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A sequence ,,, … of real numbers is called a Cauchy sequence if for every positive real number , there is a positive integer N such that for all natural numbers, >, | | <, where the vertical bars denote the absolute value.
It is also possible to replace Cauchy sequences in the definition of completeness by Cauchy nets or Cauchy filters. If every Cauchy net (or equivalently every Cauchy filter) has a limit in , then is called complete. One can furthermore construct a completion for an arbitrary uniform space similar to the completion of metric spaces.
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 completeness is related to the construction of the real numbers using Cauchy sequences. Essentially, this method defines a real number to be the limit of a Cauchy sequence of rational numbers. In mathematical analysis, Cauchy completeness can be generalized to a notion of completeness for any metric space. See complete metric space.
It is possible to prove the least-upper-bound property using the assumption that every Cauchy sequence of real numbers converges. Let S be a nonempty set of real numbers. If S has exactly one element, then its only element is a least upper bound. So consider S with more than one element, and suppose that S has an upper bound B 1.
To make this precise: a sequence (x n) in a metric space M is Cauchy if for every ε > 0 there is an integer N such that for all m, n > N, d(x m, x n) < ε. By the triangle inequality, any convergent sequence is Cauchy: if x m and x n are both less than ε away from the limit, then they are less than 2ε away from each other.
In particular, the test that a sequence is a Cauchy sequence allows proving that a sequence has a limit, without computing it, and even without knowing it. For example, the standard series of the exponential function = =! converges to a real number for every x, because the sums
In the real numbers every Cauchy sequence converges to some limit. A Cauchy sequence is a sequence whose terms become arbitrarily close together as n gets very large. 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 ...