Pointwise convergence and uniform convergence Suppose that $f$ has a uniformly continuous derivative. We define $\ f_n: \Bbb R\to\Bbb R  $  by
$$\ f_n(x) = n \left( f \left(x + \frac{1}{n}\right) - f(x)\right) $$
Find a pointwise convergence $\ f_n$. Prove that the sequence $\ f_n$ converges uniformly to its limit.

What have you done so far?
Hint: for the pointwise convergence, write 
$$
f_n(x) = \frac{f(x+h_n)-f(x)}{h_n}
$$
with $h_n=\frac{1}{n}\to 0^+$
