Corollary of the M-test

Corollary of the M-test: Let $E$ be a set and let $f_n:E\to\mathbb{R}$ (or $\mathbb{C}$) be bounded functions. If the numerical series

converges, where $\|f_n\|_\infty=\sup_{x\in E}|f_n(x)|$, then the series $\sum_{n=1}^\infty f_n$ converges uniformly on $E$ (and absolutely at each point of $E$).

This is the Weierstrass M-test applied with $M_n=\|f_n\|_\infty$, and it is a convenient criterion in settings involving the supremum norm .