% If you are working from this template to just write up your own
% solutions, I'm trying to keep it simple for you and there are only a
% few places you need to edit. On line 43 there's a place where the
% phrase "YOUR NAME HERE" appears; change it to your actual
% name. Several places below there are places where the phrase "YOUR
% ANSWER HERE" appears, and directly below (or replacing) those lines
% are good places to put your answers.
%
% One important thing you need to do, however (due to the complicated
% way I use a single form to automatically produce both the question
% sheets and solution writeups) is to make sure that you keep the word
% 'solution' in your filename, or else your answers won't be printed
% at all.
\documentclass[12pt]{article}

\usepackage[margin=0.7in,dvips]{geometry}
\usepackage{fancyhdr,lastpage}
\usepackage{amsmath,amsfonts,amsthm}
\usepackage{stix}
\usepackage{tikz}
\usepackage{substr}

\IfSubStringInString{\detokenize{solution}}{\jobname}{%
  % Solutions
  \newcommand\qfill{}
  \newcommand\qpage{}
  \newcommand\qonly[1]{}
  \newcommand\answer[1]{#1}
  \newcommand\question[1]{{\em #1}}
  \newcommand\rubric[1]{}
}
{%
  % Question sheets
  \newcommand\qfill{\vfill}
  \newcommand\qpage{\newpage}
  \newcommand\qonly[1]{#1}
  \newcommand\answer[1]{}
  \newcommand\question[1]{#1}
  \newcommand\rubric[1]{}
}

\pagestyle{fancy}
\lhead{MATH 311-02}
\chead{Problem Set \#2\answer{ solutions}}
\rhead{\answer{YOUR NAME HERE}}
\lfoot{}
\cfoot{Page \thepage\ of \pageref{LastPage}}
\rfoot{{\bf due February 2, 2018}}

\begin{document}
\begin{enumerate}
\item \question{In calculus, you probably encountered the following
    formal definition of a limit: $\lim_{x\rightarrow a} f(x)=L$ means
    that for every positive real $\epsilon$ there is a positive real
    $\delta$ such that for any $x$, if $0<|x-a|<\delta$, then
    $|f(x)-L|<\epsilon$.}
  \begin{enumerate}
  \item \question{Write this definition symbolically.}
    
    \answer{
      % YOUR ANSWER HERE
    }

  \item \question{Quantify the negation of this statement: come up
      with a quantified, easy-to-use statement of what it means to say
      that $\lim_{x\rightarrow a} f(x)\neq L$.}
    
    \answer{
      % YOUR ANSWER HERE
    }

  \item \question{Use the results of the previous part to come up with
      an argument that $\lim_{x\rightarrow 3}x^2\neq 5$.}
    
    \answer{
      % YOUR ANSWER HERE
    }

    
  \end{enumerate}

\item \question{For each of the following statements, either
    demonstrate that it is false with a counterexample or that it is
    true with a formal proof.}
  \begin{enumerate}
  \item \question{For all integers $a$, $b$, $c$, and $d$ and natural
      number $n$, if $a\equiv b\pmod n$ and $c\equiv d\pmod n$, then
      $ac\equiv bd\pmod n$.}

  \item \question{For all integers $a$, $b$, and $c$, if $a\mid (b-1)$
      and $a\mid (c-1)$, then $a\mid (bc-1)$.}

  \item \question{For all integers $a$, $b$, and $c$, if $(ab)\mid c$
      then $a\mid c$.}
  \end{enumerate}
\end{enumerate}

\end{document}
