Cauchy convergence criterion
E239286
The Cauchy convergence criterion is a fundamental concept in mathematical analysis that characterizes convergence of sequences (and series) by requiring that their terms become arbitrarily close to each other beyond some index.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Cauchy convergence criterion canonical | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T2171647 — 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 convergence criterion Context triple: [Augustin-Louis Cauchy, knownFor, Cauchy convergence criterion]
-
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.
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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C.
epsilon–delta definition of limit
The epsilon–delta definition of limit is the rigorous formalization of the intuitive notion of a function approaching a value, forming the foundation of modern analysis and calculus.
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D.
Weierstrass approximation theorem
The Weierstrass approximation theorem is a fundamental result in real analysis stating that any continuous function on a closed interval can be uniformly approximated by polynomials.
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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 convergence criterion Target entity description: The Cauchy convergence criterion is a fundamental concept in mathematical analysis that characterizes convergence of sequences (and series) by requiring that their terms become arbitrarily close to each other beyond some index.
-
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.
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.
-
C.
epsilon–delta definition of limit
The epsilon–delta definition of limit is the rigorous formalization of the intuitive notion of a function approaching a value, forming the foundation of modern analysis and calculus.
-
D.
Weierstrass approximation theorem
The Weierstrass approximation theorem is a fundamental result in real analysis stating that any continuous function on a closed interval can be uniformly approximated by polynomials.
-
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 (45)
| Predicate | Object |
|---|---|
| instanceOf |
concept in mathematical analysis
ⓘ
criterion for convergence ⓘ mathematical criterion ⓘ |
| appliesTo |
complex sequences
ⓘ
metric spaces ⓘ normed vector spaces ⓘ real sequences ⓘ sequences ⓘ series ⓘ |
| assumes | underlying metric or norm to measure distance between terms ⓘ |
| basedOn | Cauchy sequence ⓘ |
| category | theorem in analysis ⓘ |
| characterizes |
convergence of sequences in complete metric spaces
ⓘ
convergence of series in complete metric spaces ⓘ |
| contrastsWith | pointwise definition of convergence via limit point ⓘ |
| ensures | stability of limits under completion of metric spaces ⓘ |
| equivalentTo | definition of completeness of a metric space ⓘ |
| failsIn | incomplete metric spaces ⓘ |
| field | mathematical analysis ⓘ |
| formalizedBy | epsilon–N definition ⓘ |
| generalizedTo |
topological vector spaces
ⓘ
uniform spaces ⓘ |
| historicalPeriod | 19th-century mathematics ⓘ |
| holdsIfAndOnlyIf | every Cauchy sequence converges in a complete metric space ⓘ |
| implies | every convergent sequence is a Cauchy sequence in any metric space ⓘ |
| importance | fundamental for rigorous foundations of calculus ⓘ |
| isPartOf | standard undergraduate analysis curriculum ⓘ |
| logicalForm | biconditional between convergence and Cauchy property in complete spaces ⓘ |
| namedAfter | Augustin-Louis Cauchy ⓘ |
| relatedTo |
Bolzano–Weierstrass theorem
ⓘ
Cauchy completeness ⓘ Cauchy sequence ⓘ completeness of the real numbers ⓘ |
| requires | terms of the sequence become arbitrarily close to each other beyond some index ⓘ |
| role | provides epsilon–N formulation of convergence ⓘ |
| statesThat |
a sequence converges if and only if it is Cauchy in a complete metric space
ⓘ
for every epsilon greater than zero there exists an N such that for all m,n greater than or equal to N the distance between x_m and x_n is less than epsilon ⓘ |
| teaches | internal characterization of convergence without reference to limit value ⓘ |
| usedIn |
complex analysis
ⓘ
construction of real numbers from rationals via Cauchy sequences ⓘ functional analysis ⓘ real analysis ⓘ topology of metric spaces ⓘ |
| usedToProve |
convergence of numerical series
ⓘ
existence of limits of 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 convergence criterion Description of subject: The Cauchy convergence criterion is a fundamental concept in mathematical analysis that characterizes convergence of sequences (and series) by requiring that their terms become arbitrarily close to each other beyond some index.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject linked to:
Augustin-Louis