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\lhead{\bf Advanced Calculus (MAA 4211)}
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\rhead{Due Friday 11/16}
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\begin{enumerate}

\Problem Let $X$ be a compact metric space and $\mathcal{F}$ a family of closed subsets of $X$. Prove that if
\[
	\bigcap_{C\in F} C\neq \emptyset
\]
for all finite subfamilies $F\subseteq \mathcal{F}$ (this condition is called the \emph{finite intersection property}) then actually
\[
	\bigcap_{C\in \mathcal{F}} C\neq \emptyset.
\]
\emph{Hint: take complements, create an open cover, invoke the definition of compactness.}

\Problem State and prove the converse to the previous problem.

\Problem Prove that every sequentially compact metric space is also complete. (\emph{This is Proposition 6.15 in the lecture notes.})

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