Uniformly Cauchy sequence

In mathematics, a sequence of functions ${\displaystyle \{f_{n}\}}$ from a set S to a metric space M is said to be uniformly Cauchy if:

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

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

${\displaystyle d_{u}(f,g):=\sup _{x\in S}d(f(x),g(x)).}$

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

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

See also

• Modes of convergence (annotated index)