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

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.

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 $\{f_{n}\}$ from a set S to a uniform space U is said to be uniformly Cauchy if:

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

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