Uniformly Cauchy sequence

In mathematics, a sequence of functions $\{f_{n}\}$ from a set S to a metric space M is said to be uniformly Cauchy if:

• For all $x\in S$ and for all $\varepsilon >0$, there exists $N>0$ such that $d(f_{n}(x),f_{m}(x))<\varepsilon$ whenever $m,n>N$.

Another way of saying this is that $d_{u}(f_{n},f_{m})\to 0$ as $m,n\to \infty$, where the uniform distance $d_{u}$ between two functions is defined by

$d_{u}(f,g):=\sup _{x\in S}d(f(x),g(x)).$

Generalization to uniform spaces

A sequence of functions $\{f_{n}\}$ from a set S to a metric space U is said to be uniformly Cauchy if:

• For all $x\in S$ and for any entourage $\varepsilon$, there exists $N>0$ such that $(f_{n}(x),f_{m}(x))\in \varepsilon$ whenever $m,n>N$.