Couple of fixes, simplify an exercise a bit

ca.tex

@@ -14594,7 +14594,7 @@ \subsection{Harnack's inequality}
 \frac{1}{2\pi}
 \int_{-\pi}^\pi f(p+Se^{it}) \, dt
 \right)
-\leq
+=
 \frac{S+r}{S-r} 
 f(p) .
 \end{equation*}
@@ -14839,11 +14839,11 @@ \subsection{Harnack's principle}
 
 \begin{exercise}
 Prove a Montel-like theorem for harmonic functions.  Suppose $U \subset \C$
-is open and $\{ f_n \}$ is a sequence of nonnegative harmonic functions.
+is a domain and $\{ f_n \}$ is a sequence of nonnegative harmonic functions.
 Show that at least one (or both) of the following are true:
 \begin{enumerate}[(i)]
 \item
-There exists a subsequence converging to $\infty$ uniformly on compact subsets.
+There exists a subsequence converging to $+\infty$ uniformly on compact subsets.
 \item
 There exists a subsequence converging to a harmonic function
 uniformly on compact subsets.

changes-draft.html

@@ -78,4 +78,9 @@
   The way it was previously was only asking for the hard part (the if),
   and so it wasn't giving a good parallel to the theorem.
   The easy part is actually a good way to start the exercise anyway.
+<li>On page 180, the last displayed inequality in the proof of Harnack is
+  actually an equality.
+<li><b>Simplify Exercise 7.2.26 a little</b> by assuming that \(U\) is
+  connected to having to think about the technicality of countably many
+  components which is not really important.
 <li>Clarify the proof of Rado's theorem.

slides/7.2.3-7.2.4.tex

@@ -89,7 +89,7 @@
 \int_{-\pi}^\pi f(p+Se^{it}) \, dt
 \right)
 \pause
-\leq
+=
 \frac{S+r}{S-r} 
 \,
 f(p) .
@@ -329,7 +329,7 @@
 \medskip
 \pause
 
-Suppose $f(z) = \lim f_n(p) < +\infty$ for every $z \in U$.
+Suppose $f(z) = \lim f_n(z) < +\infty$ for every $z \in U$.
 
 \pause
 Let $K \subset U$ be compact, take the $C$ from Harnack's, and take any $p \in K$.
@@ -382,11 +382,11 @@
 
 \textbf{Exercise:}
 Prove a Montel-like theorem for harmonic functions.  Suppose $U \subset \C$
-is open and $\{ f_n \}$ is a sequence of nonnegative harmonic functions.
+is a domain and $\{ f_n \}$ is a sequence of nonnegative harmonic functions.
 Show that at least one (or both) of the following are true:
 \begin{enumerate}[(i)]
 \item
-$\exists$ a subsequence converging to $\infty$ uniformly on compact subsets.
+$\exists$ a subsequence converging to $+\infty$ uniformly on compact subsets.
 \item
 $\exists$ a subsequence converging to a harmonic function
 uniformly on compact subsets.