Bolzano–Weierstrass theorem
E825428
The Bolzano–Weierstrass theorem is a fundamental result in real analysis stating that every bounded infinite sequence in ℝⁿ has a convergent subsequence.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Bolzano–Weierstrass theorem canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T9843665 — 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: Bolzano–Weierstrass theorem Context triple: [Cauchy convergence criterion, relatedTo, Bolzano–Weierstrass theorem]
-
A.
Banach–Alaoglu theorem
The Banach–Alaoglu theorem is a fundamental result in functional analysis stating that the closed unit ball in the dual of a normed space is compact in the weak-* topology.
-
B.
Stone–Weierstrass theorem
The Stone–Weierstrass theorem is a fundamental result in functional analysis that characterizes when a subalgebra of continuous functions on a compact space is dense, thereby generalizing classical polynomial approximation results.
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C.
Krein–Milman theorem
The Krein–Milman theorem is a fundamental result in functional analysis and convex geometry stating that a compact convex set in a locally convex topological vector space is the closed convex hull of its extreme points.
-
D.
Banach–Saks theorem
The Banach–Saks theorem is a result in functional analysis stating that every bounded sequence in a reflexive Banach space has a subsequence whose Cesàro means converge in norm.
-
E.
Banach–Steinhaus theorem
The Banach–Steinhaus theorem is a fundamental result in functional analysis that characterizes when a family of continuous linear operators is uniformly bounded, with major implications for the behavior of sequences of operators on Banach spaces.
- 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: Bolzano–Weierstrass theorem Target entity description: The Bolzano–Weierstrass theorem is a fundamental result in real analysis stating that every bounded infinite sequence in ℝⁿ has a convergent subsequence.
-
A.
Banach–Alaoglu theorem
The Banach–Alaoglu theorem is a fundamental result in functional analysis stating that the closed unit ball in the dual of a normed space is compact in the weak-* topology.
-
B.
Stone–Weierstrass theorem
The Stone–Weierstrass theorem is a fundamental result in functional analysis that characterizes when a subalgebra of continuous functions on a compact space is dense, thereby generalizing classical polynomial approximation results.
-
C.
Krein–Milman theorem
The Krein–Milman theorem is a fundamental result in functional analysis and convex geometry stating that a compact convex set in a locally convex topological vector space is the closed convex hull of its extreme points.
-
D.
Banach–Saks theorem
The Banach–Saks theorem is a result in functional analysis stating that every bounded sequence in a reflexive Banach space has a subsequence whose Cesàro means converge in norm.
-
E.
Banach–Steinhaus theorem
The Banach–Steinhaus theorem is a fundamental result in functional analysis that characterizes when a family of continuous linear operators is uniformly bounded, with major implications for the behavior of sequences of operators on Banach spaces.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf | mathematical theorem ⓘ |
| appliesTo |
bounded sequences in ℝ
ⓘ
bounded sequences in ℝⁿ ⓘ |
| assumes | standard Euclidean topology on ℝⁿ ⓘ |
| category |
compactness theorems
ⓘ
limit theorems ⓘ |
| conclusion | existence of a convergent subsequence ⓘ |
| coreConcept |
bounded sequence
ⓘ
convergent subsequence ⓘ limit point ⓘ |
| dependsOnProperty |
closed and bounded subsets of ℝⁿ are compact
ⓘ
completeness of Euclidean space ⓘ |
| doesNotRequire | monotonicity of the sequence ⓘ |
| equivalentFormulation |
Every bounded infinite subset of ℝⁿ has at least one accumulation point.
ⓘ
Every infinite subset of a compact set has a limit point in that set. ⓘ |
| equivalentTo | sequential compactness of closed and bounded subsets of ℝⁿ ⓘ |
| field |
real analysis
ⓘ
topology ⓘ |
| generalizationOf | fact that closed intervals in ℝ are compact ⓘ |
| hasGeneralization |
compactness in topological spaces
ⓘ
sequential compactness in metric spaces ⓘ |
| holdsFor | closed and bounded subsets of ℝⁿ ⓘ |
| holdsIn | Euclidean space ℝⁿ NERFINISHED ⓘ |
| implies | compact subsets of ℝⁿ are sequentially compact ⓘ |
| isFundamentalResultIn |
metric space theory courses
ⓘ
undergraduate real analysis ⓘ |
| isToolFor |
establishing existence of limits
ⓘ
extracting convergent subsequences from bounded sequences ⓘ |
| namedAfter |
Bernard Bolzano
NERFINISHED
ⓘ
Karl Weierstrass NERFINISHED ⓘ |
| relatedTo |
Cauchy sequence
ⓘ
Heine–Borel theorem NERFINISHED ⓘ compactness ⓘ completeness of ℝ ⓘ sequential compactness ⓘ |
| requires |
sequence is bounded
ⓘ
sequence is infinite ⓘ |
| statement |
Every bounded infinite sequence in ℝⁿ has a convergent subsequence.
ⓘ
Every bounded sequence in ℝ has a convergent subsequence. ⓘ |
| typicalProofMethod |
diagonal argument in ℝⁿ
ⓘ
nested intervals argument ⓘ use of Heine–Borel theorem ⓘ |
| usedIn |
analysis of series and sequences
ⓘ
functional analysis ⓘ metric space theory ⓘ |
| usedInProofOf | Heine–Borel theorem NERFINISHED ⓘ |
| yearIntroducedApprox | 19th century ⓘ |
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: Bolzano–Weierstrass theorem Description of subject: The Bolzano–Weierstrass theorem is a fundamental result in real analysis stating that every bounded infinite sequence in ℝⁿ has a convergent subsequence.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.