Proving Rudin Theorem 7.17: Real Analysis

[algekkk]

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.

 

[ystael]

Please follow the outline for posts given in the sticky -- in particular, you should (section 2) give the statement of theorem 7.16, since not everyone here has a copy of Rudin, and (section 3) tell us what you've done so far.