Let $(f_n)$ be a sequence of functions that are all differentiable on an interval A, and suppose the sequence of derivatives $(f_n')$ converges uniformly on A to a limit function $g$. Does it follow that $(f_n)$ converges to a limit function f on A?
What I tried:
As $(f_n')$ converges uniformly to $g$, we may write that the limit of integral of $(f_n')$ is the integral of the limit of $(f_n')$. Hence, $(f_n)$ converges point wise to the integral of $g$.
How does this sound?
Best Answer
If, for each $n\in\Bbb N$, $f_n(x)=n$, then you always have $f_n'(x)=0$. Therefore, $(f_n')_{n\in\Bbb N}$ converges uniformly, but there is no $x\in\Bbb R$ such that $\bigl(f_n(x)\bigr)_{n\in\Bbb N}$ converges.