diff --git a/src/op/c-star/quotients.tex b/src/op/c-star/quotients.tex
index da7816b..d1de6f3 100644
--- a/src/op/c-star/quotients.tex
+++ b/src/op/c-star/quotients.tex
@@ -23,7 +23,7 @@
     \]
 \end{lemma}
 \begin{proof}[Proof, {{\cite[Lemma 15.6]{Zhu}}}. ]
-    For each $y \in I$, $e_\beta y \to y$. Thus
+    For each $y \in I$, $ye_\beta \to y$. Thus
     \[
     \limsup_{\beta \in B}\norm{xe_\beta - x}_A = \limsup_{\beta \in B}\norm{(x - y)(1 - e_\beta)}_A \le \norm{x - y}_A
     \]
@@ -44,11 +44,11 @@
 
     For each $x \in A$ and $y \in I$, 
     \begin{align*}
-        \norm{x + I}_{A/I}^2 &= \lim_{\beta \in B}\norm{x - e_\beta x}_A^2 = \lim_{\beta \in B}\norm{(x - e_\beta x)(x - e_\beta x)^*}_A \\
-        &= \lim_{\beta \in B}\norm{(1_A - e_\beta)xx^*(1_A - e_\beta)}_A \\ 
-        &= \lim_{\beta \in B}\norm{(1_A - e_\beta)(xx^* + y)(1_A - e_\beta)}_A \le \norm{xx^* + y}_{A}
+        \norm{x + I}_{A/I}^2 &= \lim_{\beta \in B}\norm{x - xe_\beta }_A^2 = \lim_{\beta \in B}\norm{(x - xe_\beta )^*(x - x e_\beta)}_A \\
+        &= \lim_{\beta \in B}\norm{(1_A - e_\beta)x^*x(1_A - e_\beta)}_A \\ 
+        &= \lim_{\beta \in B}\norm{(1_A - e_\beta)(x^*x + y)(1_A - e_\beta)}_A \le \norm{x^*x + y}_{A}
     \end{align*}
 
-    As the above holds for all $y \in I$, $\norm{x + I}_{A/I}^2 \le \norm{xx^* + I}_{A}$. Thus $A/I$ equipped with the quotient norm is a $C^*$-algebra. 
+    As the above holds for all $y \in I$, $\norm{x + I}_{A/I}^2 \le \norm{x^*x + I}_{A/I}$. Thus $A/I$ equipped with the quotient norm is a $C^*$-algebra. 
 \end{proof}