From 66d09aaff211309256c3f5e126790ddddd7e893b Mon Sep 17 00:00:00 2001
From: Trance-0 <60459821+Trance-0@users.noreply.github.com>
Date: Mon, 3 Mar 2025 12:34:00 -0600
Subject: [PATCH] update and fix typo

---
 pages/Math4121/Math4121_L1.md  |   2 +-
 pages/Math4121/Math4121_L12.md |   2 +-
 pages/Math4121/Math4121_L13.md |   2 +-
 pages/Math4121/Math4121_L17.md |   2 +-
 pages/Math4121/Math4121_L18.md |   6 +-
 pages/Math4121/Math4121_L19.md |   8 +--
 pages/Math4121/Math4121_L2.md  |   2 +-
 pages/Math4121/Math4121_L20.md | 108 ++++++++++++++++++++++++++++++++-
 pages/Math4121/Math4121_L3.md  |   2 +-
 pages/Math4121/Math4121_L4.md  |   2 +-
 pages/Math4121/Math4121_L5.md  |   2 +-
 pages/Math4121/Math4121_L6.md  |   2 +-
 pages/Math4121/Math4121_L7.md  |   2 +-
 pages/Math4121/Math4121_L8.md  |   2 +-
 14 files changed, 123 insertions(+), 21 deletions(-)

diff --git a/pages/Math4121/Math4121_L1.md b/pages/Math4121/Math4121_L1.md
index e68fd8f..b9b4878 100644
--- a/pages/Math4121/Math4121_L1.md
+++ b/pages/Math4121/Math4121_L1.md
@@ -1,4 +1,4 @@
-# Math4121 Lecture 1
+# Math 4121 Lecture 1
 
 ## Chapter 5: Differentiation
 
diff --git a/pages/Math4121/Math4121_L12.md b/pages/Math4121/Math4121_L12.md
index d200007..7c44821 100644
--- a/pages/Math4121/Math4121_L12.md
+++ b/pages/Math4121/Math4121_L12.md
@@ -1,4 +1,4 @@
-# Math4121 Lecture 12
+# Math 4121 Lecture 12
 
 ## Chapter 7: Uniform Convergence and Integrals
 
diff --git a/pages/Math4121/Math4121_L13.md b/pages/Math4121/Math4121_L13.md
index 1d18f9b..4300e5d 100644
--- a/pages/Math4121/Math4121_L13.md
+++ b/pages/Math4121/Math4121_L13.md
@@ -1,4 +1,4 @@
-# Math4121 Lecture 13
+# Math 4121 Lecture 13
 
 ## Hidden Chapter 1
 
diff --git a/pages/Math4121/Math4121_L17.md b/pages/Math4121/Math4121_L17.md
index 836af8a..d0e2454 100644
--- a/pages/Math4121/Math4121_L17.md
+++ b/pages/Math4121/Math4121_L17.md
@@ -1,4 +1,4 @@
-# Math4121 Lecture 17
+# Math 4121 Lecture 17
 
 ## Continue on Last lecture
 
diff --git a/pages/Math4121/Math4121_L18.md b/pages/Math4121/Math4121_L18.md
index 2b5d665..887baec 100644
--- a/pages/Math4121/Math4121_L18.md
+++ b/pages/Math4121/Math4121_L18.md
@@ -1,4 +1,4 @@
-# Math4121 Lecture 18
+# Math 4121 Lecture 18
 
 ## Continue
 
@@ -10,7 +10,7 @@ By modifying this example, we can find similar with any outer content between 0
 
 #### Definition: Perfect Set
 
-$S\subsetes[0,1]$ is perfect if $S=S'$.
+$S\subseteq[0,1]$ is perfect if $S=S'$.
 
 Example:
 
@@ -28,7 +28,7 @@ Let $C_0=[0,1]$, $C_1=[0,\frac{1}{3}]\cup[\frac{2}{3}]$ ...
 Continuing this process indefinitely, we define the Cantor set as
 
 $$
-C=\Bigcap_{n=0}^{\infty}C_n
+C=\bigcap_{n=0}^{\infty}C_n
 $$
 
 1. $C_n\subseteq C_{n-1}$
diff --git a/pages/Math4121/Math4121_L19.md b/pages/Math4121/Math4121_L19.md
index a9a5f2d..6998058 100644
--- a/pages/Math4121/Math4121_L19.md
+++ b/pages/Math4121/Math4121_L19.md
@@ -34,11 +34,7 @@ If $C$ is cover of $T$, then $S\subseteq T\subseteq C$, so $C$ is a cover of $S$
 
 QED
 
-#### Theorem Osgorod's Lemma
+#### Theorem Osgood's Lemma
 
-If $S$ is closed and bounded, then
-
-$$
-\lim_{k\to \infty} c_e(S_k)=c_e(S)
-$$
+Let $S$ be a closed, bounded set in $\mathbb{R}$, and $S_1\subseteq S_2\subseteq \ldots$, and $S=\bigcup_{n=1}^{\infty} S_n$. Then $\lim_{k\to\infty} c_e(S_k)=c_e(S)$.
 
diff --git a/pages/Math4121/Math4121_L2.md b/pages/Math4121/Math4121_L2.md
index 33fc200..6b8556c 100644
--- a/pages/Math4121/Math4121_L2.md
+++ b/pages/Math4121/Math4121_L2.md
@@ -1,4 +1,4 @@
-# Math4121 Lecture 2
+# Math 4121 Lecture 2
 
 ## Chapter 5: Differentiation
 
diff --git a/pages/Math4121/Math4121_L20.md b/pages/Math4121/Math4121_L20.md
index ff21618..3a5db79 100644
--- a/pages/Math4121/Math4121_L20.md
+++ b/pages/Math4121/Math4121_L20.md
@@ -1 +1,107 @@
-# Lecture 20
\ No newline at end of file
+# Math 4121 Lecture 20
+
+## Continue on Chapter 4
+
+### Properties of the Cantor Set
+
+Monotonicity: If $S\subseteq T$, then $c_e(S)\leq c_e(T)$.
+
+Sub-additivity: $c_e(S\cup T)\leq c_e(S)+c_e(T)$.
+
+Example: $S=\mathbb{Q}\cap[0,1]$, $T=[0,1]\setminus\mathbb{Q}$.
+
+Then $c_e(S)=1$, $c_e(T)=1$, even though $S\cap T=\emptyset$.
+
+$S\cup T=[0,1]$, $c_e(S\cup T)=1\leq 1+1=c_e(S)+c_e(T)$
+
+The above example shows that:
+
+> The following is **not true**: $c_e(S\cup T)=c_e(S)+c_e(T)$ if $S\cap T=\emptyset$.
+
+However, the following is true:
+
+(In $\mathbb{R}$)
+
+If $S=\bigcup_{n=1}^{\infty} I_n$, $T=\bigcup_{n=1}^{\infty} J_n$, where $I_n$ and $J_n$ are intervals, and $S\cap T=\emptyset$, then $c_e(S\cup T)=c_e(S)+c_e(T)$.
+
+### Back to Osgood's Lemma
+
+#### Osgood's Lemma
+
+Let $S$ be a closed, bounded set in $\mathbb{R}$, and $S_1\subseteq S_2\subseteq \ldots$, and $S=\bigcup_{n=1}^{\infty} S_n$. Then $\lim_{k\to\infty} c_e(S_k)=c_e(S)$.
+
+Proof:
+
+Trivial that $c_e(S_k)\leq c_e(S)$.
+
+We need to show that $\forall \epsilon>0, \exists K$ such that $c_e(S_k)>c_e(S)-\epsilon$ for all $k\geq K$.
+
+Let $U_k$ be finite union of open intervals containing $S_k$ such that $c_e(U_k)<c_e(S_k)+\frac{\epsilon}{2^k}$.
+
+So $\{U_k\}_{k=1}^{\infty}$ are an open cover of $S$.
+
+Since $S$ is closed and bounded in $\mathbb{R}$, it is compact.
+
+So, $\exists N$ such that $S\subseteq \bigcup_{k=1}^{N} U_k$.
+
+Then we split the $U_k$ into two parts:
+
+$U_k=(U_k\cap U_N)\cup (U_k\setminus U_N)$, we denote $U_k^{(1)}=U_k\cap U_N$, $U_k^{(2)}=U_k\setminus U_N$, for $k\leq N$.
+
+So, since $U_k^{(1)}, U_k^{(2)}$ disjoint intervals, and $S_k\subseteq U_k^{(1)}$, we have
+
+$$
+\begin{aligned}
+c_e(U_k^{(1)})+c_e(U_k^{(2)})&=c_e(U_k)\\
+c_e(S_k)+c_e(U_k^{(2)})&<c_e(S_k)+\frac{\epsilon}{2^k}\\
+c_e(U_k^{(2)})&<\frac{\epsilon}{2^k}\\
+\end{aligned}
+$$
+
+So, 
+
+$$
+\begin{aligned}
+c_e(S)&\leq c_e(U)\\
+&\leq c_e(U_N)+\sum_{k=1}^{N-1} c_e(U_k^{(2)})\\
+&<c_e(S_N)+\frac{\epsilon}{2^{N}}+\sum_{k=1}^{N-1}\frac{\epsilon}{2^k}\\
+&\leq c_e(S_N)+\epsilon\\
+&<c_e(S_N)
+\end{aligned}
+$$
+
+QED
+
+### Convergence Theorems for sequences of functions
+
+Is
+
+$$
+\lim_{n\to\infty}\int f_n(x)\ dx=\int \lim_{n\to\infty} f_n(x)\ dx
+$$
+
+?
+
+Yes when $f_n\to f$ uniformly.
+
+Uniform convergence also means $\lim_{n\to\infty} \sup_{x\in [a,b]}|f_n(x)-f(x)|=0$.
+
+But there exists some cases that does not converge to the limit but still satisfies the above condition.
+
+#### Theorem 4.5 (Arzela-Osgood Theorem)
+
+If $\{f_n\}_{n=1}^{\infty}$ is a sequence of continuous, uniformly bounded function and $f(x)=\lim_{n\to\infty} f_n(x)$ exists for all $x\in [a,b]$ (pointwise convergence), then
+
+$$
+\lim_{n\to\infty}\int_a^b f_n(x)\ dx=\int_a^b f(x)\ dx
+$$
+
+Proof:
+
+Define $\Gamma_{\alpha}=\{x:\forall m\in \mathbb{N} \textup{ and }\forall \delta>0, \exists n\geq m \textup{ s.t. } |y-x|<\delta \textup{ and } |f_n(y)-f_m(y)|>\alpha\}$.
+
+_$\Gamma_{\alpha}$ is the negation of $(\alpha,\delta)$ definition of limit._
+
+$\Gamma_{\alpha}$ is closed and nowhere dense.
+
+Continue on next lecture.
diff --git a/pages/Math4121/Math4121_L3.md b/pages/Math4121/Math4121_L3.md
index c60eb25..bf7af18 100644
--- a/pages/Math4121/Math4121_L3.md
+++ b/pages/Math4121/Math4121_L3.md
@@ -1,4 +1,4 @@
-# Math4121 Lecture 3
+# Math 4121 Lecture 3
 
 ## Continue on Differentiation
 
diff --git a/pages/Math4121/Math4121_L4.md b/pages/Math4121/Math4121_L4.md
index e02fe9e..3e64cd8 100644
--- a/pages/Math4121/Math4121_L4.md
+++ b/pages/Math4121/Math4121_L4.md
@@ -1,4 +1,4 @@
-# Math4121 Lecture 4
+# Math 4121 Lecture 4
 
 ## Chapter 5. Differentiation
 
diff --git a/pages/Math4121/Math4121_L5.md b/pages/Math4121/Math4121_L5.md
index bde8258..98dd913 100644
--- a/pages/Math4121/Math4121_L5.md
+++ b/pages/Math4121/Math4121_L5.md
@@ -1,4 +1,4 @@
-# Math4121 Lecture 5
+# Math 4121 Lecture 5
 
 ## Continue on differentiation
 
diff --git a/pages/Math4121/Math4121_L6.md b/pages/Math4121/Math4121_L6.md
index 6767b7b..7becfd2 100644
--- a/pages/Math4121/Math4121_L6.md
+++ b/pages/Math4121/Math4121_L6.md
@@ -1,4 +1,4 @@
-# Math4121 Lecture 6
+# Math 4121 Lecture 6
 
 ## Chapter 6: Riemann-Stieltjes Integral
 
diff --git a/pages/Math4121/Math4121_L7.md b/pages/Math4121/Math4121_L7.md
index 998b50f..bc04ae9 100644
--- a/pages/Math4121/Math4121_L7.md
+++ b/pages/Math4121/Math4121_L7.md
@@ -1,4 +1,4 @@
-# Math4121 Lecture 7
+# Math 4121 Lecture 7
 
 ## Continue on Chapter 6
 
diff --git a/pages/Math4121/Math4121_L8.md b/pages/Math4121/Math4121_L8.md
index c918c63..e192470 100644
--- a/pages/Math4121/Math4121_L8.md
+++ b/pages/Math4121/Math4121_L8.md
@@ -1,4 +1,4 @@
-# Math4121 Lecture 8
+# Math 4121 Lecture 8
 
 ## Continue on Riemann-Stieltjes Integral