![]() 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 has the real limit x. Equivalently, a sequence (x n) is de ned by a function f: N R where x n f(n). A sequence (x n) (of real numbers) is an ordered list of real numbers x n2R, indexed by n2N. The notions above are not as unfamiliar as they might at first appear. 1I 2 I n is a nested sequence of nonempty, closed, bounded intervals I n a n b n, then T 1 n1I nis nonempty. 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. We have already been working with sequences, but to be sure were on the same page: Definition 2.1. The Cauchy Criterion (Theorem 2.9), (6) the de nition of an in nite series, (7) the Comparison Test (Theorem 2. 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. THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e: This chapter contains the beginnings of the most important, and probably the most subtle, notion in mathematical analysis, i.e., the concept of a limit. A metric space with the property that any Cauchy sequence has a limit is called complete, see also Cauchy criteria. 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. To continue the discussion, we need to know the definition of a subsequence. 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. Every monotonic and bounded sequence of real numbers is convergent. ![]() ![]() ![]() Freebase Rate this definition: 0.0 / 0 votes ![]()
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