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Home > Cauchy sequence

Formally, a Cauchy sequence is a sequence

in a metric space (M, d) such that for every positive real number r > 0, there is an integer N such that for all integers m and n greater than N the distance

is less than r. Roughly speaking, the terms of the sequence are getting closer and closer together in a way that suggests that the sequence ought to have a limit in M. Nonetheless, this may not be the case.

A metric space X in which every Cauchy sequence has a limit (in X) is called complete. The real numbers are complete, and the standard construction of the real numbers involves Cauchy sequences of rational numbers. The rational numbers themselves are not complete: a sequence of rational numbers can have the square root of two as its limit, for example. See Complete space for an example of a Cauchy sequence of rational numbers that does not have a rational limit.

Every convergent sequence is a Cauchy sequence, and every Cauchy sequence is bounded. If is a uniformly continuous map between the metric spaces M and N and (x_n) is a Cauchy sequence in M, then is a Cauchy sequence in N. If and are two Cauchy sequences in the rational, real or complex numbers, then the sum and the product are also Cauchy sequences.

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Topics: Cauchy Sequence Numbers Limit Rational Complete Real Sequences...

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Cauchy sequenceIn mathematical analysis, a Cauchy sequence is a sequence whose terms become arbitrarily close to each other as the sequence progresses. They are named after the French mathematician Augustin Louis Cauchy. They are of interest because, given certain condi Cauchy distributionThe Cauchy distribution is a probability distribution with probability density function : where x is the location parameter and s is the scale parameter''. The special case when x 0 and s 1 is called the standard Cauchy distribution with the probability d Cauchy-Riemann equationsPartial differential equations Complex analysis In complex analysis, the Cauchy-Riemann differential equations are two partial differential equations which provide a necessary and sufficient condition for a function to be holomorphic. Let f u + iv be a fu Cauchy-Schwarz inequalityIn mathematics, the Cauchy-Schwarz inequality also known as the Schwarz inequality or the Cauchy-Bunyakovski-Schwarz inequality is a useful inequality encountered in many different settings, such as linear algebra applied to vectors, in analysis applied t Cauchy integral theoremComplex analysis Theorems The Cauchy integral theorem in complex analysis is an important statement about path integrals for holomorphic functions in the complex plane. Essentially, it says that if two different paths connect the same two points, and a fu Cauchy's integral formulaCauchy's integral formula is a central statement in complex analysis. It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary of the disk. It can also be used to formulate integral formula Cauchy-Binet formula Cauchy-à-la-Tour Cauchy principal value Cauchy horizon Cauchy product Cauchy (crater) Cauchy space Cauchy-Euler equation