\documentclass[11pt]{article}
%Options: draft shows overfull lines, reqno-leqno puts eq numbers on
% right/left side of equations.
%\documentclass[11pt,draft, reqno,a4paper,psamsfonts]{amsart}

\usepackage{amsmath,amssymb}
\usepackage{graphicx}           % replaces epsfig
\newcommand{\comments}[1]{}     % to insert a comment
\usepackage{hyperref}           % for a URL

%                       *** CHANGE DIMENSIONS ***
\voffset=-0.3truein % LaTeX has too much space at page top
\addtolength{\textheight}{0.3truein}
\addtolength{\textheight}{\topmargin}
\addtolength{\topmargin}{-\topmargin}
\textwidth  6.0in               % LaTeX article default 360pt=4.98''
\oddsidemargin 0pt % \oddsidemargin .35in % default is 21.0 pt
\evensidemargin 0pt % \evensidemargin .35in % default is 59.0 pt
%\parindent=20pt

%                       *** MACROS ***
\newcommand{\Cal}{\mathcal} % Calligraphic - caps only
\newcommand{\mbf}{\mathbf}					 % math bold
\newcommand{\R}{\mathbb{R}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\N}{\mathbb{N}}

\newcommand{\abs}[1]{\lvert #1 \rvert}          % absolute value
\newcommand{\norm}[1]{\lVert #1 \rVert}         % norm
\newcommand{\ip}[2]{\langle #1,\, #2\rangle}    % ip = inner product
\newcommand{\sstrut}{\rule[-3pt]{0pt}{15pt}}

%             ======================================
% The remainder of this preface is for sets of problems and their %parts.
\newcounter{numb}
\newcounter{bean}
\newenvironment{problems}{\begin{list}{\arabic{numb}.}{\usecounter{numb}
\setlength{\leftmargin}{20pt}
\setlength{\labelwidth}{15pt}
\setlength{\labelsep}{5pt}
\setlength{\itemsep}{ 15.0pt plus 2.5pt minus 1.0pt}
}}{\end{list}}
%The next environment is for the parts a), b), ... of a problem.
 \newenvironment{parts}{\begin{list}{\alph{bean}) }{\usecounter{bean}
\setlength{\leftmargin}{20pt}
\setlength{\labelwidth}{15pt}
\setlength{\labelsep}{5pt}
\setlength{\topsep}{0pt}
\setlength{\partopsep}{0pt}
}}{\end{list}}


%==================== END PREAMBLE ==============
\begin{document}
%\pagestyle{empty}
\parindent=0pt
\vspace*{-50pt}
{\bf  Math 361, Fall 2015\hfill Jerry L. Kazdan}
\medskip

\begin{center}
{\large\bf Problem Set 1}
\smallskip

%\fbox{{\sc \large Draft}}
\end{center}

%\comments{
{\sc Due:} Thurs. Sept.~3 in class. [Late papers will be accepted (without penalty) until 1:00 PM Friday.]
%}
\medskip

 Please review all of Chapters 1--5 in the Marsden text. 

\bigskip

\begin{problems}

\item Let $\{\vec{v}_j\}$ and $\{\vec{w}_j\}$ be sequences of points in $\R^n$ and let $\vec{v}_j\cdot\vec{w}_j$ be their inner (``dot'') product.  If $\vec{v}_j\to\vec{v}$ and $\vec{w}_j\to\vec{w}$,  show that $\vec{v}_j\cdot\vec{w}_j\to \vec{v}\cdot\vec{w}$.


\item Let $f:[0,2] \to [0,\,2]$. be a continuous function. 
\begin{parts}
\item Show that there is at least one point $c\in [0,\,2]$ so that $f(c)=c$.
\item Give an example showing that there may be more than one such point.
\item Say $\abs{ f'(x)}<1$ for all $x\in [0,\,2]$.  Show there is exactly one such point,
\end{parts}


\item \begin{parts}
\item Show that at any time, there are always at least two antipodal
points on the equator with the same temperature.
\item Generalize.
\end{parts}


\item  \begin{parts}
  \item Let $\{a_j\}\in\R$ be a sequence of real numbers with the
    property that
$$\abs{a_{j+1}-a_j}\le \tfrac{1}{2}\abs{a_{j}-a_{j-1}}.
$$  
Show that the $\{a_j\}$ are a Cauchy sequence and hence converge to some real number $A$.  

{\sc Hint: }If $n>k$, then \quad
$
a_n-a_k = (a_n-a_{n-1})+(a_{n-1}-a_{n-2})+\cdots+(a_{k+1}-a_k).
$
\item Let $\{\vec{v}_j\}\in\R^n$ be a sequence of points with the property that 
$$
\norm{\vec{v}_{j+1}-\vec{v}_j}\le \tfrac{1}{2}
\norm{\vec{v}_{j}-\vec{v}_{j-1}}.
$$
  Show that the $\{\vec{v}_j\}$ are a Cauchy sequence and hence converge to some point $p\in \R^n$.
\end{parts}

\item Find a continuous function $f$ and a constant $C$ so that
$$
\int_0^x f(t)\,dt = e^{\sin x} + C.
$$

\item 
Let $f:(0,\infty)\to (0,\infty),$ be a continuous and decreasing
function. Prove that the sequence
\[
S_n := f(1)+\cdots+f(n)-\int_{1}^{n+1}f(t)\, dt
\]
is convergent and that its limit lies in the closed interval $[0,f(1)]$.


\item   Let $f:=\R \to \R$ be a differentiable function such that
 \[|f'(x)| \le 2015 \qquad \mbox{ for all } x \in \R.\]
 Prove that $f$ is uniformly continuous on $\R$.


\item  Let $f(x)$ be a twice continuously differentiable function with the property that $f''(x)\ge 0$ for all $x\in\R$.  If $f$ is bounded from above
show that $f(x)$ must be a constant.

\item Suppose that $g: \R \to \R$ is a differentiable function whose
derivative $g'$ satisfies the inequality $\abs{g '(x)}<M$ for all
$x\in\R$.

Show that if $\varepsilon> 0$ is small enough, then the function 
$f: \R \to \R$ defined by $f(x) = x + \varepsilon g(x)$ 
is one-to-one and onto, so for every $y\in\R$ the equation $f(x)=y$ has
one and only one solution.



\end{problems}

\comments{
\vskip 20pt

\begin{center}{\large \bf Bonus Problem}
			
[Please give this directly to Professor Kazdan]
\end{center}

\begin{problems}

\item [B-1] 
 \end{problems}
}


\medskip
[Last revised: \today]
\end{document}

