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\begin{document}

\title{MTH 427/527 Homework 1}

\author{Pavel Urysohn}

\date{September 4, 2018}

\maketitle


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% EXERCISE 2.1
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\section*{Exercise 2.1}

In order to solve this problem it will be convenient to start with the following fact:

\begin{lemma}
\label{HELPFUL LEMMA}
If $(X, \varrho)$ is a metric space then the identity function $f\colon X \to X$, $f(x) = x$ is continuous
(with respect to the metric $\varrho$). 
\end{lemma}


\begin{proof}
Here is the argument...
\end{proof}

Using Lemma \ref{HELPFUL LEMMA} we will prove that the following is true:

\begin{theorem}
\label{BIG THEOREM}
Some important statement here...
\end{theorem}

We will leave here 2 inches of vertical  space for a hand drawn picture that  illustrates the statement of 
this theorem:

\vskip 2in

\begin{proof}[Proof of Theorem \ref{BIG THEOREM}]
We argue by induction with respect to $n$. For $n=1$ ...
\end{proof}

Exercise 2.1 follows immediately from Theorem \ref{BIG THEOREM}. Indeed, assume that... 


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% EXERCISE 2.3
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\section*{Exercise 2.3}

We will prove this by contradiction. Assume that....





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