\documentclass[11pt]{amsart} 
\usepackage{amssymb, amsmath, amsthm} \usepackage{color}
\usepackage{graphicx}
\usepackage{bbold}

%\usepackage[shortalphabetic]{amsrefs}  %this package allows a certain kind of bibliography style. You won't need it for typical homework assignments.


\usepackage[hidelinks=true]{hyperref} 
%\usepackage{pinlabel} %this is useful package for adding labels to figures.

%Define theorem styles

\theoremstyle{theorem} 
\newtheorem{theorem}{Theorem} 
\newtheorem*{theorem*}{Theorem} 
\newtheorem{lemma}[theorem]{Lemma} 
\newtheorem{definition}[theorem]{Definition}

\newcommand{\spacing}{\parskip 6.6pt \parindent 0pt} 
\spacing

\newcommand{\R}{{\mathbb R}} 
\newcommand{\C}{\mathbb C}
\newcommand{\Z}{\mathbb Z}
\newcommand{\N}{\mathbb N}
\newcommand{\Q}{\mathbb Q}
\newcommand{\1}{\mathbb 1} 
\newcommand{\nil}{\varnothing} 

\newcommand{\mc}[1]{\mathcal{#1}} 
\newcommand{\defn}[1]{\textbf{#1}} 

\newcommand{\range}{\operatorname{range}} 

\newcommand{\co}{\mskip0.5mu\colon\thinspace} 


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

\begin{document} 


%\title[short title]{long title}
 %\author{Scott A. Taylor}
 %\maketitle

%Use the following for homework assignments. Comment out these lines for your paper.
\textbf{Assignment:} 5 \\
\textbf{Due: } Feb 14, 2024 \\
Scott A. Taylor

%Here is an example of how you might format a homework assignment. The comments are for your benefit, you can remove them in your own document.

\hrulefill 

\textbf{Exercise 2.5: } 
Prove the following theorem:   

\begin{theorem} 
\label{Theorem in Exercise 2.5}
Suppose that $n \in \N$. If $n^2$ is even then $n$ is even. 
\end{theorem}

\hrulefill

Before proving the theorem we recall the definition of even and odd.

\begin{definition}\label{Def: even/odd} 
A number $n \in \Z$ is \defn{even} if there exists $m \in \Z$ such that $n = 2m$. An integer $n \in \Z$ is \defn{odd} if there exists $m \in \Z$ such that $n = 2m + 1$.
\end{definition}

We will also need the following lemma. Its proof requires induction, so we omit it.

\begin{lemma}\label{Even or Odd}
If $n \in \Z$ then $n$ is even or $n$ is odd, but not both.
\end{lemma}


\begin{proof}[Proof of Theorem \ref{Theorem in Exercise 2.5}] 
We will prove the contrapositive. We assume that $n$ is not even and we will show that $n^2$ is not even. Since $n$ is not even, by Lemma \ref{Even or Odd}, $n$ is odd. By Definition \ref{Def: even/odd} , there exists $m \in \Z$ such that $n = 2m + 1$. By algebra we have:
\[ 
\begin{array}{rcl} 
n^2 &=& (2m+1)^2 \\ 
&=& 4m^2 + 4m + 1 \\
&=& 2(2m^2 + 2m) + 1.
\end{array}
\]
Since there exists $k = 2m^2 + 2m \in \N$ such that $n^2 = 2k + 1$, by Definition \ref{Def: even/odd} $n^2$ is odd. Thus, by Lemma \ref{Even or Odd}, $n^2$ is not even. Thus, we have shown that if $n$ is not even then $n^2$ is not even and, by contraposition, that if $n^2$ is even then $n$ is even.
\end{proof}

Theorem \ref{Theorem in Exercise 2.5} appears in \cite{ref label}. 

Here are a few other examples of environments you might find useful:
\begin{itemize}\spacing
\item This
\item is a bulleted list
\end{itemize}

\begin{enumerate}\spacing
\item This is
\item a numbered list
\end{enumerate}

In the comments in the .tex file there is an example of how to include a figure.
% Here is how to include a figure. It's commented out so that you don't have to download a sample image file

%\begin{figure}[ht]
%\centering
%\includegraphics[scale=0.5]{ImageFile} % As written the file with the image has to be in the same directory as the LaTex document. The image should be in PDF or EPS format. You do not need to write the extension .pdf or .eps when you include the graphic.
%\caption{This is the caption}
%\label{FavoriteFigure} %The label always has to go after the caption.
%\end{figure}

\begin{thebibliography}{99}

\bibitem{ref label} Last Name, First Name  \emph{A Math Book}, date, publisher. 

\end{thebibliography}


\end{document}