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\def\classtitle{Real Analysis}
\def\classnumber{Math 70100}
\def\examtitle{Homework 3}
\def\duedate{September 23, 2019}

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\begin{document}
\noindent
{\large \bf \classnumber: \classtitle (Prof. Hooper) \hfill Due \duedate}\\
{\large \bf \examtitle}\\

\ifprintanswers 
\noindent
{\bf Solutions:}
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\noindent
%{\bf Instructions:}
%Think about how to write clear and concise answers. Write the solution in your own words.
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\begin{questions}
\question {\em (Rudin 3.5)} For any two sequences of real numbers $\{a_n\}$ and $\{b_n\}$, prove that
$$\limsup_{n \to \infty}(a_n + b_n) \leq \limsup_{n \to \infty} a_n +  \limsup_{n \to \infty} b_n$$
provided the sum on the right is not of the form $\infty - \infty$.
\begin{solution}
\end{solution}

\question {\em (Rudin 3.7)} Prove that the convergence of the series $\sum a_n$ implies the convergence of 
$$\sum \frac{\sqrt{a_n}}{n}$$
if $a_n \geq 0$. ({\em Review basic facts about series if necessary before completing this problem.})

\begin{solution}
\end{solution}


\question {\em (Rudin 3.14; Ces\`aro summation)} If $\{s_n\}_{n=1}^\infty$ is a sequence of complex numbers, define its {\em arithmetic means} by 
$$\sigma_n= \frac{s_1+s_2+\ldots+s_n}{n} \quad \text{for $n=1,2,3, \ldots$}.$$
\begin{parts}
\part If $\lim s_n = s$, prove $\lim \sigma_n=s$.

\begin{solution}
\end{solution}

\part  Construct a sequence $\{s_n\}$ which does not converge, although $\lim \sigma_n= 0$.

\begin{solution}
\end{solution}

\part Can it happen that $s_n > 0$ for all $n$ and that $\limsup s_n = +\infty$, although $\lim \sigma_n = 0$?

\begin{solution}
\end{solution}

\end{parts}
({\em Remark:} Rudin has some additional parts to this problem related to partial converses to (a).)


\question {\em (Rudin 3.20)} Suppose $\{p_n\}$ is a Cauchy sequence in a metric space X, and some subsequence
$\{p_n\}$ converges to a point $p \in X$, Prove that the full sequence $\{p_n\}$ converges to $p$.

\begin{solution}
\end{solution}


\question {\em (Based on Pugh 2.92)} Let $X$ be a topological space. 
Recall that a {\em neighborhood} $N$ of $x \in X$ is a subset $N \subset X$ so that there is an open set $U \subset X$ with $x \in U$ and $U \subset N$. A {\em boundary point} of a set $A \subset X$ is a point $x \in X$ so that every neighborhood of $x$ intersects both $A$ and $X \smallsetminus A$. The boundary of $A$, $\partial A$, is the set of all boundary points of $A$.
\begin{parts}
\part Show that $\partial A=X \smallsetminus \big(\mathrm{Int}(A) \cup \mathrm{Int}(X \smallsetminus A)\big)$. 
\begin{solution}
\end{solution}
\part Explain why $\partial A$ is closed.
\begin{solution}
\end{solution}
\part Show that $\partial \partial A \subset \partial A$.

\begin{solution}
\end{solution}

\part Show that $\partial \partial \partial A=\partial \partial A$. 
\begin{solution}
\end{solution}
\part Given an example which illustrates that
$\partial \partial A$ may not equal $\partial A$.
\begin{solution}
\end{solution}
\end{parts}

\question {\em (Lang II.5.5c)} Let $X$ be a metric space. For each $x \in X$, define the function $f_x$ on $X$
by $f_x(y)=d(x,y)$. Let $\| \cdot \|$ be the sup norm.
\begin{parts}
\part Show that $d(x,y)=\|f_x-f_y\|$. 
\begin{solution}
\end{solution}
\part Let $a$ be a fixed element of $X$, and let $g_x=f_x-f_a$. Show that the map $x \mapsto g_x$ is a distance-preserving embedding of $X$ into the normed space of bounded functions on $X$. ({\em Remark:} This shows that every metric space is isometric to a subset of a normed vector space.)
\begin{solution}
\end{solution}
\end{parts}

\question {\em (Modified from Lang II.5.3a)} 
Let $\ell^1$ be the set of all sequences $\alpha=\{a_n\}_{n \in \N}$ of real numbers such that $\sum_{n \in \N} |a_n|$ converges. (If $\sum_{n \in \N} |a_n|$ converges, we say the series $\sum_{n \in \N} a_n$ {\em converges absolutely}.) Define 
$$|\alpha|=\sum_{n \in \N} |a_n|.$$
\begin{parts}
\part Prove that $| \cdot |$ is a norm on $\ell^1$.
\begin{solution}
\end{solution}

\part Recall that a sequence $\{\alpha_n\}$ in a normed vector space is {\em Cauchy} if given any $\epsilon>0$,
there is an $N \in \N$ so that $|\alpha_m-\alpha_n|<\epsilon$ for $m,n \geq N$. A normed vector space is {\em complete}
if all Cauchy sequences converge. Show that $\ell^1$ is complete with the norm $|\cdot |$. 
\begin{solution}
\end{solution}
\end{parts}

\end{questions}
\end{document}
