\documentclass{amsart}

\usepackage{amsthm}

    \newtheorem{theorem}    {Theorem}       [section]
    \newtheorem{lemma}      [theorem]       {Lemma}
    \newtheorem{cor}        [theorem]       {Corollary}
    \newtheorem{prop}       [theorem]       {Proposition}
    \newtheorem{example}    [theorem]       {Example}
    \newtheorem{definition} [theorem]       {Definition}

\begin{document}

    \author{Daniel R. Grayson}
    \title{A sample paper}
    \address{University of Illinois at Urbana-Champaign}
    \email{dan\char`\@math.uiuc.edu}
    \urladdr{http://www.math.uiuc.edu/\char`\~dan/}
    \date{\today}

\maketitle

\section*{Introduction}         % the * means the section is unnumbered

Every math paper has an introduction that tells what the paper is
about.  It doesn't contain any proofs.

\section{Theorems and such things}

In your expositions, please be aware of the meanings of words.  For
example, an {\em equation}\index{equation} is a pair of expressions
separated by an equal sign, and one may ask whether an equation is
true or false, or for which values of the variables it holds.  An
example of an equation is $ax^2+bx+c=0$.  A {\em symbol}\index{symbol}
is usually a single letter.  Examples of symbols are $x$ and $c$.  If
you use symbols as abbreviated notation for something else, then it's
a good idea to define them before using them.  An {\em
  expression}\index{expression} is something (without an equal sign)
which represents a mathematical entity, often a number.  An example of
an expression is $b^2 - 4ac$ or $x^{n+1}$.  A {\em
  formula}\index{formula} is an equation that provides a solution to
another equation.  An example of a formula is \[x = \frac{-b \pm
  \sqrt{b^2-4ac}}{2a},\] which provides a solution to the quadratic
equation above.  One {\em proves} a statement and one {\em constructs}
a geometric figure.  A {\em term}\index{term} is a member of a sum,
and a {\em factor}\index{factor} is a member of a product.  Be careful
not to start a sentence with a mathematical symbol, variable, or
number.

A {\em theorem}\index{theorem} is an important true statement that the reader may
want to use or refer to.  Theorem \ref{my-theorem} below is an example.

A {\em lemma}\index{lemma} is a mathematical fact worth presenting
separately that plays a role in the proof of a later theorem.  It's
usually not deemed to be of independent interest to the reader.

A {\em proposition}\index{proposition} is intermediate between a lemma
and a theorem.  Perhaps it is useful in proving a theorem, and also of
some independent interest.

Theorems, lemmas, and propositions, should come with proofs.

\begin{theorem}\label{my-theorem}
  If $x=1$, then $x+x=2$.
\end{theorem}

\begin{proof}
  We compute as follows: $x+x = 1+1 = 2$.
\end{proof}

\section{Citations}\index{citations}

Some papers might even include references to the literature, such as
\cite{MR8:499g} or \cite[Theorem 1]{MR13:526c}.  The bibliographic information,
in {\em bibtex} format, can be downloaded from MathSciNet at
http://e-math.ams.org/mathscinet/ and placed into a separate file
we'll call {\tt sample.bib} for processing by {\tt bibtex}.

You probably won't need this in most of the papers for our class.

% remove the next two lines if you have no references in your paper
\bibliographystyle{amsplain}    % the style to use
\bibliography{sample}           % the information is in sample.bib

\end{document}
