From 0ef220bba5e03af78a3b30dc541e005e21071f5f Mon Sep 17 00:00:00 2001
From: Bokuan Li <bokuan.li@mail.mcgill.ca>
Date: Fri, 1 May 2026 18:33:10 -0400
Subject: [PATCH] Added proof for the BV claim.

---
 src/fa/rs/bv.tex | 2 ++
 1 file changed, 2 insertions(+)

diff --git a/src/fa/rs/bv.tex b/src/fa/rs/bv.tex
index 7b8529c..adb1fec 100644
--- a/src/fa/rs/bv.tex
+++ b/src/fa/rs/bv.tex
@@ -74,6 +74,8 @@
     \end{enumerate}
 \end{definition}
 \begin{proof}[Proof {{\cite[Proposition X.1.1]{Lang}}}. ]
+    (3): For each $P \in \scp([a, b])$, the mapping $V_{P, \rho}: E^{[a, b]} \to [0, \infty]$ is continuous. Since $[\cdot]_{\text{var}, \rho} = \sup_{P \in \scp([a, b])}V_{P, \rho}$, $[\cdot]_{\text{var}, \rho}$ is lower semicontinuous by \autoref{proposition:semicontinuous-properties}. 
+
     (5): For each $n \in \nat^+$, let
     \[
     D_n = \bracs{x \in [a, b]|\forall \eps > 0, \exists y \in (x - \eps, x + \eps): \norm{f(x) - f(y)}_E \ge 1/n}