\documentclass[11pt]{article}
% The part of the tex file between the documentclass and \begin{document} lines is
% called the preamble. Commands to set up the document appear in the preamble.
% You can probably use the same preamble as in this sample document, except
% that you will change the title and author.
\title{Work from 2011-10-13}
\author{MCS-236 class}
% These packages from the American Mathematical Society often come in handy:
\usepackage{amssymb, amsmath}
% I've defined a bunch of theorem-like environments here; you'll probably only ever
% use a few of them.
\newtheorem{prop}{Proposition}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{definition}{Definition}
\newtheorem{remark}{Remark}
\newtheorem{example}{Example}
\newtheorem{corollary}{Corollary}
\newtheorem{terminology}{Terminology}
\newtheorem{notation}{Notation}
\newtheorem{exercise}{Exercise}
\newtheorem{note}{Note}
\newtheorem{comment}{Comment}
\newtheorem{claim}{Claim}
% The next mess defines an environment for proofs.
\newenvironment{proofhelper}[1]%
{\begin{trivlist}\item\textbf{Proof.}#1\quad\ignorespaces}%
{\hfill\rule{1ex}{1ex}\end{trivlist}}
\newenvironment{proof}%
{\begin{proofhelper}{}}%
{\end{proofhelper}}
% The following is a variant for when the method of proof is indicated.
\newenvironment{proofmethod}[1]%
{\begin{proofhelper}{ [#1]}}%
{\end{proofhelper}}
% Below I define some shortcuts for some common symbols.
\newcommand{\R}{\mathbf{R}} % boldface R for the real numbers
\newcommand{\Z}{\mathbf{Z}} % boldface Z for the integers
\newcommand{\Q}{\mathbf{Q}} % boldface Q for the rational numbers
\begin{document}
\maketitle
`
\begin{theorem}
An edge $e$ of a connected graph $G$ is a bridge if and only if it lies on no cycle.
\end{theorem}
\begin{proof}
We will begin by showing that if $e$ is a bridge, then it is on no cycle. We will prove the contrapositive, that if $e$ is on a cycle, then it is not a bridge.
Let $e=sv$. If $e$ lies on the cycle $s, v, v_1, \ldots, v_k, s$, then $v, v_1, \ldots, v_k, s$ is an $s-v$ path in $G-e$, so $e$ is not a bridge.
Next, we need to show that if $e$ is not on a cycle, then it is a bridge. Once again, we can prove the contrapositive, which is that if $e$ is not a bridge, then it lies on a cycle. Because $e$ is not a bridge, we know that $G-e$ is still connected, and in particular, that there is an $sv$ path in $G-e$. That path together with the edge $e$ forms a cycle in $G$.
\end{proof}
\end{document}