Uniformly Cauchy sequence
In mathematics, a sequence of functions [math]\displaystyle{ \{f_{n}\} }[/math] from a set S to a metric space M is said to be uniformly Cauchy if:
- For all [math]\displaystyle{ \varepsilon \gt 0 }[/math], there exists [math]\displaystyle{ N\gt 0 }[/math] such that for all [math]\displaystyle{ x\in S }[/math]: [math]\displaystyle{ d(f_{n}(x), f_{m}(x)) \lt \varepsilon }[/math] whenever [math]\displaystyle{ m, n \gt N }[/math].
Another way of saying this is that [math]\displaystyle{ d_u (f_{n}, f_{m}) \to 0 }[/math] as [math]\displaystyle{ m, n \to \infty }[/math], where the uniform distance [math]\displaystyle{ d_u }[/math] between two functions is defined by
- [math]\displaystyle{ d_{u} (f, g) := \sup_{x \in S} d (f(x), g(x)). }[/math]
Convergence criteria
A sequence of functions {fn} from S to M is pointwise Cauchy if, for each x ∈ 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 → M tends uniformly to a unique continuous function f : S → M.
Generalization to uniform spaces
A sequence of functions [math]\displaystyle{ \{f_{n}\} }[/math] from a set S to a uniform space U is said to be uniformly Cauchy if:
- For all [math]\displaystyle{ x\in S }[/math] and for any entourage [math]\displaystyle{ \varepsilon }[/math], there exists [math]\displaystyle{ N\gt 0 }[/math] such that [math]\displaystyle{ d(f_{n}(x), f_{m}(x)) \lt \varepsilon }[/math] whenever [math]\displaystyle{ m, n \gt N }[/math].
See also
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