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\noindent
\parbox{2.25in}{\bf Math 3220}
\hfill {\Large \bf Homework \#7} \hfill
\parbox{2.25in}{\bf \hfill Due: Fri., Apr. $12^\text{th}$, 2019}

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%\noindent {\bf\large Name:} \underline{\qquad\sc\large This is my fabulous name! \qquad}
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\begin{description}
\item[\#1 Rolle's Rolle's] Let $f$ be twice differentiable on $(a,b)$ and suppose that both $f$ and $f'$ are continuous on $[a,b]$. 
Suppose that $a<c<b$ and that $f(a)=f(c)=f(b)$. Prove that there exists some point in $a<x_0<b$ such that $f''(x_0)=0$.

{\it Hint:} What does $f(a)=f(b)$ say about $f'$?

\item[\#2 Contractions] If we have a function $f:\mathbb{R} \to \mathbb{R}$ such that $|f(x)-f(y)| \leq |x-y|$ for all $x,y \in \mathbb{R}$, then we say that $f$ is a {\it contraction}.
\begin{enumerate}[(a)]
\item Let $f$ be a contraction. Show that $f$ is uniformly continuous.

\item Suppose that $f$ is differentiable everywhere and that $|f'(x)| \leq 1$ for all $x \in \mathbb{R}$. Show that $f$ is a contraction.

{\it Suggestion:} MVT.
\end{enumerate}

\item[\#3 Bounded and Uniform] Let $\{f_n\}$ be a sequence of functions defined on $I$ and suppose that $\{f_n\}$ converges uniformly to $f$ on $I$. In addition, assume that for each $n \in \mathbb{N}$, we have that $f_n$ is bounded on $I$. Show that $f$ must be bounded on $I$.

\item[\#4 Composition] {\color{red}\sc\bf [EXTRA CREDIT]} Let $\{f_n\}$ be a sequence of {\it continuous} functions on some interval $I$ and suppose that $\{f_n\}$ converges uniformly to $f$ on $I$. Show that given some sequence $\{ x_n \} \subseteq I$ such that $x_n \to x \in I$ we have that the sequence $\{f_n(x_n)\}$ converges to $f(x)$.

\item[RESUBMIT] Type up Homework \#6 Problem \#4 and its solution in \LaTeX. 

\end{description}

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