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math104-s21:s:martinzhai [2021/05/06 08:51]
66.154.105.2 [Week 12] 		math104-s21:s:martinzhai [2026/02/21 14:41] (current)
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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.     * __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:     * __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       - arbitrary union of open subsets is still open
       - finite intersections of open sets are open       - finite intersections of open sets are open
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   ***Uniform Continuity**:   ***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.     *__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 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.       *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**:   * **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 $\lim_{n\to\infty} x_{ni} = x_i$.     * __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 $d_{\infty}(x_n, x)$).     * __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).       * 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 $\forall x\isin\Reals$, $\lim_{n\to\infty} f_n(x) = f(x) \iff \lim_{n\to\infty} \lvert f_n(x) - f(x) \rvert = 0$.     * __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.       * 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 === === 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 7.8__: 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 $\iff \lim_{n\to\infty} (\sup \lvert f_n(x) - f(x) \rvert) = 0$.     * __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.     * __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.       * Example: See last question on Midterm 2 version A.
   ***Uniform Convergence and Continuity**:   ***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 7.12__: 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     * __Rudin 7.13__: Suppose $K$ compact and
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       * Example: $[10, 20]\subset \Reals$, then a partition would be $P = \{10, 15, 18, 19, 20\}$.       * 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 $m_i= \inf \{f(x), x\isin [x_{i-1},x_i]\}$.     * __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 $L(f)= \sup_{P} L(P,f)$.     * __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 $m(b-a)\leq L(P,f) leq U(P,f) \leq M(b-a)$.       * 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 $m(b-a)\leq L(P,f) leq U(P,f) \leq M(b-a)$.
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     *__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 $\int_{a}^{b} fd\alpha = \lim_{n\to\infty} \int_{a}^{b} f_n d\alpha$.     *__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 $\int_{a}^{b} F(x)d\alpha = \sum_{n=1}^{\infty} \int_{a}^{b} f_n(x)d\alpha$.     *__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 ==== ==== 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?   - 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 |}}
math104-s21/s/martinzhai.1620291078.txt.gz · Last modified: 2026/02/21 14:44 (external edit)

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