\documentclass{amsart}[20pt]
\usepackage{geometry}
\usepackage{amssymb,mathrsfs,multicol}
 \setlength\multicolsep{1pt}
\geometry{top=1.5cm}
\newcommand{\R}{\mathbb{R}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Q}{\mathbb{Q}}
\pagenumbering{gobble}
\theoremstyle{definition}
\newtheorem{que}{Question}[section]
\renewcommand{\thesection}{\Alph{section}}
\title{MTH435 - Homework 1}
\renewcommand{\labelenumi}{(\roman{enumi})}

\begin{document}
 \maketitle
 
 Solutions to the questions in Section B should be submitted by the start of class on 09/19/18. %You are encouraged to work in groups, but the solutions to the submitted work must be written up on your own. As per University policy, if you did a significant amount of work with another person, this should be indicated on your assignment. You may find the questions in Section A helpful in the later questions. Since this is an upper level course, your answers are supposed to be clearly written. Unclear or poorly written proofs may not receive full credit.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 \section{Warm-up Questions}
 
 

 
 
 \begin{que}
  Solve the following inequalities.
  \begin{multicols}{2}
  \begin{enumerate}
   \item $x < \frac{1}{x}$.
   \item $|x+1| + |x-2| \leq 2$.\\
   \item $x - 1 \leq |x -2|$.
   \item $|x^2 - 2| < 3$.
  \end{enumerate}
\end{multicols}
 \end{que}

 \vspace{-0.5cm}
 
 \begin{que}
  Let $a,b \in \R$. Prove the following.
  \begin{enumerate}
   \item $\max \{a,b\} = \frac{1}{2} (a + b + |a-b|)$.
   \item $\min \{a,b\} = \frac{1}{2} (a + b - |a-b|)$.
   \item $\max \{ \max \{a,b\},c\}$.
  \end{enumerate}

 \end{que}
 
 \begin{que}
  Prove the following for subsets of $\R$.
  \begin{multicols}{2}
  \begin{enumerate}
   \item $\R$ is open and closed.
   \item $\varnothing$ is open and closed.
   \item $[0,1)$ is neither open nor closed.\\
   \item $\{ x_1, x_2, \dotsc, x_n \}$ is closed.
   \item $\left\{ \frac{1}{n} \mid n \in \N \right\}$ is neither open nor closed.
  \end{enumerate}
\end{multicols}
 \end{que}

\vspace{-0.5cm}
 
\begin{que}
 Let $U_1$ and $U_2$ be open sets in $\R$. Prove the following.
 \begin{enumerate}
  \item $U_1 \cup U_2$ is open.
  \item $U_1 \cap U_2$ is open.
 \end{enumerate}

\end{que}

\begin{que}
 Compute $\sup X$ and $\inf X$ (if they exist) for the following sets.
  \begin{multicols}{2}
  \begin{enumerate}
   \item $(0,1) \cup (2,3]$.
   \item $\left\{ \frac{1}{n} \mid n \in \N \right\}$ \\
   \item $\{x \in R \mid x < \frac{1}{x} \}$
   \item $\{x \in \R \mid \sin \frac1x = 0 \}$.
  \end{enumerate}
\end{multicols}
\end{que}


\begin{que}
 Let $S$ and $T$ be non-empty bounded subsets of $\R$.  Define $-S = \{ -s \mid s \in S \}$ and $S+T = \{ s + t \mid s \in S, t \in T\}$. Prove the following.
 \begin{enumerate}
  \item $\sup (-X) = - \inf X$.
  \item $\sup (S+T) \leq \sup S + \sup T$.
\end{enumerate}
\end{que}

 \vskip 0.5cm
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 \section{Submitted Questions}

\begin{que}
 Let $X = \left\{ \frac{1}{n} \mid n \in \N \right\} \cup \{0\}$. Is $X$ open, closed, both or neither? Justify your answer.
\end{que}

 \begin{que}
  Let $\varnothing \neq S \subseteq T \subseteq \R$ and suppose $T$ is bounded. Prove that
  \[
   \inf T \leq \inf S \leq \sup S \leq \sup T.
  \]

 \end{que}

 \vskip 0.5cm
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 \section{Challenge Questions}
 
 \begin{que}
  Using the properties of $\R$ given on pages 24 and 26, prove the following. (This exercise is not really important for analysis but it is good for the soul). Let $a,b,c,d \in \R$.
  \begin{enumerate}
   \item If $a + b = 0$ then $b=-a$.
   \item $-(-a) = a$.
   \item If $a^2 = a$ then $a \in \{0,1\}$.
   \item If $a \leq b$ and $c \leq d$ then $a + c \leq b + d$.
  \end{enumerate}

 \end{que}
 
 \begin{que}
  Let $(U_\alpha)_{\alpha \in A}$ be an indexed family of open sets. Prove or disprove the following.
  \begin{enumerate}
   \item $\bigcup_{\alpha \in A} U_\alpha$ is open.
   \item $\bigcap_{\alpha \in A} U_\alpha$ is open.
  \end{enumerate}

 \end{que}


 
\end{document}
