See Cauchy sequence on Wiktionary
{
"etymology_text": "Named after French mathematician Augustin-Louis Cauchy (1789–1857), who made pioneering contributions to analysis.",
"forms": [
{
"form": "Cauchy sequences",
"tags": [
"plural"
]
}
],
"head_templates": [
{
"args": {},
"expansion": "Cauchy sequence (plural Cauchy sequences)",
"name": "en-noun"
}
],
"lang": "English",
"lang_code": "en",
"pos": "noun",
"senses": [
{
"categories": [
{
"kind": "other",
"name": "English entries with incorrect language header",
"parents": [
"Entries with incorrect language header",
"Entry maintenance"
],
"source": "w"
},
{
"kind": "other",
"name": "Entries with translation boxes",
"parents": [],
"source": "w"
},
{
"kind": "other",
"name": "Pages with 1 entry",
"parents": [],
"source": "w"
},
{
"kind": "other",
"name": "Pages with entries",
"parents": [],
"source": "w"
},
{
"kind": "other",
"name": "Terms with Danish translations",
"parents": [],
"source": "w"
},
{
"kind": "other",
"name": "Terms with Finnish translations",
"parents": [],
"source": "w"
},
{
"kind": "other",
"name": "Terms with German translations",
"parents": [],
"source": "w"
},
{
"kind": "other",
"name": "Terms with Italian translations",
"parents": [],
"source": "w"
},
{
"kind": "other",
"name": "Terms with Serbo-Croatian translations",
"parents": [],
"source": "w"
},
{
"kind": "other",
"name": "Terms with Swedish translations",
"parents": [],
"source": "w"
},
{
"kind": "topical",
"langcode": "en",
"name": "Functional analysis",
"orig": "en:Functional analysis",
"parents": [
"Mathematical analysis",
"Mathematics",
"Formal sciences",
"Sciences",
"All topics",
"Fundamental"
],
"source": "w"
},
{
"kind": "topical",
"langcode": "en",
"name": "Mathematical analysis",
"orig": "en:Mathematical analysis",
"parents": [
"Mathematics",
"Formal sciences",
"Sciences",
"All topics",
"Fundamental"
],
"source": "w"
}
],
"derived": [
{
"tags": [
"adjective"
],
"word": "Cauchy"
}
],
"examples": [
{
"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": "quote"
},
{
"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": "quote"
},
{
"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": "quote"
}
],
"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."
],
"id": "en-Cauchy_sequence-en-noun-cSaZasNj",
"links": [
[
"mathematical analysis",
"mathematical analysis"
],
[
"metric space",
"metric space"
]
],
"raw_glosses": [
"(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"
},
{
"word": "Cauchy net"
},
{
"word": "Cauchy space"
}
],
"topics": [
"mathematical-analysis",
"mathematics",
"sciences"
],
"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": [
"feminine"
],
"word": "Cauchyfolge"
},
{
"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"
],
"word": "Cauchyjev niz"
},
{
"code": "sv",
"lang": "Swedish",
"sense": "sequence in a normed vector space",
"word": "Cauchyföljd"
}
],
"wikipedia": [
"Augustin-Louis Cauchy",
"Cauchy sequence"
]
}
],
"word": "Cauchy sequence"
}
{
"derived": [
{
"tags": [
"adjective"
],
"word": "Cauchy"
}
],
"etymology_text": "Named after French mathematician Augustin-Louis Cauchy (1789–1857), who made pioneering contributions to analysis.",
"forms": [
{
"form": "Cauchy sequences",
"tags": [
"plural"
]
}
],
"head_templates": [
{
"args": {},
"expansion": "Cauchy sequence (plural Cauchy sequences)",
"name": "en-noun"
}
],
"lang": "English",
"lang_code": "en",
"pos": "noun",
"related": [
{
"word": "Cauchy convergence"
},
{
"word": "Cauchy filter"
},
{
"word": "Cauchy net"
},
{
"word": "Cauchy space"
}
],
"senses": [
{
"categories": [
"English countable nouns",
"English entries with incorrect language header",
"English eponyms",
"English lemmas",
"English multiword terms",
"English nouns",
"English terms with quotations",
"Entries with translation boxes",
"Pages with 1 entry",
"Pages with entries",
"Quotation templates to be cleaned",
"Terms with Danish translations",
"Terms with Finnish translations",
"Terms with German translations",
"Terms with Italian translations",
"Terms with Serbo-Croatian translations",
"Terms with Swedish translations",
"en:Functional analysis",
"en:Mathematical analysis"
],
"examples": [
{
"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": "quote"
},
{
"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": "quote"
},
{
"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": "quote"
}
],
"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."
],
"links": [
[
"mathematical analysis",
"mathematical analysis"
],
[
"metric space",
"metric space"
]
],
"raw_glosses": [
"(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."
],
"topics": [
"mathematical-analysis",
"mathematics",
"sciences"
],
"wikipedia": [
"Augustin-Louis Cauchy",
"Cauchy sequence"
]
}
],
"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": [
"feminine"
],
"word": "Cauchyfolge"
},
{
"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"
],
"word": "Cauchyjev niz"
},
{
"code": "sv",
"lang": "Swedish",
"sense": "sequence in a normed vector space",
"word": "Cauchyföljd"
}
],
"word": "Cauchy sequence"
}
Download raw JSONL data for Cauchy sequence meaning in All languages combined (3.7kB)
This page is a part of the kaikki.org machine-readable All languages combined dictionary. This dictionary is based on structured data extracted on 2024-11-06 from the enwiktionary dump dated 2024-10-02 using wiktextract (fbeafe8 and 7f03c9b). 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.
If you use this data in academic research, please cite Tatu Ylonen: Wiktextract: Wiktionary as Machine-Readable Structured Data, Proceedings of the 13th Conference on Language Resources and Evaluation (LREC), pp. 1317-1325, Marseille, 20-25 June 2022. Linking to the relevant page(s) under https://kaikki.org would also be greatly appreciated.