A uniformly summable numerical majorant forces uniform convergence of the function series.Suggest a slogan
Let \(E\) be a set, let \(f_n\colon E\to\RR\) for \(n\geq1\), and let \((M_n)_{n\geq1}\) be a sequence of real numbers such that \(\abs{f_n(x)}\leq M_n\) for all \(n\geq1\) and all \(x\in E\). If \(\sum_{n=1}^{\infty}M_n\) converges, then \(\sum_{n=1}^{\infty}f_n\) converges uniformly on \(E\). [@Rud76].