8. (a) Prove that Cantor's Intersection Theorem holds for nested compact sets in an arbitrary metric space;
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8.
(a) Prove that Cantor's Intersection Theorem holds for nested compact sets in an arbitrary metric space; i.e., if Hl , H2 , ... is a nested sequence of nonempty compact sets in X, then 00 k=l
(b) Prove that (y'2, v'3)nQ is closed and bounded but not compact in the metric space Q introduced in Example 10.5.
(c) Show that Cantor's Intersection Theorem does not hold in an arbitrary metric space if "compact" is replaced by "closed and bounded."
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