Cauchy sequence

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In mathematics, a Cauchy sequence, named after Augustin Cauchy, is a sequence whose elements become close 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 maximum distance between any two remaining elements arbitrarily small.

Because Cauchy sequences require the notion of distance, 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 the Cauchy property without knowing the value of the limit (if it exists), in contrast to the definition of convergence. They are also significant in constructing algebraic structures with completeness properties, such as the real numbers.

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[edit] Cauchy sequence of real numbers

A sequence

x_1, x_2, x_3, \ldots

of real numbers is called Cauchy, if for every positive real number r > 0 there is a positive integer N such that for all integers m,n > N one has

| xm − xn | < r,

where the vertical bars denote the absolute value.

In a similar way one can define Cauchy sequences of complex numbers.

[edit] Cauchy sequence in a metric space

To define Cauchy sequences in any metric space, the absolute value | xm − xn | is replaced by the distance d(xm,xn) between xm and xn.

Formally, given a metric space (M, d), a sequence

x_1, x_2, x_3, \ldots

is Cauchy, if for every positive real number r > 0 there is an integer N such that for all integers m,n > N, the distance

d(xm,xn)

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.

[edit] Completeness

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

[edit] Example: real numbers

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

[edit] 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; however it converges to the irrational square root of two, see Babylonian method of computing square root.
  • The sequence xn = Fn / Fn − 1 of ratios of consecutive Fibonacci numbers which, if it converges at all, converges to a limit φ satisfying φ2 = φ + 1, and no rational number has this property. If one considers this as a sequence of real numbers, however, it converges to the real number \phi = (1+\sqrt5)/2, the Golden ratio, which is irrational.
  • The values of the exponential, sine and cosine functions, exp(x), sin(x), cos(x), are known to be irrational for any rational value of x≠0, but each is defined as the limit of a rational sequence from the Maclaurin series.

[edit] 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(xn)) is a Cauchy sequence in N. If (xn) and (yn) are two Cauchy sequences in the rational, real or complex numbers, then the sum (xn + yn) and the product (xnyn) are also Cauchy sequences.

[edit] Generalizations

[edit] 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.

[edit] 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) if and only if for any r there is N such that \forall m,n > N, x_n x_m^{-1} \in H_r.

The set C of such Cauchy sequences forms a group (for the componentwise product), and the set C0 of null sequences (s.th. \forall r, \exists N, \forall n > N, x_n \in H_r) is a normal subgroup of C. The factor group C / C0 is called the completion of G with respect to 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".

[edit] In constructive mathematics

In constructive mathematics, Cauchy sequences often must be given with a modulus of Cauchy convergence to be useful. If (x1,x2,x3,...) is a Cauchy sequence in the set X, then a modulus of Cauchy convergence for the sequence is a function α from the set of natural numbers to itself, such that \forall k \forall m, n > \alpha(k), |x_m - x_n| < 1/k.

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 α(k) be the smallest possible N in the definition of Cauchy sequence, taking r to be 1 / k). 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 α(k) = k or α(k) = 2k). 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.

[edit] References

  • Bourbaki, Nicolas (1972). Commutative Algebra, English translation, Addison-Wesley. ISBN 0-201-00644-8. 
  • Lang, Serge (1997). Algebra, 3rd ed., reprint w/ corr., Addison-Wesley. ISBN 978-0-201-55540-0. 
  • Troelstra, A. S. and D. van Dalen. Constructivism in Mathematics: An Introduction.  (for uses in constructive mathematics)
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