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In any metric space, a Cauchy sequence is bounded (since for some N, all terms of the sequence from the N-th onwards are within distance 1 of each other, and if M is the largest distance between and any terms up to the N-th, then no term of the sequence has distance greater than + from ).
A metric space is said to be totally bounded if every sequence admits a Cauchy subsequence; in complete metric spaces, a set is compact if and only if it is closed and totally bounded. [2] Each totally bounded space is bounded (as the union of finitely many bounded sets is bounded).
This defines two Cauchy sequences of rationals, and so the real numbers l = (l n) and u = (u n). It is easy to prove, by induction on n that u n is an upper bound for S for all n and l n is never an upper bound for S for any n. Thus u is an upper bound for S. To see that it is a least upper bound, notice that the limit of (u n − l n) is 0 ...
(This limit exists because the real numbers are complete.) This is only a pseudometric, not yet a metric, since two different Cauchy sequences may have the distance 0. But "having distance 0" is an equivalence relation on the set of all Cauchy sequences, and the set of equivalence classes is a metric space, the completion of M.
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.
In a general metric space, however, a Cauchy sequence need not converge. In addition, for real-valued sequences that are monotonic, it can be shown that the sequence is bounded if and only if it is convergent.
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.
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.