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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.

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    Hint: Use the mean value theorem on $\phi_n(x)$ for each $x$.2011-09-20

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