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Line 1: − A '''Cauchy sequence''' is an infinite series of numbers that approach some other number. These sequences are often used in [[real analysis]]. For example, the sequence 1/2, 1/4, 1/8, 1/16, 1/32... is a Cauchy sequence that approaches 0.+ + + + + + + + + +
Correct incorrect definition - the definition before was for convergent sequences, not Cauchy sequences - Also expand article
In [[Mathematics]], a '''Cauchy sequence''' is an infinite sequence the members of which get progressively closer to each other. More formally, a Cauchy sequence, <math>a_1, a_2, ... a_n</math>, in a [[metric space]] M, with distance function d, is a sequence such that for any positive real number e, there is some integer N, such that <math>d(a_n, a_m) < e</math>, whenever n and m are greater than N.
All [[convergent]] sequences, those sequences that get progressively closer to a [[limit]], are Cauchy sequences. However, the reverse is not true. For example, the sequence 3,3.1, 3.14 (approximations of the value of [[Pi]]), is Cauchy, however, they are not convergent in the [[rational number]]s. The sequence is convergent in the real numbers, with the limit Pi.
A metric space where all Cauchy sequences are convergent is said to be [[complete]]. For any metric space, there exists a complete metric space, containing it. Thus, in one sense every Cauchy sequence is convergent to a limit, but in a larger set than that considered, when defining the sequence. In the case of the rational numbers, <math>\mathbb{Q}</math>, which is not complete, the larger set (called the [[completion]]) of <math>\mathbb{Q}</math>, is the set of real numbers, <math>\mathbb{R}</math>. The sequence above has a limit, in the set of real numbers, but not in the set of rational numbers.
==External Links==
*[http://mathworld.wolfram.com/CauchySequence.html "Cauchy Sequence" From MathWorld]
*[http://mathworld.wolfram.com/Completion.html "Completion" From MathWorld]
[[Category:Mathematics]]
[[Category:Mathematics]]