a) Prove that Cantor 's Intersection Theorem holds for nested compact sets in an arbitrary metric space that is, if H1, H2, is a nested sequence of nonempty compact sets in X, then b) Prove that (2, 3) Q 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 Hi 0 V) k 1

Question: a) Prove that Cantor's Intersection Theorem holds for nested compact sets in an arbitrary metric space; that is, if H1, H2, ... is a nested

a) Prove that Cantor's Intersection Theorem holds for nested compact sets in an arbitrary metric space; that is, if H1, H2, ... is a nested sequence of nonempty compact sets in X, then

b) Prove that (ˆš2, ˆš3) ˆ© Q 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.

Hi # 0. V). k=1

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