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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, as opposed to the definition of convergence, which uses the limit value as well as the terms.
Cauchy's convergence test can only be used in complete metric spaces (such as and ), which are spaces where all Cauchy sequences converge. This is because we need only show that its elements become arbitrarily close to each other after a finite progression in the sequence to prove the series converges.
The notion of a Cauchy sequence is important in the study of sequences in metric spaces, and, in particular, in real analysis. One particularly important result in real analysis is the Cauchy criterion for convergence of sequences: a sequence of real numbers is convergent if and only if it is a Cauchy sequence.
By construction, every real number x is represented by a Cauchy sequence of rational numbers. This representation is far from unique; every rational sequence that converges to x is a Cauchy sequence representing x. This reflects the observation that one can often use different sequences to approximate the same real number. [6]
Cauchy sequences are closely related to convergent sequences. In fact, for sequences of real numbers they are equivalent: any Cauchy sequence is convergent. In general metric spaces, it continues to hold that convergent sequences are also Cauchy. But the converse is not true: not every Cauchy sequence is convergent in a general metric space.
This is also known as the nth root test or Cauchy's criterion. Let = | |, where denotes the limit superior (possibly ; if the limit exists it is the same value). If r < 1, then the series converges absolutely.
Even more generally, Cauchy spaces are spaces in which Cauchy filters may be defined. Convergence implies "Cauchy convergence", and Cauchy convergence, together with the existence of a convergent subsequence implies convergence. The concept of completeness of metric spaces, and its generalizations is defined in terms of Cauchy sequences.
It states that for a converging sequence the sequence of the arithmetic means of its first members converges against the same limit as the original sequence, that is () with implies (+ +) / . [ 1 ] [ 2 ] The theorem was found by Cauchy in 1821, [ 1 ] subsequently a number of related and generalized results were published, in particular by Otto ...