% HMC Math dept HW template example
% v0.04 by Eric J. Malm, 10 Mar 2005
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\name{}
\class{Math 315}
\assignment{Homework \#9}
\duedate{5/9/2017}

% macros
\newcommand{\NN}{\mathbb{N}}
\newcommand{\ZZ}{\mathbb{Z}}
\newcommand{\QQ}{\mathbb{Q}}
\newcommand{\RR}{\mathbb{R}}
\newcommand{\II}{\mathbb{I}}

\begin{document}

\problemlist{Abbott: 3.2.1, 3.2.3, 3.2.4, 3.2.7}

\begin{problem}[3.2.1]
\begin{enumerate}
\item Where in the proof of Theorem 3.2.3 part (ii) does the assumption that the collection of open sets be \emph{finite} get used?

\item Give an example of a countable collection of open sets $\{O_1, O_2, O_3, \ldots\}$ whose intersection $\bigcap_{n=1}^{\infty} O_n$ is closed, not empty and not all of $\RR$.
\end{enumerate}
\end{problem}

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\newpage

\begin{problem}[3.2.3]
Decide whether the following sets are open, closed, or neither. If a set is not open, find a point in the set for which there is no $\epsilon$-neighborhood contained in the set. If a set is not closed, find a limit point that is not contained in the set.
\begin{enumerate}
\item $\QQ$.
\item $\NN$.
\item $\{x \in \RR : x \neq 0\}$.
\item $\{1 + 1/4 + 1/9 + \cdots + 1/n^2 : n \in \NN\}$.
\item $\{1 + 1/2 + 1/3 + \cdots + 1/n : n \in \NN\}$.
\end{enumerate}
\end{problem}

%\begin{solution}
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\newpage

\begin{problem}[3.2.4]
Let $A$ be nonempty and bounded above so that $s = \sup A$ exists.
\begin{enumerate}
\item Show that $s \in \overline{A}$.
\item Can an open set contain its supremum?
\end{enumerate}
\end{problem}

%\begin{solution}
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\newpage

\begin{problem}[3.2.7]
Given $A \subseteq \RR$, let $L$ be the set of all limit points of $A$.
\begin{enumerate}
\item Show that the set $L$ is closed.
\item Argue that if $x$ is a limit point of $A \cup L$, then $x$ is a limit point of $A$. Use this observation to furnish a proof for Theorem 3.2.12.
\end{enumerate}
\end{problem}

%\begin{solution}
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\end{document}
