Given function f(x) which is differentiable on R , and f'(x) is continuous on R, and a sequence (f_n(x)) such as :
I need to prove that f_n(x) is uniformly converges on each interval [a,b].
I have managed to show that (f_n(x)) approaches f'(x), but I couldn't get any further.
I know nothing about the values of f(x), so I can't know if f_n(x) is monotonic or not , and therefore I can't use Dini's theorem. Plus , I can't say much about |f_n(x)-f'(x)| , so I can't figure out how do I show that for every epsilon > 0 , there is N such as for every n>N , |f_n(x)-f'(x)| < epsilon
Any ideas how to get any further? Thanks people!