Cauchy sequence

Definition

In mathematics, a Cauchy sequence is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all excluding a finite number of elements of the sequence are less than that given distance from each other. Cauchy sequences are named after Augustin-Louis Cauchy; they may occasionally be known as fundamental sequences.

Related concepts

1/2 + 1/4 + 1/8 + 1/16 + ⋯1/2 − 1/4 + 1/8 − 1/16 + ⋯1/4 + 1/16 + 1/64 + 1/256 + ⋯1 + 1 + 1 + 1 + ⋯1 + 2 + 3 + 4 + ⋯1 + 2 + 4 + 8 + ⋯1 − 1 + 2 − 6 + 24 − 120 + ⋯1 − 2 + 3 − 4 + ⋯1 − 2 + 4 − 8 + ⋯Absolute convergence Absolute difference Absolute value Adequality Algebraic geometry Algorithm Alternating series Anne Sjerp Troelstra Approach space Arithmetic progression Augustin-Louis Cauchy Baire category theorem Ball (mathematics)Banach fixed-point theorem Banach space Bing metrization theorem Bolzano–Weierstrass theorem Borel set Bounded function Bounded set Canonical form Cantor space Category (mathematics)Category of metric spaces Cauchy filter Cauchy net Cauchy space Chebyshev distance Closed interval Coarse structure Cofinal (mathematics)Complete metric space Complete sequence Completion (metric space)Complex number Conditional convergence Construction of the real numbers Contraction mapping Convergence (mathematics)Convergent series Convex metric space Cosmic space Countable choice Cube (algebra)Dedekind cut Delone set Dense subspace Diameter of a set Dilation (metric space)Dirichlet series Dirk van Dalen Discrete space Distance set Divergence of the sum of the reciprocals of the primes Divergent series Diversity (mathematics)Doubling space Element (mathematics)Encyclopedia of Mathematics Equicontinuity Equivalence of metrics Equivalence relation Errett Bishop Euclidean distance European Mathematical Society Factor group Factorial Fibonacci number Fibonacci sequence Figurate number Formal power series Fourier series Function (mathematics)Functional analysis General topology Generalised metric Generalized hypergeometric series Generating series Geometric progression Golden ratio Grandi's series Gromov product Gromov–Hausdorff convergence Group (mathematics)Harmonic progression (mathematics)Harmonic series (mathematics)Hausdorff distance Heine–Borel theorem Heptagonal number Hexagonal number Hyperbolic metric space Hypergeometric function Hypergeometric function of a matrix argument Hypernatural Hyperreal number Identity element Index of a subgroup Infinite arithmetic series Infinite series Infinitesimal Injective metric space Inner product space Integer Integer sequence Intrinsic metric Inverse limit Irrational number Isometry Iterative method Kuratowski convergence Kuratowski embedding Laakso space Laurent series Lauricella hypergeometric series Least upper bound axiom Lebesgue's number lemma Length metric space Limit (mathematics)Limit of a sequence Lipschitz continuity List of OEIS sequences List of mathematical series Local base Lucas number Lévy metric Lévy–Prokhorov metric Maclaurin series Manifold Mathematics Measure space Measure theory Methods of computing square roots Metric (mathematics)Metric derivative Metric map Metric outer measure Metric projection Metric space Metric space aimed at its subspace Metrizable space Metrizable topological vector space Metrization theorem Meyer set Michael Spivak Modes of convergence (annotated index)Modular hypergeometric series Monotonic function Morphism Motion (geometry)Nagata–Smirnov metrization theorem Natural number Natural numbers Neighbourhood (mathematics)Nicolas Bourbaki Normal subgroup Normed linear space Normed space Number theory Object (category theory)Open interval P-adic number Packing dimension Partial sum Pell number Pentagonal number Periodic sequence Polish space Polygonal number Porous set Positive and negative numbers Positively separated sets Power of 10 Power of three Power of two Power series Prime number Probabilistic metric space Product metric Proximity space Pseudometric space Puiseux series Quasi-isometry Quasisymmetric map Rational number Real number Riemann's differential equation Riemann zeta function Riemannian manifold Sequence Sequence (mathematics)Serge Lang Series (mathematics)Space (mathematics)Square number Square root of 2 Standard part function Stretch factor Taxicab geometry Taylor series Telescoping series Theta hypergeometric series Tight span Topological group Topological vector space Totally bounded space Transitive relation Translation-invariant metric Tree-graded space Triangle inequality Triangular array Triangular number Trigonometric series Ultrametric space Uniform continuity Uniform convergence Uniform isomorphism Uniform space Uniform spaces Uniformly continuous Uniformly disconnected space Urysohn's metrization theorem Urysohn universal space Wasserstein metric Well-ordering property

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