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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.
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,
By selecting this open set to be contained in a coordinate chart, one can reduce the claim to the well-known fact that, in Euclidean geometry, the shortest curve between two points is a line. In particular, as seen by the Euclidean geometry of a coordinate chart around p , any curve from p to q must first pass though a certain "inner radius."
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 ...
The Cauchy distribution, named after Augustin-Louis Cauchy, is a continuous probability distribution.It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) function, or Breit–Wigner distribution.
It is also possible to replace Cauchy sequences in the definition of completeness by Cauchy nets or Cauchy filters. If every Cauchy net (or equivalently every Cauchy filter) has a limit in , then is called complete. One can furthermore construct a completion for an arbitrary uniform space similar to the completion of metric spaces.
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.