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\begin{document}
\begin{center}
\Large{Module 4: Metric Spaces and Sequences II}
\end{center}

1. Show that the infinite intersection of open sets may not be open and that the infinite union of closed sets may not be closed.
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2.  Find the closure, interior, and boundary of the following sets using Euclidean distance:
    \begin{enumerate}
        \item[(i)] $\{(x,y)\in \R^2 : y < x^2 \} \subseteq \R^2$
        \item[(ii)]  $[0,1)\times[0,1) \subseteq \R^2$
        \item[(iii)] $\{0 \} \cup  \{1/n \colon n \in \N\} \subseteq \R$ 
    \end{enumerate}

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3. Prove the following: Let $(x_n)_{n\in\N}$ be a sequence in a metric space $(X,d)$ that converges to a point $x \in X$. Then $x$ is unique.

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4. Let $(x_n)_{n \in \N}$ and $(y_n)_{n \in \N}$ be sequences in $\R$ such that $x_n \to x$ and and $y_n \to y$, with $\alpha, x,y, \in \R$. 
    \begin{enumerate}
        \item[(i)] Show that $\alpha \, x_n \to \alpha \, x$.
        \item[(i)] Show that $x_n + y_n \to x + y$.
    \end{enumerate}

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5. Show that discrete metric spaces (i.e. those with the metric defined as define $d\colon X \times X \to \R$ by $d(x,x) = 0$ and $d(x,y)=1$ for $x\neq y \in X$) are complete. 
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