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\author{Math 25}

\title{Homework 4 --- due Oct 17th}

\begin{document}

\maketitle
\setlength{\baselineskip}{18pt}

There are three sections: A,B and C 
(plus a fourth ``optional''
one which won't be graded). 
Please return the three parts
\emph{separately} to the proper CA.

\renewcommand{\thesection}{\Alph{section}}

\section{Problems graded by Jennifer}

\subsection{A regularization operator} Let $\{x_n\}_{n \geq 1}$ be a sequence
of real numbers. Set \[ y_n = \frac{1}{n} \left( x_1 + x_2 + \cdots + x_n \right) \] 
If $y_n \ra y$ for some $y \in \RR$, we say that $\{x_n\}_{n \geq 1}$
Ces\`aro-converges to $y$. 

\begin{enumerate}
\item Show that if $x_n$ converges to $x$, then 
it also Ces\`aro-converges to $x$. 
\item Find the Ces\`aro-limit of the sequence $\{x_n=(-1)^n\}$. 
\end{enumerate}

\subsection{Extending continuous functions} Let $f:[0,1) \ra \RR$ be a
continuous function. Show that one can extend $f$ to a continuous
function on $[0,1]$ if and only if $f$ is uniformly continuous on $[0,1)$.

\subsection{Discontinuities of non-decreasing functions} 
Let $f: \RR \ra \RR$ be a non-decreasing
function. 

\begin{enumerate}
\item Show that $f$ is continuous at $x \in \RR$ if and only if
$\sup_{y < x} f(y) = f(x)$ and $\inf_{y > x} f(y) = f(x)$ (draw a picture!).
\item
Prove that the set of $x \in \RR$ at which $f$ is not continuous is
countable. 
\end{enumerate}

\section{Problems graded by Benjamin}

\subsection{Writing a number in base $b$} Let $b \geq 2$ be an integer,
and choose $n \in \NN$. 

\begin{enumerate}
\item
Show that one can write $n=\sum_{k=0}^{\ell}
x_k b^k$ with $x_k \in \{0,1,\cdots,b-1\}$ and some $\ell \geq 0$
in exactly one way.
\item
Choose $x \in [0,1)$ now. Show that one can write $x =
\sum_{k=1}^{+\infty} x_k b^{-k}$ with $x_k \in \{0,1,\cdots,b-1 \}$ in
exactly one way, if we impose the additional condition that for every
$N  \geq 0$, there exists $n \geq N$ such that $x_n \neq b-1$.
\end{enumerate}

\subsection{The digits of $2^n$} Let $\log_2$ stand for the logarithm in base $2$: if $x
>0$, $\log_2(x)$ is the unique real number such that
$2^{\log_2(x)}=x$. 

\begin{enumerate}
\item Prove that $\log_2(10)$ is irrational (hint: what
can you say if $2^{p/q}=10$?).
\item
Show that there is a number $n \geq 1$ such that when one writes $2^n$
in base $10$, the first few digits are $121\,219\,76...$.
\end{enumerate}

%\subsection{A remarkable partition} Let $\alpha$ and $\beta$ be two real numbers $>1$. If $x
%\in \RR$, let $\lfloor x \rfloor$ denote the integer part of $x$. Show
%that the following two statement are equivalent:
%\begin{enumerate}
%\item the two sets $P_\alpha=\{ \lfloor n \alpha \rfloor,\ n \geq 1 \}$
%  and $P_\beta=\{ \lfloor n \beta \rfloor,\ n \geq 1 \}$ satisfy:
%  $P_\alpha \cap P_\beta = \varnothing$ and $P_\alpha \cup P_\beta =
%  \ZZ_{\geq 1}$.
%\item $\alpha$ and $\beta$ are irrational and $1/\alpha+1/\beta=1$.
%\end{enumerate}

\subsection{L\'evy's chords lemma} Let $f:[0,1] \ra \RR$ be a
continuous function such that $f(0)=f(1)$ and choose $q \in \ZZ_{\geq 1}$. 

Prove that there exists $x \in
[0,1-1/q]$ such that $f(x+1/q)=f(x)$.

\section{Problems graded by Inna}

\subsection{Lim sup and lim inf} Let $\{x_n\}_{n \geq 1}$ be a bounded sequence of real
numbers, and set $y_m = \sup_{n \geq m} x_n$. 

\begin{enumerate}
\item Prove that $\{y_m\}_{m
\geq 1}$ is a non-increasing sequence bounded from below, 
so that it converges. Define $\lim\sup x_n$ to be the 
limit of $y_m$; 
\item Give a similar definition of $\lim\inf x_n$; 
\item Prove that $\lim\sup x_n \geq \lim\inf x_n$ and that
$\{x_n\}_{n \geq 1}$ has a limit if and only if
$\lim\sup x_n = \lim\inf x_n$;
\item Recall that if
$A(x)$ is the set of accumulation values of $\{x_n\}_{n \geq 1}$, then $A(x)$ is
closed. Why does it have a largest element?
Prove that $\ell = \lim\sup x_n$ is the largest element of $A(x)$
and that for any $\epsilon > 0$ the set $\{ n \in \ZZ_{\geq 1} \ \  x_n
> \ell + \epsilon \}$ is finite;
\item Give the corresponding statements for $\lim\inf$.
\end{enumerate}

\subsection{A special class of sequences} 
Let $\{\alpha_n\}_{n \geq 1}$ be a sequence of real numbers
bounded from below and such that $(n+m)\alpha_{n+m} \leq n \alpha_n + m \alpha_m$. 

Show that there exists $\alpha \in \RR$ such that $\alpha_n \ra
\alpha$. Hint: use the preceding exercise.

\subsection{Rearranging a series}
Let $\sum_{n=1}^{\infty} a_n$ be a convergent series, which converges
to $\ell \in \RR$. If $\phi : \ZZ_{\geq
1} \ra \ZZ_{\geq 1}$ is a bijection, we consider the rearranged
series $\sum_{n=1}^{\infty} a_{\phi(n)}$.

\begin{enumerate} 
\item 
Show that if $\sum_{n=1}^{\infty} a_n$ is absolutely convergent,
then any rearrangement $\sum_{n=1}^{\infty} a_{\phi(n)}$ also converges
to $\ell$.
\item Assume now that $\sum_{n=1}^{\infty} a_n$ is not absolutely convergent
Let $P$ and $N$ be the subsets of $\ZZ_{\geq 1}$ such that $m \in P$
if $a_m \geq 0$ and $m \in N$ if $a_m < 0$. 
Prove that $P$ and $N$ both contain infinitely many elements $\neq 0$,
and that furthermore $\sum_{k \in P} a_k \ra +\infty$ and $\sum_{m \in N} a_m \ra
-\infty$. 
\item 
Let $z$ be any real number.
Set $\phi(1)=1$ and define
$\phi(n)$ by recurrence in the following way: if $\sum_{k=1}^{n-1}
a_{\phi(k)} > z$ then set $\phi(n)$ to be any element of $N$ which has
not already been chosen. Otherwise, set $\phi(n)$ to be any element of 
$P$ which has not already been chosen.
\item
Prove that  the rearranged
series $\sum_{n=1}^{\infty} a_{\phi(n)}$ converges to $z$.
\end{enumerate}

\section{Supplementary problems}

\subsection{Dilating maps} Let $(E,d)$ be a compact metric space and let $f:E \ra
E$ be a continuous map such that $d(f(x),f(y)) \geq d(x,y)$ for all
$x,y \in E$. Show that $f$ is a bijection and that 
$d(f(x),f(y)) = d(x,y)$ for all $x,y \in E$.

\subsection{Steinitz' theorem} Let $\{a_n\}_{n \geq 1}$ and
$\{b_n\}_{n \geq 1}$ be two sequences of real numbers such that
$\sum_{n \geq 1} (a_n,b_n) = (0,0)$. Let $S$ be the set of all
possible limits in $\RR^2$ of $\sum_{n \geq 1}
(a_{\phi(n)},b_{\phi(n)})$ when  $\phi$ runs among all bijections 
$\phi : \ZZ_{\geq 1} \ra \ZZ_{\geq 1}$. Show that $S$ is either
$\{(0,0)\}$ or a line or $\RR^2$ itself. Generalize to higher
dimensions.
\end{document}