Dini theorem

on uniform convergence

If the functions $u_n$, $n=1,2,\ldots,$ are continuous and non-negative on a segment $[a,b]$ and if the sum of the series $\sum_{n=1}^\infty u_n$ is a continuous function on this segment, then the series converges uniformly on $[a,b]$. Dini's theorem can be generalized to the case when an arbitrary compactum is the domain of definition of the functions $u_n$.