6. Let X be metric space, Y be a Euclidean space, and H be a nonempty compact subset of X.

(a) Suppose that I : H -+ Y is continuous. Prove that IIIIIH := sup 111(x)lly xEH is finite and there exists an Xo E H such that 111(xo)lly = IIIIIH.

(b) A sequence of functions !k : H -+ Y is said to converge uniformly on H to a function I : H -+ Y if and only if given f: > 0 there is an N E N such that k ~ N and x E H imply 11!k(x) - l(x)lly < f:.

Show that Illk - IIIH -+ 0 as k -+ 00 if and only if!k -+ I uniformly on H as k -+ 00.

(c) Prove that a sequence of functions Ik converges uniformly on H if and only if, given f: > 0, there is an N E N such that k,j ~ N implies Illk -hlIH < f:.