format updates

content/Math4121/Math4121_L20.md

@@ -30,7 +30,8 @@ If $S=\bigcup_{n=1}^{\infty} I_n$, $T=\bigcup_{n=1}^{\infty} J_n$, where $I_n$ a
 
 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:
+<details>
+<summary>Proof of Osgood's Lemma</summary>
 
 Trivial that $c_e(S_k)\leq c_e(S)$.
 
@@ -70,7 +71,7 @@ c_e(S)&\leq c_e(U)\\
 \end{aligned}
 $$
 
-QED
+</details>
 
 ### Convergence Theorems for sequences of functions
 
@@ -96,7 +97,8 @@ $$
 \lim_{n\to\infty}\int_a^b f_n(x)\ dx=\int_a^b f(x)\ dx
 $$
 
-Proof:
+<details>
+<summary>Proof of Arzela-Osgood Theorem (incomplete)</summary>
 
 Define $\Gamma_{\alpha}=\{x:\forall m\in \mathbb{N} \textup{ and }\forall \delta>0, \exists n\geq m \textup{ s.t. } |y-x|<\delta \textup{ and } |f_n(y)-f_m(y)|>\alpha\}$.
 
@@ -105,3 +107,4 @@ _$\Gamma_{\alpha}$ is the negation of $(\alpha,\delta)$ definition of limit._
 $\Gamma_{\alpha}$ is closed and nowhere dense.
 
 Continue on next lecture.
+</details>