I already proved the following:

"Suppose ${f_j(x)}$ is a sequence of functions, where (1) $f_j$ is differentiable across open interval $I$, (2) $f_j \to f$ pointwisely, and (3) $f'_j \to g$ uniformly. Then:
(a) $f$ is differentiable on $I$, and (b) $f'(x) = g(x) \; \forall x \in I$.

with help from the book. But then it says that the same result can be proven if we relax condition #2 to only say $f_j \to f$ at $x_0 \in I$.