-
Notifications
You must be signed in to change notification settings - Fork 0
Expand file tree
/
Copy pathanalysis_ii_lecture_2_notes.html
More file actions
657 lines (581 loc) · 31.5 KB
/
analysis_ii_lecture_2_notes.html
File metadata and controls
657 lines (581 loc) · 31.5 KB
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="UTF-8">
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<title>Analysis II Lecture 2 Notes</title>
<script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script>
<script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
<script>
window.MathJax = {
tex: {
inlineMath: [['$', '$'], ['\\(', '\\)']],
displayMath: [['$$', '$$'], ['\\[', '\\]']],
processEscapes: true,
processEnvironments: true
},
options: {
skipHtmlTags: ['script', 'style', 'textarea', 'pre']
}
};
</script>
<style>
/* ===== GLOBAL STYLES ===== */
body {
font-family: 'Times New Roman', serif;
line-height: 1.6;
margin: 0;
padding: 0;
background-color: #f9f9f9;
color: #333;
font-size: 18px;
overflow-x: hidden;
}
/* ===== MAIN LAYOUT CONTAINER ===== */
.main-container {
display: flex;
min-height: 100vh;
}
/* ===== LEFT SIDE - CONTENT ===== */
.content-side {
flex: 1;
padding: 20px;
max-width: 50%;
display: flex;
justify-content: center;
position: relative;
}
.content-side.full-width {
max-width: 100%;
justify-content: center;
}
.scrollable-content {
width: 100%;
max-width: 1000px;
overflow-y: auto;
max-height: 95vh;
padding-right: 10px;
}
.content-side.full-width .scrollable-content {
max-height: none;
}
/* ===== RIGHT SIDE - VIDEO ===== */
.video-side {
flex: 1;
max-width: 50%;
background-color: #000;
position: relative;
display: none;
overflow: hidden;
height: 100vh;
}
/* ===== CONTENT CONTAINER ===== */
.container {
background-color: white;
padding: 40px;
border-radius: 8px;
box-shadow: 0 2px 15px rgba(0,0,0,0.1);
max-width: 1000px;
margin: 0 auto;
}
h1 {
text-align: center;
font-size: 2.2em;
margin-bottom: 10px;
color: #2c3e50;
}
.subtitle {
text-align: center;
color: #666;
font-style: italic;
margin-bottom: 40px;
}
h2, h3, h4, h5, h6 {
color: #2c3e50;
margin-top: 30px;
}
/* ===== CONTENT BOX STYLING ===== */
.content-box {
background-color: white;
padding: 20px;
margin: 20px 0;
border-left: 4px solid #3498db;
border-radius: 0 5px 5px 0;
position: relative;
box-shadow: 0 1px 3px rgba(0,0,0,0.1);
}
.content-box h3 {
margin-top: 0;
color: #000000;
padding-bottom: 10px;
}
/* Color coding for different content types */
.definition {
border-left-color: #27ae60;
background-color: #f0f9f4;
}
.definition h3::before {
content: "Definition: ";
font-weight: bold;
color: #000000;
}
.theorem {
border-left-color: #8e44ad;
background-color: #f8f5fb;
}
.theorem h3::before {
content: "Theorem: ";
font-weight: bold;
color: #000000;
}
.corollary {
border-left-color: #8e44ad;
background-color: #f8f5fb;
}
.corollary h3::before {
content: "Corollary: ";
font-weight: bold;
color: #000000;
}
.claim {
border-left-color: #8e44ad;
background-color: #f8f5fb;
}
.claim h3::before {
content: "Claim: ";
font-weight: bold;
color: #000000;
}
.example {
border-left-color: #e67e22;
background-color: #fef5e7;
}
.example h3::before {
content: "Example: ";
font-weight: bold;
color: #000000;
}
.proof {
border-left-color: #e91e63;
background-color: #fce4ec;
}
.proof h3::before {
content: "Proof: ";
font-weight: bold;
color: #000000;
}
.non-examinable {
border-left-color: #95a5a6;
background-color: #f8f9fa;
}
.non-examinable h3::before {
content: "Non-examinable: ";
font-weight: bold;
color: #000000;
}
ul, ol {
padding-left: 25px;
font-size: 1.1em;
}
li {
margin: 8px 0;
}
p {
font-size: 1.1em;
}
/* ===== TIMESTAMP STYLING ===== */
.timestamp {
position: absolute;
top: 10px;
right: 15px;
font-size: 0.8em;
color: #666;
background-color: rgba(255,255,255,0.8);
padding: 2px 6px;
border-radius: 3px;
cursor: pointer;
transition: all 0.2s ease;
display: flex;
align-items: center;
gap: 3px;
}
.timestamp:hover {
background-color: rgba(255,255,255,0.95);
color: #333;
transform: scale(1.05);
}
.play-icon {
font-size: 0.7em;
}
/* Math equation styling */
.MathJax {
font-size: 1.1em !important;
}
/* ===== VIDEO SIDE STYLES ===== */
.video-container {
position: relative;
width: 100%;
height: 100vh;
}
.video-container iframe {
position: absolute;
top: 0;
left: 0;
width: 100%;
height: 100%;
border: none;
}
/* ===== CLOSE BUTTON ===== */
.close-button {
position: absolute;
top: 20px;
right: 20px;
background-color: rgba(0,0,0,0.7);
color: white;
border: none;
font-size: 24px;
font-weight: bold;
cursor: pointer;
line-height: 1;
padding: 10px 15px;
border-radius: 50%;
z-index: 10;
transition: all 0.2s ease;
}
.close-button:hover {
background-color: rgba(0,0,0,0.9);
transform: scale(1.1);
}
/* Responsive design */
@media (max-width: 768px) {
.main-container {
flex-direction: column;
}
.content-side {
max-width: 100%;
padding: 10px;
}
.video-side {
max-width: 100%;
height: 50vh;
}
.video-container {
height: 100%;
}
.container {
padding: 20px;
}
h1 {
font-size: 1.8em;
}
.timestamp {
position: static;
display: block;
margin-bottom: 10px;
}
.content-side {
overflow-y: auto;
}
}
</style>
</head>
<body>
<div class="main-container">
<!-- ===== LEFT SIDE - CONTENT ===== -->
<div class="content-side full-width">
<div class="scrollable-content">
<div class="container">
<h1>Analysis II Lecture 2</h1>
<p class="subtitle">Uniform Convergence and the Cauchy Criterion</p>
<!-- Opening Remark -->
<div class="content-box">
<div class="timestamp" onclick="openVideoSide('00:04', 'Notation Clarification', 'default', this)">
<span class="play-icon">▶</span>00:04
</div>
<h3>Notation Clarification</h3>
<p>A clarification about notation from the previous lecture: When writing a sequence like $(f_n)_{n \in \mathbb{N}}$, the subscript notation indicates the running index. However, for a single sequence (not a subsequence), this extra notation is unnecessary and will be omitted from now on.</p>
<p>The notation $(f_{n_k})_{k \in \mathbb{N}}$ is more useful when dealing with subsequences, where we need to specify what the running index is.</p>
</div>
<!-- Recall: Uniform Convergence Definition -->
<div class="definition content-box">
<div class="timestamp" onclick="openVideoSide('01:02', 'Uniform Convergence (Recall)', 'definition', this)">
<span class="play-icon">▶</span>01:02
</div>
<h3>Uniform Convergence (Recall)</h3>
<p>Let $(f_n)$ be a sequence of functions defined on a set $E$, and let $f$ be a function on $E$. We say that <strong>$f_n$ tends to $f$ uniformly on $E$</strong> if:</p>
<p>$$\sup_{x \in E} |f_n(x) - f(x)| \to 0 \quad \text{as } n \to \infty$$</p>
<p>This means the supremum of the difference over all $x \in E$ tends to zero. This is equivalent to saying: for every $\epsilon > 0$, there exists $N$ (depending only on $\epsilon$) such that for all $n \ge N$ and all $x \in E$:</p>
<p>$$|f_n(x) - f(x)| < \epsilon$$</p>
</div>
<!-- Remark: Uniform implies Pointwise -->
<div class="content-box">
<div class="timestamp" onclick="openVideoSide('02:35', 'Uniform Implies Pointwise', 'default', this)">
<span class="play-icon">▶</span>02:35
</div>
<h3>Uniform Convergence Implies Pointwise Convergence</h3>
<p>If $f_n \to f$ uniformly on $E$, then $f_n \to f$ pointwise on $E$.</p>
<p><strong>Proof:</strong> If we fix a point $x \in E$, then for any $\epsilon > 0$, uniform convergence gives us an $N$ such that for all $n \ge N$:</p>
<p>$$|f_n(x) - f(x)| \le \sup_{y \in E} |f_n(y) - f(y)| < \epsilon$$</p>
<p>This shows pointwise convergence at $x$. Since $x$ was arbitrary, pointwise convergence holds everywhere on $E$.</p>
</div>
<!-- Strategy for Computing Uniform Limits -->
<div class="content-box">
<div class="timestamp" onclick="openVideoSide('04:29', 'Strategy for Finding Uniform Limits', 'default', this)">
<span class="play-icon">▶</span>04:29
</div>
<h3>Strategy for Finding Uniform Limits</h3>
<p>This remark gives us a useful strategy for computing or checking uniform limits for explicitly defined sequences:</p>
<ol>
<li><strong>First, check if a pointwise limit exists.</strong> Compute $\lim_{n \to \infty} f_n(x)$ for each fixed $x$. If no pointwise limit exists, then there is no uniform limit either.</li>
<li><strong>If a pointwise limit $f$ exists,</strong> it is the only possible candidate for the uniform limit (since limits are unique).</li>
<li><strong>Then check the supremum condition:</strong> Verify whether $\sup_{x \in E} |f_n(x) - f(x)| \to 0$.</li>
</ol>
</div>
<!-- Definition: Uniformly Cauchy -->
<div class="definition content-box">
<div class="timestamp" onclick="openVideoSide('08:22', 'Uniformly Cauchy', 'definition', this)">
<span class="play-icon">▶</span>08:22
</div>
<h3>Uniformly Cauchy Sequence</h3>
<p>Let $(f_n)$ be a sequence of functions defined on a set $E$. We say the sequence is <strong>uniformly Cauchy on $E$</strong> if:</p>
<p>For every $\epsilon > 0$, there exists $N \in \mathbb{N}$ (depending on $\epsilon$) such that for all $n, m \ge N$:</p>
<p>$$\sup_{x \in E} |f_n(x) - f_m(x)| < \epsilon$$</p>
<p>This is analogous to the Cauchy criterion for sequences of real numbers, but now we require the condition to hold uniformly over all $x \in E$ (hence the supremum).</p>
</div>
<!-- Theorem: Cauchy Criterion -->
<div class="theorem content-box">
<div class="timestamp" onclick="openVideoSide('10:56', 'Cauchy Criterion for Uniform Convergence', 'theorem', this)">
<span class="play-icon">▶</span>10:56
</div>
<h3>Cauchy Criterion for Uniform Convergence</h3>
<p>Let $(f_n)$ be a sequence of functions $f_n : E \to \mathbb{R}$ defined on a common domain $E$. Then the following are equivalent:</p>
<ol>
<li>$(f_n)$ converges uniformly on $E$ (to some function $f$)</li>
<li>$(f_n)$ is uniformly Cauchy on $E$</li>
</ol>
<p>In other words: A sequence of functions converges uniformly if and only if it is uniformly Cauchy.</p>
<p><em>This is extremely useful because it allows us to check convergence without knowing the limit function in advance.</em></p>
</div>
<!-- Proof: Forward Direction -->
<div class="proof content-box">
<div class="timestamp" onclick="openVideoSide('13:21', 'Proof of Cauchy Criterion (Forward)', 'proof', this)">
<span class="play-icon">▶</span>13:21
</div>
<h3>Cauchy Criterion (Forward Direction)</h3>
<p><strong>To prove:</strong> If $f_n \to f$ uniformly on $E$, then $(f_n)$ is uniformly Cauchy.</p>
<p><strong>Proof:</strong> Suppose $f_n \to f$ uniformly. Let $\epsilon > 0$ be given. By uniform convergence, there exists $N$ such that for all $n \ge N$:</p>
<p>$$\sup_{x \in E} |f_n(x) - f(x)| < \epsilon$$</p>
<p>Now for any $n, m \ge N$ and any $x \in E$, by the triangle inequality:</p>
<p>$$|f_n(x) - f_m(x)| \le |f_n(x) - f(x)| + |f_m(x) - f(x)|$$</p>
<p>Since $x$ was arbitrary, we can take the supremum over $x \in E$:</p>
<p>$$\sup_{x \in E} |f_n(x) - f_m(x)| \le \sup_{x \in E} |f_n(x) - f(x)| + \sup_{x \in E} |f_m(x) - f(x)| < \epsilon + \epsilon = 2\epsilon$$</p>
<p>Since $\epsilon$ was arbitrary (we could have started with $\epsilon/2$), this shows $(f_n)$ is uniformly Cauchy. $\square$</p>
</div>
<!-- Proof: Reverse Direction -->
<div class="proof content-box">
<div class="timestamp" onclick="openVideoSide('19:00', 'Proof of Cauchy Criterion (Reverse)', 'proof', this)">
<span class="play-icon">▶</span>19:00
</div>
<h3>Cauchy Criterion (Reverse Direction)</h3>
<p><strong>To prove:</strong> If $(f_n)$ is uniformly Cauchy on $E$, then $(f_n)$ converges uniformly on $E$.</p>
<p><strong>Proof:</strong> Suppose $(f_n)$ is uniformly Cauchy. </p>
<p><strong>Step 1: Find a pointwise limit.</strong> For each fixed $x \in E$, the sequence $(f_n(x))$ is a Cauchy sequence of real numbers (since uniform Cauchy implies pointwise Cauchy). By completeness of $\mathbb{R}$, this sequence converges to some limit. Define:</p>
<p>$$f(x) = \lim_{n \to \infty} f_n(x)$$</p>
<p>This gives us a function $f : E \to \mathbb{R}$, and we have $f_n \to f$ pointwise on $E$.</p>
<p><strong>Step 2: Show uniform convergence.</strong> Let $\epsilon > 0$ be given. Since $(f_n)$ is uniformly Cauchy, there exists $N$ such that for all $n, m \ge N$:</p>
<p>$$\sup_{x \in E} |f_n(x) - f_m(x)| < \epsilon$$</p>
<p>This means that for every $x \in E$: $|f_n(x) - f_m(x)| < \epsilon$ for all $n, m \ge N$.</p>
<p>Now fix $n \ge N$ and let $m \to \infty$. By pointwise convergence, $f_m(x) \to f(x)$, so:</p>
<p>$$|f_n(x) - f(x)| = \lim_{m \to \infty} |f_n(x) - f_m(x)| \le \epsilon$$</p>
<p>This holds for all $x \in E$, so:</p>
<p>$$\sup_{x \in E} |f_n(x) - f(x)| \le \epsilon$$</p>
<p>Since $\epsilon$ was arbitrary and this holds for all $n \ge N$, we have shown $f_n \to f$ uniformly. $\square$</p>
</div>
<!-- Example 1.3: Triangle spike function (recall from Lecture 1) -->
<div class="example content-box">
<div class="timestamp" onclick="openVideoSide('27:52', 'Example: Triangle spike function', 'example', this)">
<span class="play-icon">▶</span>27:52
</div>
<h3>Triangle Spike Function (Recall from Lecture 1)</h3>
<p>Recall Example 1.3 from Lecture 1: the sequence of triangle functions on $[0,1]$ defined by:</p>
<p>$$f_n(x) = \begin{cases}
n^2 x & \text{if } 0 \le x \le \frac{1}{n} \\
n^2 \left(\frac{2}{n} - x\right) & \text{if } \frac{1}{n} < x \le \frac{2}{n} \\
0 & \text{otherwise}
\end{cases}$$</p>
<p>This is an isosceles triangle with base $[0, \frac{2}{n}]$ and height $n$.</p>
<p><strong>Pointwise limit:</strong> For any fixed $x > 0$, eventually the triangle is entirely to the left of $x$ (since $\frac{2}{n} \to 0$), so $f_n(x) \to 0$. Also $f_n(0) = 0$ for all $n$. Therefore $f_n \to 0$ pointwise on $[0,1]$.</p>
<p><strong>Uniform convergence:</strong> However, the supremum is:</p>
<p>$$\sup_{x \in [0,1]} |f_n(x) - 0| = \sup_{x \in [0,1]} |f_n(x)| = n$$</p>
<p>The maximum value is $n$ (the height of the triangle), which tends to infinity, not zero.</p>
<p><strong>Conclusion:</strong> The sequence converges pointwise to 0 but <em>does not</em> converge uniformly on $[0,1]$. This example was previously discussed to show that integrals need not converge: $\int_0^1 f_n(x) \, dx = 1$ for all $n$, but $\int_0^1 f(x) \, dx = 0$.</p>
</div>
<!-- Example 1.4: x/n on R -->
<div class="example content-box">
<div class="timestamp" onclick="openVideoSide('29:32', 'Example: x/n on R', 'example', this)">
<span class="play-icon">▶</span>29:32
</div>
<h3>$f_n(x) = \frac{x}{n}$ on $\mathbb{R}$</h3>
<p>Consider $f_n(x) = \frac{x}{n}$ defined on all of $\mathbb{R}$.</p>
<p><strong>Pointwise limit:</strong> For any fixed $x \in \mathbb{R}$:</p>
<p>$$\lim_{n \to \infty} \frac{x}{n} = 0$$</p>
<p>So $f_n \to 0$ pointwise on $\mathbb{R}$.</p>
<p><strong>Uniform convergence:</strong> Check the supremum:</p>
<p>$$\sup_{x \in \mathbb{R}} \left|\frac{x}{n} - 0\right| = \sup_{x \in \mathbb{R}} \frac{|x|}{n} = +\infty$$</p>
<p>Since the domain is unbounded, for any fixed $n$, we can make $|x|/n$ arbitrarily large by choosing $x$ large enough.</p>
<p><strong>Conclusion:</strong> The sequence converges pointwise to 0 but <em>does not</em> converge uniformly on $\mathbb{R}$.</p>
</div>
<!-- Example 1.4': x/n on bounded interval -->
<div class="example content-box">
<div class="timestamp" onclick="openVideoSide('31:07', 'Example: x/n on Bounded Interval', 'example', this)">
<span class="play-icon">▶</span>31:07
</div>
<h3>$f_n(x) = \frac{x}{n}$ on $[-a, a]$</h3>
<p>Now consider the same sequence $f_n(x) = \frac{x}{n}$ but restricted to a bounded interval $[-a, a]$ for some finite $a > 0$.</p>
<p><strong>Pointwise limit:</strong> As before, $f_n \to 0$ pointwise (restriction doesn't change pointwise limits).</p>
<p><strong>Uniform convergence:</strong> Now the supremum is:</p>
<p>$$\sup_{x \in [-a,a]} \left|\frac{x}{n}\right| = \frac{a}{n} \to 0$$</p>
<p><strong>Conclusion:</strong> The sequence <em>does</em> converge uniformly to 0 on any bounded interval $[-a, a]$.</p>
<p><em>This example illustrates that the domain matters! The same sequence can converge uniformly on one domain but not on another.</em></p>
</div>
<!-- Theorem: Continuity Preserved -->
<div class="theorem content-box">
<div class="timestamp" onclick="openVideoSide('34:07', 'Continuity Preserved Under Uniform Convergence', 'theorem', this)">
<span class="play-icon">▶</span>34:07
</div>
<h3>Continuity is Preserved Under Uniform Convergence</h3>
<p>Let $(f_n)$ and $f$ be functions defined on an interval $[a,b]$. Suppose:</p>
<ol>
<li>$f_n \to f$ uniformly on $[a,b]$</li>
<li>Each $f_n$ is continuous at a point $x_0 \in [a,b]$</li>
</ol>
<p>Then $f$ is continuous at $x_0$.</p>
<p><em>Note: We only need continuity at a single common point $x_0$ for all the $f_n$. If all $f_n$ are continuous everywhere, then $f$ is continuous everywhere.</em></p>
</div>
<!-- Proof: Continuity Preserved -->
<div class="proof content-box">
<div class="timestamp" onclick="openVideoSide('37:15', 'Proof: Continuity Preserved', 'proof', this)">
<span class="play-icon">▶</span>37:15
</div>
<h3>Continuity is Preserved Under Uniform Convergence</h3>
<p><strong>Proof:</strong> Let $\epsilon > 0$ be given. We need to show that $f$ is continuous at $x_0$.</p>
<p><strong>Step 1:</strong> By uniform convergence of $f_n \to f$, there exists $N$ such that for all $n \ge N$:</p>
<p>$$\sup_{y \in [a,b]} |f_n(y) - f(y)| < \epsilon$$</p>
<p>In particular, this holds for $n = N$.</p>
<p><strong>Step 2:</strong> Since $f_N$ is continuous at $x_0$, there exists $\delta > 0$ such that for all $y \in [a,b]$ with $|y - x_0| < \delta$:</p>
<p>$$|f_N(y) - f_N(x_0)| < \epsilon$$</p>
<p><strong>Step 3:</strong> Now for any $y$ with $|y - x_0| < \delta$, we use the triangle inequality:</p>
<p>$$|f(y) - f(x_0)| \le |f(y) - f_N(y)| + |f_N(y) - f_N(x_0)| + |f_N(x_0) - f(x_0)|$$</p>
<p>By Step 1, the first and third terms are each less than $\epsilon$ (since $n = N \ge N$). By Step 2, the middle term is less than $\epsilon$. Therefore:</p>
<p>$$|f(y) - f(x_0)| < \epsilon + \epsilon + \epsilon = 3\epsilon$$</p>
<p>Since $\epsilon$ was arbitrary, this shows $f$ is continuous at $x_0$. $\square$</p>
<p><em><strong>Where would this fail for pointwise convergence?</strong> The issue is that with only pointwise convergence, the $N$ in Step 1 would depend on $y$, and we couldn't use the same $N$ for all $y$ near $x_0$.</em></p>
</div>
<!-- Corollary: Uniform Limits of Continuous Functions -->
<div class="corollary content-box">
<div class="timestamp" onclick="openVideoSide('45:28', 'Uniform Limits of Continuous Functions', 'corollary', this)">
<span class="play-icon">▶</span>45:28
</div>
<h3>Uniform Limits of Continuous Functions are Continuous</h3>
<p>Let $(f_n)$ and $f$ be functions defined on an interval $[a,b]$. Suppose:</p>
<ol>
<li>$f_n \to f$ uniformly on $[a,b]$</li>
<li>Each $f_n$ is continuous on $[a,b]$ (for every $n$)</li>
</ol>
<p>Then $f$ is continuous on $[a,b]$.</p>
<p><em>This is a direct consequence of the previous theorem applied at every point in $[a,b]$.</em></p>
<p><strong>Significance:</strong> We've found a notion of convergence (uniform convergence) that preserves the property of continuity! This is what we set out to achieve. The space of continuous functions is "closed" under taking uniform limits.</p>
</div>
<!-- Theorem 1.3: Completeness -->
<div class="theorem content-box">
<div class="timestamp" onclick="openVideoSide('50:18', 'Completeness of C[a,b]', 'theorem', this)">
<span class="play-icon">▶</span>50:18
</div>
<h3>Completeness of $C[a,b]$</h3>
<p>Let $C[a,b]$ denote the space of all continuous real-valued functions on the interval $[a,b]$. This is a vector space (we can add continuous functions and multiply by scalars).</p>
<p>If $(f_n)$ is a sequence in $C[a,b]$ that is uniformly Cauchy, then $(f_n)$ converges uniformly to some function $f \in C[a,b]$.</p>
<p>In other words: <em>$C[a,b]$ is complete with respect to uniform convergence</em>.</p>
<p><strong>Proof:</strong> Combine Theorem 1.1 (Cauchy criterion) and Corollary 1.1 (continuity preservation):</p>
<ul>
<li>By the Cauchy criterion, if $(f_n)$ is uniformly Cauchy, then $f_n \to f$ uniformly for some function $f$.</li>
<li>By Corollary 1.1, since each $f_n$ is continuous and convergence is uniform, the limit $f$ is also continuous.</li>
<li>Therefore $f \in C[a,b]$, and the sequence converges to a function within the space.</li>
</ul>
<p>$\square$</p>
</div>
<!-- Closing Remarks -->
<div class="content-box">
<div class="timestamp" onclick="openVideoSide('53:40', 'Completeness Analogy', 'default', this)">
<span class="play-icon">▶</span>53:40
</div>
<h3>Completeness Analogy</h3>
<p>Compare this result to the completeness of $\mathbb{R}$:</p>
<ul>
<li><strong>For real numbers:</strong> Every Cauchy sequence of real numbers converges to a real number. There are "no holes" in $\mathbb{R}$.</li>
<li><strong>For $C[a,b]$:</strong> Every uniformly Cauchy sequence of continuous functions converges uniformly to a continuous function. There are "no holes" in $C[a,b]$ with respect to uniform convergence.</li>
</ul>
<p>This is remarkable because $C[a,b]$ is an <em>infinite-dimensional</em> vector space, yet it has this completeness property.</p>
<p><em>If we removed a single point from $\mathbb{R}$ (e.g., $\mathbb{R} \setminus \{0\}$), it would no longer be complete—there would be Cauchy sequences (like $1/n$) that want to converge to the missing point. Similarly, if we removed a function from $C[a,b]$, we could have uniformly Cauchy sequences trying to converge to it.</em></p>
</div>
</div>
</div>
</div>
<!-- ===== RIGHT SIDE - VIDEO ===== -->
<div id="videoSide" class="video-side">
<div class="video-container">
<iframe
id="videoFrame"
src=""
allowfullscreen
allow="autoplay">
</iframe>
</div>
<!-- ===== CLOSE BUTTON ===== -->
<button class="close-button" onclick="closeVideoSide()">×</button>
</div>
</div>
<script>
// ===== VIDEO CONFIGURATION =====
const PANOPTO_BASE_URL = "https://cambridgelectures.cloud.panopto.eu/Panopto/Pages/Viewer.aspx?id=449ebeba-de5a-41e4-a4fe-b33700b5682f&autoplay=false&offerviewer=true&showtitle=true&showbrand=false&captions=false&interactivity=all";
// ===== OPEN VIDEO SIDE FUNCTION =====
function openVideoSide(timestamp, title, contentType, timestampElement) {
const videoSide = document.getElementById('videoSide');
const videoFrame = document.getElementById('videoFrame');
const contentSide = document.querySelector('.content-side');
// Convert timestamp to seconds for Panopto
const timeInSeconds = convertTimestampToSeconds(timestamp);
// Set the video URL with the timestamp
const videoUrl = PANOPTO_BASE_URL + `&start=${timeInSeconds}`;
videoFrame.src = videoUrl;
// Remove full-width class and show video
contentSide.classList.remove('full-width');
videoSide.style.display = 'block';
}
// ===== CLOSE VIDEO SIDE FUNCTION =====
function closeVideoSide() {
const videoSide = document.getElementById('videoSide');
const videoFrame = document.getElementById('videoFrame');
const contentSide = document.querySelector('.content-side');
// Hide the video side
videoSide.style.display = 'none';
// Clear the video source to stop playback
videoFrame.src = '';
// Add full-width class back
contentSide.classList.add('full-width');
}
// ===== TIMESTAMP CONVERSION FUNCTION =====
function convertTimestampToSeconds(timestamp) {
const parts = timestamp.split(':');
const minutes = parseInt(parts[0]);
const seconds = parseInt(parts[1]);
return minutes * 60 + seconds;
}
// Re-render MathJax when page loads
document.addEventListener('DOMContentLoaded', function() {
if (window.MathJax) {
MathJax.typesetPromise();
}
});
</script>
</body>
</html>