Let $E$ be a nonempty set, and let $(f_n)_{n=1}^{\infty}$ be a sequence of bounded functions $f_n:E \to \mathbb{R}$. For each [bounded function](/page/Bounded%20Function) $h:E \to \mathbb{R}$, define its sup norm by
Then $(f_n)_{n=1}^{\infty}$ is uniformly Cauchy on $E$ if and only if $(f_n)_{n=1}^{\infty}$ is Cauchy with respect to the sup norm; that is, if and only if for every $\varepsilon > 0$ there exists $N \in \mathbb{N}$ such that, for all $m,n \geq N$,