From 0b1ea591c1f2964fd20b2666dcb96f4e83a063ba Mon Sep 17 00:00:00 2001
From: Trance-0 <60459821+Trance-0@users.noreply.github.com>
Date: Mon, 10 Feb 2025 15:19:11 -0600
Subject: [PATCH] Update Math4121_L12.md

---
 pages/Math4121/Math4121_L12.md | 42 +++++++++++++++++++++++++++++++++-
 1 file changed, 41 insertions(+), 1 deletion(-)

diff --git a/pages/Math4121/Math4121_L12.md b/pages/Math4121/Math4121_L12.md
index 7835175..d169718 100644
--- a/pages/Math4121/Math4121_L12.md
+++ b/pages/Math4121/Math4121_L12.md
@@ -96,4 +96,44 @@ Define $\epsilon_n=\sup_{x\in[a,b]}|f_n(x)-f(x)|$.
 
 By uniform convergence, $\epsilon_n\to 0$ as $n\to\infty$.
 
-CONTINUE HERE
+$$
+f_n(x)-\epsilon_n\leq f(x)\leq f_n(x)+\epsilon_n
+$$
+
+$$
+\int_a^b f_n-\epsilon_nd\alpha\leq\overline{\int_a^b}fd\alpha\leq\int_a^bf_n+\epsilon_nd\alpha
+$$
+
+
+$$
+\int_a^b f_n-\epsilon_n d\alpha\leq\underline{\int_a^b}fd\alpha\leq\int_a^b f_n+\epsilon_n d\alpha
+$$
+
+So,
+
+$$
+0\leq\overline{\int_a^b}fd\alpha-\underline{\int_a^b}fd\alpha\leq\int_a^b \epsilon_n d\alpha+\int_a^b \epsilon_n d\alpha=2\epsilon_n[\alpha(b)-\alpha(a)]
+$$
+
+So $f\in\mathscr{R}(\alpha)$ and
+
+$$
+\int_a^b fd\alpha\leq \int_a^b f_n d\alpha+\int_a^b \epsilon_n d\alpha\leq \int_a^b fd\alpha+2\epsilon_n d\alpha
+$$
+
+So,
+
+$$
+\int_a^b fd\alpha-\int_a^b \epsilon_n d\alpha\leq \int_a^b f_n d\alpha\leq \int_a^b fd\alpha+\int_a^b \epsilon_n d\alpha
+$$
+
+Since $\int_a^b \epsilon_n d\alpha\to 0$ as $n\to\infty$, by the squeeze theorem, we have
+
+by the squeeze theorem, we have
+
+$$
+\lim_{n\to\infty}\int_a^b f_nd\alpha=\int_a^b fd\alpha
+$$
+
+_Key is that $\int_a^b (f-f_n)d\alpha\leq \sup_{x\in[a,b]}|f-f_n|(\alpha(b)-\alpha(a))$_
+