The terms of a uniformly convergent series tend to zero uniformly

Let \(S\) be a set and let \(f_n\colon S\to\CC\) for \(n\geq1\). If the series \(\sum_{n\geq1}f_n\) converges uniformly on \(S\), then \[ \sup_{x\in S}\abs{f_n(x)}\convergesto{n\to\infty}0 . \]