Difference between revisions of "Cauchy sequence"

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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.
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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.
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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.
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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.     
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==External Links==
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*[http://mathworld.wolfram.com/CauchySequence.html "Cauchy Sequence" From MathWorld]
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*[http://mathworld.wolfram.com/Completion.html "Completion" From MathWorld]
  
 
[[Category:Mathematics]]
 
[[Category:Mathematics]]

Revision as of 19:40, 24 September 2008

In Mathematics, a Cauchy sequence is an infinite sequence the members of which get progressively closer to each other. More formally, a Cauchy sequence, , 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 , 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 numbers. 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, , which is not complete, the larger set (called the completion) of , is the set of real numbers, . The sequence above has a limit, in the set of real numbers, but not in the set of rational numbers.

External Links