Uniform convergence is exactly convergence to zero in the sup norm.Suggest a slogan
Let \(S\) be a set and let \(f_n,f\colon S\to\CC\) for \(n\geq1\). Then \(f_n\to f\) uniformly on \(S\) if and only if there are real numbers \(M_n\) with \[ \sup_{x\in S}\abs{f_n(x) - f(x)} \leq M_n \text{ for all } n \quad\text{and}\quad M_n\convergesto{n\to\infty}0 . \]