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In mathematicsa Cauchy sequence French pronunciation: It is not sufficient for each term to become arbitrarily close to the preceding term.
For instance, in the sequence of square roots of natural numbers:. As a result, despite how far one goes, the remaining terms of the sequence never get close to each otherhence the sequence is not Cauchy.
The utility of Cauchy sequences lies in the fact that in babach complete metric space one where all such sequences are known to converge to a limitthe 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 algorithmsboth 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.
The notions above are not as unfamiliar as they might at first appear. The customary acceptance of the fact that any real banacch x has a decimal expansion is an implicit acknowledgment that a particular Cauchy sequence of rational numbers whose terms are the successive truncations of the decimal expansion of x has the real limit x. In some cases it may be difficult to describe x independently of such a limiting process involving rational numbers.
Generalizations of Cauchy sequences in more abstract uniform spaces exist in the form of Cauchy filters and Cauchy nets.
In a similar way one can define Cauchy sequences of rational or complex numbers. 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 value x m – x n is replaced by the distance d x mx n where d denotes a metric between x m and x n. Formally, given a metric space Xda sequence.
Roughly speaking, the terms of the sequence are getting closer and closer together in a way that suggests that the sequence ought to d a limit in X.
Nonetheless, such a limit does not always exist within X. A metric space X banadh, d in which every Cauchy sequence converges to an element of X is called complete. 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 edpacio beyond some fixed point, and converges to the eventually repeating term.
Normed vector space – Wikipedia
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. In fact, if a real number x is irrational, then the sequence x nwhose n -th term is the truncation to n bansch places of the decimal expansion of xgives a Cauchy sequence of rational numbers with irrational limit x.
Irrational numbers certainly exist in Rfor example:. Espacoo last two properties, together with the Bolzano—Weierstrass theoremyield 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. It should be noted, though, that this proof of the completeness of the real numbers implicitly makes use of the least upper bound axiom.
The alternative approach, mentioned above, of constructing the real numbers as the completion of the rational numbers, espscio 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 es;acio normed linear spaceor Banach space.
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: This relation is an equivalence relation: It is reflexive since the sequences are Cauchy sequences.
An example of this construction, familiar in number theory and algebraic geometry is the construction of edpacio p -adic completion of the integers with respect to a prime p. In this case, G is the integers under addition, and H r is the additive subgroup consisting d integer multiples of p r. Esacio further details, see ch. In constructive mathematicsCauchy sequences often must be given with a modulus of Cauchy convergence to be useful.
Clearly, any sequence with a modulus of Cauchy banaxh is a Cauchy sequence. However, this well-ordering property does not hold in constructive mathematics it is equivalent to the principle of excluded middle.
Normed vector space
On the other hand, this converse also follows directly from the principle of dependent choice in fact, it will follow from the weaker AC 00which 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.
Any Cauchy sequence with a modulus espaclo Cauchy espavio 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.
However, Bridges also works on mathematical constructivism; the concept has not spread far outside of that milieu. Krause introduced a notion of Cauchy espacuo of a category.
From Wikipedia, the free encyclopedia.
Banach space – Wikipedia
If the space containing the sequence is complete, the “ultimate destination” of this sequence that is, the limit exists. The elements of the sequence fail to get arbitrarily close to each other as the sequence progresses.
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