\documentclass[12pt]{article}
\usepackage{fancyhdr,fancybox,amssymb,epsfig, amsmath, latexsym}
\topmargin=-.5in
\headsep=0.2in
\oddsidemargin=0in
\textwidth=6.7in
\textheight=9.2in
\footskip=.5in

\newcommand{\A}{\mathbb{A}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\F}{\mathbb{F}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\Z}{\mathbb{Z}}
\def\e{\mathbf{e}}
\def\v{\mathbf{v}}
\def\w{\mathbf{w}}
\def\<{\langle}
\def\>{\rangle}
\DeclareMathOperator{\Per}{Per}
\DeclareMathOperator{\Poly}{Poly}
\DeclareMathOperator{\Span}{Span}
\DeclareMathOperator{\diam}{diam}

\renewcommand{\footrulewidth}{1pt}
\newenvironment{proof}[1][Proof.]{\smallskip\begin{trivlist}
    \item[\hskip \labelsep {\sffamily #1}]}{\qed\end{trivlist}\bigskip}
\newenvironment{sol}[1][Solution.]{\smallskip\begin{trivlist}
    \item[\hskip \labelsep {\sffamily #1}]}{\qed\end{trivlist}\bigskip}
\newenvironment{fminipage}%
{\setlength{\fboxsep}{15pt}\begin{Sbox}\begin{minipage}}%
{\end{minipage}\end{Sbox}\fbox{\TheSbox}}
\newcommand{\qed}{\hfill \ensuremath{\Box}}
\newcommand{\disp}{\displaystyle}


\pagestyle{fancy} \lhead{{\sf Homework 9 $|$ Due June 15 (Friday) }}
\rhead{\thepage} \cfoot{{\sf MthSc 453 $|$ Real Analysis I $|$ Summer
    I 2012 $|$ M.~Macauley}}


\begin{document}

$\;$ 

\noindent\emph{Read}: Rudin, Chapter 3, pages 65--78 and exercises 24,
25. \\

\begin{enumerate}
  
  
\item Let $X$ be a complete metric space. For a nonempty set
  $E\subset X$, define the \emph{diameter} of $E$ by
  \[
  \diam E=\sup\{d(p,q):p,q\in E\}\,.
  \]
  \begin{enumerate}
  \item Let $\{E_n\}$ be a sequence of nested closed nonempty bounded
    sets. Prove that if $\displaystyle\lim_{n\to\infty}\diam E_n=0$, then
    $\cap_{n=1}^\infty E_n$ consists of exactly one point. [\emph{Hint}:
      First show that it is nonempty, then show that if it contains
      $p,q\in X$, then $p=q$.]
  \item Prove the \emph{Baire category theorem}: If $\{G_n\}$ is a
    sequence of dense open subsets of $X$, then $\cap_{n=1}^\infty
    G_n$ is nonempty. (In fact, it is dense in $X$, but you do not
    need to prove this.) [\emph{Hint}: Find a nested sequence of
      neighborhoods $E_n$ such that $\bar{E}_n\subset G_n$ and appeal
      to Part~(a).]
  \end{enumerate}  
  
  
  
\item Suppose $\{p_n\}$ and $\{q_n\}$ are Cauchy sequences in a metric
  space $X$. Show that the sequence $\{d(p_n,q_n)\}$
  converges. [\emph{Hint}: You are trying to show that for any
    $\epsilon>0$, there is some $N$ such that for all $n,m\geq N$, 
    \[
    |d(p_n,q_n)-d(p_m,q_m)|<\epsilon\,.
    \]
    Notice that by the triangle inequality, 
    \[
    d(p_n,q_n)\leq d(p_n,p_m)+d(p_m,q_m)+d(q_m,q_n)
    \]
    holds for all $n$ and $m$.]
    
    
  
\item Let $X$ be a metric space. 
  \begin{enumerate}
  \item Call two Cauchy sequences $\{p_n\}$, $\{q_n\}$ \emph{equivalent} if
    \[
    \lim_{n\to\infty} d(p_n,q_n)=0\,.
    \]
    Prove that this is an equivalence relation.
  \item Let $X^*$ be the set of all such equivalence classes. If
    $P,Q\in X^*$, where $\{p_n\}\in P$ and $\{q_n\}\in Q$, define
    a distance function by
    \[
    \Delta(P,Q)=\lim_{n\to\infty} d(p_n,q_n)\,,
    \]
    which exists by the previous exercise. Show that this distance
    function is \emph{well-defined}. That is, it does \emph{not}
    depend on the choice of representative from $P$ or $Q$.
  \item We call $X^*$ the \emph{completion} of $X$. In a few
    sentences, summarize this construction and its significance.  Your
    explanation should be understandable to a student who has taken
    taken MthSc 453 and thus knows what a Cauchy sequence and a
    complete metric space is, but did not learn about completions.
  \end{enumerate}
  
  
  
\item Suppose $f\colon\R\to\R$ satisfies
  \[
  \lim_{h\to 0}[f(x+h)-f(x-h)]=0
  \]
  for every $x\in\R$. Does this imply that $f$ is continuous? \\ 



\item If $f\colon X\to Y$ is a continuous mapping between metric spaces,
  prove that
  \[
  f(\bar{E})\subset\overline{f(E)}
  \]
  for every set $E\subset X$. Show by an example that $f(\bar{E})$ can
  be a proper subset of $\overline{f(E)}$. \\



\item Let $f\colon X\to\R$ be a continuous real-valued function on a
  metric space $X$. Let $Z(f)$ (the \emph{zero set} of $f$) be the set
  of all $p\in X$ at which $f(p)=0$. Prove that $Z(f)$ is closed. \\

  
\end{enumerate}

\end{document}
