I wish to find the best way to prove that $\phi_n:= [\psi(x+{1\over n})-\psi(x)]n$ where $\psi$ is continuously differentiable on $(a,b)$, converges uniformly to \psi'(x) on all closed subintervals of $(a,b)$.
It is clear that $(\phi_n)$ converges to \psi'. But not so clear that (\phi_n') converges uniformly on every closed subinterval. I am thinking of using the continuity property of the derivative...? If I could show this then it follows that $\phi_n$ converges uniformly to \psi'(x) on all closed subintervals of $(a,b)$.
Is there a better way to show this though?
Thanks.