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- Cauchy_sequence abstract "In mathematics, a Cauchy sequence (French pronunciation: [koʃi]; English pronunciation: /ˈkoʊʃiː/ KOH-shee), named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all but a finite number of elements of the sequence are less than that given distance from each other.The utility of Cauchy sequences lies in the fact that in a complete metric space (one where all such sequences are known to converge to a limit), the criterion for convergence depends only on the terms of the sequence itself (as opposed to the definition of convergence, which uses the limit value as well as the terms). This is often exploited in algorithms, both theoretical and applied, where an iterative process can be shown relatively easily to produce a Cauchy sequence, consisting of the iterates, thus fulfilling a logical condition, such as termination.The notions above are not as unfamiliar as they might at first appear. The customary acceptance of the fact that any real number x has a decimal expansion is an implicit acknowledgment that a particular Cauchy sequence of rational numbers (whose terms are the successive truncations of the decimal expansion of x) has the real limit x. In some cases it may be difficult to describe x independently of such a limiting process involving rational numbers.Generalizations of Cauchy sequences in more abstract uniform spaces exist in the form of Cauchy filters and Cauchy nets.".
- Cauchy_sequence thumbnail Cauchy_sequence_illustration.svg?width=300.
- Cauchy_sequence wikiPageExternalLink books?id=Z7I-AAAAIAAJ&dq=intitle:Foundations+intitle:of+intitle:constructive+intitle:analysis&lr=&as_brr=0&pgis=1.
- Cauchy_sequence wikiPageExternalLink cal.htm.
- Cauchy_sequence wikiPageID "6085".
- Cauchy_sequence wikiPageRevisionID "603073155".
- Cauchy_sequence align "right".
- Cauchy_sequence caption "A sequence that is not Cauchy. The elements of the sequence fail to get arbitrarily close to each other as the sequence progresses.".
- Cauchy_sequence caption "The plot of a Cauchy sequence shown in blue, as versus If the space containing the sequence is complete, the "ultimate destination" of this sequence exists.".
- Cauchy_sequence direction "vertical".
- Cauchy_sequence hasPhotoCollection Cauchy_sequence.
- Cauchy_sequence id "p/f042240".
- Cauchy_sequence image "Cauchy sequence illustration.svg".
- Cauchy_sequence image "Cauchy sequence illustration2.svg".
- Cauchy_sequence title "Fundamental sequence".
- Cauchy_sequence width "250".
- Cauchy_sequence subject Category:Abstract_algebra.
- Cauchy_sequence subject Category:Convergence_(mathematics).
- Cauchy_sequence subject Category:Metric_geometry.
- Cauchy_sequence subject Category:Sequences_and_series.
- Cauchy_sequence subject Category:Topology.
- Cauchy_sequence type Abstraction100002137.
- Cauchy_sequence type Arrangement107938773.
- Cauchy_sequence type Group100031264.
- Cauchy_sequence type Ordering108456993.
- Cauchy_sequence type Sequence108459252.
- Cauchy_sequence type SequencesAndSeries.
- Cauchy_sequence type Series108457976.
- Cauchy_sequence comment "In mathematics, a Cauchy sequence (French pronunciation: [koʃi]; English pronunciation: /ˈkoʊʃiː/ KOH-shee), named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses.".
- Cauchy_sequence label "Cauchy sequence".
- Cauchy_sequence label "Cauchy-Folge".
- Cauchy_sequence label "Cauchyrij".
- Cauchy_sequence label "Ciąg Cauchy'ego".
- Cauchy_sequence label "Successione di Cauchy".
- Cauchy_sequence label "Sucesión de Cauchy".
- Cauchy_sequence label "Sucessão de Cauchy".
- Cauchy_sequence label "Suite de Cauchy".
- Cauchy_sequence label "Фундаментальная последовательность".
- Cauchy_sequence label "متتالية كوشي".
- Cauchy_sequence label "コーシー列".
- Cauchy_sequence label "柯西序列".
- Cauchy_sequence sameAs Cauchyovská_posloupnost.
- Cauchy_sequence sameAs Cauchy-Folge.
- Cauchy_sequence sameAs Ακολουθίες_του_Cauchy.
- Cauchy_sequence sameAs Sucesión_de_Cauchy.
- Cauchy_sequence sameAs Suite_de_Cauchy.
- Cauchy_sequence sameAs Successione_di_Cauchy.
- Cauchy_sequence sameAs コーシー列.
- Cauchy_sequence sameAs 코시_열.
- Cauchy_sequence sameAs Cauchyrij.
- Cauchy_sequence sameAs Ciąg_Cauchy'ego.
- Cauchy_sequence sameAs Sucessão_de_Cauchy.
- Cauchy_sequence sameAs m.01tpf.
- Cauchy_sequence sameAs Q217847.
- Cauchy_sequence sameAs Q217847.
- Cauchy_sequence sameAs Cauchy_sequence.
- Cauchy_sequence wasDerivedFrom Cauchy_sequence?oldid=603073155.
- Cauchy_sequence depiction Cauchy_sequence_illustration.svg.
- Cauchy_sequence isPrimaryTopicOf Cauchy_sequence.
