Definition:Uniform Cauchy Criterion

Definition
Let $S \subseteq \mathbb R$.

Let $\sequence {f_n}$ be a sequence of real functions $S \to \R$.

We say that $\sequence {f_n}$ satisfies the uniform Cauchy criterion or is uniformly Cauchy on $S$ if for all $\varepsilon \in \R_{> 0}$, there exists $N \in \N$ such that:


 * $\size {\map {f_n} x - \map {f_m} x} < \varepsilon$

for all $x \in S$ and $n, m > N$.

By Sequence of Functions is Uniformly Cauchy iff Uniformly Convergent, this criterion gives a necessary and sufficient condition for a sequence of real functions to be uniformly convergent.