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\flushleft\underline{Homework 3 --- MATH 4590 Spring 2018} \\
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\item Prove that if $M=\mathbb{R}$ and $d(x,y)=|x-y|$, then $(M,d)$ is a metric space. 
\item Prove the following ``$\epsilon$-principle": if for all $\epsilon > 0$ it holds that $|x-y| < \epsilon$, then $x=y$.
\item Prove that if $M=\mathbb{R}^2$ and $d((x_1,y_1),(x_2,y_2))= \mathrm{max}\{|x_1-x_2|,|y_1-y_2|\}$, then $(M,d)$ is a metric space. Draw the ``unit circle" in this metric space, i.e. the set $\{x \in \mathbb{R}^2 \colon d(x,0)=1\}$.

\item Let $(M,d)$ be any metric space and let $(x_n)$ and $(y_n)$ be convergent sequences in $M$ with $x_n \rightarrow x$ and $y_n \rightarrow y$. Prove that the sequence $(d(x_n,y_n))$ is a convergent sequence in $\mathbb{R}$ with $d(x_n,y_n) \rightarrow d(x,y)$. \\
(\textit{note: you don't have a formula for the metric $d$ in $M$ nor do you know much about $M$ -- it could be a lot of things -- but you should be thinking of the metric in $\mathbb{R}$ as $d_{\mathbb{R}}(x,y)=|x-y|$})

\item Consider the metric space $(\mathbb{R},d)$ where $d(x,y)=|x-y|$. Prove that the function $f \colon \mathbb{R} \rightarrow \mathbb{R}$ with $f(x)=x$ is continuous.
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