For instance, in the sequence of square roots of natural numbers:
the consecutive terms become arbitrarily close to each other – their differences
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.
In a similar way one can define Cauchy sequences of rational or complex numbers.
The existence of a modulus for a Cauchy sequence follows from the well-ordering property of the natural numbers (let
Moduli of Cauchy convergence are used by constructive mathematicians who do not wish to use any form of choice.
Using a modulus of Cauchy convergence can simplify both definitions and theorems in constructive analysis.
Regular Cauchy sequences were used by Bishop (2012) and by Bridges (1997) in constructive mathematics textbooks.
A metric space (X, d) in which every Cauchy sequence converges to an element of X is called complete.
In this construction, each equivalence class of Cauchy sequences of rational numbers with a certain tail behavior—that is, each class of sequences that get arbitrarily close to one another— is a real number.
Any Cauchy sequence of elements of X must be constant beyond some fixed point, and converges to the eventually repeating term.
are not complete (for the usual distance): There are sequences of rationals that converge (in
In fact, if a real number x is irrational, then the sequence (xn), whose n-th term is the truncation to n decimal places of the decimal expansion of x, gives a Cauchy sequence of rational numbers with irrational limit x. Irrational numbers certainly exist in
This proof of the completeness of the real numbers implicitly makes use of the least upper bound axiom.
One of the standard illustrations of the advantage of being able to work with Cauchy sequences and make use of completeness is provided by consideration of the summation of an infinite series of real numbers (or, more generally, of elements of any complete normed linear space, or Banach space).
It is a routine matter to determine whether the sequence of partial sums is Cauchy or not, since for positive integers
is a uniformly continuous map between the metric spaces M and N and (xn) is a Cauchy sequence in M, then
are two Cauchy sequences in the rational, real or complex numbers, then the sum
There is also a concept of Cauchy sequence for a topological vector space
Since the topological vector space definition of Cauchy sequence requires only that there be a continuous "subtraction" operation, it can just as well be stated in the context of a topological group: A sequence
As above, it is sufficient to check this for the neighbourhoods in any local base of the identity in
As in the construction of the completion of a metric space, one can furthermore define the binary relation on Cauchy sequences in
of such Cauchy sequences forms a group (for the componentwise product), and the set
One can then show that this completion is isomorphic to the inverse limit of the sequence
is a cofinal sequence (that is, any normal subgroup of finite index contains some
), then this completion is canonical in the sense that it is isomorphic to the inverse limit of
has a natural hyperreal extension, defined for hypernatural values H of the index n in addition to the usual natural n. The sequence is Cauchy if and only if for every infinite H and K, the values
are infinitely close, or adequal, that is, where "st" is the standard part function.
Krause (2020) introduced a notion of Cauchy completion of a category.