Uniform metric on bounded functions

Example

Let $S$ be a nonempty set. Let $B(S)$ be the set of bounded functions $f \colon S \to \mathbb{R}$, that is, the functions for which there is a real $M$ with $|f(s)| \le M$ for all $s \in S$. For $f, g \in B(S)$, let $$d_\infty(f, g) = \sup_{s \in S} |f(s) - g(s)|.$$ Then $d_\infty$ is a metric on $B(S)$, the uniform metric.