Intuitive understanding a theorem in analysis Is there a way to intuitively understand/visualize the following theorem in analysis?

Let $(f_n)$ be a sequence of real functions differentiable in a finite/infinite open interval $(a,b)$. Suppose that $(f_n(x))$ converges for at least one $x\in(a,b)$ and that $(f_n')$ converges uniformly on every finite closed subinterval of $(a,b)$. Then 

(i) $(f_n)$ converges uniformly on every finite closed subinterval of $(a,b)$
(ii)$f=\lim_{n\to\infty}f_n$ is differentiable in $(a,b)$ and $\forall x\in(a,b)$, $f'(x)=\lim_{n\to\infty}f_n'(x)$


Thanks.
 A: I'm not entirely sure what counts as intuition, but here's one way of thinking about it.  To avoid technicalities, I'm going to be completely vague as to how things converge, etc.  
As every calculus student knows, if $g(x)$ is a function and $G(x)$ is some antiderivative of $g(x)$, then all other antiderivatives of $g(x)$ are of the form $G(x)+C$.  This has the following consequence:
FACT : Let $g(x)$ be a function and let $G_1(x)$ and $G_2(x)$ be two antiderivatives of $g(x)$.  Assume that there exists a single point $p$ such that $G_1(p)=G_2(p)$.  Then $G_1(x)=G_2(x)$ for all $x$.
Now let's assume that we are in the situation of the theorem and we have a sequence $\{f_n(x)\}$ of functions such that $\{f'_n(x)\}$ converges to a function $g(x)$.  If the functions $f_n(x)$ were to converge to some function $G(x)$, then $G(x)$ would be an antideriviative for $g(x)$.  The one problem that might be happening is that (and here I'm going to be totally vague) "the functions $f_n(x)$ don't know what antiderivative they want to converge to".  By the fact, pinning down one single point eliminates that ambiguity.
