Cauchy sequence

In 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.

Formally, a sequence x₁, x₂, x₃, ... in a metric space (M, d) is called a Cauchy sequence (or Cauchy for short) if for every positive real number r, there is an integer N such that for all integers m and n greater than N the distance d(x_m, x_n) 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 does not need to be the case.

A metric space in which every Cauchy sequence has a limit 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 f : M -> N is a uniformly continuous map between the metric spaces M and N and (x_n) is a Cauchy sequence in M , then (f(x_n)) is a Cauchy sequence in N. If (x_n) and (y_n) are two Cauchy sequences in the rational, real or complex numbers, then the sum (x_n + y_n) and the product (x_ny_n) are also Cauchy sequences.

Generalisation

A net (x_α) in a uniform space X is a Cauchy net if for every entourage V there exists an α₀ such that for all α, β > α₀ we have (x_α, x_β) in V.

See also

• Dedekind cut