\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 2011-10-14}
\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}
In an acyclic graph with $m$ edges, $n$ vertices, and $k$ components, $k=n-m$.
\end{theorem}
\begin{proof}
Each component is a tree, so in each component, there is one more vertex than edge. Summing all of the components, $k=n-m$.
\end{proof}
We could also prove this in another, more elementary way. (Note that this still could use polishing.)
\begin{proof}
We can proceed by induction on $m$. If $m=0$, then each vertex is a component and $k=n-0$. Assume, then, that with $m-1$ edges there are $n-(m-1)$ components, that is, $n-m+1$ components. Adding another edge brings the total edges to $m$ and combines two components. To see that it combines two components, consider the alternative possibility that the new edge, $sv$, is within one of the components. Because that component is connected, it already had an $s-v$ path. Thus, adding the edge $sv$ would complete a cycle, which is disallowed by the premise that the graph is acyclic.
\end{proof}
\end{document}