a) Prove that Cantor's Intersection Theorem holds for nested compact sets in an arbitrary metric space; that

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 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.