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 |
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 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.