\documentclass[12pt]{amsart}
\usepackage{amsmath, amssymb, amsthm}
\usepackage[pdflang={en-US}]{hyperref}
\usepackage[margin=.75in]{geometry} % Adjust margins to your preference
%%% The following sets the numbering scheme.
\renewcommand{\thesubsection}{\Alph{subsection}}
%% This sets up some definition and theorem environments.
\newtheorem*{thm}{Theorem}
\newtheorem*{lem}{Lemma}
\newtheorem*{stm}{Statement}
\theoremstyle{definition}
\newtheorem{dfn}{Definition}
%%% Add any useful abbreviations here
\newcommand{\lcm}{\text{lcm}}
\title{Math 301 Portfolio}
\author{YOUR NAME}
\date{\today}
\begin{document}
\maketitle
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{section}{Expository entries}
%%%
\begin{subsection}{Response to \textbf{TITLE} by AUTHOR}
Paste your first portfolio entry here.
\end{subsection}
%%%
\begin{subsection}{Response to \textbf{TITLE} by AUTHOR}
Paste your second portfolio entry here.
\end{subsection}
%%%
\begin{subsection}{Response to WHATEVER YOU DID FOR THE THIRD EXPOSITORY PORTFOLIO}
Paste your third portfolio entry here.
\end{subsection}
%%%
\end{section}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{section}{Proofs}
\begin{subsection}{Direct and contrapositive proofs}
Paste your first proof here.
I recommend putting your proof in the following format.
If you wish to include a relevant definition, use a definition environment:
\begin{dfn} \label{limit} % This label allows me to easily refer to the definition
$\lim_{x \to a} f(x) = L$ if for every number $\epsilon > 0$ there is a number $\delta > 0$ such that \[ 0 < \lvert x - a \rvert < \delta \implies \lvert f(x) - L \rvert < \epsilon. \]
\end{dfn}
State the result you proved as a theorem, then give the proof.
If you'd like to include a lemma, then use a lemma environment.
\begin{lem}
$\lim_{x \to 0} f(x) \neq 0$ if there is a number $\epsilon > 0$ such that for every number $\delta > 0$ there is a number $x$ such that \[ 0 < \lvert x \rvert < \delta \text{ and } \lvert f(x) \rvert \geq \epsilon. \]
\end{lem}
\begin{proof}
This is the negation of the definition of the limit (definition \ref{limit}) with $a = 0$ and $L=0$.
All quantifiers have been swapped, even the implicit quantifier for the conditional statement at the end of the definition.
\end{proof}
\begin{thm}
$\displaystyle \lim_{x \to 0} \sin\left(1 / x \right) \neq 0$.
\end{thm}
\begin{proof}
Let $\delta$ be a real number greater than $0$.
Let $n$ be an odd natural number such that $\frac{1}{n} < \delta$.
Because $0 < \frac{2}{\pi} < 1$, it follows that $0 < \frac{2}{ n \pi} < \delta$.
Moreover, because $n$ is odd \[
\Big{\lvert} \sin \left( \frac{1}{\left( 2/ ( n \pi) \right) } \right) \Big{\rvert} = \Big{\lvert} \sin \left( \frac{n \pi}{2} \right) \Big{\rvert} = 1 .
\]
Thus we have shown that there is a number $x = \frac{2}{ n \pi}$, such that \[
0 < \lvert x \rvert < \delta \text{ and } \lvert \sin\left(1 / x \right) \rvert \geq 1 .
\]
Since $\delta$ was arbitrary, we can conclude that this holds for every positive real number $\delta$.
Therefore, by the lemma, $\displaystyle \lim_{x \to 0} \sin\left(1 / x \right) \neq 0$
\end{proof}
\end{subsection}
\vspace{.1in}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{subsection}{Proofs by contradiction and non-conditional statements}
Paste your second proof here.
\end{subsection}
\vspace{.1in}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{subsection}{Proofs and disproofs}
Paste your third proof (or disproof) here.
It may not be be appropriate to call the statement in question a theorem, since you may be providing a disproof.
I suggest the following.
\begin{stm}
Suppose $A$, $B$, and $C$ are sets. If $A \times C \subseteq B \times C$, then $A \subseteq B$.
\end{stm}
The statement is \textbf{false}.
\begin{proof}
To prove that the statement is false, we produce sets $A$, $B$, and $C$, such that $A \times C \subseteq B \times C$ and $A \nsubseteq B$.
Et cetera.
\end{proof}
\end{subsection}
\vspace{.1in}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{subsection}{Induction I}
Paste your fourth proof here.
\end{subsection}
\vspace{.1in}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{subsection}{Induction II}
Paste your fifth proof here.
\end{subsection}
\vspace{.1in}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{subsection}{Uncategorized proofs}
Paste your sixth proof here.
\end{subsection}
\end{section}
\end{document}