diff --git a/src/cat/tricks/dyadic.tex b/src/cat/tricks/dyadic.tex
index 190cc0b..61f3958 100644
--- a/src/cat/tricks/dyadic.tex
+++ b/src/cat/tricks/dyadic.tex
@@ -44,10 +44,10 @@
         \item[(b)] For each $x, y \in G$, $x + y \ge x, y$. 
     \end{enumerate}
     
-    For each $x \in \mathbb{D} \cap [0, 1)$, let $\phi(x) = \sum_{n \in M(x)}g_n$, then
+    For each $x \in \mathbb{D} \cap (0, 1)$, let $\phi(x) = \sum_{n \in M(x)}g_n$, then
     \begin{enumerate}
-        \item For any $x, y \in \mathbb{D} \cap [0, 1)$ such that $x + y \in [0, 1)$, $\phi(x) + \phi(y) \le \phi(x + y)$.
-        \item For any $x, y \in \mathbb{D} \cap [0, 1)$ with $x \le y$, $\phi(x) \le \phi(y)$.
+        \item For any $x, y \in \mathbb{D} \cap (0, 1)$ such that $x + y \in (0, 1)$, $\phi(x) + \phi(y) \le \phi(x + y)$.
+        \item For any $x, y \in \mathbb{D} \cap (0, 1)$ with $x \le y$, $\phi(x) \le \phi(y)$.
     \end{enumerate}
 \end{proposition}
 \begin{proof}
diff --git a/src/fa/rs/bv.tex b/src/fa/rs/bv.tex
index 74fbb99..e63f94f 100644
--- a/src/fa/rs/bv.tex
+++ b/src/fa/rs/bv.tex
@@ -68,3 +68,10 @@
 
     Therefore there exists pairs $\bracs{(x_k, y_k)|1 \le k \le N}$ such that $\norm{f(x_k) - f(y_k)}_E \ge 1/n$ for all $n$, and the smallest interval containing each $(x_k, y_k)$ are pairwise disjoint. Thus $[f]_{\text{var}} \ge N/n$ for all $N \in \nat^+$, so $[f]_{\text{var}} = \infty$.
 \end{proof}
+
+\begin{proposition}
+\label{proposition:bounded-variation-one-side-limit}
+    
+\end{proposition}
+
+