Let $X$ be any set, $\{f_n\}_{n\in\N}$ a sequence of real or complex valued functions on $X$ and $\{M_n\}_{n\in\N}$ a sequence of non-negative real numbers. Suppose that, for each $n \in \N$ and $x \in X$ , we have $|f_n(x)| \le M_n$ . Then $f=\sum_{n=1}^{\infty} f_n$converges uniformly if $\sum_{n=1}^{\infty} M_n$converges.
