![]() ![]() These isomorphisms allow identifying the results of the constructions, and, in practice, to forget which construction has been chosen.Īn axiomatic definition of the real numbers consists of defining them as the elements of a complete ordered field. This results from the above definition and is independent of particular constructions. They are equivalent in the sense that, given the result of any two such constructions, there is a unique isomorphism of ordered field between them. The article presents several such constructions. The number of elements (possibly infinite) is called the length of the sequence. Like a set, it contains members (also called elements, or terms ). Such a definition does not prove that such a complete ordered field exists, and the existence proof consists of constructing a mathematical structure that satisfies the definition. In mathematics, a Cauchy sequence (French pronunciation: English: / k o i / KOH-shee), named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses. In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. One of them is that they form a complete ordered field that does not contain any smaller complete ordered field. In the metric space of real numbers with the absolute value function, the sequence xn 1. In mathematics, there are several equivalent ways of defining the real numbers. Lets look at some examples of Cauchy sequences in different metric spaces. But many Cauchy sequences do not have multiplicative inverses. So Cauchy sequences form a commutative ring. The constant sequences 0 (0 0 :::) and 1 (1 1 :::) are additive and multiplicative identities, and every Cauchy sequence (x n) has an additive inverse ( x n). Axiomatic definitions of the real numbers Thus we can add and multiply Cauchy sequences.
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