In mathematics, a Cauchy sequence, named after Augustin Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses. To be more precise, by dropping enough (but still only a finite number of) terms from the start of the sequence, it is possible to make the maximum of the distances from any of the remaining elements to any other such element smaller than any preassigned positive value.
In other words, suppose a pre-assigned positive real value is chosen. However small is, starting from a Cauchy sequence and eliminating terms one by one from the start, after a finite number of steps, any pair chosen from the remaining terms will be within distance of 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), they give a criterion for convergence which depends only on the terms of the sequence itself. 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.
The notions above are not as unfamiliar as might at first appear. The customary acceptance of the fact that any real number x has a decimal expansion is an implicit acknowledgment that a particular Cauchy sequence of rational numbers (whose terms are the successive truncations of the decimal expansion of x) has the real limit x. In some cases it may be difficult to describe x independently of such a limiting process involving rational numbers.
where the vertical bars denote the absolute value.
In a similar way one can define Cauchy sequences of complex numbers.
To define Cauchy sequences in any metric space, the absolute value is replaced by the distance between and .
Formally, given a metric space (M, d), a sequence
is less than ε. 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, such a limit does not always exist within M.
A rather different type of example is afforded by a metric space X which has the discrete metric (where any two distinct points are at distance 1 from each other). Any Cauchy sequence of elements of X must be constant beyond some fixed point, and converges to the eventually repeating term.
These last two properties, together with a lemma used in the proof of the Bolzano-Weierstrass theorem, yield one standard proof of the completeness of the real numbers, closely related to both the Bolzano-Weierstrass theorem and the Heine–Borel theorem. The lemma in question states that every bounded sequence of real numbers has a convergent subsequence. Given this fact, every Cauchy sequence of real numbers is bounded, hence has a convergent subsequence, hence is itself convergent. It should be noted, though, that this proof of the completeness of the real numbers implicitly makes use of the least upper bound axiom. The alternative approach, mentioned above, of constructing the real numbers as the of the rational numbers, makes the completeness of the real numbers tautological.
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). Such a series is considered to be convergent if and only if the sequence of partial sums is convergent, where . It is a routine matter to determine whether the sequence of partial sums is Cauchy or not, since for positive integers p > q ,
If is a uniformly continuous map between the metric spaces M and N and (xn) 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.
Since the topological vector space definition of Cauchy sequence only requires that there is a continuous "subtraction" operation, it can just as well be stated in the context of a topological group: A sequence in a topological group is a Cauchy sequence if there for every open neighbourhood of the identity in exists some number such that whenever it follows that . 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 that and are equivalent if there for every open neighbourhood of the identity in exists some number such that whenever it follows that . This relation is an equivalence relation. More precisely, it is reflexive since the sequences are Cauchy sequences. It is symmetric since which by continuity of the inverse is another open neighbourhood of the identity. It is transitive since where and are open neighbourhoods of the identity such that ; such pairs exist by the continuity of the group operation.
There is also a concept of Cauchy sequence in a group : Let be a decreasing sequence of normal subgroups of of finite index. Then a sequence in is said to be Cauchy (w.r.t. ) if and only if for any there is such that .
Technically, this is the same thing as a topological group Cauchy sequence for a particular choice of topology on , namely that for which is a local base.
The set of such Cauchy sequences forms a group (for the componentwise product), and the set of null sequences (s.th. ) is a normal subgroup of . The factor group is called the completion of with respect to .
One can then show that this completion is isomorphic to the inverse limit of the sequence .
An example of this construction, familiar in number theory and algebraic geometry is the construction of the p-adic completion of the integers with respect to a prime p. In this case, G is the integers under addition, and Hr is the additive subgroup consisting of integer multiples of pr.
If is a cofinal sequence (i.e., 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 , where varies over all normal subgroups of finite index. For further details, see ch. I.10 in Lang's "Algebra".
In constructive mathematics, Cauchy sequences often must be given with a modulus of Cauchy convergence to be useful. If is a Cauchy sequence in the set , then a modulus of Cauchy convergence for the sequence is a function from the set of natural numbers to itself, such that .
Clearly, any sequence with a modulus of Cauchy convergence is a Cauchy sequence. The converse (that every Cauchy sequence has a modulus) follows from the well-ordering property of the natural numbers (let be the smallest possible in the definition of Cauchy sequence, taking to be ). However, this well-ordering property does not hold in constructive mathematics (it is equivalent to the principle of excluded middle). On the other hand, this converse also follows (directly) from the principle of dependent choice (in fact, it will follow from the weaker AC00), which is generally accepted by constructive mathematicians. Thus, moduli of Cauchy convergence are needed directly only by constructive mathematicians who (like Fred Richman) do not wish to use any form of choice.
That said, using a modulus of Cauchy convergence can simplify both definitions and theorems in constructive analysis. Perhaps even more useful are regular Cauchy sequences, sequences with a given modulus of Cauchy convergence (usually or ). Any Cauchy sequence with a modulus of Cauchy convergence is equivalent (in the sense used to form the completion of a metric space) to a regular Cauchy sequence; this can be proved without using any form of the axiom of choice. Regular Cauchy sequences were used by Errett Bishop in his Foundations of Constructive Analysis, but they have also been used by Douglas Bridges in a non-constructive textbook (ISBN 978-0-387-98239-7). However, Bridges also works on mathematical constructivism; the concept has not spread far outside of that milieu.