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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.
By construction, every real number x is represented by a Cauchy sequence of rational numbers. This representation is far from unique; every rational sequence that converges to x is a Cauchy sequence representing x. This reflects the observation that one can often use different sequences to approximate the same real number. [6]
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 Cauchy sequence is a Cauchy net that is a sequence. If B {\displaystyle B} is a subset of an additive group X {\displaystyle X} and N {\displaystyle N} is a set containing 0 , {\displaystyle 0,} then B {\displaystyle B} is said to be an N {\displaystyle N} -small set or small of order N {\displaystyle N} if B − B ⊆ N . {\displaystyle B-B ...
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.
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 ...
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.
Cauchy's convergence test can only be used in complete metric spaces (such as and ), which are spaces where all Cauchy sequences converge. This is because we need only show that its elements become arbitrarily close to each other after a finite progression in the sequence to prove the series converges.