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 $xin\; S$ and for all $varepsilon\; >\; 0$, there exists $N>0$ such that whenever $m,\; n\; >\; N$.

Another way of saying this is that $d\_u\; (f\_\{n\},\; f\_\{m\})\; o\; 0$ as $m,\; n\; o\; infty$, where the uniform distance $d\_u$ between two functions is defined by

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

Convergence criteria

A sequence of functions {"f"_n} from "S" to "M" is pointwise Cauchy if, for each "x" ∈ "S", the sequence {"f"_n("x")} is a Cauchy sequence in "M". This is a weaker condition than being uniformly Cauchy. Nevertheless, if the metric space "M" is complete, then any pointwise Cauchy sequence converges pointwise to a function from "S" to "M". Similarly, any uniformly Cauchy sequence will tend uniformly to such a function.

The uniform Cauchy property is frequently used when the "S" is not just a set, but a topological space, and "M" is a complete metric space. The following theorem holds:

* Let "S" be a topological space and "M" a complete metric space. Then any uniformly Cauchy sequence of continuous functions "f"_n : "S" → "M" tends uniformly to a unique continuous function "f" : "S" → "M".

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 $xin\; S$ and for any entourage $varepsilon$, there exists $N>0$ such that $(f\_\{n\}(x),\; f\_\{m\}(x))\; in\; varepsilon$ whenever $m,\; n\; >\; N$.

ee also

*Modes of convergence (annotated index)