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\lhead{\bf Advanced Calculus (MAA 4211)}
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\rhead{Due Friday 9/21}
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\begin{enumerate}

\Problem Let $(X,d)$ be a metric space. Define the function $d_\ast$ by
\[
	d_\ast(x,y)=\frac{d(x,y)}{1+d(x,y)}.
\]
Prove that $(X,d_\ast)$ is also a metric space.

\Problem For the following subsets of the metric space $(\mathbb{R}^n, d_2)$, determine (with justification) whether they are open and/or closed (you may use any results we have proved in class):
\[\begin{array}{l}
	\{(x,y)\st x,y>0\},\\
	\{(x,y)\st xy=1\},\\
	\{(x,y)\st |x-y|>1\}.
\end{array}\]

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