Showing uniform convergence of a series of functions

Use the Weierstrass \(M\)-test. Let \(S\) be a set and \(f_n\colon S\to\CC\) for \(n\geq1\). Find constants \(M_n\geq0\), independent of \(x\), such that \(\sup_{x\in S}\abs{f_n(x)} \le M_n\) for every \(n\) and \(\sum_{n\geq1} M_n < \infty\). Then \(\sum_{n\geq1} f_n\) converges absolutely at every point of \(S\), and its partial sums converge uniformly on \(S\), since for \(N>N'\) and all \(x\in S\), \[ \abs{\sum_{n=N'+1}^{N} f_n(x)} \le \sum_{n>N'} M_n \xrightarrow{N'\to\infty} 0. \]