\documentclass[reqno]{amsart}
% amsart is a particular set of macros added on top of LaTeX for writing articles in the American Mathematical Society style. (The % at the beginning of this line tells tex to ignore the line, so you can write comments like this without them appearing in the final document.)
% the [reqno] is an option: its effect s to get the numbers on displayed equations on the right, rather than o the left which is the default.
% This document has to be "typeset" using latex.
\numberwithin{equation}{section}
\newtheorem{theorem}[equation]{Theorem}
\newtheorem{proposition}[equation]{Proposition}
\newtheorem{lemma}[equation]{Lemma}
\newtheorem{corollary}[equation]{Corollary}
\theoremstyle{definition}
\newtheorem{definition}[equation]{Definition}
\newtheorem{remark}[equation]{Remark}
\newtheorem{example}[equation]{Example}
% the part of this file between \documentclass and \begin{document} is the preamble
\begin{document}
\title{Sample \TeX{} document}
\author{One Math Student}
\author{Another Math Student}
\begin{abstract}
The abstract goes here\dots. This is a very brief document to get you started using \TeX{} with the macro package \LaTeX.
\end{abstract}
\email{student1@temple.edu}
\address{Department of Mathematics\\
Miskatonic University\\
Arkham, MA 314159}
\email{student2@temple.edu}
\address{Department of Mathematics\\
Miskatonic University\\
Arkham, MA 314159}
\maketitle
\section{Title of the first section}
The best approach to \TeX{} is to just try it out. After installing whatever version of \TeX{} you can get, try ``typesetting'' this document. After that, just add stuff, typeset again, and see the effect. The document \cite{lshort} is a very useful guide to \LaTeX. I suggest you download it and keep it handy. The manual for \TeX, by its creator, Donald Knuth, is \cite{Knuth}. The manual for the macro package \LaTeX{} is \cite{Lamport}. Usually \cite{lshort} is quite sufficient; you would need these other manuals only for very special typesetting needs.
Usually the first section is an overview of what the paper is about, mat be stating the main theorems already. Also, typically this section places the results being presented in the context of the existing research literature.\footnote{By the way, Miskatonic University, in Arkham, MA, is a fictional university in a fictional town, invented by H. P. Lovecraft}
\medskip %\medskip adds some extra space between paragraphs; there is also \smallskip and \bigskip
This is how a theorem is stated:
\begin{theorem}
Let this be the statement of some theorem.
\end{theorem}
The proof comes next, perhaps:
\begin{proof}
Pretend this is the proof.
\end{proof}
\section{Title of the second section}
And so it goes. May be you want to start with a lemma, as in
\begin{lemma}
Let $a$, $b$ be real numbers with $a