Cauchy sequence


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 term. For instance, in the sequence of square roots of natural numbers:
the consecutive terms become arbitrarily close to each other – their differences
tend to zero as the index grows. However, with growing values of, the terms become arbitrarily large. So, for any index and distance, there exists an index big enough such that 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, 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

A sequence
of real numbers is called a Cauchy sequence if for every positive real number 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. Cauchy formulated such a condition by requiring to be infinitesimal for every pair of infinite m, n.
For any real number r, the sequence of truncated decimal expansions of r forms a Cauchy sequence. For example, when this sequence is. The mth and nth terms differ by at most when m < n, and as m grows this becomes smaller than any fixed positive number

Modulus of Cauchy convergence

If is a 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 for all natural numbers and 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 of the natural numbers. The existence of a modulus also follows from the principle of countable choice. Regular Cauchy sequences are sequences with a given modulus of Cauchy convergence. 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 is replaced by the distance between and
Formally, given a metric space a sequence of elements of
is Cauchy, if for every positive real number there is a positive integer such that for all positive integers the distance
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 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. 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 :

There are sequences of rationals that converge to irrational numbers; these are Cauchy sequences having no limit in In fact, if a real number x is irrational, then the sequence, 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. Irrational numbers certainly exist in for example:
  • The sequence defined by consists of rational numbers, which is clear from the definition; however it converges to the irrational square root of 2, see Babylonian method of computing square root.
  • The sequence of ratios of consecutive Fibonacci numbers which, if it converges at all, converges to a limit satisfying and no rational number has this property. If one considers this as a sequence of real numbers, however, it converges to the real number the Golden ratio, which is irrational.
  • The values of the exponential, sine and cosine functions, exp, sin, cos, are known to be irrational for any rational value of but each can be defined as the limit of a rational Cauchy sequence, using, for instance, the Maclaurin series.
Completion turns into

Non-example: open interval

The open interval in the set of real numbers with an ordinary distance in is not a complete space: there is a sequence in it, which is Cauchy, however does not converge in —its 'limit', number 0, does not belong to the space
Completion turns the open interval into the closed interval

Other properties

  • Every convergent sequence is a Cauchy sequence, since, given any real number beyond some fixed point, every term of the sequence is within distance of s, so any two terms of the sequence are within distance of each other.
  • In any metric space, a Cauchy sequence is bounded.
  • In any metric space, a Cauchy sequence which has a convergent subsequence with limit s is itself convergent, 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
. 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
If is a uniformly continuous map between the metric spaces M and N and 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.

Generalizations

In topological vector spaces

There is also a concept of Cauchy sequence for a topological vector space : Pick a local base for about 0; then is a Cauchy sequence if for each member there is some number such that whenever
is an element of If the topology of 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: A sequence in a topological group is a Cauchy sequence if for every open neighbourhood of the identity in there 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 for every open neighbourhood of the identity in there exists some number such that whenever it follows that This relation is an equivalence relation: 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.