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\begin{center}\textbf{Problem Set 5 \hspace{.7in} Real Analysis 1 \hspace{.7in} Due 10/12/16} \end{center}

\noindent Four of these problems will be graded, with each problem worth 5 points.  Clear and complete justification is required for full credit.  You are welcome to discuss these problems with anyone and everyone, but must write up your own final submission without reference to any sources other than the textbook and instructor. 

\begin{enumerate}
\item If a set $E \subseteq \mathbb{R}$ is closed, then $\mathbb{R} \backslash E$ is open.

\item If $\mathbb{R} \backslash E$ is open, then $E \subseteq \mathbb{R}$ is closed.

\item Show that if $f$ is continuous at $c \in [a,b]$ then for any sequence $\{x_n\}$ converging to $c$, the sequence
$\{f(x_n)\}$ converges to $f(c)$.

\item Give an example of a function $f$ which is continuous on a bounded set $D$, but not bounded on $D$.

\end{enumerate}

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