Cauchy sequence

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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.

Cauchy sequences require the notion of distance so they can only be defined in a metric space. Generalizations to more abstract uniform spaces exist in the form of Cauchy filter and Cauchy net.

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

Formal definition

Formally, a Cauchy sequence is a sequence

x_{1},x_{2},x_{3},\ldots

in a metric space (M, d) such that for every positive real number r > 0, there is an integer N such that for all integers m,n > N, the distance

d(x_{m},x_{n})

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.

Completeness

A metric space X in which every Cauchy sequence has a limit (in X) is called complete.

Example: real numbers

The real numbers are complete, and the standard construction of the real numbers involves Cauchy sequences of rational numbers.

Counter-example: rational numbers

The rational numbers Q are not complete (for the usual distance): There are sequences of rationals that converge (in R) to irrational numbers; these are Cauchy sequences having no limit in Q.

For example:

  • 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.

Other properties

Every convergent sequence is a Cauchy sequence, and every Cauchy sequence is bounded. If f\colon M\rightarrow N is a uniformly continuous map between the metric spaces M and N and (xn) is a Cauchy sequence in M, then (f(x_{n})) is a Cauchy sequence in N. If (x_{n}) and (y_{n}) are two Cauchy sequences in the rational, real or complex numbers, then the sum (x_{n}+y_{n}) and the product (x_{n}y_{n}) 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".

References

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