Uniform convergence in supremum norm

Uniform convergence in supremum norm: Let $E$ be a set and let $f_n,f:E\to\mathbb{R}$ be bounded functions. Then $f_n\to f$ uniformly on $E$ if and only if

\|f_n-f\|_\infty \longrightarrow 0,

where $\|g\|_\infty=\sup_{x\in E}|g(x)|$ is the supremum norm.

Remarks

In particular, uniform convergence is exactly convergence in the metric induced by the supremum norm, which underlies the space of continuous functions equipped with $\|\cdot\|_\infty$ .