21. Essentially, this method defines a real number to be the limit of a Cauchy sequence of rational numbers. 22. By construction, every real number " x " is represented by a Cauchy sequence of rational numbers. 23. Then the reals are defined to be the sequences of rationals that have the Cauchy sequence property using this distance. 24. In mathematical analysis, the rational numbers form a completion, using Cauchy sequences , Dedekind cuts, or infinite decimals. 25. The vector space structure allows one to relate the behavior of Cauchy sequences to that of converging absolutely convergent series in converges, 26. The Banach space is "'weakly sequentially complete "'if every weakly Cauchy sequence is weakly convergent in. 27. A metric space M is said to be "'complete "'if every Cauchy sequence converges in M. 28. More formally, for a given prime, the complete in the sense that every Cauchy sequence converges to a point in. 29. A Cauchy sequence is a sequence whose terms ultimately become arbitrarily close together, after sufficiently many initial terms have been discarded. 30. There are at least two popular ways to achieve this step, both published in 1872 : Dedekind cuts and Cauchy sequences .
