Added proof for the BV claim.

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}