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--- math104-s21:s:martinzhai [2021/05/06 01:51]
66.154.105.2 [Week 12]
+++ math104-s21:s:martinzhai [2022/01/11 18:31] (current)
24.253.46.239 ↷ Links adapted because of a move operation
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     * __Theorem 13.5 (Bolzano-Weiestrass Theorem for  $\Reals^n$ )__: Every bounded sequence  $(s_m)_m \isin \Reals^n$  has a convergent subsequence.
     * __Topology__: Let  $S$  be a set. A topological structure on  $S$  is the data of a collection of subsets in S. This collection needs to satisfy:
-      -  $S$  and $\text{\o} are open
+      -  $S$  and $\text{\o}$ are open
       - arbitrary union of open subsets is still open
       - finite intersections of open sets are open
@@ Line 309: / Line 309: @@
   ***Uniform Continuity**:
     *__Uniform Continuous Function__: $f:X\rightarrow Y $. Suppose for all$ \epsilon >0 $,$ \exists \delta >0 $such that$ \forall p,q\isin X $with$ d_X(p,q)<\delta$, we have $d_Y(f(p),f(q)) <\epsilon $. Then we say$ f$ is a uniform continuous function.
-      *Example:  $f:[0,1] \rightarrow \Reals$  and  $f(x)=x^2$ . Then  $f$  is uniformly continuous function.  $\forall \epsilon >0$ , we can take  $\delta=\frac{\epsilon}{2}$ , then  $\forall p,q \isin [0,1]$ ,  $\lvert p-q \rvert < \delta$  we have $\lvert p^2-q^2 \rvert = \lvert (p-q)(p+q) \rvert = \lvert (p-q) \rvert \lvert (p+q) \rvert < \delta * 2 = \epsilon.
+      *Example:  $f:[0,1] \rightarrow \Reals$  and  $f(x)=x^2$ . Then  $f$  is uniformly continuous function.  $\forall \epsilon >0$ , we can take  $\delta=\frac{\epsilon}{2}$ , then  $\forall p,q \isin [0,1]$ ,  $\lvert p-q \rvert < \delta$  we have $\lvert p^2-q^2 \rvert = \lvert (p-q)(p+q) \rvert = \lvert (p-q) \rvert \lvert (p+q) \rvert < \delta * 2 = \epsilon$.
       *Non-Example 1:  $f:\Reals \rightarrow \Reals$  and  $f(x)=x^2$ . Then  $f$  is not uniformly continuous.  $\delta > 0$ , we could always find  $p,q\isin \Reals$  and  $\lvert p-q \rvert < \delta$  such that  $\lvert f(p) - f(q) \rvert = 1$ . If we take  $p-q = \frac{\delta}{2}$  and  $p+q = \frac{2}{\delta}$ , then  $\lvert f(p) - f(q)\rvert = \lvert p^2-q^2 \rvert = \lvert (p-q)(p+q) \rvert = \lvert (p-q) \rvert \lvert (p+q) \rvert = \frac{\delta}{2} * \frac{2}{\delta} = 1$ .
       *Non-Example 2:  $f:(0, \infty) \rightarrow \Reals$ ,  $f(x)=\frac{1}{x}$  is not uniformly continuous. Intuition: when  $x\rightarrow 0$ , then distance between two points  $p,q$  may be close enough but  $\lvert \frac{1}{p} - \frac{1}{q} \rvert$  may be large.
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   * **Sequence and Convergence of Functions**:
     * __Pointwise Convergence of Sequence of Sequences__: Let  $(x_n)_n$  be a sequence of sequences,  $x_n\isin \Reals^{\natnums}$ , we say  $(x_n)_n$  converges to  $x\isin \Reals^{\natnums}$  pointwise if  $\forall i\isin \natnums$ , we have  $\lim_{n\to\infty} x_{ni} = x_i$ .
-      * Example:  $x_{ni} = \frac{i}{n+i}$ , then this sequence of sequences converge to  $0$  pointwise, since for arbitrary fixed  $i$ , we have  $\lim_{n\to\infty} x_{ni} = \lim_{n\to\infty} \frac{i}{n+i} = 0$ .
+      * Example:  $x_{ni} = \frac{i}{n+i}$ , then this seq to  $0$  pointwise, since for arbitrary fixed  $i$ , we have  $\lim_{n\to\infty} x_{ni} = \lim_{n\to\infty} \frac{i}{n+i} = 0$ .
     * __Uniform Convergence of Sequence of Sequences__: Let  $(x_n)_n$  be a sequence of sequences,  $x_n\isin \Reals^{\natnums}$ , we say  $x_n \rightarrow x$  uniformly if  $\forall \epsilon >0$ ,  $\exists N>0$  such that  $\forall n>N$ , $\sup\{\lvert x_{ni} - x_i \rvert :\, i\isin\natnums\} <\epsilon $(also known as$ d_{\infty}(x_n, x)$).
       * Non-Example:  $x_{ni} = \frac{i}{n+i}$  failed to converge uniformly to  $0$ , since $d_{\infty}(x_n,0) = \sup \{\lvert x_{ni} \rvert :\, i\isin\natnums\} = \sup \{\frac{i}{n+i}:\, i\isin\natnums\} = 1$ (n is fixed).
     * __Pointwise Convergence of Sequence of Functions__: Given a sequence of functions  $f_n \isin$  Map$(\Reals, \Reals) $, we say$ f_n $converge to$ f $pointwise if$ \forall x\isin\Reals$,  $\lim_{n\to\infty} f_n(x) = f(x) \iff \lim_{n\to\infty} \lvert f_n(x) - f(x) \rvert = 0$ .
       * Examples：Running and Shrinking Bumps.
-{{ :math104:s:img_d0796d4b49e0-1.jpeg?400 |}}
+{{ math104-s21:s:img_d0796d4b49e0-1.jpeg?400 |}}
 === Lecture 18 (Mar 18) - Covered Rudin Chapter 7 ===
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     * __Rudin 7.8__: Suppose  $f_n: X\rightarrow \Reals$  satisfies that $\forall \epsilon >0,\exists N>0 $such that$ \forall x\isin X, \lvert f_n(x) - f_m(x) \rvert < \epsilon $, then$ f_n $converges uniformly (Uniform Cauchy$ \iff$ Uniform Convergence).
     * __Rudin 7.9__: Suppose  $f_n\rightarrow f$  pointwise, then  $f_n \rightarrow f$  uniformly  $\iff \lim_{n\to\infty} (\sup \lvert f_n(x) - f(x) \rvert) = 0$ .
+    * A sequence of functions  $\{ f_n\}$  is uniformly convergent to $f:D\to\Reals\iff \lim_{n\to\infty} \sup \{\lvert f_n(x) - f(x) \rvert : x\isin D\}$.
     * __Rudin 7.10 (Weiestrass M-Test)__: Suppose  $f(x) = \sum_{n=1}^{\infty} f_n(x)\, \forall x\isin X$ . If  $\exists M_n>0$  such that  $\sup_x \lvert f_n(x) \rvert \leq M_n$  and  $\sum_{n} M_n < \infty$ , then the partial sum  $F_N(x)=\sum_{n=1}^{N} f_n(x)$  converges to  $f(x)$  uniformly.
       * Example: See last question on Midterm 2 version A.
   ***Uniform Convergence and Continuity**:
-    * __Rudin 7.11__: Suppose  $f_n \rightarrow f$  uniformly on set  $E$  in a metric space. Let  $x$  be a limit point of  $E$ , and suppose that  $\lim_{t\to x} f_n(t) = A_n$ . Then  $\{ A_n\}$  converges and  $\lim_{t\to x} f(t) = \lim_{n\to\infty} A_n$ . In conclusion,  $\lim_{t\to x} \lim_{n\to\infty} f_n(t) = \lim_{n\to\infty} \lim_{t\to x} f_n(t)$ .
+    * __Rudin 7.11__: Suppose  $f_n \rightarrow f$  uniformly on set  $E$  in a metric space.  of  $E$ , and suppose that  $\lim_{t\to x} f_n(t) = A_n$ . Then  $\{ A_n\}$  converges and  $\lim_{t\to x} f(t) = \lim_{n\to\infty} A_n$ . In conclusion,  $\lim_{t\to x} \lim_{n\to\infty} f_n(t) = \lim_{n\to\infty} \lim_{t\to x} f_n(t)$ .
     * __Rudin 7.12__: If  $\{f_n\}$  is a sequence of continuous functions on  $E$ , and if  $f_n\rightarrow f$  uniformly on  $E$ , then  $f$  is continuous on  $E$ .
     * __Rudin 7.13__: Suppose  $K$  compact and
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       * Example:  $[10, 20]\subset \Reals$ , then a partition would be  $P = \{10, 15, 18, 19, 20\}$ .
     * __U(P,f) and L(P,f)__: Given  $f:[a,b]\to \Reals$  bounded, and partion  $p = \{x_0 \leq x_1 \leq ... \leq x_n\}$ , we define  $U(P,f) = \sum_{i=1}^{n} \Delta x_i M_i$  where $M_i= \sup \{f(x), x\isin [x_{i-1},x_i]\}$;  $L(P,f) = \sum_{i=1}^{n} \Delta x_i m_i$  where $m_i= \inf \{f(x), x\isin [x_{i-1},x_i]\}$.
-{{ :math104:s:img_0a0c56a64ed8-1.jpeg?400 |}}
+{{ math104-s21:s:img_0a0c56a64ed8-1.jpeg?400 |}}
     * __U(f) and L(f)__: Define  $U(f) = \inf_{P} U(P,f)$  and  $L(f)= \sup_{P} L(P,f)$ .
       * Since  $f$  is bounded, hence  $\exists m,M\isin \Reals$  such that  $m\leq f(x)\leq M$  for all  $x\isin [a,b]$ , then  $\forall P$  partition of  $[a,b]$ ,  $U(P,f) \leq \sum_{i=1}^{n} \Delta x_i M = M(b-a)$ , and  $L(P,f) \geq m(b-a)$ , and  $m(b-a)\leq L(P,f) leq U(P,f) \leq M(b-a)$ .
@@ Line 535: / Line 536: @@
     *__Rudin 7.16__: Let  $\alpha$  be monotone increasing on  $[a,b]$ . Suppose  $f_n\isin\mathscr{R}(\alpha)$ , and  $f_n\to f$  uniformly on  $[a,b]$ . Then  $f$  is integrable and  $\int_{a}^{b} fd\alpha = \lim_{n\to\infty} \int_{a}^{b} f_n d\alpha$ .
     *__Corollary__: Suppose  $f_n\isin\mathscr{R}(\alpha)$  and  $F(x) = \sum_{n=1}^{\infty} f_n(x)$ , the series converges uniformly, then  $F\isin\mathscr{R}(\alpha)$  and  $\int_{a}^{b} F(x)d\alpha = \sum_{n=1}^{\infty} \int_{a}^{b} f_n(x)d\alpha$ .
-    *__Theorem__: Suppose $\{ f_N \} $is a sequence of differentiable functions on$ [a,b] $such that$ f_n'(x) $converges uniformly to$ g(x) $and$ \exists x_o\isin [a,b] $such that$ \{f_n(x_o)\} $converges. Then$ f_n(x) $converges to some function$ f $uniformly and$ f'(x)=g(x)=\lim_{n\to\infty} f_n'(x)$.
+    *__Theorem__: Suppose $\{ f_n \} $is a sequence of differentiable functions on$ [a,b] $such that$ f_n'(x) $converges uniformly to$ g(x) $and$ \exists x_o\isin [a,b] $such that$ \{f_n(x_o)\} $converges. Then$ f_n(x) $converges to some function$ f $uniformly and$ f'(x)=g(x)=\lim_{n\to\infty} f_n'(x)$.
 ==== Questions ====
-  - {{:math104:s:img_769dec51bb88-1.jpeg?400|}} I understand the visualization of this recursive sequence, but how to prove it converges to  $\sqrt{5}$ ?
+  - {{math104-s21:s:img_769dec51bb88-1.jpeg?400|}} I understand the visualization of this recursive sequence, but  to  $\sqrt{5}$ ?
-  - In general, how to prove a set is infinite (in order to use theorem 11.2 in Ross)?
+  - In general, how to prove a set is infinite(in order to use theorem 11.2 in Ross)?
-  - Is there a way/analogy to understand/visualize the closure of a set? The definition is quite vague.
+  - Is there a way/analogy to understand/visualize the closure of a set?
-  - When should we use strong induction instead of regular induction? Will we get different results after using strong induction instead of induction?
+  - Is there a way to actually test if a set is compact or not instead of merely finding finite subcovers for all open covers?
-  - Is there a way to actually test if a set is compact or not instead of merely coming up with some open covers where the set is not finitely covered?
+  - Rudin 4.6 states that if  $p\isin E$ , a limit point, and  $f$  is continuous at  $p$  if and only if  $\lim_{x\to p} f(x) = f(p)$ . Does this theorem hold for  $p\isin E$  but not a limit point of  $E$ ?
-  - Rudin 4.6 states that if  $p\isin E$  as a limit point of  $E$ , then  $f$  is continuous at  $p$  if and only if  $\lim_{x\to p} f(x) = f(p)$ . Does this theorem hold for  $p\isin E$  but  $p$  is not a limit point of  $E$ ?
+  - {{math104-s21:s:img_0324.jpg?400|}} How is the claim at the bottom proved?
-  - {{:math104:s:img_0324.jpg?400|}} How is the claim at the bottom proved?
+  - Could we regard the global maximum as the maximum of all local minimums?
-  - Could we regard global maxima as the maximum of all local maximums?
   - Under what circumstance would Taylor Theorem be essential to our proof, since for the question appeared in HW11 Q2, I really do not think using Taylor Theorem is a better way to prove the claim?
-  - In order for a Taylor series to converge ( $\sum_{n} c_n z^n$ ),  $\lvert z \rvert < R$  where  $R$  is the radium of convergence. But if  $\lvert z \rvert = R$ , how can we determine whether the series is convergent or divergent?
+  - In order for a Taylor series to converge ( $\sum_{n} c_n z^n$ ),  $\lvert z \rvert < R$  where  $R$  is the radium of convergence. But if  $\lvert z \rvert = R$ , how can we tell?
-  - If we are claiming  $f$  is continuous on $[a,b]$, how can we prove that  $f$  is continuous at the endpoints, i.e. do we just extend our interval to the left side of  $a$  and right side of  $b$  to do so?
+  - If we are claiming  $f$  is continuous on  $[a,b]$ , , i.e. do we just extend our interval to the left side of  $a$  and right side of  $b$  to do so?
-  - What information could we extract from the line " $f$  has a bounded first derivative (i.e.  $\lvert f' \rvert \leq M$  for some $M>0$)"?
+  - What information can we extract from the line " $f$  has a bounded first derivative (i.e.  $\lvert f' \rvert \leq M$  for some $M>0$)"?
-  - How do we prove a set is sequentially compact without proving that it is compact? (Starting from its definition seems too complicated to take into account all sequences in the set)
+  - How sequentially compact without proving that it is compact? (Starting from ms too complicated to take into account all sequences in the set)
-  -
+  - If  $a_{n+1} = \cos (a_n)$  and choose  $a_1$  such that  $0 < a_1 < 1$ , is  $a_n$  a ?
+  - Does uniform convergence on a sequence of functions  $\{f_n\}$  in  $F$  to  $f$  imply ?
+  - If  $\sum f_n$  converges uniformly, does it imply  $f_n$  satisfies Weiestrass M-test?
+  - For the alternating series test, if instead of sequence of numbers we have sequence of functions and those functions  $\{ f_n \}$  satisfies  $f_1 \geq f_2 \geq f_3 ...$  and  $f_n \geq 0$  for all  $x\isin X$ ,  $\lim f_n = 0$ , does that mean  $\sum_{n} (-1)^n f_n$  converges uniformly?
+  - What is measure zero? (Related to Lebesgue measure and volume of open balls)
+  - Is there any covering for a compact set that satisfies total length of the finite subcovers for the set is less than  $\lvert b-a \rvert$ ? (Answer is no)
+  - Question 16 on Prof Fan's practice exam.
+  - This is my solutions towards the practice exam: {{ math104-s21:s:practice_solutions.pdf |}}

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