Theorem 1   Let $T$ be a topological space, $f$ be a continuous function from $T$ to $[0,\infty)$ , and let $\{f_k\}_{k=0}^\infty$ be a sequence of continuous functions from $T$ to $[0,\infty)$ such that, for all $x \in T$ , the sum $\sum_{k=0}^\infty f_k (x)$ converges to $f(x)$ . Then the convergence of this sum is uniform on compact subsets of $T$ .

Proof. Let $X$ be a compact subset of $T$ and let $\epsilon$ be a positive real number. We will construct an open cover of $X$ . Because the series is assumed to converge pointwise, for every $x \in X$ , there exists an integer $n_x$ such that $\sum_{k=n_x}^\infty f_k (x) < \epsilon / 3$ . By continuity, there exists an open neighborhood $N_1$ of $x$ such that $|f(x) - f(y)| < \epsilon /3$ when $y \in N_1$ and an open neighborhood $N_2$ of $x$ such that $\left| \sum_{k=0}^{n_x} f_k (x) - \sum_{k=0}^n f_k (y) \right| < \epsilon / 3$ when $y \in N_2$ . Let $N_x$ be the intersection of $N_1$ and $N_2$ . Then, for every $y \in N$ , we have $$ f(y) - \sum_{k=0}^{n_x} f_k (y) < |f(y) - f(x)| + \left| f(x) - \sum_{k=0}^{n_x} f_k (x) \right| + \left| \sum_{k=0}^{n_x} f_k (x) - \sum_{k=0}^{n_x} f_k (y) \right| < \epsilon . $$ In this way, we associate to every point $x$ an neighborhood $N_x$ and an integer $n_x$ . Since $X$ is compact, there will exist a finite number of points $x_1, \ldots x_m$ such that $X \subseteq N_{x_1} \cup \cdots \cup N_{x_m}$ . Let $n$ be the greatest of $n_{x_1}, \ldots, n_{x_m}$ . Then we have $f(y) - \sum_{k=0}^n f_k (y) < \epsilon$ for all $y \in X$ , so, the functions $f_k$ being positive, $f(y) - \sum_{k=0}^h f_k (y) < \epsilon$ for all $h \ge n$ , which means that the sum converges uniformly.