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\begin{document}

\centerline {\bf Math 523a - Homework 1} 

\bigskip

\begin{enumerate}

\item Let $E_n$ be a sequence of subsets of some set $X$. 
Define 
\bea 
\limsup_{n \ra \infty} E_n = \bigcap_{n=1}^\infty \bigcup_{k=n}^\infty E_k
\eea
The characteristic function, $\chi_E(x)$, of a set $E$ is the function  
that is $1$ when $x \in E$ and is $0$ when $x \notin E$. 
Letting $E=\limsup_{n \ra \infty} E_n$,
prove that 
\bea
\chi_E(x) = \limsup_{n \ra \infty} \chi_{E_n}(x)
\eea

\item For each of the following prove that it is 
countable or that it is not:
\begin{enumerate}
  \item The set of binary sequences that converge. (A binary sequence is 
     a sequence containing only $0$'s and $1$'s.)
  \item The set of sequence of rational numbers  that converge. 
  \item The set of sequences of rational numbers that are eventually constant.
  \end{enumerate}

\item Let $a_{n,m} \ge 0$ for $n,m \in \naturals$. 
Using only what we did in class (or in the book), prove that 
\bea
\sum_{n=1}^\infty \left[ \sum_{m=1}^\infty a_{n,m} \right]
= 
\sum_{m=1}^\infty \left[ \sum_{n=1}^\infty a_{n,m} \right]
\eea
Hint: show both sides are equal to the infinite sum we defined in class. 

\item Let $(X,\rho)$ be a metric space. 
Fix $x_0 \in X$. Define $f(x)=\rho(x,x_0)$.
  \begin{enumerate}
  \item Prove $f$ is continuous on $X$.
  \item Is $f$ uniformly continuous on $X$ ? Prove your answer.
  \end{enumerate}
  
\item Let $l^\infty$ be the set of bounded sequences of real numbers. 
We define a metric on it by 
\bea
\rho((x_n)_{n=1}^\infty, (y_n)_{n=1}^\infty) 
= \sup_{1 \le n < \infty} |x_n-y_n|
\eea
Let $U$ be the set of sequences in $l^\infty$ that do not converge. Prove
$U$ is open. 

\end{enumerate}

\end{document}
\end

