Cauchy sequence

(a) The plot of a Cauchy sequence ${\displaystyle (x_{n}),}$ shown in blue, as ${\displaystyle x_{n}}$ versus ${\displaystyle n.}$ If the space containing the sequence is complete, then the sequence has a limit.

(b) A sequence that is not Cauchy. The elements of the sequence do not get arbitrarily close to each other as the sequence progresses.

In mathematics, a Cauchy sequence is a sequence whose elements become arbitrarily close to each other as the sequence progresses.^[1] 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.^[2]

It is not sufficient for each term to become arbitrarily close to the preceding term. For instance, in the sequence of square roots of natural numbers: $${\displaystyle a_{n}={\sqrt {n}},}$$ the consecutive terms become arbitrarily close to each other – their differences $${\displaystyle a_{n+1}-a_{n}={\sqrt {n+1}}-{\sqrt {n}}={\frac {1}{{\sqrt {n+1}}+{\sqrt {n}}}}<{\frac {1}{2{\sqrt {n}}}}}$$ tend to zero as the index n grows. However, with growing values of n, the terms ${\displaystyle a_{n}}$ become arbitrarily large. So, for any index n and distance d, there exists an index m big enough such that ${\displaystyle a_{m}-a_{n}>d.}$ As a result, no matter how far one goes, the remaining terms of the sequence never get close to each other; hence the sequence is not Cauchy.

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), the criterion for convergence depends only on the terms of the sequence itself, as opposed to the definition of convergence, which uses the limit value as well as the terms. This is often exploited in algorithms, both theoretical and applied, where an iterative process can be shown relatively easily to produce a Cauchy sequence, consisting of the iterates, thus fulfilling a logical condition, such as termination.

Generalizations of Cauchy sequences in more abstract uniform spaces exist in the form of Cauchy filters and Cauchy nets.