In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a Cauchy sequence , named after Augustin-Louis Cauchy, is a sequence

Sequence

In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...

 whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all but a finite number of elements of the sequence are less than that given distance from each other.

The utility of Cauchy sequences lies in the fact that in a complete metric space (one where all such sequences are known to converge to a limit

Limit of a sequence

The limit of a sequence is, intuitively, the unique number or point L such that the terms of the sequence become arbitrarily close to L for "large" values of n...

), the criterion for convergence depends only on the terms of the sequence itself.