More prose corrections.

src/fa/convex/legendre.tex

@@ -167,7 +167,7 @@
     \sup_{(y, \beta) \in \text{epi}(f)}\dpn{y, \phi}{E} - \mu \beta < \dpn{x, \phi}{E} - \mu \alpha
     \]
 
-    Since for any $(y, \beta) \in A$, $\beta$ may be arbitrarily small by \autoref{lemma:closed-convex-epigraph}, $\mu > 0$. Thus for each $y \in \bracs{f < \infty}$, 
+    For any $(y, \beta) \in \text{epi}(f)$, $\beta$ may be arbitrarily large, $\mu \ge 0$. In particular, since $(x, f(x)) \in \text{epi}(f)$, the strict inequality implies that $\mu > 0$. Thus for each $y \in \bracs{f < \infty}$, 
     \begin{align*}
         \dpn{x, \phi}{E} - \mu\alpha &> \dpn{y, \phi}{E} - \mu f(y) \\
         -\dpn{y, \phi}{E} + \dpn{x, \phi}{E} - \mu\alpha &> -\mu f(y)\\
@@ -231,7 +231,7 @@
     \sup_{(y, \beta) \in A}\dpn{y, \phi}{\lambda} - \mu \beta \le \dpn{x, \phi}{\lambda} - \mu\alpha_0
     \]
 
-    Since for any $(y, \beta) \in A$, $\beta$ may be arbitrarily large by \autoref{lemma:closed-convex-epigraph}, $\mu \ge 0$. 
+    For any $(y, \beta) \in A$, $\beta$ may be arbitrarily large by \autoref{lemma:closed-convex-epigraph}, so $\mu \ge 0$. 
 
     In the case that $\mu > 0$, for each $y \in \bracs{f < \infty}$, 
     \begin{align*}