Let \(S\) be a set, and let \(f_n\colon S\to\CC\) for \(n\geq1\) and \(f\colon S\to\CC\) be functions. To show that \((f_n)\) does not converge uniformly to \(f\) on \(S\), exhibit \(\varepsilon>0\) and, for infinitely many \(n\), a point \(x_n\in S\) with \(\abs{f_n(x_n) - f(x_n)} \geq \varepsilon\).
On \(S=(0,\infty)\), the functions \(f_n(x)=\frac{1}{1+nx}\) converge pointwise to \(f=0\), but not uniformly: with \(\varepsilon=\frac12\) and \(x_n=\frac1n\), \(\abs{f_n(x_n)-f(x_n)}=\frac{1}{2}\) for every \(n\).