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 {fn} from S to M is pointwise Cauchy if, for each x &isin; S, the sequence {fn(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 fn : S &rarr; M tends uniformly to a unique continuous function f : S &rarr; 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 $$m, n > N$$.