diff --git a/tex/bounds-on-semistabilisers.tex b/tex/bounds-on-semistabilisers.tex
index 59030e9e293a52b3ab965b4553765a01b6484d6e..06fd0f65a61cf0a5fccdaf316fbab77838b33a47 100644
--- a/tex/bounds-on-semistabilisers.tex
+++ b/tex/bounds-on-semistabilisers.tex
@@ -979,7 +979,7 @@ Next, we seek to find a larger $\epsilon$ to use in place of $\epsilon_v$ in the
 proof of Theorem \ref{thm:rmax_with_uniform_eps}:
 
 \begin{lemmadfn}[%
-	Finding a better alternative to $\epsilon_v$:
+	A better alternative to $\epsilon_v$:
 	$\epsilon_{v,q}$
 	]
 	\label{lemdfn:epsilon_q}
diff --git a/tex/setting-and-problems.tex b/tex/setting-and-problems.tex
index 9a6cf0820fee4828c924fefe2ee435303e7b1bc0..d9c8a3fb03bd9d8e0b9367bf1cde433abbbfc5da 100644
--- a/tex/setting-and-problems.tex
+++ b/tex/setting-and-problems.tex
@@ -167,7 +167,7 @@ are equivalent to the following more numerical conditions:
 
 \begin{proof}
 First, consider the case where $\chern_0(v)>0$.
-Let $u,v$ be Chern characters with
+Let $u,v$ be Chern characters such that
 $\Delta(u),\Delta(v) \geq 0$, and $v$ has positive rank.
 
 For the forwards implication, assume that the suppositions of the Lemma are
@@ -461,7 +461,7 @@ problem using Lemma \ref{lem:pseudo_wall_numerical_tests}.
 
 \begin{proof}
 	Consider the context of $v$ being a Chern character with non-negative rank
-	(and $\chern_1(v)>0$ if rank 0)
+	(and with $\chern_1(v)>0$ if rank 0)
 	and
 	$\Delta \geq 0$, and $u$ being a Chern character with $\Delta(u) \geq 0$.
 	Lemma \ref{lem:pseudo_wall_numerical_tests} gives that the remaining