diff --git a/src/dg/complex/runge.tex b/src/dg/complex/runge.tex
index ae0e099..afb0c7b 100644
--- a/src/dg/complex/runge.tex
+++ b/src/dg/complex/runge.tex
@@ -153,12 +153,12 @@
     \abs{\int_{\gamma_j}\frac{f(z)}{z - z_0}dz - R_j(z)} < \frac{\eps}{n}
     \]
 
-    so
+    Thus
     \[
     \abs{f(z_0) - \sum_{j = 1}^n R_j(z)} < \eps
     \]
     
-    for all $z_0 \in K$. By the \hyperref[pole pushing lemma]{lemma:pole-pushing}, there exists $S \in \complex(z) \cap H(\complex_\infty \setminus P)$ such that $|S(z_0) - \sum_{j = 1}^n R_j(z_0)| < \eps$ for all $z_0 \in K$. 
+    for all $z_0 \in K$. By the \hyperref[pole pushing lemma]{lemma:pole-pushing}, there exists $S \in \complex(z) \cap H(\complex_\infty \setminus P; \complex)$ such that $|S(z_0) - \sum_{j = 1}^n R_j(z_0)| < \eps$ for all $z_0 \in K$. Therefore $|S(z_0) - f(z_0)| < 2\eps$ for all $z_0 \in K$.
 \end{proof}
 
 \begin{corollary}