In mathematics, a Cauchy sequence is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all excluding a finite number of elements of the sequence are less than that given distance from each other. Cauchy sequences are named after Augustin-Louis Cauchy; they may occasionally be known as fundamental sequences.

It is not sufficient for each term to become arbitrarily close to the tend to zero as the index grows. However, with growing values of become arbitrarily large. So, for any index and distance , there exists an index As a result, no matter how far one goes, the remaining terms of the sequence never get close to ; hence the sequence is not Cauchy.

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.

Generalizations of Cauchy sequences in more abstract uniform spaces exist in the form of Cauchy filters and Cauchy nets.

In real numbers

of real numbers is called a Cauchy sequence if for every positive there is a positive integer N such that for all natural numbers where the vertical bars denote the absolute value. In a similar way one can define Cauchy sequences of rational or complex numbers to be infinitesimal for every pair of infinite m, n.

For any real number r, the sequence of truncated decimal expansions of r this sequence is (3, 3.1, 3.14, 3.141, ...). The mth and n when m < n, and as m

Modulus of Cauchy convergence

then a modulus of Cauchy convergence for the sequence is a function from the set of natural numbers

Any sequence with a modulus of Cauchy convergence is a Cauchy sequence. The existence of a modulus for a Cauchy sequence follows from the well-ordering property). The existence of a modulus also follows from the principle of countable choice. Regular Cauchy sequences). Any Cauchy sequence with a modulus of Cauchy convergence is equivalent to a regular Cauchy sequence; this can be proven without using any form of the axiom of choice.

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 and by in constructive mathematics textbooks.

In a metric space

Since the definition of a Cauchy sequence only involves metric concepts, it is straightforward to generalize it to any metric space X. To do so, the absolute difference (where d denotes a metric

Formally, given a metric space is Cauchy, if for every positive real number there is a positive integer

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 X. Nonetheless, such a limit does not always exist within X: the property of a space that every Cauchy sequence converges in the space is called completeness, and is detailed below.

Completeness

A metric space (X, d) in which every Cauchy sequence converges to an element of X is called complete. For any metric space M, it is possible to construct a complete metric space M′ that contains M as a dense subspace; see .

Examples of complete metric spaces

The real numbers are complete under the metric induced by the usual absolute value, and one of the standard constructions of the real numbers involves Cauchy sequences of rational numbers. 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.

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.

Non-example: rational numbers

The rational numbers are not complete (for the usual distance):) to irrational numbers 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 for example:

• consists of rational numbers (1, 3/2, 17/12,...), which is clear from the definition; however it converges to the irrational square root of 2, see Babylonian method of computing square root.

• of ratios of consecutive Fibonacci numbers the Golden ratio, which is irrational.

• The values of the exponential, sine and cosine functions, exp(x), sin(x), cos(x but each can be defined as the limit of a rational Cauchy sequence, using, for instance, the Maclaurin series.

Non-example: open interval

The open interval

into the closed interval

Other properties

• Every convergent sequence (with limit s of s of each other.

• is bounded (since for some N, all terms of the sequence from the N-th onwards are within distance 1 of each other, and if M and any terms up to the N).

• In any metric space, a Cauchy sequence which has a convergent subsequence with limit s is itself convergent (with the same limit), since, given any real number r > 0, beyond some fixed point in the original sequence, every term of the subsequence is within distance r/2 of s, and any two terms of the original sequence are within distance r/2 of each other, so every term of the original sequence is within distance r of s.

These last two properties, together with 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. Every Cauchy sequence of real numbers is bounded, hence by Bolzano–Weierstrass has a convergent subsequence, hence is itself convergent. This proof of the completeness of the real numbers implicitly makes use of the least upper bound axiom. The alternative approach, mentioned above, of the real numbers as the completion 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 is considered to be convergent if and only if the sequence of partial sums

is a uniformly continuous map between the metric spaces M and N and (xn) is a Cauchy sequence in M is a Cauchy sequence in N are also Cauchy sequences.

Generalizations

In topological vector spaces

There is also a concept of Cauchy sequence for a topological vector space: Pick a local base is compatible with a translation-invariant metric the two definitions agree.

In topological groups

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 of the identity

As in the construction of the completion of a metric space are equivalent if for every open neighbourhood This relation is an equivalence relation which by continuity of the inverse is another open neighbourhood of the identity. It is transitive; such pairs exist by the continuity of the group operation.

In groups

There is also a concept of Cauchy sequence in a group be a decreasing sequence of normal subgroups of finite index

is a local base.

The factor group

One can then show that this completion is isomorphic to the inverse limit

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

is a cofinal), then this completion is canonical varies over normal subgroups of finite index. For further details, see Ch. I.10 in Lang's "Algebra".

In a hyperreal continuum

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 are infinitely close, or adequal, that is,

where "st" is the standard part function.

Cauchy completion of categories

introduced a notion of Cauchy completion of a category (the category whose objects are rational numbers, and there is a morphism from x to y (again interpreted as a category using its natural ordering).

See also

References

Further reading

• (for uses in constructive mathematics)

External links