| 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.
|
