Cauchy Criterion for Uniform Convergence

Theorem

A sequence of functions $\left. \left( f_{n} \right) \right.$ defined on a set $A \subseteq \mathbb{R}$ converges uniformly on $A$ if and only if for every $\epsilon > 0$ there exists an $N \in \mathbb{N}$ such that $\left| f_{n}\left. (x) \right. - f_{m}\left. (x) \right. \right| < \epsilon$ whenever $m,n \geq N$ and $x \in A$.

Proof: See Exercise 6.2.5 in Abbott.