% LaTeX file -- AM/PM 331, Fall 2011, Assignment 4

\documentclass[11pt]{amsart}
\usepackage{amssymb, amstext, amscd, amsmath}
\pagestyle{empty}
%\topmargin 1cm
%\textheight 9in
\textwidth 5.5in
\oddsidemargin .5in
\evensidemargin .5in
%\voffset -1.5cm

\newcommand{\bN}{{\mathbb N}}
\newcommand{\bQ}{{\mathbb Q}}
\newcommand{\bR}{{\mathbb R}}
\newcommand{\bC}{{\mathbb C}}
\newcommand{\bZ}{{\mathbb Z}}

\newcommand{\Bu}{{\mathbf{u}}}
\newcommand{\Bv}{{\mathbf{v}}}
\newcommand{\Bw}{{\mathbf{w}}}
\newcommand{\Bx}{{\mathbf{x}}}
\newcommand{\By}{{\mathbf{y}}}
\newcommand{\Bz}{{\mathbf{z}}}

\newcommand{\rC}{{\mathrm{C}}}

\newcommand{\FOR}{\text{ for }}
\newcommand{\FORAL}{\text{ for all }}
\newcommand{\AND}{\text{ and }}
\newcommand{\IF}{\text{ if }}
\newcommand{\OR}{\text{ or }}
\newcommand{\qand}{\quad\text{and}\quad}
\newcommand{\qor}{\quad\text{or}\quad}
\newcommand{\qfor}{\quad\text{for}\quad}
\newcommand{\qforal}{\quad\text{for all}\quad}
\newcommand{\qif}{\quad\text{if}\quad}

\newcommand{\ol}{\overline}
\newcommand{\ep}{\varepsilon}
\newcommand{\dlim}{\displaystyle\lim\limits}
\renewcommand{\phi}{\varphi}
\newcommand{\mt}{\emptyset}
\newcommand{\dint}{\displaystyle\int}

\newcounter{exerci}[section]
\renewcommand{\theexerci}{\arabic{exerci}.}
\newenvironment{Ex}{\begin{list}%
 {\theexerci\hfill}{\usecounter{exerci}\rightmargin=0pt\leftmargin=12pt%
 \labelwidth=12pt\labelsep=6pt\itemsep=3ex}}{\end{list}\medbreak}
\newenvironment{parts}{\vspace{1ex}\begin{enumerate}\renewcommand{\itemsep}{1ex}
 \renewcommand{\labelenumi}{(\alph{enumi})}}{\end{enumerate}}
\newcommand{\hint}{\textbf{Hint:} }

\begin{document}
\thispagestyle{empty}

%%%%%%%   TITLE   %%%%%%
\begin{center}
\textbf{\large AMath/PMath 331 \\ Assignment 4\\
Due Friday October 28}
\end{center}
\vskip 2em 

\begin{Ex}

\item Let $f_n(x) = nxe^{-nx}$ on $[0,\infty)$. 
\begin{parts}
\item Prove that this sequence converges pointwise.
\item Is the convergence uniform?
\end{parts}


\item  Let $g_n(x) = \dfrac{x}{1+nx^2}$ on $[0,\infty)$. 
\begin{parts}
\item Prove that this sequence converges pointwise.
\item Is the convergence uniform?
\end{parts}


\item Let $h_n(x) = \dfrac{n+x}{4n+x}$ on $[0,\infty)$. 
\begin{parts}
\item Prove that this sequence converges pointwise.
\item Prove that  it converges uniformly on $[0,N]$.
\item Show that it does not converge uniformly on $[0,\infty)$.
\end{parts}


\item  Let $K$ be a compact subset of $\bR^n$. Suppose that $f_n \in \rC(K)$
are all Lipschitz with Lipschitz constant $C$, and they converge uniformly to $f$.
Prove that $f$ is Lipschitz with Lipschitz constant $C$.\vspace{-2ex}

%\item Let $K$ be a compact subset of $\bR^n$. If $f_n \in \rC(K)$ converge
%uniformly to $f$, prove that $\sup_{n\ge1} \|f_n\|_\infty < \infty$.

\item Suppose that $f(x,t)$ is continuous on $[a,b]\times[c,d]$.
Define $F(x) = \dint_c^d f(x,t) \,dt$. Show that $F$ is continuous.
\hint $f$ is uniformly continuous.

\item Define the $L^1$ norm on $\rC[0,2]$ by $\|f\|_1 = \dint_0^2 |f(x)|\,dx$.
\begin{parts}
\item Prove that this is a norm.
\item Let
\[ f_n(x)= \begin{cases}
 x^n &\qif 0 \le x \le 1\\ 1 &\qif 1 \le x \le 2 \end{cases}
 \qand
 f(x) = \begin{cases}
 0 &\qif 0 \le x < 1\\ 1 &\qif 1 \le x \le 2 \end{cases} .
\]
Show that the sequence $f_n(x)$ 
converges in the $L^1$ norm to $f(x)$.
\item \textbf{Bonus.} Hence show that $\rC[0,2]$ is not complete in the $L^1$ norm.\\
\hint why is there no \textit{continuous} function which is the limit of $(f_n)$?
\end{parts}

%\item \textbf{Bonus.} 


\end{Ex}

\newpage
\centerline{\textbf{What is covered on the Midterm on Tuesday October 25?}}
\bigskip

\noindent
Tuesday October 25, 7:00--8:30.  Arrive 10 minutes early to get set up.\\
\strut\ \ \ \ If you are enrolled in AMATH 331, go to room MC 4020.\\
\strut\ \ \ \ If you are enrolled in PMATH 331, go to room MC 4045.\\
Spread out so that no-one is sitting right beside you.
The exam is the same in both rooms. This is just a method to split the class properly.

\bigskip

\begin{itemize}
\setlength{\itemsep}{2ex}
\item You should know all the definitions and how to use them.
\item You should know the statements of all theorems that have names.
\item You should know the proofs of the following theorems:\\
\begin{enumerate}
\setlength{\itemsep}{1ex}
\item The Monotone Convergence Theorem.\\ (deduced from the Least Upper Bound Principle.)
\item Lemma \textit{A compact subset of $\bR^n$ is closed and bounded.}\\
(This is the easy half of the Heine-Borel Theorem.)
\item Cantor Intersection Theorem.
\item Extreme Value Theorem.
\end{enumerate}
\item You should be able to do problems like the homework exercises (although
because of time considerations, the midterm problems will be a bit shorter).

\end{itemize}
\end{document}
