Pointwise convergence

Let \(S\) be a set, and let \(f_n\colon S\to\CC\) for \(n\geq 1\) and \(f\colon S\to\CC\) be functions. The sequence \((f_n)\) converges pointwise to \(f\) on \(S\) if \[ (\forall \varepsilon>0)\,(\forall x \in S)\,(\exists n_0 = n_0(x, \varepsilon))\,(\forall n>n_0)\quad \abs{f_n(x)-f(x)}<\varepsilon. \]