Bolzano–Weierstrass theorem

E825428

mathematical theorem

The Bolzano–Weierstrass theorem is a fundamental result in real analysis stating that every bounded infinite sequence in ℝⁿ has a convergent subsequence.

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All labels observed (3)

Label	Occurrences
Bolzano–Weierstrass theorem canonical	1
Bolzano–Weierstrass theorem (early formulation)	1
Bolzano–Weierstrass theorem for functions	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.
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 ⓘ linked to: Euclidean space
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 ⓘ Karl Weierstrass ⓘ
relatedTo	Cauchy sequence ⓘ Heine–Borel theorem ⓘ linked to: Borel–Lebesgue theorem 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 ⓘ linked to: Borel–Lebesgue theorem
yearIntroducedApprox	19th century ⓘ

How these facts were elicited

Referenced by (3)

Full triples — surface form annotated when it differs from this entity's canonical label.

Cauchy convergence criterion → relatedTo → Bolzano–Weierstrass theorem ⓘ

Bernard Bolzano → notableIdea → Bolzano–Weierstrass theorem (early formulation) ⓘ

linked to: Bolzano–Weierstrass theorem

Arzelà–Ascoli theorem → generalizationOf → Bolzano–Weierstrass theorem for functions ⓘ