Let S be a compact subset of a metric space (X, d). Show that there exists a countable subset P of S such that cl(P) = S. [Hint: Recall the first lemma used in class to show the Arzel`a-Ascoli theorem.]

Exercise 3 (10 points). Let S be a compact subset of a metric space (X , 01). Show that there exists a countable subset P of S such that cl(P) = S. [Hint Recall the rst lemma used in class to show the ArzelaAscoli theorem]