Uniform Cauchy criterion for sequences and series of functions

Let \(S\) be a set, let \(f_n\colon S\to\CC\) for \(n\geq1\), and for \(h\colon S\to\CC\) put \(\norm{h}_\infty\coloneqq\sup_{x\in S}\abs{h(x)}\). The sequence \((f_n)\) converges uniformly on \(S\) to some function \(f\colon S\to\CC\) if and only if it is uniformly Cauchy: \[ \forall\varepsilon>0\ \exists N\ \forall m,n\geq N:\quad \norm{f_n-f_m}_\infty<\varepsilon . \] In particular, a series \(\sum_{n\geq1}f_n\) converges uniformly on \(S\) if and only if \[ \forall\varepsilon>0\ \exists N\ \forall n>m\geq N:\quad \sup_{x\in S}\abs{\sum_{k=m+1}^{n}f_k(x)}<\varepsilon . \]