The real numbers are complete, and one of the standard constructions of the real numbers involves Cauchy sequences of rational numbers.Ī 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). Nonetheless, such a limit does not always exist within M.Ī metric space X in which every Cauchy sequence has a limit in X is called complete. 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 M. Is Cauchy, if for every positive real number ε > 0 there is a positive integer N such that for all natural numbers m, n > N, the distance
įormally, given a metric space ( M, d), a sequence To define Cauchy sequences in any metric space, the absolute value is replaced by the distance between and. In a similar way one can define Cauchy sequences of complex numbers. Where the vertical bars denote the absolute value. Of real numbers is called Cauchy, 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 filter and Cauchy net. 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. 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), they give a criterion for convergence which depends only on the terms of the sequence itself. However small is, starting from a Cauchy sequence and eliminating terms one by one from the start, after a finite number of steps, any pair chosen from the remaining terms will be within distance of each other. In other words, suppose a pre-assigned positive real value is chosen. To be more precise, by dropping enough (but still only a finite number of) terms from the start of the sequence, it is possible to make the maximum of the distances from any of the remaining elements to any other such element smaller than any preassigned, necessarily positive, value.
In mathematics, a Cauchy sequence, named after Augustin Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses. The elements of the sequence fail to get close to each other as the sequence progresses.
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