\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}

\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 8 $|$ Due June 12 (Tuesday) }}
\rhead{\thepage} \cfoot{{\sf MthSc 453 $|$ Real Analysis I $|$ Summer
    I 2012 $|$ M.~Macauley}}


\begin{document}

$\;$ 

\noindent\emph{Read}: Rudin, Chapter 3, pages 47--65. \\

\begin{enumerate}

\item For this problem, consider the metric space $X=\C$.
  \begin{enumerate}
  \item Show that $\big||z|-|w|\big|\leq|z-w|$ for all
    $z,w\in\C$. [\emph{Hint}: By the triangle inequality,
    $|z|=|z-w+w|\leq|z-w|+|w|$.]
  \item Prove that convergence of $\{z_n\}$ implies convergence of
    $\{|z_n|\}$. Show by example that the converse need not hold. \\
  \end{enumerate}
  
  
  
\item Put $p_1=\sqrt{2}$ and recursively define a sequence $\{p_n\}$ by
  \[
  p_{n+1}=\sqrt{2+\sqrt{p_n}}\qquad (n=1,2,3,\dots)\,.
  \]
  Prove that $\{p_n\}$ is monotonically increasing and bounded above
  by $2$, from which we can deduce that it converges. \\
  
  
  
\item Consider the sequence $\{a_n\}$ defined by 
  \[
  a_1=0\,,\qquad a_{2m}=\frac{a_{2m-1}}{2}\,,\qquad
  a_{2m+1}=\frac{1}{2}+a_{2m}\,.
  \]
  \begin{enumerate}
  \item Write out the first 10 terms of this sequence. Make a
    conjecture for what $a_{2n}$ and $a_{2n+1}$ are for all $n$.
  \item Prove your conjectures by induction.
  \item Find all subsequential limits of $\{a_n\}$, and determine
    $\limsup a_n$ and $\liminf a_n$. \\
  \end{enumerate}
  
  

\item For any two sequences $\{a_n\}$ and $\{b_n\}$ of real numbers,
  prove that
  \[
  \limsup(a_n+b_n)\leq\limsup a_n+\limsup b_n\,,
  \]
  provided that the sum on the right is not of the form
  $\infty-\infty$. Give an explicit example of where equality does not
  hold. \\
  
  

\item If $\sum a_n$ converges, and if $\{b_n\}$ is monotonic and
  bounded, prove that $\sum a_nb_n$ converges. [\emph{Hint}: Define
    $c_n=|b_n-b|$, where $b_n\to b$ (how do you know that $b$ exists?)
    and use the comparison test.]  Additionally, give examples to show
  how this can fail if \emph{either} the ``monotonic'' or ``bounded''
  condition is dropped from the hypothesis. \\
  


\item Let $\{p_n\}$ be a sequence of real numbers, and define its
  arithmetic means $\sigma_n$ by
  \[
  \sigma_n=\frac{p_0+p_1+\cdots+p_n}{n+1}\qquad (n=0,1,2,\dots)\,.
  \]
  \begin{enumerate}
  \item If $\lim p_n=p$, prove that $\lim\sigma_n=p$.
  \item Construct a sequence $\{p_n\}$ which does not converge,
    although $\lim\sigma_n=0$.
  \item Can it happen that $p_n>0$ for all $n$ and that $\limsup
    p_n=\infty$, although $\lim\sigma_n=0$?
  \end{enumerate}
  
  
\end{enumerate}

\end{document}
