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In mathematics, a Cauchy sequence is a sequence whose elements become arbitrarily close to each other as the sequence progresses. [1] More precisely, given any small positive distance, all excluding a finite number of elements of the sequence are less than that given distance from each other.
Criteri de Cauchy; Usuari:Alex Gómez/Test de convergència de Cauchy; Usage on cs.wikipedia.org Cauchyovská posloupnost; Usage on de.wikipedia.org Cauchy-Folge; Cauchy-Kriterium; Usage on de.wikibooks.org Mathe für Nicht-Freaks: Cauchy-Folgen und das Cauchy-Kriterium; Benutzer:Dirk Hünniger/mnfana; Serlo: EN: Cauchy sequences; Usage on de ...
In mathematics, a contraction mapping, or contraction or contractor, on a metric space (M, d) is a function f from M to itself, with the property that there is some real number < such that for all x and y in M,
The following other wikis use this file: Usage on ar.wikipedia.org متتالية كوشي; Usage on ar.wikiversity.org نهاية متتالية
In the real numbers every Cauchy sequence converges to some limit. A Cauchy sequence is a sequence whose terms become arbitrarily close together as n gets very large. The notion of a Cauchy sequence is important in the study of sequences in metric spaces, and, in particular, in real analysis. One particularly important result in real analysis ...
(This limit exists because the real numbers are complete.) This is only a pseudometric, not yet a metric, since two different Cauchy sequences may have the distance 0. But "having distance 0" is an equivalence relation on the set of all Cauchy sequences, and the set of equivalence classes is a metric space, the completion of M.
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.
Here the nth term in the sequence is the nth decimal approximation for pi. Though this is a Cauchy sequence of rational numbers, it does not converge to any rational number. (In this real number line, this sequence converges to pi.) Cauchy completeness is related to the construction of the real numbers using Cauchy sequences.