Weierstrass M-test for continuous functions

When the set $X$ in the statement of the Weierstrass M-test is a topological space, a strengthening of the hypothesis produces a stronger result. When the functions $f_n$ are continuous, then the limit of the series $f={\sum }_{n=1}^{\mathrm{\infty }}{f}_n$ is also continuous.

The proof follows directly from the fact that the limit of a uniformly convergent sequence of continuous functions is continuous.