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{\bf Supplementary problems (assigned 11/17/03)}
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\begin{enumerate}
\item Prove that $\cos(2 x)$ is analytic at every point.
\vspace*{.2in}

\item 
\vspace*{.2in} 
(Not entirely irrelevant sequence problem)
Let $\{a_{i,j}\}_{i,j\in\N}\subset\R^+$ be a double sequence of positive
numbers.  Suppose that 
\begin{itemize}
\item for each fixed $i\in \N$ the sequence $\{a_{i,j}\}_{j\in\N}$ is 
      increasing and converges to to a number $A_i\in\R$; and
\item $\sum_{i=0}^\infty A_i$ converges (call the value of the sum $A$).
\end{itemize}
Show that 
\begin{itemize}
\item $\sum_{i=0}^\infty a_{i,j}$ converges for each fixed $j\in\N$ (call
the value of the sum $A_j$); and
\item $\lim_{j\to\infty} A_j = A$.
\end{itemize} 
The main point here is the second item, because it involves switching
two limits

\end{enumerate}
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