Let H be a nonempty, closed, bounded subset of Rn.
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 ε > 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 ε > 0 there is an N ˆˆ N such that

Transcribed Image Text:

reH k, j > N implies |f-fill H < ε.