكب عادى وشكب عادى وشكب عادى
+
+55
+00:04:19,310 --> 00:04:19,350
+وشكب عادى وشكب عادى وشكب عادى وشكب عادى وشكب عادى
+
+56
+00:04:19,350 --> 00:04:23,050
+وشكب عادى وشكب عادى وشكب عادى وشكب عادى وشكب عادى
+
+57
+00:04:23,050 --> 00:04:32,960
+وشكب عادى وشكب عادى وشاللي فوق الخمسين سنة على
+
+58
+00:04:32,960 --> 00:04:36,980
+سبيل المثال اللي هي في موضوع الـ Prostate هذه أمور
+
+59
+00:04:36,980 --> 00:04:41,640
+يعني بيقدر الواحد من بدري يكتشفها لو دار باله
+
+60
+00:04:41,640 --> 00:04:46,660
+أحاله مجموعة من مئات مخلوقات مختلفة موجودة يعني
+
+61
+00:04:46,660 --> 00:04:53,220
+حواليها Cancer are a large family of diseases that
+
+62
+00:04:53,220 --> 00:04:56,960
+involve abnormal cell growth with the potential to
+
+63
+00:04:56,960 --> 00:05:02,410
+invade or spread to other parts of the bodyوهي
+
+64
+00:05:02,410 --> 00:05:06,230
+بتمثل نوع من أنواع الـ Neoplasm نفس القضية هي
+
+65
+00:05:06,230 --> 00:05:10,050
+نفسها نيوبلازم ممكن تكون cancer a neoplasm or
+
+66
+00:05:10,050 --> 00:05:13,230
+tumor is a group of cells that have undergone
+
+67
+00:05:13,230 --> 00:05:19,610
+unregulated growth and will often form a mass or
+
+68
+00:05:19,610 --> 00:05:23,710
+lump بيعمل زي كتلة أو lump برضه كتلة but maybe
+
+69
+00:05:23,710 --> 00:05:28,530
+distributed diffusely وهذه مشكلة الـ Neoplasm
+
+70
+00:05:28,530 --> 00:05:33,030
+بصورة عامةالـ Classification وهذا يمكن المهم اللي
+
+71
+00:05:33,030 --> 00:05:38,410
+بدنا يهيأه الآن أنه Classified by the type of
+
+72
+00:05:38,410 --> 00:05:43,150
+cells that the tumor cells resemble and is
+
+73
+00:05:43,150 --> 00:05:47,910
+therefore presumed to be the origin of the tumor
+
+74
+00:05:47,910 --> 00:05:53,950
+الـ Type هاي Carcinoma رقم واحد، ما بنقول هوكتشف
+
+75
+00:05:53,950 --> 00:05:55,850
+وما بنقول Carcinoma
+
+76
+00:05:58,040 --> 00:06:01,960
+طبعاً لما أقول Carcinoma معناته أنا بحكي عن Benign
+
+77
+00:06:01,960 --> 00:06:07,080
+ولا Malignant؟ Malignant هي عبارة عن الـCancers
+
+78
+00:06:07,080 --> 00:06:11,200
+اللي بتيجي من الـepithelial cells يعني المصدر
+
+79
+00:06:11,200 --> 00:06:14,420
+تبعها، المخرج تبعها الـepithelial cells
+
+80
+00:06:22,660 --> 00:06:26,060
+وهذه بتمثّر many of the most common cancers
+
+81
+00:06:26,060 --> 00:06:30,300
+particularly in the aged and include nearly all
+
+82
+00:06:30,300 --> 00:06:34,300
+those developing in the breast, prostate, lung,
+
+83
+00:06:34,760 --> 00:06:40,080
+pancreas and colon نقول عنها Carcinoma وزي ما قلنا
+
+84
+00:06:40,080 --> 00:06:42,500
+بتطلع من وين خدوا بالكوا من هالكلمات هاي
+
+85
+00:06:42,500 --> 00:06:48,840
+Epithelial cells وفي أن الساركومة هذه cancers
+
+86
+00:06:48,840 --> 00:06:51,860
+بتطلع من ال connective tissues إيش يعني connective
+
+87
+00:06:51,860 --> 00:06:58,700
+tissue؟الـ bone cartilage الـ fat الـ nerves وهذه
+
+88
+00:06:58,700 --> 00:07:03,400
+طبعا تطلع من الـ organating in the mesenchymal
+
+89
+00:07:03,400 --> 00:07:06,240
+cells outside the bone marrow، مش من الـ bone
+
+90
+00:07:06,240 --> 00:07:10,060
+marrow، من فرع من الـ mesenchymal itself، نقول
+
+91
+00:07:10,060 --> 00:07:16,380
+عنها Sarcoma، من أسوأ أنواع الـ cancer كمل، في
+
+92
+00:07:16,380 --> 00:07:21,310
+عندنا الـ lymphoma and leukemiaأو في الـ cancers
+
+93
+00:07:21,310 --> 00:07:24,710
+هنحكي عن اللوكيميا بعد شوية أو في المحاضرة اللي
+
+94
+00:07:24,710 --> 00:07:27,250
+جاي these two classes of cancer arise from
+
+95
+00:07:27,250 --> 00:07:31,410
+hematopoietic blood-forming cells يعني هما بيطلعوا
+
+96
+00:07:31,410 --> 00:07:37,210
+من الأماكن اللي بتصليعلي اللي هي خلايا الدم الأسس
+
+97
+00:07:37,210 --> 00:07:42,330
+تبع خلاية الدم blood-forming cells that leave the
+
+98
+00:07:42,330 --> 00:07:45,130
+marrow من ال bone marrow يعني and tend to mature
+
+99
+00:07:45,130 --> 00:07:49,250
+in the lymph node and bloodالليوكيميا هي الـmost
+
+100
+00:07:49,250 --> 00:07:53,530
+common type of cancer في الأطفال تعتمد على حوالي
+
+101
+00:07:53,530 --> 00:07:58,150
+30% في الأطفال، الليوكيميا، بس طبعا الإشارة الحلوة
+
+102
+00:07:58,150 --> 00:08:03,090
+فيها إنه كتير منهم بيصير له healing لكن في بعضهم
+
+103
+00:08:03,090 --> 00:08:06,410
+ما بيصيروش، يمكن اللينفومة الليوكيميا هي عبارة عن
+
+104
+00:08:06,410 --> 00:08:10,830
+cancer بيطلع من الـhematopoietic، hematoderm
+
+105
+00:08:10,830 --> 00:08:17,820
+poietic معناته يصنع، blood formingأو مكوّن cells
+
+106
+00:08:17,820 --> 00:08:22,100
+that leave the marrow وبتروح على الدم أو على الـ
+
+107
+00:08:22,100 --> 00:08:27,310
+lymph node وهناك بيصير لها عملية cancerفيه جرم
+
+108
+00:08:27,310 --> 00:08:31,250
+cell tumor هي عبارة عن cancers derived من
+
+109
+00:08:31,250 --> 00:08:36,810
+pluripotent cells عبارة بتتخرج يعني بتطلعلي من
+
+110
+00:08:36,810 --> 00:08:41,270
+المناطق اللي مخلوطة الخلايا زي عنا على سبيل المثال
+
+111
+00:08:41,270 --> 00:08:45,970
+اللي موجودة في ال testicles أو ال ovary testicles
+
+112
+00:08:45,970 --> 00:08:52,590
+اللي هي الخصيتين ال ovary المبيضين بنقول عنها
+
+113
+00:08:52,590 --> 00:08:55,350
+seminoma أو dysgerminoma
+
+114
+00:08:57,050 --> 00:09:01,030
+وفينا الـ Plastoma هي عبارة عن Cancer بتطلع من الـ
+
+115
+00:09:01,030 --> 00:09:07,190
+Immature precursor cells أو الـ Embryonic tissue
+
+116
+00:09:07,190 --> 00:09:10,930
+لسه الخلايا ما كبرتش، بيقول عنها إيش؟ Plastoma
+
+117
+00:09:10,930 --> 00:09:18,850
+وهدي common في الأطفال أكتر منه في الـ Adult الـ
+
+118
+00:09:18,850 --> 00:09:21,830
+Cancer عادة ماحدش بيجي ويقول مثلا و الله عنده
+
+119
+00:09:21,830 --> 00:09:25,750
+liver، carcinoma، لأ، فيه إلها اسموهذا بتتكون
+
+120
+00:09:25,750 --> 00:09:29,850
+عادةً، فهذه تكميلة الـ classification تتكون من
+
+121
+00:09:29,850 --> 00:09:33,410
+كلمة مثلاً كارسينوما، ساركوما، بلاستوما، اللي
+
+122
+00:09:33,410 --> 00:09:38,750
+قررناه متى وبنحط في الأول اسم العضو أو اسم الـ
+
+123
+00:09:38,750 --> 00:09:46,330
+tissue اللي طالع منه الـ cancer كيف يعني؟ هجيت
+
+124
+00:09:46,330 --> 00:09:49,510
+عندنا cancer of the liver parenchyma ال liver
+
+125
+00:09:49,510 --> 00:09:54,360
+parenchyma بنقول عنهاوهي طالعة من الـ Epithelial
+
+126
+00:09:54,360 --> 00:10:00,340
+cells وكبد
+
+127
+00:10:00,340 --> 00:10:01,760
+Hepato
+
+128
+00:10:09,740 --> 00:10:13,800
+طيب لو نفس الكلام هو طالع من الكبد لكن طالع من
+
+129
+00:10:13,800 --> 00:10:18,040
+الـPrimitive Liver Precursor Cells وهي
+
+130
+00:10:18,040 --> 00:10:21,360
+Hepatoplastoma إحنا قولنا اللي Plastoma هي عبارة
+
+131
+00:10:21,360 --> 00:10:24,940
+عن الـImmature Precursor Cells أو الـEmbryonic
+
+132
+00:10:24,940 --> 00:10:31,840
+Cells A cancer arising from fat cells، بيقول عنه
+
+133
+00:10:31,840 --> 00:10:38,200
+برضه fatالفات قبل قليل صار كومة رقم اتنين صار كومة
+
+134
+00:10:38,200 --> 00:10:42,820
+من الفات لأنه connective tissue صار كومة وهي fat
+
+135
+00:10:42,820 --> 00:10:47,220
+cells بيقول عنها Lipo Lipo-Sarcoma فيه طبعا الـ
+
+136
+00:10:47,220 --> 00:10:51,240
+Benign tumors وهذه ليس كانسر
+
+137
+00:10:57,760 --> 00:11:05,900
+بنفس الكلام عادةً لما يكون بنائي مثلًا لو في شيء
+
+138
+00:11:05,900 --> 00:11:11,920
+طالع من خلايا العضلات العادية نقول عنها Lymyoma
+
+139
+00:11:12,430 --> 00:11:18,570
+أوما اللي هي أنه بيناين ليما يوما معناته الاسم
+
+140
+00:11:18,570 --> 00:11:21,830
+العام للأشخاص اللي يحدث بشكل عام بيناين تمور في
+
+141
+00:11:21,830 --> 00:11:26,050
+الجلد هي الفيبرويد هذه مشهورة عند الفميل خاصة
+
+142
+00:11:26,050 --> 00:11:29,670
+الأصغار في العمر بيصير عندهم فيبرويد يوتراس وهذا
+
+143
+00:11:29,670 --> 00:11:34,790
+بنقول عنه ليما يوما هذه مش كانسر لكن بيناين تمور
+
+144
+00:11:34,790 --> 00:11:41,280
+بعض الأحيان من الكانسر يسمونمرة تانية for the size
+
+145
+00:11:41,280 --> 00:11:45,540
+and shape of that cells حسب ما هي مبينة في ال
+
+146
+00:11:45,540 --> 00:11:48,860
+microscope زي اللي بنقولها giant cell carcinoma أو
+
+147
+00:11:48,860 --> 00:11:52,360
+spindle cell carcinoma أو small cell carcinoma هذه
+
+148
+00:11:52,360 --> 00:11:56,560
+الأسماء برضه واردة إنها تكون موجودة كمان مرة
+
+149
+00:11:56,560 --> 00:12:02,280
+namingاللي هو الـ prefix اللي هو المقدمة تبعته لو
+
+150
+00:12:02,280 --> 00:12:05,640
+كان أنا بحكي عن gland طلع الصورة يا جماعة لو بحكي
+
+151
+00:12:05,640 --> 00:12:10,280
+عن gland بقول adeno cartilage chondro red blood
+
+152
+00:12:10,280 --> 00:12:16,920
+cells erythro blood vessels hemangio liver hepato
+
+153
+00:12:16,920 --> 00:12:22,570
+fat lipo lymphocyte lymphoBig Men Cells Milano
+
+154
+00:12:22,570 --> 00:12:29,730
+Bone Marrow Myelo Muscle Myoc Bone Osteo حسب فبقول
+
+155
+00:12:29,730 --> 00:12:36,030
+Osteo Sarcoma Myo Myoma هيك طب أيش أسباب الـ
+
+156
+00:12:36,030 --> 00:12:45,710
+Cancer حقيقة أنه في أسباب كتيرة في
+
+157
+00:12:45,710 --> 00:12:52,410
+جزئية منها تتعلق بالـ Geneticوهذا يتمثل 5 إلى 10%
+
+158
+00:12:52,410 --> 00:12:57,030
+من الحالات وإن الواحد يكون لاياب اللي يصير عنده
+
+159
+00:12:57,030 --> 00:13:03,190
+وال environment تمثل 90 إلى 95% يعني هذا ما صنعناه
+
+160
+00:13:03,190 --> 00:13:09,410
+أنفسنا إحنا اللي جيبناه لنفسنا ال environment
+
+161
+00:13:09,410 --> 00:13:15,070
+الجزء الكبير تمثل ال tobacco التدخين وال diet
+
+162
+00:13:15,070 --> 00:13:21,630
+والأكل اللي بناكلهالناس اللي بتاخد الكحول الناس
+
+163
+00:13:21,630 --> 00:13:24,250
+الـ obesity ليبل يكون أكتر من غيرها ال infections
+
+164
+00:13:24,250 --> 00:13:28,690
+وحنا حكينا مثلا على سبيل المثال الهيباطيتس C صحيح؟
+
+165
+00:13:28,690 --> 00:13:31,270
+كنا بيصير cancer، الهيباطيتس B بيصير cancer،
+
+166
+00:13:31,270 --> 00:13:35,930
+tobacco، الدخان حكينا عنه، هذه اللي هي ال
+
+167
+00:13:35,930 --> 00:13:38,630
+environmental causes of cancer
+
+168
+00:13:48,850 --> 00:13:52,750
+Tobacco use is the cause of about 22% of cancer
+
+169
+00:13:52,750 --> 00:13:58,350
+deaths رقم كتير مهم 22% من حالات الوفيات بتاعة ال
+
+170
+00:13:58,350 --> 00:14:03,290
+cancer بمثلها التدخين The incidence of lung cancer
+
+171
+00:14:03,290 --> 00:14:06,970
+is highly correlated with smoking كتير من الأحيان
+
+172
+00:14:06,970 --> 00:14:11,130
+حتى لو كان الناس اللي حضرت عن smoking and cancer
+
+173
+00:14:11,130 --> 00:14:16,150
+حتى لو كان passive smoker اش يعني passive smoker؟
+
+174
+00:14:17,600 --> 00:14:21,960
+يعني هو مش هو اللي بدخن نفسه اللي بدخن .. اللي
+
+175
+00:14:21,960 --> 00:14:28,240
+بدخن حد تاني بس هو بشمعه عشرة في المية ناتج عن
+
+176
+00:14:28,240 --> 00:14:32,020
+طريق الـ obesity بوردية طريقة الأكل المستسلمة lack
+
+177
+00:14:32,020 --> 00:14:36,380
+of physical activity consumption of alcohol other
+
+178
+00:14:36,380 --> 00:14:40,620
+factors include certain infections ionizing
+
+179
+00:14:40,620 --> 00:14:43,940
+radiation environmental pollutants
+
+180
+00:14:46,270 --> 00:14:53,030
+التي هي الحاجات للبيئة
+
+181
+00:14:53,030 --> 00:14:56,090
+في التطور
+
+182
+00:15:01,910 --> 00:15:07,710
+والعالم التالت 20% نتيجة انفكشن على رأسها
+
+183
+00:15:07,710 --> 00:15:13,810
+الهيباطايتس بي وهيباطايتس سي وهيومان بابيلوما
+
+184
+00:15:13,810 --> 00:15:14,570
+فيروس
+
+185
+00:15:18,640 --> 00:15:25,820
+تعمل لـ Cancer في العالم التالت وممكن أكتر من الـ
+
+186
+00:15:25,820 --> 00:15:29,400
+Smoking لكن بدأنا نلحقهم وهنا بدنا نفوتهم
+
+187
+00:15:29,400 --> 00:15:33,560
+للأوروبيين These factors act at least partly by
+
+188
+00:15:33,560 --> 00:15:36,960
+changing the genes of cells يعني ممكن مايكونش هو
+
+189
+00:15:36,960 --> 00:15:41,480
+السبب الأساسي لكن على الأقل بيلعب دور Genetic
+
+190
+00:15:41,480 --> 00:15:46,430
+changes are acquired before cancer developsخمسة
+
+191
+00:15:46,430 --> 00:15:48,790
+لعشرة في المية of cancers are due to genetic
+
+192
+00:15:48,790 --> 00:15:54,850
+defects inherited from a person's parents وهذه
+
+193
+00:15:54,850 --> 00:15:59,310
+كلها بنعرف جثة الممثلة اللي شالت لبريست نتيجة أن
+
+194
+00:15:59,310 --> 00:16:03,430
+أمها وستها واختها كان عندهم عاملة فحص جيني لاجت أن
+
+195
+00:16:03,430 --> 00:16:07,440
+هي liable يصير عندهايصير عندها breast cancer فقبل
+
+196
+00:16:07,440 --> 00:16:10,820
+ما يصير عندها breast cancer راحت شالت ال breast
+
+197
+00:16:10,820 --> 00:16:13,660
+طبعا أنا مش بدأ أقول إن هذا صح أو غلط، لأ إنما
+
+198
+00:16:13,660 --> 00:16:18,860
+الموضوع هذا بلعب دور ويبدو إنه في المستقبل هياخد
+
+199
+00:16:18,860 --> 00:16:27,500
+دور أكبر، طيب إيه ال incidence of cancer؟ طبعا عند
+
+200
+00:16:27,500 --> 00:16:31,340
+رجالالعدد تقريبًا مشابه إحنا بنحكي عن أمريكا في
+
+201
+00:16:31,340 --> 00:16:36,500
+2015 أمريكا في 2015 تقريبًا 848 مليون صار عندهم
+
+202
+00:16:36,500 --> 00:16:42,680
+cancer ال woman الرجال 848 و ال woman 810 يعني
+
+203
+00:16:42,680 --> 00:16:47,720
+جريمين من بعض مش بعاد لكن مثلت لبروستيت 26% عند ال
+
+204
+00:16:47,720 --> 00:16:52,550
+men ل breast 29% عند ال woman ال lungبالصلاة وعن
+
+205
+00:16:52,550 --> 00:16:56,930
+النبي عند الرجال 14% وعند الستات بدوا يلحقونا
+
+206
+00:16:56,930 --> 00:17:02,710
+ويقعدهم ويوصلونا 13% ليش؟ لأن بدوا الستات برضه
+
+207
+00:17:02,710 --> 00:17:07,470
+يكونوا مدخنين وشيشة وهيك ال colon عند الرجال
+
+208
+00:17:07,470 --> 00:17:13,030
+والستات 8 8 وبعد هيك بيصير فيه خصوصية لموضوع ال
+
+209
+00:17:13,030 --> 00:17:16,880
+cancerأو بيبطل فيه خصوصية عفوًا لموضوع الـ Cancer
+
+210
+00:17:16,880 --> 00:17:19,760
+بيصير جاربين من بعض لكن هي المشكلة كلها الـ
+
+211
+00:17:19,760 --> 00:17:23,480
+Prostate والـ Breast اللي فيهم الخلاف والباقي كله
+
+212
+00:17:23,480 --> 00:17:25,180
+جاريب من بعض
+
+213
+00:17:35,720 --> 00:17:40,800
+وهو مجموع الارجال والستات اللي شوفناهم في الـ
+
+214
+00:17:40,800 --> 00:17:48,240
+«سلايد» هذا 800 و 800 تقريباً مليون و ستمية new
+
+215
+00:17:48,240 --> 00:17:52,700
+casesكانت برستيتكالة أكتر قصر عامة بين الرجال، 26
+
+216
+00:17:52,700 --> 00:17:56,860
+%، ثم أجابعاء الـLung، ثم الـColon and Rectum، في
+
+217
+00:17:56,860 --> 00:18:00,240
+الفتاة 29% برستكالة، ثم الـLung، ثم الـColon and
+
+218
+00:18:00,240 --> 00:18:05,560
+Rectum، كانت أكتر قصر عامة بين الرجال، الـColon
+
+219
+00:18:05,560 --> 00:18:08,500
+and Rectum كانت أكتر قصر عامة بين الرجال، الـColon
+
+220
+00:18:08,500 --> 00:18:09,440
+and Rectum كانت أكتر قصر عامة بين الرجال، الـColon
+
+221
+00:18:09,440 --> 00:18:09,680
+and Rectum كانت أكتر قصر عامة بين الرجال، الـColon
+
+222
+00:18:09,680 --> 00:18:09,880
+and Rectum كانت أكتر قصر عامة بين الرجال، الـColon
+
+223
+00:18:09,880 --> 00:18:10,220
+and Rectum كانت أكتر قصر عامة بين الرجال،
+
+224
+00:18:17,910 --> 00:18:22,150
+بتحسن الامور كل مانكشف الـ cancer بدري، بتعالج
+
+225
+00:18:22,150 --> 00:18:26,610
+بدري، بينحطله حد بدري، for example due to
+
+226
+00:18:26,610 --> 00:18:29,890
+improvement of treatment the five year relative
+
+227
+00:18:29,890 --> 00:18:35,390
+survival rate for leukemia has increased من 34% في
+
+228
+00:18:35,390 --> 00:18:40,190
+السبعيناتلـ 60% لتاريخ كتابة هذه المحاضرة والآن
+
+229
+00:18:40,190 --> 00:18:43,150
+يمكن يوصل لـ 70% أو 80% بالنسبة للـ Leukemia في
+
+230
+00:18:43,150 --> 00:18:46,410
+الأطفال، إيش يعني الـ Five Year Relative Survivor
+
+231
+00:18:46,410 --> 00:18:52,730
+Rate؟ في الأورام لا نقول إنه سيعيش عمر طويل، دائما
+
+232
+00:18:52,730 --> 00:18:56,950
+نحكم على خمس سنين، حسبوها حسبة جداش نسبة من الناس
+
+233
+00:18:56,950 --> 00:19:01,550
+بتصمت خمس سنين، تصوروا؟ هيك بتنحسب، اللي هي Five
+
+234
+00:19:01,550 --> 00:19:03,530
+Year Survivor Rate
+
+235
+00:19:06,890 --> 00:19:10,470
+الـ 2012 كان فيه 14 مليون new case of cancer
+
+236
+00:19:10,470 --> 00:19:14,110
+occurred globally بدون طبعا ال skin cancer اللي هي
+
+237
+00:19:14,110 --> 00:19:19,150
+بدون ال skin cancers العادية ما عدا ال melanoma it
+
+238
+00:19:19,150 --> 00:19:26,810
+caused about 8.2 مليون death or 14.6% of all human
+
+239
+00:19:26,810 --> 00:19:33,060
+death ال symptoms اللي بنشوفها حسبعشان هيك مقدرش
+
+240
+00:19:33,060 --> 00:19:38,200
+الواحد الآن يقول والله حسب العضو يعني أنت ممكن أي
+
+241
+00:19:38,200 --> 00:19:42,400
+symptom يصير صحيح يكون simple لكن ممكن يكون وراه
+
+242
+00:19:42,400 --> 00:19:48,060
+حاجة كبيرة زي مثلا أجا
+
+243
+00:19:48,060 --> 00:19:51,680
+واحد شاف كتلة في ال breast ست شافت كتلة في ال
+
+244
+00:19:51,680 --> 00:19:55,360
+breast maybe a sign of breast cancer ممكن مايكونش
+
+245
+00:19:55,360 --> 00:20:01,820
+ممكن دمل ممكن lipoma لكن وجود a thickening أو lump
+
+246
+00:20:02,110 --> 00:20:06,710
+في جسم، في أي مكان، وخاصة لما يكون في الـ breast
+
+247
+00:20:06,710 --> 00:20:12,970
+بتفكر في أنه يكون tumor واحد بدون سبب نزل وزنه من
+
+248
+00:20:12,970 --> 00:20:18,510
+تسعين كيلو صار ستين، لا عنده سكري ولا بلعب رياضة
+
+249
+00:20:18,510 --> 00:20:23,590
+ولا عنده اكتئاب وعادي، unexplained weight loss
+
+250
+00:20:23,590 --> 00:20:27,870
+feeling weak or very tired، دائما تعبان، دائما
+
+251
+00:20:27,870 --> 00:20:33,420
+هلكان، unusual bleeding or dischargeالستات، بنزل
+
+252
+00:20:33,420 --> 00:20:37,040
+دايما عندها، وفي الأخر بتقول استحاضة مثلا، لأ،
+
+253
+00:20:37,040 --> 00:20:40,500
+بدها تاخد بالها، بدها تفحص، خاصة كل أو discharge
+
+254
+00:20:40,500 --> 00:20:45,980
+أو سوائل، خاصة كل ما كبرت في العمر، a sore that
+
+255
+00:20:45,980 --> 00:20:49,380
+does not heal، جرح ومش راضي يطيب، ممكن يكون سكري،
+
+256
+00:20:49,380 --> 00:20:54,460
+أه، بس ممكن يكون برضه حاجة تانية، hoarseness، بحة
+
+257
+00:20:54,460 --> 00:20:58,060
+أو cough that does not go away، مش راضي تحسن الكحة
+
+258
+00:21:00,460 --> 00:21:02,000
+التغييرات في الـ bowel أو العبادات الدمية
+
+259
+00:21:05,300 --> 00:21:09,020
+discomfort after eating دائما بعد الأكل تعبان ومش
+
+260
+00:21:09,020 --> 00:21:11,800
+مرتاح a hard time swallowing يا جماعة هذه الـ
+
+261
+00:21:11,800 --> 00:21:13,880
+symptoms ممكن تكون symptoms لحاجة تانية زي بيبتك
+
+262
+00:21:13,880 --> 00:21:17,960
+ألصر زي كذا لكن مش شرط أنه يكون cancer لكن لازم
+
+263
+00:21:17,960 --> 00:21:21,360
+أحط الموضوع في بالي most often these symptoms are
+
+264
+00:21:21,360 --> 00:21:24,380
+not due to cancer they may be caused by non
+
+265
+00:21:24,380 --> 00:21:27,660
+cancers or benign tumors or other problems بس بدنا
+
+266
+00:21:27,660 --> 00:21:32,820
+ناخد بالنا منهم ال cancer يا جماعة أو ال massإله
+
+267
+00:21:32,820 --> 00:21:37,520
+أكتر من تأثير يعني هو مش بس ع جد مشبوك في حاله
+
+268
+00:21:37,520 --> 00:21:44,370
+إنما هوبعملي مشاكل Locally بعملي Mass Effect زي
+
+269
+00:21:44,370 --> 00:21:47,150
+الـ Lung Cancer بيعملي Blockage of the Bronchus
+
+270
+00:21:47,150 --> 00:21:51,110
+Resulting in Cough أو Pneumonia يعني هو بتغطلي على
+
+271
+00:21:51,110 --> 00:21:54,550
+أماكن تانية Esophageal Cancer can cause Narrowing
+
+272
+00:21:54,550 --> 00:21:58,390
+of the Esophagus Making difficult or painful to
+
+273
+00:21:58,390 --> 00:22:01,650
+swallow هذا المشكلة Correctal Cancer ممكن يعملي
+
+274
+00:22:01,650 --> 00:22:09,240
+Narrowing أو Blockage of the bowelمثلما يبين على
+
+275
+00:22:09,240 --> 00:22:13,880
+طول فيه جروح و نزيق دم عشان هيك لو ما بنلاقي دم
+
+276
+00:22:13,880 --> 00:22:18,380
+لازم ناخد بانا if the lung will lead to a coughing
+
+277
+00:22:18,380 --> 00:22:23,280
+of blood الـ hemoptysis in the bowel بيصير
+
+278
+00:22:23,280 --> 00:22:26,160
+اندوانيميا و rectal bleeding in the bladder blood
+
+279
+00:22:26,160 --> 00:22:29,580
+in the urine hematuria in the uterus بديني vaginal
+
+280
+00:22:29,580 --> 00:22:42,150
+bleeding و في الأخر بطلع ورا حاجة كبيرةأي ورم
+
+281
+00:22:42,150 --> 00:22:53,750
+بيجعش بيخوف أكتر من اللي بيجعقريب من قاعدة بعض
+
+282
+00:22:53,750 --> 00:22:57,230
+الأنواع الـ cancers بتعمل إيه؟ اللي هو زي ascites
+
+283
+00:22:57,230 --> 00:22:59,990
+اللي هو fluid in the abdomen أو pleural effusion
+
+284
+00:22:59,990 --> 00:23:05,500
+زي in the pleural cavity يكون فيه عنده fluidطبعاً
+
+285
+00:23:05,500 --> 00:23:08,720
+إلو إلو الـ Cancer إحنا بنحكي حاجات قاعدين
+
+286
+00:23:08,720 --> 00:23:11,600
+generalize يا جماعة general، معلومات عامة اللي
+
+287
+00:23:11,600 --> 00:23:14,880
+بنحكيها general symptoms occur due to distant
+
+288
+00:23:14,880 --> 00:23:18,140
+effects of the cancer that are not related to
+
+289
+00:23:18,140 --> 00:23:23,180
+direct or metastatic sprintيعني الـ General
+
+290
+00:23:23,180 --> 00:23:26,480
+Systemic Symptoms اللي هو أنا يكون عندي Weakness
+
+291
+00:23:26,480 --> 00:23:31,000
+على سبيل المثال أو Headache أو زي ما قلنا ينزل
+
+292
+00:23:31,000 --> 00:23:35,120
+وزني أو عندي Fever بدون ما يكون فيه Metastasis هذه
+
+293
+00:23:35,120 --> 00:23:40,800
+ما بيقولها أنا Systemic Symptoms بجمبها وبدنا
+
+294
+00:23:40,800 --> 00:23:44,620
+ناخد بالنا لو أجاني حد بـPersistent Unknown Cause
+
+295
+00:23:44,620 --> 00:23:50,780
+of Fever ممكن يكون وراهايكون وراها لويكيميا أو
+
+296
+00:23:50,780 --> 00:23:53,880
+cancer of the liver أو kidney ده ناخد بالنا من قبل
+
+297
+00:23:53,880 --> 00:24:03,240
+يعنى الـ metastases أو الـ mets هي عبارة can
+
+298
+00:24:03,240 --> 00:24:08,370
+spread its original sideبالـ Local Spread يعني هي
+
+299
+00:24:08,370 --> 00:24:12,870
+بتعمل الـ Mets أو الـ Metastases أو البنات أو
+
+300
+00:24:12,870 --> 00:24:16,410
+الأخوات هدول اللي بنتشروا من الـ Cancer لباقية
+
+301
+00:24:16,410 --> 00:24:20,050
+الجسم بعدة طرق إما Local Spread بتغطي على اللي
+
+302
+00:24:20,050 --> 00:24:23,470
+جنبه أو Lymphatic Spread عن طريق الـ Lymph Nodes
+
+303
+00:24:23,470 --> 00:24:31,530
+أو Hematogenous Spread اللي هو عن طريق الدم وبروحه
+
+304
+00:24:31,530 --> 00:24:36,760
+على two distant sites أماكن بعيدةيعني بلاقي عصب
+
+305
+00:24:36,760 --> 00:24:40,380
+المثال من الكبد بروح أدماغ أو من الـ lung بروح
+
+306
+00:24:40,380 --> 00:24:45,480
+أدماغ وهذه بنقول عنها metastases أو metes the new
+
+307
+00:24:45,480 --> 00:24:50,420
+tumor are called metastatic tumors اللي بيصير جديد
+
+308
+00:24:50,420 --> 00:24:55,260
+لكن ال original is called a primary tumor يعني ال
+
+309
+00:24:55,260 --> 00:25:00,860
+primary tumor هو الأصل واللي راحله بنقول عنه
+
+310
+00:25:00,860 --> 00:25:03,900
+metastatic tumor almost all cancers can
+
+311
+00:25:03,900 --> 00:25:08,080
+metastasizemost cancer deaths are due to cancer
+
+312
+00:25:08,080 --> 00:25:11,340
+that has spread from its primary site to other
+
+313
+00:25:11,340 --> 00:25:14,560
+organs يعني معظم الحالات اللي بتتوفى وبيصير عندها
+
+314
+00:25:14,560 --> 00:25:17,960
+مشاكل نتيجة الـ metastases الـ mets different
+
+315
+00:25:17,960 --> 00:25:22,060
+types of cancer tend to metastase to particular
+
+316
+00:25:22,060 --> 00:25:25,560
+organs طبعا معروف عند الأطباء والله إذا كان عندى
+
+317
+00:25:25,560 --> 00:25:29,140
+tumor في المكان اللي فلاني الأماكن المحببة اللي
+
+318
+00:25:29,140 --> 00:25:34,060
+بيبعت عليها mets هي كذا وكذا وكذا وكذا most common
+
+319
+00:25:34,060 --> 00:25:38,580
+placesوهذه الأماكن اللي بيروحلها لـLung,
+
+320
+00:25:38,680 --> 00:25:48,600
+Liver, Brain and Bone عندما
+
+321
+00:25:48,600 --> 00:25:52,540
+يتفرق الحرارة بطريقة هيموتوجينية، يتفرق بشكل عام
+
+322
+00:25:52,540 --> 00:25:59,560
+للجميع على جسمممكن يصير عندى إنلاج love node، ممكن
+
+323
+00:25:59,560 --> 00:26:02,200
+يكون عندى إنلاج liver، ممكن يكون عندى إنلاج
+
+324
+00:26:02,200 --> 00:26:07,440
+spleen، ممكن يصير عندى نتيجة متس هاي pain أو
+
+325
+00:26:07,440 --> 00:26:10,500
+fracture في ال bone، ممكن يكون عندى لو راق على ال
+
+326
+00:26:10,500 --> 00:26:15,360
+brain يعملني neurological symptoms كيف بشخص؟ chest
+
+327
+00:26:15,360 --> 00:26:18,420
+x-ray، showing lung cancer
+
+328
+00:26:20,980 --> 00:26:24,080
+عشان المثال، أكثر المصابين بالنسبة لي هم المصابين
+
+329
+00:26:24,080 --> 00:26:29,540
+بسبب وجود إشارات أو إشارات أو من خلال الـ
+
+330
+00:26:29,540 --> 00:26:34,800
+Screening طبعاً بدنا ندّي دايما، ندّي دايما نعمل
+
+331
+00:26:34,800 --> 00:26:38,860
+صور حسب وين الشكوى، يعني واحد بيشكي من severe
+
+332
+00:26:38,860 --> 00:26:42,020
+headache وrecurrent و projectile vomiting، بده يشك
+
+333
+00:26:42,020 --> 00:26:44,720
+في brain tumor، بده يعمله CT مثلا
+
+334
+00:26:52,010 --> 00:26:56,790
+إذا أي واحد من هؤلاء لم يصبح لديه نتيجة تقريبًا
+
+335
+00:26:56,790 --> 00:26:57,890
+لديه نتيجة تقريبًا لديه نتيجة تقريبًا لديه نتيجة
+
+336
+00:26:57,890 --> 00:27:00,130
+تقريبًا لديه نتيجة تقريبًا لديه نتيجة تقريبًا لديه
+
+337
+00:27:00,130 --> 00:27:00,530
+نتيجة تقريبًا لديه نتيجة تقريبًا لديه نتيجة
+
+338
+00:27:00,530 --> 00:27:00,710
+تقريبًا لديه نتيجة تقريبًا لديه نتيجة تقريبًا لديه
+
+339
+00:27:00,710 --> 00:27:01,350
+نتيجة تقريبًا لديه نتيجة تقريبًا لديه نتيجة
+
+340
+00:27:01,350 --> 00:27:04,650
+تقريبًا لديه نتيجة تقريبًا لديه نتيجة تقريبًا لدي
+
+341
+00:27:07,380 --> 00:27:11,460
+والذين يشبهوا القنصر يتم تحقيقهم بمجال مجال مجال
+
+342
+00:27:11,460 --> 00:27:12,420
+مجال مجال مجال مجال مجال مجال مجال مجال مجال مجال
+
+343
+00:27:12,420 --> 00:27:16,840
+مجال مجال مجال مجال مجال مجال مجال مجال مجال مجال
+
+344
+00:27:16,840 --> 00:27:17,260
+مجال مجال مجال مجال مجال مجال مجال مجال مجال مجال
+
+345
+00:27:17,260 --> 00:27:17,540
+مجال مجال مجال مجال مجال مجال مجال مجال مجال مجال
+
+346
+00:27:17,540 --> 00:27:17,540
+مجال مجال مجال مجال مجال مجال مجال مجال مجال مجال
+
+347
+00:27:17,540 --> 00:27:18,180
+مجال مجال مجال مجال مجال مجال مجال مجال مجال مجال
+
+348
+00:27:18,180 --> 00:27:33,680
+مجال مجال مجال مجال مج
+
+349
+00:27:35,000 --> 00:27:38,350
+لو فيه symptoms مش screening بيصير diagnosisيعني
+
+350
+00:27:38,350 --> 00:27:41,590
+واحد ماعندهوش ولي حاجة صار عمره مثلا ست صار عمرها
+
+351
+00:27:41,590 --> 00:27:45,090
+خمسة وعشرين أو تلاتين الـ Screening تبعها إنها
+
+352
+00:27:45,090 --> 00:27:48,930
+تعمل التحسيس لحالها أو الدكتورة تشوفيها تحت القفات
+
+353
+00:27:48,930 --> 00:27:52,410
+وكذا وهيك هذا بقول عنه Screening لكن والله أجد
+
+354
+00:27:52,410 --> 00:27:54,890
+الست تقول والله يا دكتورة أنا حاسة زي كتلة اندي في
+
+355
+00:27:54,890 --> 00:27:57,930
+سدري وفحصة الدكتورة هذا ما اسمهش Screening هذا
+
+356
+00:27:57,930 --> 00:28:01,130
+اسمه Diagnosis بيصير خلاص Screening may involve
+
+357
+00:28:01,130 --> 00:28:05,050
+physical examination lab test or test to look at
+
+358
+00:28:05,050 --> 00:28:12,730
+internal organsHow is cancer treated؟ في كتير
+
+359
+00:28:12,730 --> 00:28:16,690
+علاجات، Surgery, Radiation therapy, Chemotherapy،
+
+360
+00:28:16,690 --> 00:28:21,710
+Anticancer drugs and Palliative for some cancers
+
+361
+00:28:29,990 --> 00:28:34,290
+Okay يعني أنا هوقف عند الـ slide هاي وإن شاء الله
+
+362
+00:28:34,290 --> 00:28:38,370
+تعالى آخر محاضرة هنعملها على ال weekend هبعتلكوا
+
+363
+00:28:38,370 --> 00:28:40,870
+إياها إن شاء الله تعالى اللي هي هنحكي عن ال
+
+364
+00:28:40,870 --> 00:28:46,950
+leukemia خمس تاشر slide يمكن بالنسبة للمحاضرة هي
+
+365
+00:28:46,950 --> 00:28:49,110
+يا جماعة إحنا لما أديتكوا إياها أديتكوا إياها
+
+366
+00:28:49,110 --> 00:28:52,770
+كاملة بس إحنا هنضلنا بس ماشيين لحد slide تلاتة
+
+367
+00:28:52,770 --> 00:29:00,620
+وتلاتين بس ل slide تلاتة وتلاتينومش هدخل على
+
+368
+00:29:00,620 --> 00:29:06,100
+التفاصيل بس هنوقف لحد slide تلاتة و تلاتين في هذه
+
+369
+00:29:06,100 --> 00:29:11,400
+وبيفهم بدي أكمل على الـ Leukemia برضه عشر، خمسة
+
+370
+00:29:11,400 --> 00:29:14,660
+أشهر slide وبعدين هيك بيبجنتها إن شاء الله تعالى
+
+371
+00:29:14,660 --> 00:29:20,680
+المنهج بإذن الله تعالى يعني أشكركم على حسن
+
+372
+00:29:20,680 --> 00:29:24,900
+الاستماع في الفترة الجاي هديكوا شوية تفاصيل كيف
+
+373
+00:29:24,900 --> 00:29:30,980
+الامتحانومن وين هيكون و هيك ان شاء الله انتظرونا
+
diff --git a/PL9fwy3NUQKwaf4aPlmtIQXiASpDxVLEnO/xJGkNKjINVE.srt b/PL9fwy3NUQKwaf4aPlmtIQXiASpDxVLEnO/xJGkNKjINVE.srt
new file mode 100644
index 0000000000000000000000000000000000000000..8eacfe8d1d6b4fb97610d96c545d68eeed57e198
--- /dev/null
+++ b/PL9fwy3NUQKwaf4aPlmtIQXiASpDxVLEnO/xJGkNKjINVE.srt
@@ -0,0 +1,1767 @@
+1
+00:00:00,000 --> 00:00:02,800
+بسم الله الرحمن الرحيم الحمد لله رب العالمين
+
+2
+00:00:02,800 --> 00:00:06,280
+والصلاة والسلام علي سيدنا محمد سيد المرسلين أجمعين
+
+3
+00:00:06,280 --> 00:00:10,020
+يعني اليوم إن شاء الله تعالى هندخل على موضوع ثاني
+
+4
+00:00:10,020 --> 00:00:17,320
+موضوع طويل موضوع مهم موضوع عمليًا في كل بيت موجود
+
+5
+00:00:17,320 --> 00:00:21,200
+إن ما كانش في كل بيت في البيت اللي جنبه أوي يعني
+
+6
+00:00:21,200 --> 00:00:24,560
+يعني بمعنى إنه من كل بيتين هتلاقوا في بيت حد عنده
+
+7
+00:00:24,560 --> 00:00:30,240
+سكري كانوا زمان يقولوا عنه سكر البول، لكن الآن هو
+
+8
+00:00:30,240 --> 00:00:34,520
+السكري، سكر البول، لإنه كانوا زمان أول ما يكتشفوه 
+
+9
+00:00:34,520 --> 00:00:39,020
+يكتشفوه من كثرة التبول وإنه كان فيه سكر في البول
+
+10
+00:00:39,020 --> 00:00:44,260
+نتيجة تهريب الكلى للسكر لزيادة على أي حال الـ
+
+11
+00:00:44,260 --> 00:00:50,640
+diabetes mellitus الديابتس ميليتس، طيب هو أنا لما
+
+12
+00:00:50,640 --> 00:00:54,760
+أقول والله فلان عنده الديابتس، هل بقصد فيه بس الـ
+
+13
+00:00:54,760 --> 00:00:59,300
+diabetes mellitus السكري العادي تبعنا؟ أه، لكن في
+
+14
+00:00:59,300 --> 00:01:05,660
+ديابتس انسبيدس، نوع ثاني، ديابتس انسبيدس حاجة ثانية هي 
+
+15
+00:01:05,660 --> 00:01:11,740
+صحيح مرتبطة بكثرة البول برضه، لكن شيء ثاني يختلف عن
+
+16
+00:01:11,740 --> 00:01:18,020
+الـ diabetes mellitus، صح كلامكم مية في المية، الـ
+
+17
+00:01:18,020 --> 00:01:24,140
+assignment تبعنا مجسوم من حاجتين اليوم، أول حاجة في
+
+18
+00:01:24,140 --> 00:01:28,860
+نصف صفحة مصورة، تصوير مكتوبة بالإيد ومصورة، تصوير
+
+19
+00:01:28,860 --> 00:01:35,000
+تحكولي عن الـDiabetes Encephalitis، ما هو أنواعه
+
+20
+00:01:35,000 --> 00:01:39,180
+وطرق علاجه، كلمات يا جماعة، بديش شرح، شرح ملزمنيش
+
+21
+00:01:39,180 --> 00:01:44,500
+موجود في الكتب، الشرح، اجروا أنتم كثير واطلعولي بملخص 
+
+22
+00:01:44,500 --> 00:01:47,220
+صغير عن الـDiabetes Encephalitis بالحاجات اللي أنا
+
+23
+00:01:47,220 --> 00:01:50,320
+ذكرتها، عن الـEncephalitis، الـDiabetes Melitus أنا
+
+24
+00:01:50,320 --> 00:01:53,780
+هحكي عنه اللي هو السكري، هذا الجزء الأول من الـ
+
+25
+00:01:53,780 --> 00:01:57,660
+Assignment، هنحكي عن الجزء الثاني بعد شوية من الـ
+
+26
+00:01:57,660 --> 00:02:02,280
+Assignment، الـ Diabetes Militas الاختصار تبعه DM
+
+27
+00:02:02,280 --> 00:02:08,880
+Diabetes DM Militas وهو عبارة عن Metabolic Disease
+
+28
+00:02:08,880 --> 00:02:14,320
+إيش يعني Metabolic؟ Metabolic المقصود فيه إنه يختص
+
+29
+00:02:14,320 --> 00:02:21,050
+بموضوع الاستقلاب الخلية، مش حاجة من بعيد ورم ولا إيد
+
+30
+00:02:21,050 --> 00:02:26,750
+مقصورة ولا دهر ولا desk لأ، إنما حاجة جوا الخلية
+
+31
+00:02:26,750 --> 00:02:33,570
+بتسير، هذا مقصود في metabolic، واللي أهم معلم بنشوفه 
+
+32
+00:02:33,570 --> 00:02:38,970
+فيها وبدونه ما ينفعش أقول diabetes high blood sugar
+
+33
+00:02:38,970 --> 00:02:44,250
+level في الدم over، خدوا بالكم الجمعة الجملة هاي 
+
+34
+00:02:44,250 --> 00:02:48,200
+prolonged period، نفس الكلام اللي حكيناه عن مين عن 
+
+35
+00:02:48,200 --> 00:02:50,720
+الـ hypertension، مش مابقول أنا أجيس لواحد مرة ضغط 
+
+36
+00:02:50,720 --> 00:02:53,340
+علاجية عالية أقول له يا أبو الشباب والله شكلك عندك
+
+37
+00:02:53,340 --> 00:02:57,660
+ضغط، أو والله عملتله فحص سكر اللي جاته مرة عالي لأ،
+
+38
+00:02:57,660 --> 00:03:00,500
+لها شروط، هذه الجذيع وهذا هنحكي عنه في المحاضرة 
+
+39
+00:03:00,500 --> 00:03:06,660
+يبقى prolonged period، لازم عدة قياسات في عدة أوقات
+
+40
+00:03:06,660 --> 00:03:11,720
+مختلفة وعدة للسكر طبعًا ساعتها بقول عنه diabetes
+
+41
+00:03:11,720 --> 00:03:15,730
+and diabetes، هنحكي عن الموضوع هذا خلال خلال
+
+42
+00:03:15,730 --> 00:03:23,770
+المحاضرة، الـ diabetes مشكلته multi system بمعنى
+
+43
+00:03:23,770 --> 00:03:29,490
+ما فيش مكان في الجسم إلا بيغزوه، هذه خطورته إنه هو
+
+44
+00:03:29,490 --> 00:03:39,750
+بيأثر لي تأثيرات مباشرة، إنه
+
+45
+00:03:39,750 --> 00:03:44,710
+هو بيأثر لي تأثيرات مباشرة على العين،
+
+46
+00:03:46,660 --> 00:03:50,720
+بيأثر لي تأثيرات مباشرة على الدورة الدموية، على 
+
+47
+00:03:50,720 --> 00:03:54,960
+القلب، على الكلى، على الشرايين التاجية، على
+
+48
+00:03:54,960 --> 00:03:58,080
+الأعصاب التاجية، طب وإيش ضال في البني آدم؟ أه ما
+
+49
+00:03:58,080 --> 00:04:03,240
+هي مشكلة السكر، السكر دمار، السكري دمار، طيب هو
+
+50
+00:04:04,600 --> 00:04:08,680
+شوّقتنا، يعني السكري يعني هو بيصيب كل واحد ولا كيف
+
+51
+00:04:08,680 --> 00:04:11,820
+الموضوع ولا إيش اللي بيصير، هذا اللي هنحكي عنه وهذا
+
+52
+00:04:11,820 --> 00:04:15,380
+اللي لازم أنتم تكونوا جزء من الموضوع، أنا بحكي في 
+
+53
+00:04:15,380 --> 00:04:19,920
+ثاني أهم موضوع في الباطنة اللي هو الـ Diabetes
+
+54
+00:04:19,920 --> 00:04:23,740
+Mellitus، كمان مرة، طيب اللي هو إيش اللي بيصير
+
+55
+00:04:23,740 --> 00:04:27,760
+بالظبط، حاجة من حاجتين، هنحكي بعد هيك إحنا كيف الـ
+
+56
+00:04:27,760 --> 00:04:31,600
+Type 1 و Type 2، لكن من الأول إنه بيكون عند
+
+57
+00:04:31,600 --> 00:04:37,740
+الإنسولين، الإنسولين عبارة عن هرمون هذا شغله ينظم 
+
+58
+00:04:37,740 --> 00:04:47,200
+دخول السكر للخلية، للخلية الجسدية ينظم دخول السكر
+
+59
+00:04:47,200 --> 00:04:52,460
+للخلية، الان ممكن يكون عنده الانسولين production
+
+60
+00:04:52,460 --> 00:04:56,420
+abnormal، قليل
+
+61
+00:04:56,420 --> 00:05:01,780
+كثير أو ما فيش، فمين ينظم دخول السكر للخلية؟ يبقى
+
+62
+00:05:01,780 --> 00:05:06,200
+السكر وين بيضل؟ بيضل في الدم، فاهمين عليه كيف؟ يعني
+
+63
+00:05:06,200 --> 00:05:12,740
+وكأنه الخلية غرفة ولها باب وفي مفتاح، المفتاح عشان 
+
+64
+00:05:12,740 --> 00:05:16,860
+يدخل السكر الغرفة هو الـ Insulin، طب وإذا المفتاح
+
+65
+00:05:16,860 --> 00:05:20,280
+ضاع وما فيش مفتاح كيف بدي يدخل الغرفة السكر؟ مش
+
+66
+00:05:20,280 --> 00:05:24,000
+هيدخل الغرفة، هيضل وين؟ هيضل برة، وين برة؟ في
+
+67
+00:05:24,000 --> 00:05:27,540
+الشرايين يعني، وعشان هيك لما نعمل فحص السكر بيقول لك
+
+68
+00:05:27,540 --> 00:05:35,180
+السكر عالي لإنه الجسم ما أخذش، ما أخذش السكر، الحاجة
+
+69
+00:05:35,180 --> 00:05:38,320
+الثانية اللي ممكن تحصل الـ Impaired إحنا بنحكي عن
+
+70
+00:05:38,320 --> 00:05:41,820
+Abnormal Insulin Production أو Impaired Insulin
+
+71
+00:05:41,820 --> 00:05:48,600
+Utilization، يعني عملية إنه كيف يدبر نفسه السكر مع 
+
+72
+00:05:48,600 --> 00:05:51,920
+الخلية في فشل أو الاثنين أو الموضوعين الموجودين
+
+73
+00:05:51,920 --> 00:05:56,240
+يعني نقص في السكر وإنه مش قادر يدخل نتيجة مشكلة
+
+74
+00:05:56,240 --> 00:05:59,020
+الـ Receptor مثلًا على سبيل المثال، هنحكي عن الموضوع
+
+75
+00:05:59,020 --> 00:06:05,560
+يا جماعة موضوع السكر ما هواش بس إنه والله فلان دمه
+
+76
+00:06:05,560 --> 00:06:11,000
+حلو وخفيف وعنده سكر، لأ، لأ، لأ المشكلة إنه هو
+
+77
+00:06:11,000 --> 00:06:14,700
+الـleading cause of heart disease، اتصوروا إنه
+
+78
+00:06:14,700 --> 00:06:18,500
+مشاكل القلب، تلاحظين الآن وإحنا بنحكي blood
+
+79
+00:06:18,500 --> 00:06:20,840
+pressure ال hypertension المرة اللي فاتت، المرة
+
+80
+00:06:20,840 --> 00:06:24,060
+اللي هي السكر وبنربطها في heart disease وبنربطها
+
+81
+00:06:24,060 --> 00:06:30,080
+في ال stroke، حتى الضغط ربطنا بال stroke، الآن
+
+82
+00:06:30,080 --> 00:06:33,380
+بدخلني على الـ Adult Blindness العمى تبع الناس
+
+83
+00:06:33,380 --> 00:06:36,860
+الكبار، من إيش إنعمى؟ كان زمان العمى يصير نتيجة
+
+84
+00:06:36,860 --> 00:06:40,760
+التهابات وكذا، الآن لأ، نتيجة سكري، ليش؟ لإنه
+
+85
+00:06:40,760 --> 00:06:44,300
+بيصير عندنا Retinal Detachment، هذا شفناه وين؟ قبل 
+
+86
+00:06:44,300 --> 00:06:46,980
+أكمّل المحاضرة اللي فاتت أو اللي قبلها، برضه في
+
+87
+00:06:46,980 --> 00:06:51,360
+الضغط، وفيه كمان حاجة جديدة، Non-traumatic Lower
+
+88
+00:06:51,360 --> 00:06:55,500
+limb amputation، طول عمرنا بنسمع والله مسيرات
+
+89
+00:06:55,500 --> 00:06:58,940
+العودة، جسوله رجله، لأ، الآن إحنا بنقول إنه non
+
+90
+00:06:58,940 --> 00:07:02,440
+-traumatic، لا حادث سيارة، ولا مسيرات عودة، ولا
+
+91
+00:07:02,440 --> 00:07:07,040
+وجع من فوق وانقطعت رجله، ولا حاجة، لأ، هذه نتيجة
+
+92
+00:07:07,040 --> 00:07:10,540
+لما لاقيت راجل كبير انقطعت جديد رجله، بقول هذا في
+
+93
+00:07:10,540 --> 00:07:13,840
+الغالب، في الغالب يبدو إن عندك سكر، بيبقى عنده سكر
+
+94
+00:07:14,650 --> 00:07:18,790
+إذا السكر، طب يعني هو كل الدنيا كل واحد عنده سكر
+
+95
+00:07:18,790 --> 00:07:22,090
+يبتعالج، هو المفروض إنه يتعالج لإنه if left
+
+96
+00:07:22,090 --> 00:07:25,710
+untreated إذا انتركت، diabetes can cause many
+
+97
+00:07:25,710 --> 00:07:28,790
+complications، الآن طبعًا أنا يا جماعة قاعد بحكي
+
+98
+00:07:28,790 --> 00:07:33,090
+عموميات، هدخل على الخصوصيات بعد شوية من الـ Acute
+
+99
+00:07:33,090 --> 00:07:35,530
+Complication، هذه الـ Chronic Complication اللي
+
+100
+00:07:35,530 --> 00:07:38,090
+حكيناها قبل شوية اللي هو الـ Heart Disease و الـ
+
+101
+00:07:38,090 --> 00:07:40,210
+Stroke و الـ Blindness و الـ Non-Traumatic Lower
+
+102
+00:07:40,210 --> 00:07:45,590
+Limb Amputation، وهذه كلها Complication طويلة
+
+103
+00:07:45,590 --> 00:07:48,890
+الأمد، لكن فيه Complications قصيرة الأمد، بيقول
+
+104
+00:07:48,890 --> 00:07:52,290
+عنها Acute Complication، يعني أنا إذا سكر صار 500
+
+105
+00:07:52,290 --> 00:07:56,750
+ولا 600 بدخل في حاجة اسمها diabetic ketoacidosis
+
+106
+00:07:56,750 --> 00:08:02,330
+بيصير عنده حموضة في الدم نتيجة ارتفاع كمية السكر
+
+107
+00:08:02,330 --> 00:08:05,590
+أو بيقول أنا DKA كمية السكر في الدم، هنحكي عنها
+
+108
+00:08:05,590 --> 00:08:09,310
+بعد شوية، بس خلوها في بالكم acute complication
+
+109
+00:08:09,310 --> 00:08:14,350
+diabetic ketoacidosis، الـ Serious Long Term
+
+110
+00:08:14,350 --> 00:08:16,970
+Complications تحدثنا عنها قبل شوية مشاكل الـ
+
+111
+00:08:16,970 --> 00:08:22,110
+cardiovascular stroke chronic kidney failure foot
+
+112
+00:08:22,110 --> 00:08:28,130
+ulcer damage to the eyes، الوضع الطبيعي إنه أنا باجي
+
+113
+00:08:28,130 --> 00:08:32,170
+بفطر وبتغدى وبتعشى، breakfast lunch and dinner في
+
+114
+00:08:32,170 --> 00:08:35,250
+الفترة هذه بيطلع الانسولين بيزيد الانسولين في الدم
+
+115
+00:08:35,250 --> 00:08:39,210
+ليش؟ عشان الأكل اللي أكلته اللي قاعد بنهضم إحنا
+
+116
+00:08:39,210 --> 00:08:43,090
+عارفين يا جماعة إنه أي أكل أنا باكله بتكون من تلت 
+
+117
+00:08:43,090 --> 00:08:50,330
+أنواع من الأكل، ال carbohydrate والبروتينات 
+
+118
+00:08:50,330 --> 00:08:57,660
+والدهنيات، الدهنيات تتحول لـ Fatty Acids البروتينات
+
+119
+00:08:57,660 --> 00:09:02,120
+تتحول لـ Amino Acids، لكن الـ Carbohydrate تتحول
+
+120
+00:09:02,120 --> 00:09:12,970
+لإيش؟ لسكر، لجلوكوز، الجلوكوز هذا الآن بده مفاتيح
+
+121
+00:09:12,970 --> 00:09:16,510
+تدخله على الخلايا بعد ما اتحلل الأكل للآخر في 
+
+122
+00:09:16,510 --> 00:09:20,390
+الجهاز الهضمي تم امتصاصه في الدم، الدم بتودي للـ
+
+123
+00:09:20,390 --> 00:09:26,130
+cells، الآن بدها مفاتيح، مين المفاتيح، الانسولين 
+
+124
+00:09:26,130 --> 00:09:30,750
+بيسير بطريقة أو باخرة بيطلع الانسولين في الجسم و
+
+125
+00:09:30,750 --> 00:09:37,500
+بفتح الخلايا عشان السكر يدخل أو الجلوكوز يدخل على
+
+126
+00:09:37,500 --> 00:09:41,580
+الخلايا، لو ما فيش انسولين بيضل السكر وين؟ مرة
+
+127
+00:09:41,580 --> 00:09:45,820
+ثانية بيضل في الشرايين، يبقى في عندي أنا
+
+128
+00:09:45,820 --> 00:09:51,700
+breakfast، lunch، dinner، وهذه بيصير فيها زيادة أو
+
+129
+00:09:51,700 --> 00:09:58,680
+إفراز لهرمون الانسولين واللي بيحاول يحافظ لي على
+
+130
+00:09:58,680 --> 00:10:05,700
+كميات على مستوى سكر طبيعي في الدم يتراوح بين ثلاثة
+
+131
+00:10:05,700 --> 00:10:10,400
+وتسعة من عشرة لستة وسبعة من عشرة ملي مول في
+
+132
+00:10:10,400 --> 00:10:14,960
+الليتر، إيه الشاي ملي مول، طول عمرنا بنقول والله
+
+133
+00:10:14,960 --> 00:10:23,970
+فلان السكر تبعه مية وتسعين وثمانين 105 سكر طبيعي يطلع
+
+134
+00:10:23,970 --> 00:10:29,310
+من وين بـ 3.9 و 6.7 ملي مول بالليتر، هذه الحسبة 
+
+135
+00:10:29,310 --> 00:10:33,630
+الأمريكية، في الحسبة التي تبعتنا فيها طريقة للحسبة
+
+136
+00:10:33,630 --> 00:10:39,810
+كيفية تحويل الملي مول بالليتر لملي جرام 
+
+137
+00:10:39,810 --> 00:10:44,030
+بالديسليتر اللي إحنا بنحسب فيها نضربها في 17 مش
+
+138
+00:10:44,030 --> 00:10:48,950
+حتى أعبكوا كثير نضربها في 17 بيطلع الحسبة تبعتنا
+
+139
+00:10:52,350 --> 00:10:59,390
+الـ Normal Insulin Metabolism، يدخل
+
+140
+00:10:59,390 --> 00:11:04,970
+للجلوكوز يسهل للجلوكوز يتبع من مصادر الدم من خلال 
+
+141
+00:11:04,970 --> 00:11:08,710
+مصادر الجلوكوز إلى مصادر الجلوكوز من خلال مصادر
+
+142
+00:11:08,710 --> 00:11:09,390
+الجلوكوز من خلال مصادر الجلوكوز من خلال مصادر
+
+143
+00:11:09,390 --> 00:11:13,390
+الجلوكوز من خلال مصادر الجلوكوز من خلال مصادر
+
+144
+00:11:13,390 --> 00:11:16,370
+الجلوكوز من خلال مصادر الجلوكوز من خلال مصادر
+
+145
+00:11:16,370 --> 00:11:16,470
+الجلوكوز من خلال مصادر الجلوكوز من خلال مصادر
+
+146
+00:11:16,470 --> 00:11:16,710
+الجلوكوز من خلال مصادر الجلوكوز من خلال مصادر
+
+147
+00:11:16,710 --> 00:11:18,600
+الجلوكوز من خلال مصادر الجلوكوز من خلاله،  آن بعد ما
+
+148
+00:11:18,600 --> 00:11:22,160
+أكل، بيزيد الإنسولين، إيش بيعمل الإنسولين؟ شغلته 
+
+149
+00:11:22,160 --> 00:11:27,660
+الطبيعية، stimulate the storage of glucose as glycogen
+
+150
+00:11:27,660 --> 00:11:33,900
+طبعاً الخلية بتستهلك ما تحتاج، قولي بيزيد، بتروح، 
+
+151
+00:11:33,900 --> 00:11:40,520
+بتخزنه في الكبد، كـ glycogen، هذا شغلته يعني 
+
+152
+00:11:40,520 --> 00:11:45,790
+يعني العكس تمام لو ما فيش إنسولين، إيش بيعمل كمان؟
+
+153
+00:11:45,790 --> 00:11:50,510
+Inhibits gluconeogenesis، إيش يعني gluconeogenesis؟
+
+154
+00:11:50,510 --> 00:11:57,410
+Gluconeogenesis هو عبارة عن تحويل البروتين لسكر، هو 
+
+155
+00:11:57,410 --> 00:12:01,130
+بيوقف الموضوع هذا لأن عندنا سكر كفاية فهو بيوقفها 
+
+156
+00:12:01,130 --> 00:12:07,110
+الأمر الثاني بيزيد fat deposition in adipose
+
+157
+00:12:07,110 --> 00:12:13,110
+tissue، بيزيد اللي هي تخزين الدهونيات في مناطق
+
+158
+00:12:13,110 --> 00:12:17,210
+الدهنية، هذا شغل الإنسولين، increase protein
+
+159
+00:12:17,210 --> 00:12:21,870
+synthesis، اللي بيزيد يعني هو من الشغلات تبعته أنه
+
+160
+00:12:21,870 --> 00:12:30,370
+بيبني البروتين، طيب هو أنا كيف بدي أعرف أن فلان عنده
+
+161
+00:12:30,370 --> 00:12:35,690
+سكري؟ كيف بدي أعرف؟ في حاجات نعرفها في اللسان تبعنا
+
+162
+00:12:35,690 --> 00:12:41,450
+اللسان العادي، كثير من الناس بتيجي شايفة زميلتك أو
+
+163
+00:12:41,450 --> 00:12:48,430
+صاحبتك أو أي حد بتعرفيه أو بتعرفها شفته قبل فترة و
+
+164
+00:12:48,430 --> 00:12:51,830
+اليوم شفته إيه، ما وزنك نازل هيك مرة واحدة، إيش 
+
+165
+00:12:51,830 --> 00:12:57,370
+أول حاجة بتفكر فيها على طول، اه أيوة احكوا إيش لأ 
+
+166
+00:12:57,370 --> 00:13:02,390
+غلط، مش ديابتس، أي واحد بشوفه نازل وزنه بطريقة 
+
+167
+00:13:02,390 --> 00:13:06,490
+unexpected، و أنا مش عارف من إيش بفكر بالوضع الطبيعي
+
+168
+00:13:06,490 --> 00:13:09,690
+يا جماعة common is common، فكروا بعيد فكروا حتى 
+
+169
+00:13:09,690 --> 00:13:16,610
+في السكري، في إيش بدنا نفكر؟ في إيش بدنا نفكر؟
+
+170
+00:13:16,610 --> 00:13:20,190
+اتبرعوا، وكأنه الآن تعليم وجاهي، حاولوا تجاوبوني صوت 
+
+171
+00:13:20,190 --> 00:13:28,260
+عالي، علوا، آه أيوة لأ مش صح، لأ مش صح أيوة صح، احتمال 
+
+172
+00:13:28,260 --> 00:13:34,480
+بيروح الجيم و بيعمل diet، اه عشان هيك نزل وزنه، في 
+
+173
+00:13:34,480 --> 00:13:41,040
+أسباب تانية، اه للـ Unexpected وبسرعة
+
+174
+00:13:41,040 --> 00:13:46,080
+ينزل الوزن لازم أفكر في الـ Diabetes إلى جانب الجيم
+
+175
+00:13:46,080 --> 00:13:49,050
+و الحاجات التالية، لازم أفكر في مشاكل في الـ Thyroid
+
+176
+00:13:49,050 --> 00:13:53,610
+لازم أفكر في مشاكل في Cancer، على سبيل المثال، لازم 
+
+177
+00:13:53,610 --> 00:13:57,690
+أفكر في مشاكل Tuberculosis، إيش يعني Tuberculosis؟
+
+178
+00:13:57,690 --> 00:14:01,190
+Tuberculosis، مرض الـ TB، آه، مجموعة كبيرة من الأمراض
+
+179
+00:14:01,190 --> 00:14:04,990
+بتنزّل وزن البني آدم بدون ما يكون قاصد، unexpected
+
+180
+00:14:04,990 --> 00:14:09,650
+لكن إحنا هنا في إطار الـ Diabetes Mellitus، من أهم 
+
+181
+00:14:09,650 --> 00:14:12,590
+الحاجات اللي بنلاقيها الـ Classic Symptoms اللي
+
+182
+00:14:12,590 --> 00:14:19,550
+بنقول عنها is Weight Loss، فقدان الوزن، وإيش كمان 
+
+183
+00:14:19,550 --> 00:14:25,290
+الثلاثة P، إيش الثلاثة P هاي؟ أول P، و ثاني P، و ثالث
+
+184
+00:14:25,290 --> 00:14:31,030
+P، الـ Polyuria، إيش يعني Polyuria؟ يعني urine كتير
+
+185
+00:14:31,030 --> 00:14:36,690
+increase urination، كل شوية رايح جاي على الحمام، طب هو 
+
+186
+00:14:36,690 --> 00:14:39,750
+كل واحد بشوفه رايح جاي على الحمام أول حاجة بفكر فيها 
+
+187
+00:14:39,750 --> 00:14:49,180
+سكري، لأ، إيه التهاب البول، حسب إذا أنا بحكي عن راجل
+
+188
+00:14:49,180 --> 00:14:55,940
+كبير بفكر كمان فيه البروستاتا، نتفرع
+
+189
+00:14:55,940 --> 00:15:00,480
+بس هيك أنا أفكار، أنا قلت لكم موضوع الباطنة، الموضوع 
+
+190
+00:15:00,480 --> 00:15:05,100
+ممتع، مش قضية إنه بس علم، مش بس بس بدي أحفظ منهج، 
+
+191
+00:15:05,100 --> 00:15:09,980
+الموضوع مش موضوع منهج، الموضوع موضوع علم، موضوع 
+
+192
+00:15:09,980 --> 00:15:14,180
+أعرف في الطب، ماضلش restricted على حركات العلاج
+
+193
+00:15:14,180 --> 00:15:17,280
+الطبيعي والعضلات وكذا، لأ، أنا بدي أعرف البني آدم 
+
+194
+00:15:17,280 --> 00:15:22,220
+أكثر من جهازه الحركي، بدي أعرف البني آدم أجزاءه من
+
+195
+00:15:22,220 --> 00:15:22,980
+جوا كمان
+
+196
+00:15:27,740 --> 00:15:31,300
+وأفهم في الأمراض هذه لأن كلها هيك هيك لها 
+
+197
+00:15:31,300 --> 00:15:36,560
+relationship to my subject، كعلاج طبيعي، إيش كمان الـ 
+
+198
+00:15:36,560 --> 00:15:39,160
+P الثاني؟ Polydipsia، إيش يعني Polydipsia؟
+
+199
+00:15:39,160 --> 00:15:44,920
+increased thirst، بشرب كتير دائماً، عشان، وإيش كمان
+
+200
+00:15:44,920 --> 00:15:51,780
+Polyphagia، بياكل كتير، increased hunger، هناك حاجات
+
+201
+00:15:51,780 --> 00:15:55,000
+تانية يا جماعة بس هي هذه الأساسيات، Weight Loss، 
+
+202
+00:15:55,000 --> 00:15:58,080
+Polyuria، Polydipsia، و Polyphasia، زي إيش عندنا
+
+203
+00:15:58,080 --> 00:16:00,980
+حاجات؟ بيصير يشوف blurry vision، إيش يعني blurry
+
+204
+00:16:00,980 --> 00:16:06,080
+vision؟ blurry vision يعني مش بيشوف، وهتز الصورة، 
+
+205
+00:16:06,080 --> 00:16:11,670
+double vision، هدّق باعجع راسه، في تيّق، دايماً تعبان، slow 
+
+206
+00:16:11,670 --> 00:16:16,030
+healing of cuts، الجروح عنده بتطول لما بتطيب، إيش 
+
+207
+00:16:16,030 --> 00:16:19,490
+كمان؟ itchy skin، vision changes، ليش الـ vision بيصير 
+
+208
+00:16:19,490 --> 00:16:23,300
+في مشاكل؟ أكتر من حاجة، ممكن يكون نتيجة اللي هو 
+
+209
+00:16:23,300 --> 00:16:25,700
+المشاكل اللي بتصير في الـ Retina، لكن ممكن تكون 
+
+210
+00:16:25,700 --> 00:16:29,620
+نتيجة الـ Hyperglycemia، بتعمل ليه؟ Can cause 
+
+211
+00:16:29,620 --> 00:16:33,120
+glucose absorption in the lens، في العدسات، وهذه
+
+212
+00:16:33,120 --> 00:16:37,060
+بتؤدي لـ Changes in its shape، resulting in vision
+
+213
+00:16:37,060 --> 00:16:42,240
+changes، يعني تأثيرها على عدسات العين، كمان مرة الـ
+
+214
+00:16:42,240 --> 00:16:48,020
+Symptoms، باللسان العادي، اللسان الناس العادية 
+
+215
+00:16:48,020 --> 00:16:53,920
+دايماً tired، رايح جاي على الحمام، بنزل الوزن، وصدن
+
+216
+00:16:53,920 --> 00:16:59,980
+wound that once healed، فيه طبعاً sexual problems 
+
+217
+00:16:59,980 --> 00:17:07,540
+بتصير بالذات من جهة الراجل، always hungry، دايماً 
+
+218
+00:17:07,540 --> 00:17:15,480
+جعان، blurry vision، numbness، خدلان، و .. و .. و .. و
+
+219
+00:17:15,480 --> 00:17:15,740
+.. و .. و .. و .. و .. و .. و .. و .. و .. و .. و
+
+220
+00:17:15,740 --> 00:17:16,060
+.. و .. و .. و .. و .. و .. و .. و .. و .. و .. و
+
+221
+00:17:16,060 --> 00:17:16,900
+.. و .. و .. و .. و .. و .. و .. و .. و .. و .. و
+
+222
+00:17:16,900 --> 00:17:17,060
+.. و .. و .. و .. و .. و .. و .. و .. و .. و .. و
+
+223
+00:17:17,060 --> 00:17:17,180
+.. و .. و .. و .. و .. و .. و .. و .. و .. و .. و
+
+224
+00:17:17,180 --> 00:17:17,540
+.. و .. و .. و .. و .. و .. و .. و .. و ..
+
+225
+00:17:23,310 --> 00:17:28,590
+اللي هي في اللسان العادي الـ symptoms اللي بنلاقيها
+
+226
+00:17:28,590 --> 00:17:31,730
+مش لازم كلها تكون موجودة يا جماعة، بعضها موجودة أو
+
+227
+00:17:31,730 --> 00:17:34,250
+واحدة منهم موجودة لازم أفكر في الـ diabetes
+
+228
+00:17:37,690 --> 00:17:40,730
+اللي فيها إنها طبعاً للـ Diabetes Emergencies، مش 
+
+229
+00:17:40,730 --> 00:17:44,550
+هحكي عنها الآن، ذكروني فيها بس، هحكي معها بعد هيك،
+
+230
+00:17:44,550 --> 00:17:46,830
+في الآخر لما أحكي عن الـDKA، هحكي عن
+
+231
+00:17:46,830 --> 00:17:50,750
+الـHypoglycemia، فبدأ أروح على الـClassification of
+
+232
+00:17:50,750 --> 00:17:55,470
+Diabetes، حقيقة إنه في أكثر من الـClassification، 
+
+233
+00:17:55,470 --> 00:17:59,360
+لكن إحنا اللي اعتمدناها، Diabetes mellitus عنده
+
+234
+00:17:59,360 --> 00:18:04,600
+أربعة broad categories، اللي هو diabetes type 1، و 
+
+235
+00:18:04,600 --> 00:18:09,900
+type 2، و في Gestational diabetes اللي مرتبط بالحمل 
+
+236
+00:18:09,900 --> 00:18:16,580
+أو سكر الحمل، و في specific types، يعني عدد منها
+
+237
+00:18:16,580 --> 00:18:20,900
+هنتحدث عنها في حينها، مش كلها لكن هذكر بعضها حسب 
+
+238
+00:18:20,900 --> 00:18:26,200
+الواجهة اللي معناها، طيب إيش الفرق بين type 1 و
+
+239
+00:18:26,200 --> 00:18:36,900
+type 2؟ ومن أكثر بيجي الـ
+
+240
+00:18:36,900 --> 00:18:40,760
+Type، يعني الآن أنا هدي هحكي بعد شوية إيش الـ Type
+
+241
+00:18:40,760 --> 00:18:43,640
+I و Type II، لكن خدوا بالكم من الـ curve اللي موجود
+
+242
+00:18:43,640 --> 00:18:51,110
+في الدائرة هاي، إنه الأحمر هو Type II وهو غالبية يعني 
+
+243
+00:18:51,110 --> 00:18:53,750
+هنا في أمريكا على سبيل المثال، وقلنا أحنا مرة مرة 
+
+244
+00:18:53,750 --> 00:18:56,570
+تانية يا جماعة عشان ما تأخذوش .. ما أخدلش ده
+
+245
+00:18:56,570 --> 00:18:58,930
+ما بجيب إحصائيات، هم الأمريكيين عندهم إحصائيات باقي
+
+246
+00:18:58,930 --> 00:19:03,090
+الدول معظمها ما عندهاش إحصائيات، عشان هيك بنعتمد 
+
+247
+00:19:03,090 --> 00:19:11,370
+الإحصائيات المنشورة طبعة أمريكا، في أمريكا في سنة 
+
+248
+00:19:11,370 --> 00:19:16,890
+من السنوات سنة الألفين، من قد النظر عن الأرقام لكن 
+
+249
+00:19:17,340 --> 00:19:23,260
+كان عدد سكان أمريكا حدود 300 مليون، يعني وأنتم 
+
+250
+00:19:23,260 --> 00:19:27,940
+بتحسبوا، احسبوا 300 مليون عدد الناس اللي الـ type 2
+
+251
+00:19:27,940 --> 00:19:33,600
+كان عشرة مليون واحد عنده سكري type 2، type 1 لأ نص
+
+252
+00:19:33,600 --> 00:19:39,600
+مليون، لمليون أجل يعني تقريباً، أجل أجل، وفيه طبعاً
+
+253
+00:19:39,600 --> 00:19:44,260
+نسبة كبيرة شايفين الأخضر هذا الرقم، الأسود هذا الرقم
+
+254
+00:19:45,200 --> 00:19:49,020
+خمسة مليون عندهم سكري بس مش مشخصين، لا هم رايحين
+
+255
+00:19:49,020 --> 00:19:52,460
+على الدكاترة ولا الدكاترة باكتشفينهم، وهم هذا 
+
+256
+00:19:52,460 --> 00:19:56,220
+الرقم الأسود، يعني أنا بحكي قاعدة عن ستة عشر مليون أو 
+
+257
+00:19:56,220 --> 00:20:01,020
+سبعة عشر مليون أمريكي في سنة الألفين كان عندهم سكري
+
+258
+00:20:01,020 --> 00:20:07,000
+type 2، هو الماسك خمسة وتسعين في المئة من السكري
+
+259
+00:20:07,000 --> 00:20:11,020
+اللي موجود، خمسة وتسعين type 2، طب وإيش الفرق بين
+
+260
+00:20:11,020 --> 00:20:15,740
+type 1 و type 2؟ هو فيه فروق كبيرة بينهم، type one 
+
+261
+00:20:15,740 --> 00:20:19,360
+ما فيهوش 
+
+262
+00:20:19,360 --> 00:20:21,900
+insulin بالمرة، بالمناسبة الإنسولين هو من وين بيجي؟ 
+
+263
+00:20:21,900 --> 00:20:27,100
+من وين بيطلع؟ الغدة النخامية في الدماغ بيجي؟ ولا
+
+264
+00:20:27,100 --> 00:20:31,600
+من خلايا Beta تاعة الـ Langerhans Islet Cells 
+
+265
+00:20:31,600 --> 00:20:37,040
+الموجودة في البنكرياس؟ ولا من وين بيطلع؟ آه من الـ
+
+266
+00:20:37,040 --> 00:20:42,320
+Islet Cells اللي هو الـ Langerhans، جزر Langerhans
+
+267
+00:20:42,320 --> 00:20:47,460
+في البنكرياس من خلايا Beta بيطلع، الآن هذي لو 
+
+268
+00:20:47,460 --> 00:20:53,720
+اتدمرت المصنع هذا لو اتدمر مش هيطلع انسولين، طب هو 
+
+269
+00:20:53,720 --> 00:20:58,080
+إيش بدي يدمره؟ اه في عندنا أكثر من سبب لتدميره، ممكن
+
+270
+00:20:58,080 --> 00:21:03,920
+يكون autoimmune بمعنى دمر نفسه بنفسه، أو يكون نتيجة 
+
+271
+00:21:03,920 --> 00:21:09,670
+الالتهابات، المهم إنها بتطلعش Insulin، No Insulin هذا
+
+272
+00:21:09,670 --> 00:21:14,130
+الـ Type I في الـ Islet Cells، فإذا ما فيش Insulin
+
+273
+00:21:14,130 --> 00:21:18,070
+يبقى على الدنيا السلام، يبقى ما فيش مفاتيح للغرف
+
+274
+00:21:18,070 --> 00:21:23,230
+يبقى السكر هيظل برا، والخلايا مش هيروحلها سكر، طيب
+
+275
+00:21:23,230 --> 00:21:27,150
+وبدأت تعيش بدون سكر، ماهي المشكلة؟ هو ليش بيصير عندنا 
+
+276
+00:21:27,150 --> 00:21:30,590
+كل الـ symptoms؟ طب و الـ diabetes type 2، إيش اللي 
+
+277
+00:21:30,590 --> 00:21:40,330
+بيصير؟ هو في Insulin لكن المفتاح ما بيجيع الجفل أو 
+
+278
+00:21:40,330 --> 00:21:46,250
+بتحط المفتاح في الباب وبتحاول تفتح، بعصلج الباب 
+
+279
+00:21:46,250 --> 00:21:51,450
+برضه ما بيفتحش، بنقول عنه Insulin Resistance، Insulin
+
+280
+00:21:51,450 --> 00:21:58,070
+Resistance، في عدم قدرة لدخول السكر إلى داخل الخلية
+
+281
+00:21:58,070 --> 00:22:05,420
+نتيجة إنه الـ Gestational Diabetes هو تبع Insulin Resistance
+
+282
+00:22:05,420 --> 00:22:09,640
+الحمل مثل الـ Type 2 تقريباً بس هو بيجي .. هنحكي 
+
+283
+00:22:09,640 --> 00:22:13,500
+عنه بيجي في فترة الحمل .. فترات معينة في الحمل
+
+284
+00:22:24,370 --> 00:22:32,610
+النقص حقيقي أو نقص عدم وجود إنسولين حقيقي موجود في
+
+285
+00:22:32,610 --> 00:22:32,930
+الجسم
+
+286
+00:22:38,370 --> 00:22:40,450
+وهذا بنقول عنه Relatively، يعني أنا في عندي
+
+287
+00:22:40,450 --> 00:22:44,290
+Insulin لكن مش قادر يشتغل الشغل اللي بدناياه وهذا
+
+288
+00:22:44,290 --> 00:22:50,610
+بيؤدي لـ، بعد فترة، نتيجة إنه ال Insulin بحاول يفتح،
+
+289
+00:22:50,610 --> 00:22:56,670
+بيرفض، بيدي إشارة للجسم،
+
+290
+00:22:56,670 --> 00:23:00,830
+نتيجة زيادة السكر، إشارة للبنكرياس، افرز أكثر،
+
+291
+00:23:00,830 --> 00:23:05,480
+بيظل يفرز أكثر البنكرياس نتيجة الإفراز الأكثر الـ
+
+292
+00:23:05,480 --> 00:23:09,140
+beta cells برضه في الآخر بتضعف وبتقولك خلاص أنا 
+
+293
+00:23:09,140 --> 00:23:12,420
+أطلعت كل الانسولين اللي عندي فبيصير عندي انسولين
+
+294
+00:23:12,420 --> 00:23:17,460
+beta cells failure وطبعاً في عندنا الـ gestational
+
+295
+00:23:17,460 --> 00:23:25,870
+diabetes اللي هو عبارة عن subtype لـ type 2، الـ Type-2
+
+296
+00:23:25,870 --> 00:23:28,790
+هي الـ Secondary Form لـ Type-2 اللي قلنا بنقول عنها
+
+297
+00:23:28,790 --> 00:23:34,510
+بعد شوية الـ MODY و Exocrine Pancreas Disease
+
+298
+00:23:34,510 --> 00:23:38,310
+والـ Steroids هذه عبارة عن حاجات بتعمل زي Type-2،
+
+299
+00:23:38,310 --> 00:23:41,730
+هنحكي عنها بعد شوية طيب، ندخل الآن على تفاصيل
+
+300
+00:23:41,730 --> 00:23:46,470
+الـ Type-1، أيه أسماء ثانية للـ Type-1؟ بنقوله
+
+301
+00:23:46,470 --> 00:23:50,130
+juvenile onset or insulin dependent diabetes
+
+302
+00:23:50,130 --> 00:23:56,110
+جوفينايل لأنه لاحظوا أنه الغالبية أخذوا باركم الكلمة 
+
+303
+00:23:56,110 --> 00:24:03,550
+الغالبية بتصير عند الأطفال مفهوم
+
+304
+00:24:03,550 --> 00:24:07,150
+باركم تقول الغالبية لكن مش مئة في المية عند
+
+305
+00:24:07,150 --> 00:24:11,770
+الأطفال ممكن type one يجي عند ناس غير الأطفال عشان
+
+306
+00:24:11,770 --> 00:24:17,010
+هيك قالوا زمان juvenile فيه اسم ثاني Insulin
+
+307
+00:24:17,010 --> 00:24:24,030
+Dependent لأن علاجه لا يتم إلا بالانسولين بدون هيك
+
+308
+00:24:24,030 --> 00:24:27,530
+يبقى ما ينفعش بدون انسولين type one يا جماعة عفواً إن ال
+
+309
+00:24:27,530 --> 00:24:31,890
+diabetes type one لا يعالج إلا بالانسولين أبار
+
+310
+00:24:31,890 --> 00:24:34,930
+ما ينفعش يعالجه بحبوب عشان هيك جاله dependent
+
+311
+00:24:34,930 --> 00:24:40,210
+dependent يعني أنك متعلق به وهذا بيجي عند الناس
+
+312
+00:24:40,210 --> 00:24:43,170
+الأقل من ثلاثين سنة عشان هيك ربطه بالأطفال كانت مش
+
+313
+00:24:43,170 --> 00:24:48,890
+صحيح لأنه بيجي لحتى ثلاثين سنة يعني جوفينايل 
+
+314
+00:24:50,330 --> 00:24:54,470
+لكن العمر المشهور له اللي بيبين فيه سكري بين 11
+
+315
+00:24:54,470 --> 00:25:00,730
+و 13 سنة و بيصير عندنا طب ليش؟ طب ما هو 19
+
+316
+00:25:00,730 --> 00:25:05,110
+سنة ما سلك؟ آه، سلك وما عنده سكر الطفل، شو يعني
+
+317
+00:25:05,110 --> 00:25:07,670
+أنا الآن أتدمرت؟ بمجرد صراحة عندي progressive
+
+318
+00:25:07,670 --> 00:25:12,410
+destruction للـ Beta cells؟ لحد ما وصل مرحلة على عمر
+
+319
+00:25:12,410 --> 00:25:16,990
+11، 12، 13، 14، ما أعرفش وابتدى لحد ما اتدمرت كامل
+
+320
+00:25:16,990 --> 00:25:22,550
+الخلايا وصار عندنا الـ Destruction للـ Pancreatic
+
+321
+00:25:22,550 --> 00:25:26,480
+Beta Cells وهذه زي ما قلنا في الغالب Auto
+
+322
+00:25:26,480 --> 00:25:31,380
+Antibodies عبارة عن نيران صديقة هي بتعملي
+
+323
+00:25:31,380 --> 00:25:34,740
+reduction لـ 8 لـ 90% of normal beta cells function
+
+324
+00:25:34,740 --> 00:25:37,740
+قبل manifestation it occurs يعني يوم بصير عندي 
+
+325
+00:25:37,740 --> 00:25:41,700
+manifestation ببجي عارف إنه على ال beta cells اللي 
+
+326
+00:25:41,700 --> 00:25:47,320
+بتفرز الإنسولين السلام الأسباب لحد الآن تتفسر
+
+327
+00:25:47,320 --> 00:25:51,860
+بـ Genetic Predisposition وفي قال لك لأ ممكن يكون
+
+328
+00:25:51,860 --> 00:25:57,280
+نتيجة Exposure to a virus الـ Onset مش مختلف كتير 
+
+329
+00:25:57,280 --> 00:26:01,940
+الـ Symptoms في الـ Type 1 Manifestation developed
+
+330
+00:26:01,940 --> 00:26:05,380
+when the pancreas can no longer produce insulin
+
+331
+00:26:05,380 --> 00:26:10,120
+Rapid onset of symptoms present في الغالب في الغالب
+
+332
+00:26:10,120 --> 00:26:14,340
+هدولة أول ما بيبين عنده بيبين بالصورة الصعبة يعني
+
+333
+00:26:14,340 --> 00:26:18,200
+بيجيني الطفل بحاجة اسمها diabetic ketoacidosis
+
+334
+00:26:18,200 --> 00:26:25,050
+يعني عملياً أو بندوسيرو عملياً في حالة صعبة بيجونا
+
+335
+00:26:25,050 --> 00:26:28,430
+الأطفال في العادة زي ما بيجونا في الديابتس 1 ليش؟
+
+336
+00:26:28,430 --> 00:26:33,610
+لأنه مرة واحدة بينتقل من سكر معقول لـ Ketoacidosis
+
+337
+00:26:33,610 --> 00:26:38,690
+حموضة في الدم نتيجة ارتفاع حاد في السكر وطبعاً هذا
+
+338
+00:26:38,690 --> 00:26:43,070
+بيؤثر لي على باقي الجسم ممكن يجيني بـ Weight Loss
+
+339
+00:26:43,070 --> 00:26:46,270
+وكمان مرة ممكن يجيني بـ Weakness، Fatigue،
+
+340
+00:26:46,270 --> 00:26:49,950
+Polyphagia، Polydipsia و Polyuria
+
+341
+00:26:52,810 --> 00:26:57,530
+زي ما حكينا قبل شوية زي
+
+342
+00:26:57,530 --> 00:27:01,890
+ما حكينا قبل شوية is characterized by insulin
+
+343
+00:27:01,890 --> 00:27:06,350
+resistance which may be combined with relatively
+
+344
+00:27:06,350 --> 00:27:10,890
+reduced insulin secretion characterized by insulin
+
+345
+00:27:10,890 --> 00:27:15,910
+resistance إنه مش قادر الانسولين يغزو هذه الغرفة
+
+346
+00:27:15,910 --> 00:27:20,940
+لسبب من الأسباب وهذا which may be combined with
+
+347
+00:27:20,940 --> 00:27:24,060
+relatively reduced insulin secretion بعد شوية 
+
+348
+00:27:24,060 --> 00:27:26,600
+بتصير repeated cells مش قادرة تطلع انسولين ليش؟
+
+349
+00:27:26,600 --> 00:27:29,400
+لأنه طلعت كثير نتيجة إنه مش داخل الانسولين داخل 
+
+350
+00:27:29,400 --> 00:27:36,340
+الخلايا فبيقل كمية الانسولين The defective
+
+351
+00:27:36,340 --> 00:27:45,240
+responsiveness استجابة خلايا الجسم للانسولين is
+
+352
+00:27:45,240 --> 00:27:49,200
+believed to involve the insulin receptors وقال إن
+
+353
+00:27:49,200 --> 00:27:54,380
+الـ Receptors هي السبب تبع الموضوع ديابتس ميليتاس
+
+354
+00:27:54,380 --> 00:27:58,200
+is the second .. is the most common type تسعين في 
+
+355
+00:27:58,200 --> 00:28:01,100
+المية of patient تسعين أو خمسة وتسعين في المية
+
+356
+00:28:01,100 --> 00:28:05,280
+usually بتيجي بعد في الناس اللي usually وإيش معنى
+
+357
+00:28:05,280 --> 00:28:10,620
+usually مش always usually يعني غالباً occur in 
+
+358
+00:28:10,620 --> 00:28:19,090
+people over 40 الـ 98% of patients are overweight 
+
+359
+00:28:19,090 --> 00:28:25,290
+لكن ممكن واحد مش overweight برضه يصير عنده type 2
+
+360
+00:28:25,290 --> 00:28:30,190
+لكن لاحظوا إن type 2 diabetes is due to primarily
+
+361
+00:28:30,190 --> 00:28:35,910
+two lifestyle factors and genetics إحنا في لغتنا
+
+362
+00:28:35,910 --> 00:28:40,130
+بنقول عنه multifactorial إيش يعني multifactorial؟
+
+363
+00:28:40,540 --> 00:28:43,660
+يعني بتحكم فيه أكثر من حاجة هو ممكن يكون الواحد
+
+364
+00:28:43,660 --> 00:28:50,600
+genetic predisposition لكن الممارسة السيئة بتظهر
+
+365
+00:28:50,600 --> 00:28:55,860
+المرض ممكن كثير ناس تكون عندها حاجات لكن الممارسة 
+
+366
+00:28:55,860 --> 00:29:02,820
+السيئة هي اللي أظهرته اللي من الحاجات اللي .. اللي 
+
+367
+00:29:02,820 --> 00:29:05,460
+من ال .. اللي لها علاقة بالـ lifestyle مرة ثانية،
+
+368
+00:29:05,460 --> 00:29:08,980
+هنا بدينا نيجي للـ lifestyle وطريقة الحياة اللي مش
+
+369
+00:29:08,980 --> 00:29:12,520
+كويسة، ما بديش أقولها الكلمة، اللي إحنا عايشينها هي
+
+370
+00:29:12,520 --> 00:29:15,060
+اللي عاملانا المشكلة، اللي هو الـ obesity
+
+371
+00:29:18,240 --> 00:29:21,300
+وكنا حكينا عن الـ Body Mass Index of Greater Than
+
+372
+00:29:21,300 --> 00:29:25,680
+30 لأن أنا لما أقول Body Mass Index أكثر من ثلاثين
+
+373
+00:29:25,680 --> 00:29:29,480
+في الـ Obesity كلكم فاهمين لأن إحنا في الـ
+
+374
+00:29:29,480 --> 00:29:35,780
+Assignment اللي فاتت شرحنا إيش الـ BMI وقلنا إيش
+
+375
+00:29:35,780 --> 00:29:40,020
+يعني BMI ال Body Mass Index وكيف نحسب وقلنا
+
+376
+00:29:40,020 --> 00:29:43,430
+الدرجات تبعته فاكرين ولا مش فاكرين؟ ديروا بالكوا
+
+377
+00:29:43,430 --> 00:29:45,570
+أنا دائماً بركز عليه الموضوع هذا الـ Body Mass
+
+378
+00:29:45,570 --> 00:29:48,650
+Index لأنه من الحاجات الأساسية اللي ما تعلمناها
+
+379
+00:29:48,650 --> 00:29:52,250
+في المدارس لكن الآن it's time نكون عارفين إيش 
+
+380
+00:29:52,250 --> 00:29:55,990
+الموضوع هذا اللي أقل من 30 اللي أكثر من 30 الـ
+
+381
+00:29:55,990 --> 00:29:59,730
+Body Mass Index تبعه opposite أو opposite هذا من 
+
+382
+00:29:59,730 --> 00:30:04,210
+اللي هو الـ factors اللي بتساعد إنه يكون عند الواحد
+
+383
+00:30:04,210 --> 00:30:08,110
+diabetes type 2 lack of physical activity يا سلام
+
+384
+00:30:08,110 --> 00:30:12,900
+نفس الكلام عن الضغط الجعدة وجلة الحركة الـ Poor
+
+385
+00:30:12,900 --> 00:30:20,140
+Diet، الطيح رايح، إيش بيجي، بيتاكل، شبسي كل يوم،
+
+386
+00:30:20,140 --> 00:30:24,120
+كوكاكولا كل يوم، جنون كبيرة و جنون صغيرة رايح جاي،
+
+387
+00:30:24,120 --> 00:30:28,540
+حاجات فيها بيتزا عالفاضي وعالمليان، ماكرونة عالفاضي
+
+388
+00:30:28,540 --> 00:30:33,140
+وعالمليان، حاجات فاضية، طيب كله سلطات؟ لأ، ما بدناش 
+
+389
+00:30:33,140 --> 00:30:38,520
+سلطات، حاجة خضراء؟ لأ، ما بدناش حاجة خضراء، زيوت طبيعية
+
+390
+00:30:38,520 --> 00:30:41,540
+زي الزيتون وحاجات؟ لأ، لازم يكون حاجات مهدرجة و
+
+391
+00:30:41,540 --> 00:30:49,890
+حاجات يعني poor diet أكل ما له لازم نهائي بالإضافة
+
+392
+00:30:49,890 --> 00:30:53,170
+للـ stress اللي إحنا عايشين ولا فيه شغل ولا ولا 
+
+393
+00:30:53,170 --> 00:30:57,610
+ولا ولا هذا أمر طبعاً هذه الأمور كلها لما تتجمع في
+
+394
+00:30:57,610 --> 00:31:02,270
+وجود الواحد يكون predisposition يعني هو مؤهل أمه و
+
+395
+00:31:02,270 --> 00:31:04,990
+أبوه وكل عيلته عندها سكري في الآخر هيصير عنده
+
+396
+00:31:04,990 --> 00:31:05,470
+سكري
+
+397
+00:31:08,780 --> 00:31:12,680
+من الـ .. زي ما قلنا الحاجات اللي بنشوفها في الـ
+
+398
+00:31:12,680 --> 00:31:17,900
+Dietary Factors المأكولات اللي بناكلها واللي بتؤهل 
+
+399
+00:31:17,900 --> 00:31:21,380
+إنه يصير عند الواحد Type 2 Consumption of Sugar
+
+400
+00:31:21,380 --> 00:31:25,360
+-Sweetened Drinks In Excess، خدوا بالك In Excess،
+
+401
+00:31:25,360 --> 00:31:28,820
+إحنا ما بنقولش ممنوع يا جماعة، خير الأمور الوسط،
+
+402
+00:31:28,820 --> 00:31:31,140
+طول عمرنا بنحكي خير الأمور الوسط، ما تمنعش حالك ولا
+
+403
+00:31:31,140 --> 00:31:35,840
+عن حاجة، لكن مش معقول كل يوم حلويات، مش معقول كل 
+
+404
+00:31:35,840 --> 00:31:38,460
+يوم شوكولاتة، كل شوكولاتة حلوة، شوكولاتة منيحة 
+
+405
+00:31:38,690 --> 00:31:43,530
+كويسة للمزاج وكويسة لـ .. كويسة ماشي بس مش كل يوم
+
+406
+00:31:43,530 --> 00:31:47,150
+لوح يعني شو هذا الموضوع؟ يا ريت شوكولاتة من نوع
+
+407
+00:31:47,150 --> 00:31:51,010
+اللي كويس من نوع اللي تافه وهذا مشكلة consumption
+
+408
+00:31:51,010 --> 00:31:55,410
+of sugar-sweetened drinks in excess is associated
+
+409
+00:31:55,410 --> 00:32:00,370
+with increased risk type of fats نوع الدهونيات اللي 
+
+410
+00:32:00,370 --> 00:32:04,990
+بنستخدمها اللي هي saturated fats and trans fatty
+
+411
+00:32:04,990 --> 00:32:11,010
+acids increasing the risk of وأنواع الـ unsaturated
+
+412
+00:32:11,010 --> 00:32:15,870
+وأنواع الـ monounsaturated fat بتقليل الخطر خدوا
+
+413
+00:32:15,870 --> 00:32:18,810
+بالكم أنواع الـ fat وهذا الجزء الثاني من الـ
+
+414
+00:32:18,810 --> 00:32:23,950
+assignment إيش أنواع الـ fat اللي هو saturated و
+
+415
+00:32:23,950 --> 00:32:27,890
+unsaturated وأنواع الـ unsaturated والفرق بينهم
+
+416
+00:32:27,890 --> 00:32:37,050
+بدي جدول مقارنة بينهم saturated و unsaturated و ال
+
+417
+00:32:37,050 --> 00:32:42,440
+trans fat مش الفرق بينهم في جدول مش أكثر من جدول
+
+418
+00:32:42,440 --> 00:32:47,300
+وينكتب بالإيد ويتصور وينبعت لي لأن هذا غاية في 
+
+419
+00:32:47,300 --> 00:32:51,080
+الأهمية وإما هنروح في القلب في ال lung في الباطنة
+
+420
+00:32:51,080 --> 00:32:58,240
+دائماً هنتحدث عن أنواع الـ fats eating lots of white
+
+421
+00:32:58,240 --> 00:33:04,660
+rice مقارنة ورز كتير الـ Lack of Exercise ممكن في 7
+
+422
+00:33:04,660 --> 00:33:09,840
+% من الـ cases يكون سببها Lack of Exercise  جدا
+
+423
+00:33:09,840 --> 00:33:16,500
+قلة الحركة تكون هي السبب في الموضوع يا جماعة موضوع
+
+424
+00:33:16,500 --> 00:33:21,080
+السكري موضوع غاية في الأهمية اللي أنا حكيته الآن
+
+425
+00:33:21,080 --> 00:33:25,240
+حكيت عن الـ Type 2 ولسه ما دخلناش في التفاصيل 
+
+426
+00:33:25,240 --> 00:33:30,590
+تبعها أنا بتهيأ لي المحاضرة هذه احتمال تكون ثلاثة 
+
+427
+00:33:30,590 --> 00:33:38,090
+parts هحاول أسرع لأن المنهج عن جد طويل هجوموني على
+
+428
+00:33:38,090 --> 00:33:42,350
+الـ assignment اللي حكيناهم يا جماعة الـ assignment 
+
+429
+00:33:42,350 --> 00:33:46,450
+مش انتقام منكم الـ assignment حتى تجروا المادة و
+
+430
+00:33:46,450 --> 00:33:51,290
+تتبعوا معايا يعني يكفينا اللي احنا فيه  في قضية المنهج
+
+431
+00:33:51,290 --> 00:33:56,650
+و زي ما قلتلكوا أنا حريص جدا أن كل فكرة في الباطنة،
+
+432
+00:33:56,650 --> 00:33:59,730
+بغض النظر عن الامتحانات و بغض النظر عن العلامات،
+
+433
+00:33:59,730 --> 00:34:03,530
+و هي مهمة حقيقية، مش بتجلل من أهميتها، لكن بدي
+
+434
+00:34:03,530 --> 00:34:07,510
+المادة تهضمها، وهذا من خلال متابعتكم و مراجعتكم
+
+435
+00:34:07,510 --> 00:34:10,970
+معايا أول بأول، و مش نترك المنهج لحد ما يجي الامتحان
+
+436
+00:34:10,970 --> 00:34:14,370
+ بيوم، وبعدين نقعد نحكي، لأ هيك، تتابعوا معايا جدا
+
+437
+00:34:14,370 --> 00:34:20,330
+الإمكان أنا بقدر ما أنا حريص أنكوا تاخدوا علامات و
+
+438
+00:34:20,330 --> 00:34:28,390
+تتفوقوا لكن حريص جدا أنكوا تهضموا المادة طبعا مش
+
+439
+00:34:28,390 --> 00:34:32,730
+هقولكوا أنه شئتم أو أبيتم بدكوا تعرفوا باطنة و
+
+440
+00:34:32,730 --> 00:34:36,970
+بدكوا تعرفوا الأطفال يعني هذا ما فيش مجال لكوا
+
+441
+00:34:36,970 --> 00:34:39,630
+للحياة وهتتذكروني في يوم من الأيام يعطيكوا
+
+442
+00:34:39,630 --> 00:34:41,030
+العافية وسلامات