Cauchy sequence

E239283

mathematical concept sequence

A Cauchy sequence is a sequence whose terms become arbitrarily close to each other as the sequence progresses, providing a fundamental criterion for convergence in metric and normed spaces.

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All labels observed (2)

Label	Occurrences
Cauchy sequence canonical	2
A sequence (x_n) in a metric space (X,d) is Cauchy if for every ε > 0 there exists N such that for all m,n ≥ N, d(x_m,x_n) < ε.	1

Statements (46)

Predicate	Object
instanceOf	mathematical concept ⓘ sequence ⓘ
characterizes	convergence in complete metric spaces ⓘ
codomain	metric space ⓘ
contrastsWith	pointwise definition of limit ⓘ
definedIn	metric space ⓘ normed vector space ⓘ topological vector space ⓘ
domain	elements of a metric space ⓘ
exampleOf	sequence defined by internal closeness rather than external limit point ⓘ
field	functional analysis ⓘ mathematical analysis ⓘ topology ⓘ
generalizationOf	Cauchy sequence of complex numbers ⓘ Cauchy sequence of real numbers ⓘ
hasCategory	foundational concept in metric space theory ⓘ foundational concept in real analysis ⓘ
hasDefinition	Cauchy sequence self-linksurface differs ⓘ surface form: A sequence (x_n) in a metric space (X,d) is Cauchy if for every ε > 0 there exists N such that for all m,n ≥ N, d(x_m,x_n) < ε.
hasEquivalentFormulation	for every ε > 0 there exists N such that for all k ≥ 0, d(x_{N+k},x_N) < ε ⓘ for every ε > 0 there exists N such that for all n ≥ N, d(x_n,x_N) < ε ⓘ
hasHistoricalRole	formalization of convergence in analysis ⓘ
hasLogicalForm	∀ε>0 ∃N ∀m,n≥N : d(x_m,x_n)<ε ⓘ
hasProperty	every convergent sequence in a metric space is Cauchy ⓘ in a complete metric space every Cauchy sequence converges ⓘ in an incomplete metric space a Cauchy sequence may fail to converge ⓘ in ℚ with the usual metric some Cauchy sequences do not converge in ℚ ⓘ in ℝ with the usual metric every Cauchy sequence converges ⓘ
hasType	sequence indexed by natural numbers ⓘ
implies	bounded sequence in a metric space ⓘ
namedAfter	Augustin-Louis Cauchy ⓘ
relatedTo	Cauchy completion ⓘ Cauchy criterion for series ⓘ Cauchy filter ⓘ Cauchy net ⓘ complete metric space ⓘ convergent sequence ⓘ
requiresStructure	distance function ⓘ metric ⓘ
usedAs	criterion for convergence ⓘ
usedIn	Banach space theory ⓘ Hilbert space theory ⓘ analysis of series and infinite products ⓘ construction of real numbers from rationals ⓘ definition of completeness of a metric space ⓘ proofs of convergence theorems ⓘ
usedToDefine	completion of a metric space via equivalence classes of Cauchy sequences ⓘ

Referenced by (3)

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

Augustin-Louis Cauchy → knownFor → Cauchy sequence ⓘ

Augustin-Louis → notableFor → Cauchy sequence ⓘ

subject surface form: Augustin-Louis Cauchy

Cauchy sequence → hasDefinition → Cauchy sequence self-linksurface differs ⓘ

this entity surface form: A sequence (x_n) in a metric space (X,d) is Cauchy if for every ε > 0 there exists N such that for all m,n ≥ N, d(x_m,x_n) < ε.