In mathematical analysis, a Cauchy sequence, named after Augustin Cauchy, is a sequence whose elements become closer as the sequence progresses. To be more precise, by dropping a finite number of elements from the start of the sequence we can make the distance between any two remaining elements arbitrarily small.
They are of interest because in a complete space, all such sequences converge to a limit, and one can test for "Cauchiness" without knowing the value of the limit (if it exists), in contrast to the definition of convergence.
Cauchy sequence in a metric space
Formally, a Cauchy sequence is a sequence
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 may not be the case.
A metric space X in which every Cauchy sequence has a limit (in X) is called complete.
Example: real numbers
Counter-example: rational numbers
- The sequence defined by x0 = 1, xn+1 = (xn + 2/xn)/2 consists of rational numbers (1, 3/2, 17/12,...), which is clear from the definition; it converges to the irrational square root of two, see Babylonian method of computing square root.
- The values of the exponential, sine and cosine functions, exp(x), sin(x), cos(x), are irrational for any rational value of x≠0, but are defined as limit of a rational sequence which is their Maclaurin series.
Every convergent sequence is a Cauchy sequence, and every Cauchy sequence is bounded. 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.
Cauchy sequences in topological vector spaces
There is also a concept of Cauchy sequence for a topological vector space X: Pick a local base B for X about 0; then (xk) is a Cauchy sequence if for all members V of B, there is some number N such that whenever n,m > N, xn - xm is an element of V. If the topology of X is compatible with a translation-invariant metric d, the two definitions agree.
Cauchy sequences in groups
There is also a concept of Cauchy sequence in a group G: Let H=(Hr) be a decreasing sequence of normal subgroups of G of finite index. Then a sequence (xn) in G is said to be Cauchy (w.r.t. H) iff for any r there is N such that ∀m,n > N, xn xm-1 ∈ Hr.
The set C of such Cauchy sequences forms a group (for the componentwise product), and the set C0 of null sequences (s.th. ∀r, ∃N, ∀n > N, xn∈Hr) is a normal subgroup of C. The factor group C/C0 is called the completion of G w.r.t. H.
One can then show that this completion is isomorphic to the inverse limit of the sequence (G/Hr).
If H is a cofinal sequence (i.e., any normal subgroup of finite index contains some Hr), then this completion is canonical in the sense that it is isomorphic to the inverse limit of (G/H)H, where H varies over all normal subgroups of finite index. For further details, see ch. I.10 in Lang's "Algebra".
- Lang, Serge (1997). Algebra (3rd ed., reprint w/ corr.), Addison-Wesley. ISBN 0-201-55540-9.cs:Cauchyovská posloupnost
de:Cauchy-Folge es:Sucesión de Cauchy fr:Suite de Cauchy he:סדרת קושי it:Successione fondamentale hu:Cauchy-sorozat nl:Cauchyrij ja:コーシー列 pl:Ciąg Cauchy'ego ru:Фундаментальная последовательность zh:柯西序列