From 256de2ac329248497c8e5c26a616826ea7ac0381 Mon Sep 17 00:00:00 2001
From: Zheyuan Wu <60459821+Trance-0@users.noreply.github.com>
Date: Wed, 16 Apr 2025 10:52:38 -0500
Subject: [PATCH] Update Math4121_L35.md

---
 pages/Math4121/Math4121_L35.md | 21 +++++++++++++++++++++
 1 file changed, 21 insertions(+)

diff --git a/pages/Math4121/Math4121_L35.md b/pages/Math4121/Math4121_L35.md
index cef68f5..eb8a83b 100644
--- a/pages/Math4121/Math4121_L35.md
+++ b/pages/Math4121/Math4121_L35.md
@@ -89,5 +89,26 @@ $$
 
 Proof:
 
+First to show the limit exists almost everywhere. It suffices to show
+
+$$
+\mathcal{U}=\{x\in E: f_n(x) \text{ is unbounded}\}
+$$
+
+has measure 0.
+
+Let $\epsilon>0$ and write
+
+$$
+U=\bigcup_{n=1}^{\infty} E_n
+$$
+
+where $E_n=\{x\in E: |f_n(x)|\geq \epsilon\}$.
+
+Then $U\subseteq \mathcal{U}$ and $m(U)<\epsilon$.
+
+
+CONTINUE NEXT TIME.
+
 QED