Cauchy sequence
E239283
A Cauchy sequence is a sequence whose terms become arbitrarily close to each other as the sequence progresses, providing a fundamental criterion for convergence in metric and normed spaces.
All labels observed (2)
How this entity was disambiguated
This entity first appeared as the object of triple T2171643 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Cauchy sequence Context triple: [Augustin-Louis Cauchy, knownFor, Cauchy sequence]
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A.
Weierstrass M-test
The Weierstrass M-test is a criterion in real and complex analysis that provides a sufficient condition for the uniform convergence of a series of functions by comparing it to a convergent series of bounding constants.
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B.
Cauchy-à-la-Tour
Cauchy-à-la-Tour is a small commune in the Pas-de-Calais department of northern France.
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C.
Ulam sequence
The Ulam sequence is an integer sequence starting with 1 and 2 in which each subsequent term is the smallest integer that can be written uniquely as the sum of two distinct earlier terms.
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D.
Archimedean property of real numbers
The Archimedean property of real numbers is a fundamental axiom stating that for any real number, there exists a natural number larger than it, ensuring there are no infinitely large or infinitesimally small elements in the real number system.
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E.
Banach fixed-point theorem
The Banach fixed-point theorem is a fundamental result in metric space theory that guarantees the existence and uniqueness of a fixed point for any contraction mapping and provides a method for finding it via iterative approximation.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Cauchy sequence Target entity description: A Cauchy sequence is a sequence whose terms become arbitrarily close to each other as the sequence progresses, providing a fundamental criterion for convergence in metric and normed spaces.
-
A.
Weierstrass M-test
The Weierstrass M-test is a criterion in real and complex analysis that provides a sufficient condition for the uniform convergence of a series of functions by comparing it to a convergent series of bounding constants.
-
B.
Cauchy-à-la-Tour
Cauchy-à-la-Tour is a small commune in the Pas-de-Calais department of northern France.
-
C.
Ulam sequence
The Ulam sequence is an integer sequence starting with 1 and 2 in which each subsequent term is the smallest integer that can be written uniquely as the sum of two distinct earlier terms.
-
D.
Archimedean property of real numbers
The Archimedean property of real numbers is a fundamental axiom stating that for any real number, there exists a natural number larger than it, ensuring there are no infinitely large or infinitesimally small elements in the real number system.
-
E.
Banach fixed-point theorem
The Banach fixed-point theorem is a fundamental result in metric space theory that guarantees the existence and uniqueness of a fixed point for any contraction mapping and provides a method for finding it via iterative approximation.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical concept
ⓘ
sequence ⓘ |
| characterizes | convergence in complete metric spaces ⓘ |
| codomain | metric space ⓘ |
| contrastsWith | pointwise definition of limit ⓘ |
| definedIn |
metric space
ⓘ
normed vector space ⓘ topological vector space ⓘ |
| domain | elements of a metric space ⓘ |
| exampleOf | sequence defined by internal closeness rather than external limit point ⓘ |
| field |
functional analysis
ⓘ
mathematical analysis ⓘ topology ⓘ |
| generalizationOf |
Cauchy sequence of complex numbers
ⓘ
Cauchy sequence of real numbers ⓘ |
| hasCategory |
foundational concept in metric space theory
ⓘ
foundational concept in real analysis ⓘ |
| hasDefinition |
A sequence (x_n) in a metric space (X,d) is Cauchy if for every ε > 0 there exists N such that for all m,n ≥ N, d(x_m,x_n) < ε.
ⓘ
linked to:
Cauchy sequence
|
| hasEquivalentFormulation |
for every ε > 0 there exists N such that for all k ≥ 0, d(x_{N+k},x_N) < ε
ⓘ
for every ε > 0 there exists N such that for all n ≥ N, d(x_n,x_N) < ε ⓘ |
| hasHistoricalRole | formalization of convergence in analysis ⓘ |
| hasLogicalForm | ∀ε>0 ∃N ∀m,n≥N : d(x_m,x_n)<ε ⓘ |
| hasProperty |
every convergent sequence in a metric space is Cauchy
ⓘ
in a complete metric space every Cauchy sequence converges ⓘ in an incomplete metric space a Cauchy sequence may fail to converge ⓘ in ℚ with the usual metric some Cauchy sequences do not converge in ℚ ⓘ in ℝ with the usual metric every Cauchy sequence converges ⓘ |
| hasType | sequence indexed by natural numbers ⓘ |
| implies | bounded sequence in a metric space ⓘ |
| namedAfter | Augustin-Louis Cauchy ⓘ |
| relatedTo |
Cauchy completion
ⓘ
Cauchy criterion for series ⓘ Cauchy filter ⓘ Cauchy net ⓘ complete metric space ⓘ convergent sequence ⓘ |
| requiresStructure |
distance function
ⓘ
metric ⓘ |
| usedAs | criterion for convergence ⓘ |
| usedIn |
Banach space theory
ⓘ
Hilbert space theory ⓘ analysis of series and infinite products ⓘ construction of real numbers from rationals ⓘ definition of completeness of a metric space ⓘ proofs of convergence theorems ⓘ |
| usedToDefine | completion of a metric space via equivalence classes of Cauchy sequences ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Cauchy sequence Description of subject: A Cauchy sequence is a sequence whose terms become arbitrarily close to each other as the sequence progresses, providing a fundamental criterion for convergence in metric and normed spaces.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject linked to:
Augustin-Louis
Cauchy sequence
→
hasDefinition
→
A sequence (x_n) in a metric space (X,d) is Cauchy if for every ε > 0 there exists N such that for all m,n ≥ N, d(x_m,x_n) < ε.
ⓘ
linked to: Cauchy sequence