Lecture Notes

User Tools

Site Tools

You are here: Course Notes » Math 104: Introduction to Real Analysis (Spring 2021) » Student Area » Martin Zhai's Review Note
math104-s21:s:martinzhai

Differences

This shows you the differences between two versions of the page.

Link to this comparison view

Both sides previous revision Previous revision
Next revision 		Previous revision
math104-s21:s:martinzhai [2022/01/11 18:57]
pzhou ↷ Page moved from math104-2021sp:s:martinzhai to math104-s21:s:martinzhai 		math104-s21:s:martinzhai [2026/02/21 14:41] (current)
Line 365: Line 365:
     * __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.
-{{ math104-2021sp: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 ===
Line 466: Line 466:
       * 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 $m_i= \inf \{f(x), x\isin [x_{i-1},x_i]\}$.
-{{ math104-2021sp: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 $m(b-a)\leq L(P,f) leq U(P,f) \leq M(b-a)$.
Line 539: Line 539:
  
 ==== Questions ==== ==== Questions ====
-  - {{math104-2021sp: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 $E$?
-  - {{math104-2021sp: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?
Line 558: Line 558:
   - 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: {{ math104-2021sp:s:practice_solutions.pdf |}}+  - This is my solutions towards the practice exam: {{ math104-s21:s:practice_solutions.pdf |}}
math104-s21/s/martinzhai.1641927445.txt.gz · Last modified: 2026/02/21 14:44 (external edit)

Page Tools

Except where otherwise noted, content on this wiki is licensed under the following license: CC Attribution-Share Alike 4.0 International
CC Attribution-Share Alike 4.0 International Donate Powered by PHP Valid HTML5 Valid CSS Driven by DokuWiki