This will be a quick one, I know it exists somewhere, but I don't even know where to look.
Let's say we have a uniformly converging series of functions defined by $f_n$. Furthermore, we know that, for the pointwise convergent perfectly equivalent version of this series, $h_n$, it converges to some function, $g$. By this logic, can we say that the $f_n$ converges to $g$? If so, why?
Example: We see the pointwise convergence of $\sum_0^\infty x^n = \frac{1}{1-x}$ on $x \in (-1,1)$. Thus, if we prove that $f$ is uniformly convergent, can we say it also converges to $\frac{1}{1-x}$?