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 uniformly to \(f\) on \(S\) if \[ (\forall \varepsilon>0)\,(\exists n_0 = n_0(\varepsilon))\,(\forall x \in S)\,(\forall n>n_0)\quad \abs{f_n(x)-f(x)}<\varepsilon. \]
Negating the quantifiers, \((f_n)\) does not converge uniformly to \(f\) on \(S\) if and only if \[ (\exists \varepsilon>0)\,(\forall n_0)\,(\exists x = x(n_0) \in S)\,(\exists n>n_0)\quad \abs{f_n(x)-f(x)} \geq \varepsilon. \] The point \(x\) at which the estimate fails may depend on \(n_0\).