Cauchy sequence

English Etymology Named after French mathematician Augustin-Louis Cauchy (1789–1857), who made pioneering contributions to analysis. Noun en analysis Any sequence in a metric space with metric d such that for every there exists a natural number N such that for all , . en 1955 [Van Nostrand]; John L. Kelley General Topology https://books.google.com.au/books?id=-goleb9Ov3oC&pg=PA174&dq=%22Cauchy+sequence%22%7C%22Cauchy+sequences%22&hl=en&sa=X&ved=0ahUKEwi00e_RjrPWAhVryFQKHbItDgkQ6AEIbDAL#v=onepage&q=%22Cauchy%20sequence%22%7C%22Cauchy%20sequences%22&f=false 174 Springer 1975 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. en 2000 George Bachman; Lawrence Narici Functional Analysis http://books.google.com/books?id=wCHtLumoGY4C&pg=PA52&dq=%22Cauchy+sequence%22&hl=en&ei=VZUVTsHrB-jWmAWYq9Qn&sa=X&oi=book_result&ct=result&resnum=1&ved=0CCgQ6AEwAA#v=onepage&q=%22Cauchy%20sequence%22&f=false 52 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. en 2012 David Applebaum Limits, Limits Everywhere: The Tools of Mathematical Analysis https://books.google.com.au/books?id=n1UoRGDnll0C&pg=PA153&dq=%22Cauchy+sequence%22%7C%22Cauchy+sequences%22&hl=en&sa=X&ved=0ahUKEwi00e_RjrPWAhVryFQKHbItDgkQ6AEIWDAI#v=onepage&q=%22Cauchy%20sequence%22%7C%22Cauchy%20sequences%22&f=false 153 w:Oxford University Press 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. Usage notes The formal definition of Cauchy sequence represents a formulation of the notion of en convergence without reference to a supposed element to which the sequence converges. In fact, the spaces of most interest to analysis are those, called en complete, in which such limits do exist within the space. Derived terms en Cauchy adjective Related terms en Cauchy convergence en Cauchy filter en Cauchy net en Cauchy space Translations sequence in a normed vector space Bulgarian: bg редица на Коши f Danish: da Cauchyfølge c Finnish: fi Cauchyn jono French: fr suite de Cauchy f German: de Cauchy-Folge f, de Cauchyfolge f Italian: it successione di Cauchy f Polish: pl ciąg Cauchy'ego m Serbo-Croatian: sh Cauchyjev niz m Swedish: sv Cauchyföljd en eponyms en Functional analysis