# Uniformly Cauchy sequence

Uniformly Cauchy sequence

In mathematics, a sequence of functions $\left\{f_\left\{n\right\}\right\}$ 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 $d\left(f_\left\{n\right\}\left(x\right), f_\left\{m\right\}\left(x\right)\right) < varepsilon$ whenever $m, n > N$.

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

:$d_\left\{u\right\} \left(f, g\right) := sup_\left\{x in S\right\} d \left(f\left(x\right), g\left(x\right)\right).$

Convergence criteria

A sequence of functions {"f"n} from "S" to "M" is pointwise Cauchy if, for each "x" &isin; "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" &rarr; "M" tends uniformly to a unique continuous function "f" : "S" &rarr; "M".

Generalization to uniform spaces

A sequence of functions $\left\{f_\left\{n\right\}\right\}$ 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 $\left(f_\left\{n\right\}\left(x\right), f_\left\{m\right\}\left(x\right)\right) in varepsilon$ whenever $m, n > N$.

ee also

*Modes of convergence (annotated index)

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