Cauchy sequence

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In mathematics, a Cauchy sequence (pronounced Ko-shee), named after Augustin-Louis Cauchy, is a sequence whose elements become increasingly close to each other as the sequence progresses. To be more precise, given any positive number, we can always drop some terms from the start of the sequence, so that the maximum of the distances between any two of the remaining elements is smaller than that number.

In other words, suppose a pre-assigned positive real value ε is chosen. However small ε is, starting from a Cauchy sequence and eliminating terms one by one from the start, after a finite number of steps, any pair chosen from the remaining terms will be within distance ε of each other.

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. 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.

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 filter and Cauchy net.

Contents

Real numbers

A sequence

of real numbers is called Cauchy, if for every positive real number ε, there is a positive integer N such that for all natural numbers m, n > N

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