% /usr/local/doc/tex-inputs/latex/notes/paper.tex

% An example showing how to prepare an article using AMSLaTeX.
% Stephen G. Simpson, Fall 1995.

% Lines beginning with a percent sign are comments.  LaTeX ignores them.

% Begin by declaring a document class and options.

\documentclass[12pt,oneside]{amsart}

% \documentclass{amsart} says to use the AMS article document class.
% [12pt,oneside] says to use the 12pt and oneside options.
% If you don't want these options, just say \documentstyle{amsart}.

% After the document class declaration comes the preamble.
% The preamble begins here.

% First we activate any packages that we may need.
%
% The amssymb package provides \mathbb and other
% math symbols.  The amsmath package provides sophisticated math
% constructions.  The amsthm package provides \theoremstyle and
% the \proof environment.
%
% The amsmath and amsthm packages are automatically activated by
% \documentclass{amsart}, so there is no need to activate them here.

\usepackage{amssymb}

% Next we use \newtheorem to specify our theorem-like environments
% (theorem, definition, etc.) and how to display and number them.
%
% Note: The \theoremstyle declarations affect the appearance of the
% Theorems, Definitions, etc.

\theoremstyle{plain}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}

\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}

\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}

% The preamble is also a good place to define new commands and macros.
% This part of the preamble is strictly optional according to your taste.

\newcommand{\R}{{\mathbb R}}
\newcommand{\nil}{\varnothing}

% The following mysterious maneuver gets rid of AMS junk at the top
% and bottom of the first page.

\makeatletter
\def\serieslogo@{}
\makeatother

% This ends the preamble.  We now proceed to the document itself.

\begin{document}

% First we specify the top matter (author, title, etc).
%
% Note: All of the top matter items are optional and can be omitted.
% But you will probably want to specify at least the author and title
% and perhaps an abstract.

% author information

% first author

\author{David Hilbert}
\email{dh@math.gottingen.edu.de}

% second author

\author{Stephen G.~Simpson}

% the address where the research was carried out

% current address, usually not needed because it is the same as the
\curraddr{Department of Mathematics, Pennsylvania State University,
University Park, State College PA 16802}

\email{simpson@math.psu.edu}

% title

\title[The Infinite]{On \LaTeX{} and the Role of the Infinite\\
in Mathematical Reasoning}

% Note that the short title for running heads goes in square
% brackets.  This is optional.  The long title goes in curly
% braces.  In the long title, line breaks are indicated by \\.

% abstract (optional)
\begin{abstract}
This is a sample document illustrating how to use \LaTeX{} in the
Penn State Math Department.  We also attempt to clarify the
mysteries of the infinite.
\end{abstract}

% AMS subject classifications (used in AMS journals)
\subjclass{Primary 00A30; Secondary 00A22, 03E20}

% AMS keywords (used in AMS journals)
\keywords{infinite, seminar}

% acknowledge support, etc
\thanks{This research was partially supported by NSF grant
DOA-123456789.}
\thanks{We would like to thank our colleagues for their helpful
criticism.}

% dedication
\dedicatory{Dedicated to Professor Donald Knuth on the occasion
of his $100$th birthday}

% today's date, or fill in whatever date you prefer
\date{\today}

% This ends the top matter information.
% We can now tell LaTeX to display the top matter.

\maketitle

% Having displayed the top matter, we now proceed to the body of the
% article.

% The body of the article is divided into sections.
% Each section begins with a \section command.

\section{Introduction}

This document was prepared with \LaTeX{} using the AMS article
style.  By comparing this printed document with the raw document
file from which it was generated, you can learn how to use \LaTeX{}
to typeset mathematical documents.

To start a new paragraph, skip a line.

Many people regard the infinite as a great mystery.  In this
paper we shall try to end the mystery once and for all, so that
mathematicians will never again have to bother about it.

The set of real numbers is denoted $\R$.  One of its subsets is
$\nil$, \emph{i.e.} the empty set.

We end this introductory section with a brief outline of the rest
of the paper.  Section~\ref{prelim} reviews a great number of
important concepts and results.  Then Section~\ref{main} presents
our new contributions.  At the end of the paper comes a
bibliography, also known as a list of references.

% Sections can be labeled for cross referencing.

\section{Preliminaries: the many faces of $\infty$}
\label{prelim}

The purpose of this section is to review some well-known concepts
and results.  Most of these results are taken from the local
newspaper \cite{CDT}.  The O. J. Simpson trial is very important.
Cousin OJ seems to be in a lot of trouble.  See also the front page
of \cite{CDT}.

The infinite has many guises.  One popular notation for the
infinite is $\infty$.  Cantor's aleph notation provides an
important refinement: $\aleph_0$ denotes countable infinity, while
other infinities are denoted $\aleph_\alpha$ where $\alpha$ is an
ordinal number greater than $0$.

\section{The main mathematical results}
\label{main}

We begin with the following lemma.

\begin{lemma}
For all ordinal numbers $\alpha$, we have
$\aleph_\alpha<2^{\aleph_\alpha}$.
\end{lemma}

\begin{proof}
The proof is obvious by diagonalization.
\end{proof}

An immediate consequence of the previous lemma is:

% Theorems can be labeled for cross referencing.

\begin{theorem}
\label{continuum}
We have $\aleph_0<2^{\aleph_0}$.
\end{theorem}

% A special theorem name can be specified in square brackets.

\begin{corollary}[Cantor's Theorem]
The real number system, $\mathbb{R}$, is uncountable.
\end{corollary}

\begin{proof}[Proof of Cantor's Theorem]
Immediate from Theorem~\ref{continuum}.
\end{proof}

This completes our discussion of Cantor's Theorem.

% Equations can be labeled for cross referencing.

An interesting equation is
$$\label{bt} (A+B)^n = \sum_{i=0}^n \binom{n}{i} A^i B^{n-i} \,.$$
Note that equation~\eqref{bt} is sometimes called the Binomial
Theorem.

\section{Some features of \LaTeX}

\LaTeX{} has a lot of author-friendly features.  Among them are:
\begin{enumerate}
\item a well-written reference manual;
\item automatic numbering and cross referencing of
\begin{enumerate}
\item sections
\item equations
\item tables
\item figures
\item \emph{etc}.
\end{enumerate}
\item convenient bibliographic citations;
\item special environments for tables and figures.
\end{enumerate}
For instance, Table~\ref{grades} is a simple table, produced
using the \texttt{tabular} environment.

% Tables and figures can be labeled for cross referencing, but
% the \label must come after the \caption.

\begin{table}
\begin{center}
\begin{tabular}{|l|c|c|}              \hline
Last Name & Score & Letter Grade \\ \hline
Clinton   & 95    & A            \\ \hline
Dole      & 95    & A            \\ \hline
Gingrich  & 95    & A            \\ \hline
\end{tabular}
\end{center}
\end{table}

\section{Conclusions}

We hope that this article has illustrated how easy it is to use
\LaTeX.  It is also clear from Theorem~\ref{continuum} that the
infinite is a subject that may or may not be worthy of the
attention of all mathematicians.

% This is the end of the last section.

% Finally we create the bibliography or list of references.

\begin{thebibliography}{99}

\bibitem{DK}
D. K\"onig, \emph{Theorie der Endlichen und Unendlichen