Uniformly Cauchy sequence

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

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

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

:$d_ (f, g) := \sup_ d (f(x), g(x)).$

Convergence criteria

A sequence of functions from ''S'' to ''M'' is pointwise Cauchy if, for each ''x'' ∈ ''S'', the sequence is a Cauchy sequence in ''M''. This is a weaker condition than being uniformly Cauchy.

In general a sequence can be pointwise Cauchy and not pointwise convergent, or it can be uniformly Cauchy and not uniformly convergent. 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 $\$ from a set ''S'' to a uniform space ''U'' is said to be uniformly Cauchy if:

* For any entourange ''E'' of ''U'', there exists $N>0$ such that, for all $x\in S$, $(f_(x), f_(x))\in E$ whenever $m, n > N$.

See also

*Modes of convergence (annotated index)

Functional analysis
Convergence (mathematics)

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