% Comments are always preceded by a percent sign. They don't show up when you compile the document.

% The first part of a LaTex document is called the "header". In the header you specify rules for formatting and can define new commands. If you have some idea what you are doing, you can edit this part. The worst that will happen is your document won't compile. I have put several full rows of % signs between the header and the body of the document.

\documentclass[11pt]{amsart} % this command specifies the font-size and that we want to use the AMS (American Mathematical Society) article class.
\usepackage{amssymb, amsmath, amsthm} %this command loads in special ways of formatting theorems and additional mathematical symbols.
\usepackage{color} %this package allows you to use colored text
\usepackage{graphicx} %this package lets you include images
\usepackage{bbold}

%-----------------------------------------------
% Here are some optional packages you may want to use at some point. To make use of them uncomment them.

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

%\usepackage{fourier} %this command changes the font. You can find the commands for other fonts online. Be sure to use one that includes the math symbols. This font was chosen so that the blackboard bold "1" will show up correctly.
%\usepackage[T1]{fontenc}



\usepackage[hidelinks=true]{hyperref} % this command turns references and citations into links.
%\usepackage{pinlabel} %this is useful package for adding labels to figures.
%-----------------------------------------------

%Define theorem styles

\theoremstyle{theorem} % this sets the overall style for the following environments
\newtheorem{theorem}{Theorem} %this is the environment for theorems. To enter the environment you type \begin{theorem} as in the example below. The second argument is the name that appears when the document is compiled.       
\newtheorem*{theorem*}{Theorem} %This produces an unnumbered theorem
\newtheorem{lemma}[theorem]{Lemma} %This is the lemma enviornment. The middle argument specifies that Lemmas and Theorems should be given the same numbering.
\newtheorem{definition}[theorem]{Definition}
%-----------------------------------------------

\newcommand{\spacing}{\parskip 6.6pt \parindent 0pt} %This specifies that paragraphs should not be indented and that there should be a space between paragraphs
\spacing


%-----------------------------------------------
% This section contains special user-defined commands. It could be included in the header file, but I've elected to put it here so that you know that these commands exist. You should feel free to define new commands.

\newcommand{\R}{{\mathbb R}} %this is a shortcut for writing the blackboard bold R for the real numbers.
\newcommand{\C}{\mathbb C}
\newcommand{\Z}{\mathbb Z}
\newcommand{\N}{\mathbb N}
\newcommand{\Q}{\mathbb Q}
\newcommand{\1}{\mathbb 1} %This is the command for blackboard bold 1
\newcommand{\nil}{\varnothing} %this lets us use \nil to get the symbol for the empty set.

\newcommand{\mc}[1]{\mathcal{#1}} %this is a shortcut for putting text in the caligraphic style
\newcommand{\defn}[1]{\textbf{#1}} % We'll use this to bold face words being defined.

\newcommand{\range}{\operatorname{range}} %here is an example of how to get the word "range" to print while in math mode and to have it spaced and typeset correctly. 

\newcommand{\co}{\mskip0.5mu\colon\thinspace} % When specifying a function such as f:R \to R simply typing a colon will produce incorrect spacing as the colon will be too far away from the function name. This line defines a new command \co which produces the correct spacing.


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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{document} 


% For homework assignments you don't need this, but for your paper you will. Uncomment it when you want to use it.
% The short title for running heads goes in square brackets.  This is optional.  The long title goes in braces. 

%\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 %this puts a long horizontal line


\textbf{Exercise 2.5: } %Specify what exercise you are about to provide the answer to in bold.
Prove the following theorem:   % Always give the problem statement.

\begin{theorem} %This is an example of how to use the theorem environment. You don't necessarily have to put every statement that you're going to prove into the theorem environment.
\label{Theorem in Exercise 2.5} %this label will help us refer to it later
Suppose that $n \in \N$. If $n^2$ is even then $n$ is even. %Observe how math symbols are placed between dollar signs. 
\end{theorem}

\hrulefill

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

\begin{definition}\label{Def: even/odd} %You can specify whatever label you want
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}] % The text in the square brackets appears along with the word ``Proof''.
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:
\[ % Use \[ and \] to enter and leave "display equation" mode.
\begin{array}{rcl} % strings of equalities are best typeset using the array environment. We have specified an array with 3 columns. The first is right justified, the second is centered, and the third is left justified.
n^2 &=& (2m+1)^2 \\ % in an array, the "&" separates columns and the \\ separates rows.
&=& 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}. % This is an example of how to include a citation. You don't need to cite everything from the course text, but in your final paper you should err on the side of citing.

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}