If $(f_n')$ converges uniformly on an interval, does $(f_n)$ converge?
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?
Solution 1:
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.
Solution 2:
This might help in your thinking about this.
You have your simple counterexample so the real question is what extra assumption is needed so that your attempted proof would work? It would be a failed student exercise to toss such questions out the window simply because a trivial counterexample dismisses them.
Problem #1. As pointed out by José in his answer and Severin in his comment, you need to have some control over the primitive functions $f_n$. The standard assumption usually added to make this work is that $f_n(x_0)$ converges for at least one point $x_0$ in the interval $A$.
Problem #2. That is an excellent idea to convert this to a statement about integrals and then use some integration theory to solve it for you. What integration theory would you like to use?
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If each $f_n$ is continuously differentiable you can use the Riemann integral. Then $f_n'\to g$ uniformly etc. Good.
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If $f_n$ are all differentiable with each $f_n'$ Riemann integrable you can do the same. Great.
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If $f_n$ are all differentiable with $f_n'$ each Lebesgue integrable you can do the same. Great yet again.
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But (big but) what if $f_n$ are all differentiable with $f_n'$ not Lebesgue integrable. You have any more integrals hiding in your closet? (There are a few in mine but I am a collector of such esoterica.)
So the integration idea is shot down in general unless you want to prove a weak version by adding an extra hypothesis about the functions in the sequence $\{f_n\}$.
A proof of the most general version of your "derivatives converge uniformly" theorem can be found in the reference below.
See Section 9.6.1 Limits of Discontinuous Derivatives, Theorem 9.37: in this free PDF http://classicalrealanalysis.info/documents/TBB-AllChapters-Portrait.pdf
The assumptions are that $f_n:[a,b]\to \mathbb R$ are differentiable, $f'_n\to g$ uniformly on $[a,b]$ and $f_n(x_0)$ converges for some point. Then $f_n$ converges uniformly to some differentiable function $f$ and $f'(x)=\lim_{n\to\infty}f_n'(x).$
Before consulting consider trying for a proof (avoiding integrals as too special) and using instead everyone's favorite tool from your freshman calculus: the mean-value theorem.
P.S. If you are reading this and confused about the fact that derivatives do not have to be integrable (Riemann or Lebesgue) then there is something more to learn here.