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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Cauchy sequence ： ウィキペディア英語版 Cauchy sequence In mathematics, a Cauchy sequence ((:koʃi); ), named after Augustin-Louis Cauchy, is a sequence whose elements become ''arbitrarily close to each other'' as the sequence progresses.〔Lang, Serge (1993), Algebra (Third ed.), Reading, Mass.: Addison-Wesley Pub. Co., ISBN 978-0-201-55540-0, Zbl 0848.13001〕 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. Note: it is not sufficient for each term to become arbitrarily close to the preceding term. For instance, 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. The notions above are not as unfamiliar as they might at first appear. 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''. 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 real numbers== A sequence :$x\_1,\; x\_2,\; x\_3,\; \backslash ldots$ 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'' : where the vertical bars denote the absolute value. In a similar way one can define Cauchy sequences of rational or complex numbers. Cauchy formulated such a condition by requiring $x\_m\; -\; x\_n$ to be infinitesimal for every pair of infinite ''m, n''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Cauchy sequence」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.