I'd like your help with proving that for a function uniformly differentiable $f$, and a series of functions, $f_n(x)=n[f(x+\frac{1}{n})-f(x)]$ is uniformly converges in closed interval $[a,b]$, for a fixed x. I proved that the function pointwise converges to f'(x) and for the uniformly convergence I tried to use Dini's theorem but I don't see why $f_n(x)$ is monotonic for a fixed x. I tried to use the $\epsilon$ definition, but I didn't managed to show that |f_n(x)-f'(x)|< \epsilon.
Any suggestion?
Thanks a lot!