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 |
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 ⓘ |
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject surface form:
Augustin-Louis Cauchy