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\begin{document}

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\lhead{\bf HOMEWORK 7}
\chead{\bf  DUE:  Mon., May 14}

\rhead{\bf  NAME: \hspace{80} }
\noindent

\cfoot{\thepage}
 {\bf DIRECTIONS:}  

\begin{itemize}                 
         \item Turn in your homework as {\bf SINGLE-SIDED} typed or handwritten pages.  
         \item {\bf STAPLE} your homework together.  Do not use paper clips, folds, etc.
	\item {\bf STAPLE} this page to the front of your homework.
         \item {Be sure to write your name on your homework.}
	\item Show all work, {\bf clearly and in order}.
\end{itemize}

{\bf You will lose point 0.5 points for each instruction not followed.}

\bigskip\bigskip
\centerline{\begin{tabular}{|c||c|c|}\hline
& & \\
Questions & \quad Points \quad & \qquad Score \qquad\,\\ \hline\hline
& & \\
1& 2 & \\ \hline
& & \\
2& 3 & \\ \hline
& & \\
3& 1 & \\ \hline
& & \\
4& 1 & \\ \hline
& & \\
5& 1 & \\ \hline
& & \\
6& 1 & \\ \hline
& & \\
7& 1 & \\ \hline
\hline
& & \\
Total& 10 & \\
\hline
\end{tabular}}

\newpage

{\bf Problem 1:}  (2 points) Find the accumulation points of the following sets in $\mathbb{R}$.

{\bf (a)} (0.5 points) $S=(0,1)$.

{\bf (b)} (0.5 points) $S=\left\{ (-1)^n+\frac{1}{n} | n \in \mathbb{N} \right\}$.

{\bf (c)} (0.5 points) $S=\mathbb{Q}$.

{\bf (d)} (0.5 points) $S=\mathbb{Z}$.

{\bf Problem 2:}  (3 points)

{\bf (a)} (1 point) Find an infinite subset of $\mathbb{R}$ that does not have an accumulation point in $\mathbb{R}$.

{\bf (b)} (1 point) Find a bounded subset of $\mathbb{R}$ that does not have an accumulation point in $\mathbb{R}$.

{\bf (c)} (1 point) Find a bounded infinite subset of $\mathbb{Q}$ that does not have an accumulation point in $\mathbb{Q}$.

{\bf Problem 3:} (1 point) Let $S \subset \mathbb{R}$.  Suppose every neighborhood of $x \in S$ contains infinitely many points of $S$.  Prove that $x$ is an accumulation point of $S$.  {\it (This is the second half of the proof from class.)}

{\bf Problem 4:} (1 point) Show that the arbitrary union of open sets in $\mathbb{R}$ is open.  That is suppose $\left\{ U_i \right\}_{i \in \mathcal{I}}$ is a collection of open sets in $\mathbb{R}$.  Prove that $\cup_{i \in \mathcal{I}} U_i$ is also open.  {\it Note:  $\mathcal{I}$ need not be a denumerable set of indices.}

{\bf Problem 5:} (1 point) Show, by example, that an infinite intersection of open sets in $\mathbb{R}$ is not necessarily open (you will still need to prove that your example is not open).

{\bf Problem 6:} (1 point) Show, by example, that an infinite union of closed sets in $\mathbb{R}$ is not necessarily closed.

{\bf Problem 7:} (1 point) Show that a finite union of closed sets in $\mathbb{R}$ is a closed set in $\mathbb{R}$.  

\end{document}