# Let H be a nonempty, closed, bounded subset of Rn. a) Suppose that f : H Rm

## Question:

a) Suppose that f : H †’ Rm is continuous. Prove that

is finite and there exists an x0 ˆˆ H such that ||f(x0)|| = ||f||H.

b) A sequence of functions fk: H †’ Rm is said to converge uniformly on H to a function f: H †’ Rm if and only if for every Îµ > 0 there is an N ˆˆ N such that

k > N and x ˆˆ H imply ||fk(x) - f(x)|| Show that ||fk - f||H †’ 0 as k †’ ˆž if and only if fk †’ f uniformly on H as k: †’ ˆž.

c) Prove that a sequence of functions converges uniformly on fk if and only if for every Îµ > 0 there is an N ˆˆ N such that

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