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algekkk
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i am asked to prove the remark Rudin made in theorem 7.17 in his Mathematical Analysis.
Suppose {fn} is a sequence of functions, differentiable on [a,b] such that {fn(x0)} converges for some x0 in [a,b]. Assume f'n (derivative of fn) is continuous for every n. Show if {f'n} converges uniformly, then {fn} converges uniformly to some function f and f'(x)=lim(n goes to inf) f'n(x). (x in [a,b])
Rudin's hint is to use theorem 7.16 and fundamental theorem of calculus.
Thanks for any help.
Suppose {fn} is a sequence of functions, differentiable on [a,b] such that {fn(x0)} converges for some x0 in [a,b]. Assume f'n (derivative of fn) is continuous for every n. Show if {f'n} converges uniformly, then {fn} converges uniformly to some function f and f'(x)=lim(n goes to inf) f'n(x). (x in [a,b])
Rudin's hint is to use theorem 7.16 and fundamental theorem of calculus.
Thanks for any help.