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% MATH 790: Real Analysis Spring 2022
% hw_template.tex
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% The preamble that follows can be ignored. Go on
% down to the section that says "START HERE"
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\documentclass{article}
\usepackage[margin=1in]{geometry}
\usepackage{amsmath,amsthm,amssymb,hyperref}
\newcommand{\R}{\mathbf{R}}
\newcommand{\Z}{\mathbf{Z}}
\newcommand{\N}{\mathbf{N}}
\newcommand{\Q}{\mathbf{Q}}
\newenvironment{theorem}[2][Theorem]{\begin{trivlist}
\item[\hskip \labelsep {\bfseries #1}\hskip \labelsep {\bfseries #2.}]}{\end{trivlist}}
\newenvironment{lemma}[2][Lemma]{\begin{trivlist}
\item[\hskip \labelsep {\bfseries #1}\hskip \labelsep {\bfseries #2.}]}{\end{trivlist}}
\newenvironment{exercise}[2][Exercise]{\begin{trivlist}
\item[\hskip \labelsep {\bfseries #1}\hskip \labelsep {\bfseries #2.}]}{\end{trivlist}}
\newenvironment{problem}[2][Problem]{\begin{trivlist}
\item[\hskip \labelsep {\bfseries #1}\hskip \labelsep {\bfseries #2.}]}{\end{trivlist}}
\newenvironment{question}[2][Question]{\begin{trivlist}
\item[\hskip \labelsep {\bfseries #1}\hskip \labelsep {\bfseries #2.}]}{\end{trivlist}}
\newenvironment{corollary}[2][Corollary]{\begin{trivlist}
\item[\hskip \labelsep {\bfseries #1}\hskip \labelsep {\bfseries #2.}]}{\end{trivlist}}
\newenvironment{solution}{\begin{proof}[Solution]}{\end{proof}}
\begin{document}
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% START HERE %
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\title{Proof 1} % Replace with appropriate title
\author{Author's Name\\Math 790, Real Analysis} % Replace "Author's Name" with your name
\maketitle
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% The following two environments (theorem, proof) are
% where you will enter the statement and proof of your
% first problem for this assignment.
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% In the theorem environment, you can replace the word
% "theorem" in the \begin and \end commands with
% "exercise", "problem", "lemma", etc., depending on
% what you are submitting.
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\begin{theorem}{(Page 4 $\#$ 6)}
For any metric space $E$, the entire space $E$ is an open set.
\end{theorem}
\begin{proof}
Replace this text with the details of your proof. Mathematical symbols go between dollar signs, like this: $E$, \emph{not} in plain text like this: E. If there are more symbols you want, such as $\in$, you can find them at\\ \url{http://detexify.kirelabs.org/classify.html}. If you want to set expressions or equations out from the text so that they are more readable (great idea!), put them between double dollar signs: $$|x-x_0| < \delta \implies |f(x)-f(x_0)| < \epsilon.$$
The small square marks the end of the proof.
\end{proof}
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\end{document}
