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\documentclass[12pt]{article}

\usepackage{amssymb,amsmath,amsthm}
\usepackage[top=1in, bottom=1in, left=1.25in, right=1.25in]{geometry}
\usepackage{fancyhdr}
\usepackage{enumerate}

% Comment the following line to use TeX's default font of Computer Modern.
\usepackage{times,txfonts}

\newtheoremstyle{homework}% name of the style to be used
  {18pt}% measure of space to leave above the theorem. E.g.: 3pt
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\theoremstyle{homework} 

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\newtheorem*{exercisecore}{Exercise \@currentlabel}
\newenvironment{exercise}[1]
{\def\@currentlabel{#1}\exercisecore}
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\newcommand\solution{\localhead{Solution:}}

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\fancyfoot[C]{\ifnum \value{page} > 1\relax\thepage\fi}
\fancyhead[L]{\ifx\@doclabel\@empty\else\@doclabel\fi}
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\fancyhead[R]{\ifx\@docauthor\@empty\else\@docauthor\fi}
\headheight 15pt

\def\doclabel#1{\gdef\@doclabel{#1}}
\doclabel{Use {\tt\textbackslash doclabel\{MY LABEL\}}.}
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\docdate{Use {\tt\textbackslash docdate\{MY DATE\}}.}
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\makeatother

% Shortcuts for blackboard bold number sets (reals, integers, etc.)
\newcommand{\Reals}{\ensuremath{\mathbb R}}
\newcommand{\Nats}{\ensuremath{\mathbb N}}
\newcommand{\Ints}{\ensuremath{\mathbb Z}}
\newcommand{\Rats}{\ensuremath{\mathbb Q}}
\newcommand{\Cplx}{\ensuremath{\mathbb C}}
%% Some equivalents that some people may prefer.
\let\RR\Reals
\let\NN\Nats
\let\II\Ints
\let\CC\Cplx

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\doclabel{Math 401: Homework 3}
\docauthor{Your name goes here!}
\docdate{September 21, 2020}

\begin{document}

\begin{exercise}{Supplemental 1} 
Show that the sequence $(-1)^n$ does not converge.
\end{exercise}
\begin{proof}
\end{proof}

\begin{exercise}{Supplemental 2} \strut
\begin{enumerate}[(a)]
	\item Show that for all $n\in\Nats$, $2^n\ge n$.
	\item Show  that $\lim_{n\to\infty} 1/2^n = 0$.
\end{enumerate}
\end{exercise}
\begin{proof}[Part (a)]
\end{proof}
\begin{proof}[Part (b)]
\end{proof}

\begin{exercise}{2.2.2}
From the definition, compute the given limits.
\end{exercise}
\begin{proof}[Part (a)]
\end{proof}
\begin{proof}[Part (b)]
\end{proof}
\begin{proof}[Part (c)]
\end{proof}

\begin{exercise}{2.2.3}
Describe what needs to be shown to disprove the given statements.
\end{exercise}
\solution
\begin{enumerate}[(a)]
\item
\item
\item
\end{enumerate}

\begin{exercise}{2.2.6}
Prove that limits are unique.
\end{exercise}
\begin{proof}
\end{proof}

\begin{exercise}{2.2.5(a)}
Determine, with a proof, $\lim_{n\to\infty} [[5/n]]$.
\end{exercise}
\solution
Claim: The limit is ??.

\begin{proof}
\end{proof}


\begin{exercise}{2.3.9(a)(c)}
\strut
\begin{enumerate}
	\item[(a)] If $(a_n)$ is a bounded sequence and $b_n\to 0$,
	show $a_nb_n\to 0$.
	\item[(c)] Prove Theorem 2.3.3(iii) for the case $a=0$.
\end{enumerate}
\end{exercise}
\solution
\begin{enumerate}
	\item[(a)] 
	\begin{proof}
	\end{proof}
	\item[(c)]
	\begin{proof}
	\end{proof}
\end{enumerate}


\end{document}