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In Mathematics , a Cauchy sequence, named after Augustin Cauchy , is a Sequence whose elements become ''close to each other'' as the sequence progresses. To be more precise, by dropping enough ( but still only a finite number of) terms from the start of the sequence, it is possible to make the maximum of the Distances from any of the remaining elements to any other such element smaller than any preassigned positive value. In other words, suppose a pre-assigned positive real value is chosen. However small 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 of each other. Because Cauchy sequences require the notion of distance, they can only be defined in a Metric Space . Their utility lies in the fact that in a Complete Metric Space ( one where all such sequences are known to Converge To A Limit ), they give a criterion for convergence which 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 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 . CAUCHY SEQUENCE OF REAL NUMBERS A sequence : of real numbers is called Cauchy, if for every Positive real number ''ε'' > 0 there is a positive Integer ''N'' such that for all integers ''m'',''n'' > ''N'' | ||
| In | "http://wwwinformationdelightinfo/information/entry/constructive_mathematics" class="copylinks">Constructive Mathematics , Cauchy sequences often must be given with a ''modulus of Cauchy convergence'' to be useful If <math>(x_1, x_2, x_3, )</math> is a Cauchy sequence in the set <math>X</math>, then a modulus of Cauchy convergence for the sequence is a Function <math>\alpha</math> from the set of Natural Number s to itself, such that <math> orall k orall m, n > \alpha(k), x_m - x_n < 1/k</math> |
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