cauchy sequence in a sentence

Examples

1. The definition of real numbers as Cauchy sequences was first published separately by Eduard Heine and Georg Cantor, also in 1872.
2. There are computer applications of the Cauchy sequence, in which an iterative process may be set up to create such sequences.
3. Uniform spaces do not introduce distances, but still allow one to use uniform continuity, Cauchy sequences, completeness and completion.
4. Prove that in a normed field the following assertion holds : Let be a Cauchy sequence, but not a null sequence.
5. For 2 ), take a real number r and show that f _ i ( r ) is a Cauchy sequence.
6. Every Cauchy sequence of real numbers is bounded, hence by Bolzano-Weierstrass has a convergent subsequence, hence is itself convergent.
7. In constructive mathematics, Cauchy sequences often must be given with a " modulus of Cauchy convergence " to be useful.
8. We define \ overline { K } to be the set of Cauchy sequences in K modulo Cauchy sequences that converge to zero.
9. We define \ overline { K } to be the set of Cauchy sequences in K modulo Cauchy sequences that converge to zero.
10. Another approach is to define a real number as the "'limit of a Cauchy sequence of rational numbers " '.