See Cauchy sequence in All languages combined, or Wiktionary
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{
"etymology_text": "Named after French mathematician Augustin-Louis Cauchy (1789–1857), who made pioneering contributions to analysis.",
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"word": "Cauchy"
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{
"ref": "1955, [Van Nostrand], John L. Kelley, General Topology, Springer, published 1975, page 174",
"text": "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.",
"type": "quotation"
},
{
"ref": "2000, George Bachman, Lawrence Narici, Functional Analysis, page 52",
"text": "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.",
"type": "quotation"
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{
"ref": "2012, David Applebaum, Limits, Limits Everywhere: The Tools of Mathematical Analysis, Oxford University Press, page 153",
"text": "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.",
"type": "quotation"
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"glosses": [
"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."
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"(mathematical analysis) 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."
],
"related": [
{
"word": "Cauchy convergence"
},
{
"word": "Cauchy filter"
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{
"word": "Cauchy net"
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{
"word": "Cauchy space"
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{
"code": "da",
"lang": "Danish",
"sense": "sequence in a normed vector space",
"tags": [
"common-gender"
],
"word": "Cauchyfølge"
},
{
"code": "fi",
"lang": "Finnish",
"sense": "sequence in a normed vector space",
"word": "Cauchyn jono"
},
{
"code": "de",
"lang": "German",
"sense": "sequence in a normed vector space",
"tags": [
"feminine"
],
"word": "Cauchy-Folge"
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{
"code": "de",
"lang": "German",
"sense": "sequence in a normed vector space",
"tags": [
"feminine"
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"word": "Cauchyfolge"
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{
"code": "it",
"lang": "Italian",
"sense": "sequence in a normed vector space",
"tags": [
"feminine"
],
"word": "successione di Cauchy"
},
{
"code": "sh",
"lang": "Serbo-Croatian",
"sense": "sequence in a normed vector space",
"tags": [
"masculine"
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"word": "Cauchyjev niz"
},
{
"code": "sv",
"lang": "Swedish",
"sense": "sequence in a normed vector space",
"word": "Cauchyföljd"
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],
"word": "Cauchy sequence"
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"word": "Cauchy"
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"etymology_text": "Named after French mathematician Augustin-Louis Cauchy (1789–1857), who made pioneering contributions to analysis.",
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"word": "Cauchy convergence"
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"ref": "1955, [Van Nostrand], John L. Kelley, General Topology, Springer, published 1975, page 174",
"text": "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.",
"type": "quotation"
},
{
"ref": "2000, George Bachman, Lawrence Narici, Functional Analysis, page 52",
"text": "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.",
"type": "quotation"
},
{
"ref": "2012, David Applebaum, Limits, Limits Everywhere: The Tools of Mathematical Analysis, Oxford University Press, page 153",
"text": "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.",
"type": "quotation"
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],
"glosses": [
"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."
],
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"(mathematical analysis) 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."
],
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"translations": [
{
"code": "da",
"lang": "Danish",
"sense": "sequence in a normed vector space",
"tags": [
"common-gender"
],
"word": "Cauchyfølge"
},
{
"code": "fi",
"lang": "Finnish",
"sense": "sequence in a normed vector space",
"word": "Cauchyn jono"
},
{
"code": "de",
"lang": "German",
"sense": "sequence in a normed vector space",
"tags": [
"feminine"
],
"word": "Cauchy-Folge"
},
{
"code": "de",
"lang": "German",
"sense": "sequence in a normed vector space",
"tags": [
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"word": "Cauchyfolge"
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{
"code": "it",
"lang": "Italian",
"sense": "sequence in a normed vector space",
"tags": [
"feminine"
],
"word": "successione di Cauchy"
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{
"code": "sh",
"lang": "Serbo-Croatian",
"sense": "sequence in a normed vector space",
"tags": [
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"word": "Cauchyjev niz"
},
{
"code": "sv",
"lang": "Swedish",
"sense": "sequence in a normed vector space",
"word": "Cauchyföljd"
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"word": "Cauchy sequence"
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This page is a part of the kaikki.org machine-readable English dictionary. This dictionary is based on structured data extracted on 2024-05-05 from the enwiktionary dump dated 2024-05-02 using wiktextract (f4fd8c9 and c9440ce). The data shown on this site has been post-processed and various details (e.g., extra categories) removed, some information disambiguated, and additional data merged from other sources. See the raw data download page for the unprocessed wiktextract data.
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