A series of functions converges uniformly if the sequence of partial sums converges uniformly.

Question

Is there a proof or quick explanation for this statement:

A series $\sum_{k=1}^{\infty} f_k(x)$ converges uniformly if the sequence of partial sums $s_n(x) = \sum_{k=1}^n f_k(x)$ converges uniformly.

I know that convergence of sequence of partial sums implies convergence of the series but I am not sure how to extend it to uniformity.

Answer 1

This is a question of terminology/definitions. By definition, $\sum_{n=1}^\infty f_n(x)$ is the function $f(x)$ defined by the limit $\lim_N s_N(x)$ of partial sums. The sequence of functions $s_N(x)$ may or may not converge uniformly in $x$, but if it does, then we say the series defining $f(x)$ converges uniformly.