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math104-s21:s:martinzhai [2021/05/09 15:16]
173.205.95.2 [Week 15] 		math104-s21:s:martinzhai [2026/02/21 14:41] (current)
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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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 ==== Questions ==== ==== Questions ====
-  - {{:math104:s:img_769dec51bb88-1.jpeg?400|}} I understand the visualization of this recursive sequence, but  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?   - Is there a way/analogy to understand/visualize the closure of a set?
   - 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 finding finite subcovers for all open covers?
   - 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 $p\isin E$, 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 $E$?
-  - {{:math104:s:img_0324.jpg?400|}} How is the claim at the bottom proved?+  - {{math104-s21: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 the global maximum as the maximum of all local minimums?
   - 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?
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   - 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)   - 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.   - Question 16 on Prof Fan's practice exam.
-  - This is my solutions towards the practice exam: {{ :math104:s:practice_exam.pdf |}}+  - This is my solutions towards the practice exam: {{ math104-s21:s:practice_solutions.pdf |}}
math104-s21/s/martinzhai.1620573373.txt.gz · Last modified: 2026/02/21 14:44 (external edit)

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