Cauchy sequences - Definition. Was ist Cauchy sequences
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Was (wer) ist Cauchy sequences - definition

SEQUENCE WHOSE ELEMENTS BECOME ARBITRARILY CLOSE TO EACH OTHER
Cauchy sequences; Regular Cauchy sequence; Cauchy Sequences; Cauchy Sequence

Cauchy sequence         
<mathematics> A sequence of elements from some vector space that converge and stay arbitrarily close to each other (using the norm definied for the space). (2000-03-10)
Cauchy sequence         
In mathematics, a Cauchy sequence (; ), named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all but a finite number of elements of the sequence are less than that given distance from each other.
Uniformly Cauchy sequence         
SEQUENCE FUNCTION
Uniformly cauchy; Uniformly Cauchy
In mathematics, a sequence of functions \{f_{n}\} from a set S to a metric space M is said to be uniformly Cauchy if:

Wikipedia

Cauchy sequence

In mathematics, a Cauchy sequence (French pronunciation: ​[koʃi]; English: KOH-shee), named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all but a finite number of elements of the sequence are less than that given distance from each other.

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:

the consecutive terms become arbitrarily close to each other: However, with growing values of the index n, the terms a n {\displaystyle a_{n}} become arbitrarily large. So, for any index n and distance d, there exists an index m big enough such that a m a n > d . {\displaystyle a_{m}-a_{n}>d.} As a result, no matter how far one goes, the remaining terms of the sequence never get close to each other; 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.