Uniformly Cauchy sequence explained

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 inf ty , where the uniform distance d _{u} between two functions is defined byd _{u} (f, g) : = \sup _{x} 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

\{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 entourage \varepsilon , there exists N> 0 such that d(f _{n} (x), f _{m} (x)) < \varepsilon whenever m, n > N .

See also

• Modes of convergence (annotated index)