Cauchy sequence en · NOUN
Etymology
Named after French mathematician Augustin-Louis Cauchy (1789–1857), who made pioneering contributions to analysis.
Meanings
-
Any sequence x_n in a metric space with metric d such that for every ϵ>0 there exists a natural number N such that for all k,m>N, d(x_k,x_m)<ϵ.
Cantor first redefined Cauchy sequences using rational numbers only.[…]Cantor's idea was to define the real number line as the collection of all (rational) Cauchy sequences.
2012, David Applebaum, Limits, Limits Everywhere: The Tools of Mathematical Analysis, Oxford University Press, page 153:However, it is possible to derive topological results from statements about Cauchy sequences; for example, a subset A of the space of real numbers is closed if and only if each Cauchy sequence in A converges to some point of A.
1955, [Van Nostrand], John L. Kelley, General Topology, Springer, published 1975, page 174:In the case of the real line, every Cauchy sequence converges; that is, being a Cauchy sequence is sufficient to guarantee the existence of a limit. In the general case, however, this is not so. If a metric space does have the property that every Cauchy sequence converges, the space is called a complete metric space.
2000, George Bachman, Lawrence Narici, Functional Analysis, page 52:
Forms
| Spelling | Features | Labels | Source |
|---|---|---|---|
| Cauchy sequences | Number=Plur | lexicographic |
Deriveds
Relateds
Cauchy convergence · Cauchy net · Cauchy filter · Cauchy space
Translations (10)
de Cauchyfolge (sequence in a normed vector space) · bg редица на Коши (sequence in a normed vector space) · de Cauchy-Folge (sequence in a normed vector space) · fi Cauchyn jono (sequence in a normed vector space) · pl ciąg Cauchy'ego (sequence in a normed vector space) · sh Cauchyjev niz (sequence in a normed vector space) · da Cauchyfølge (sequence in a normed vector space) · sv Cauchyföljd (sequence in a normed vector space) · it successione di Cauchy (sequence in a normed vector space) · fr suite de Cauchy (sequence in a normed vector space)