\documentclass{article}
\usepackage{amsmath,amsthm,amssymb,fancyhdr,epic,eepic,multicol,graphicx}

\renewcommand{\baselinestretch}{1.1}

\theoremstyle{plain}
\theoremstyle{definition}
\newtheorem{problem}{Problem}
\newtheorem*{answer}{Answer}

\newcommand{\bp}{\begin{problem}}
\newcommand{\ep}{\end{problem}}
\newcommand{\ba}{\begin{answer}}
\newcommand{\ea}{\end{answer}}
\newcommand{\ds}{\displaystyle}
\newcommand{\ben}{\renewcommand{\theenumi}{\alph{enumi}}\renewcommand{\labelenumi}{(\theenumi)}\begin{enumerate}}
\newcommand{\een}{\end{enumerate}}

\newcommand{\N}{\mathbb{N}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\R}{\mathbb{R}}

\fancyhead[L]{\Large \textbf{Math 157 Fall 2013}}
\fancyhead[R]{\large \textbf{Homework 5}}



\begin{document}
\pagestyle{fancy}
\fancyfoot[R,L,C]{}

\bp Read the rest of chapter I (through page 43).
\ep

% \bp True or False:  
% \ben
% \item $\ds \forall n\in \N \exists m \in \N ( 2^n >m )$
% \item $\ds \forall \epsilon >0 \exists n\in \N \left (
%   \frac{1}{2^n}<\epsilon \right)$
% \item $\ds \forall x\in \R \forall y \in \R \exists z \in \R (x+y=z).$
% \item $\ds \forall x\in \R \forall y \in \R \exists z \in \R (x+z=y).$
% \item $\ds \forall x\in \R \exists z \in \R \forall y \in \R (x+z=y).$
% \item $\ds \forall x\in \R \forall y\in \R ((x\leq y )\wedge (y\leq x)
%   \Rightarrow x=y )$
% \een
% \ep


\bp Find the least upper bound of the set
$\left\{\frac{1}{2},\frac{2}{3},\frac{3}{4},\ldots \right\}$.  Prove
your answer.
\ep


\bp True or False:
\ben
\item $p\Rightarrow (q \vee r) \equiv (p \wedge \neg q )\Rightarrow r$
\item $(p \wedge q)\Rightarrow r \equiv (p \wedge \neg r)\Rightarrow
  \neg q.$
\een
\ep

\bp Prove that 
$$\text{If }a,b,c >0\text{ and }a+b+c=1\text{ then }(1-a)(1-b)(1-c)\geq
8abc.$$
\ep

\bp $n!$ may be defined inductively for $n=0,1,2,\ldots$ by 
\begin{itemize}
\item $0!=1$ and
\item $n!=n\times (n-1)!$
\end{itemize}
For $n,k =0,1,2,\ldots$, define $\binom{n}{k}$ by
$$\binom{n}{k}=\frac{n!}{k!(n-k)!}.$$
Prove that $$\binom{n+1}{k}=\binom{n}{k-1}+\binom{n}{k}.$$
\ep

\bp Prove that for all $x\in \R$, $0\leq |x|-x \leq 2|x|.$
\ep

\bp Prove that for all $n\in \N$, $\ds \sum_{k=1}^n 2k-1=n^2$.
\ep

\end{document}