\( |x_m - x_n| 0 there is a positive integer N such that for all positive integers m,n > N, the distance Of real numbers is called a Cauchy sequence, if for every positive real number ε, there is a positive integer N such that for all natural numbers m, n > N Generalizations of Cauchy sequences in more abstract uniform spaces exist in the form of Cauchy filters and Cauchy nets. In some cases it may be difficult to describe x independently of such a limiting process involving rational numbers. The customary acceptance of the fact that any real number 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. The notions above are not as unfamiliar as they might at first appear. 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.
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. 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. In mathematics, a Cauchy sequence (French pronunciation: English pronunciation: /ˈkoʊʃiː/ KOH-shee), named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses.
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