Problem:

Let X be a metric space and let $\displaystyle K \subseteq X$ be a sequentially compact subspace. Letpbe a point inX\K. Prove that there is a point $\displaystyle q_{0} $ that is closet topof all points inKin the sense that

$\displaystyle d(p,q_{0}) \leq d(p,q)$ for all pointsqinK.

Is this point unique?

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Suppose that $\displaystyle K \subseteq X$ is sequentially compact subspace of the metric spaceX. ThenKis a closed subset ofX.

Define the function $\displaystyle f:K \rightarrow R$ by $\displaystyle f(q)=d(q,p)$ for allqinKandpbe a point inX\K. I want to show that this functionfis continuous first, but I don't know how nor what theorem to use.

Thank you for your time. Much appreciated.