Cauchy Sequence

In mathematics, a Cauchy sequence (pronounced ), named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all but a finite number of elements of the sequence are less than that given distance from 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.

Read more about Cauchy SequenceIn Real Numbers, In A Metric Space, Completeness

Other articles related to "cauchy sequence, sequence, cauchy, cauchy sequences, sequences":

Cauchy Sequence - Generalizations - In A Hyperreal Continuum
... A real sequence has a natural hyperreal extension, defined for hypernatural values H of the index n in addition to the usual natural n ... The sequence is Cauchy if and only if for every infinite H and K, the values and are infinitely close, or adequal, i.e where "st" is the standard part function ...
Riesz–Fischer Theorem - History: The Note of Riesz and The Note of Fischer (1907)
... Let {φn } be an orthonormal system in L2 and {an } a sequence of reals ... in a theorem (almost with modern words) that a Cauchy sequence in L2 converges in L2-norm to some function f  in L2 ... In this Note, Cauchy sequences are called "sequences converging in the mean" and L2 is denoted by Ω ...
Completeness Of The Real Numbers - Forms of Completeness - Cauchy Completeness
... Cauchy completeness is the statement that every Cauchy sequence of real numbers converges ... The rational number line Q is not Cauchy complete ... An example is the following sequence of rational numbers Here the nth term in the sequence is the nth decimal approximation for pi ...
0.999... - Proofs From The Construction of The Real Numbers - Cauchy Sequences
... Further information Cauchy sequence Another approach is to define a real number as the limit of a Cauchy sequence of rational numbers ... Then the reals are defined to be the sequences of rationals that have the Cauchy sequence property using this distance ... That is, in the sequence (x0, x1, x2...), a mapping from natural numbers to rationals, for any positive rational δ there is an N such that
Complete Metric Space
... analysis, a metric space M is called complete (or a Cauchy space) if every Cauchy sequence of points in M has a limit that is also in M or, alternatively, if every Cauchy sequence in M converges ... is "missing" from it, even though one can construct a Cauchy sequence of rational numbers that converges to it ...

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