31. The definition of real numbers as Cauchy sequences was first published separately by Eduard Heine and Georg Cantor, also in 1872. 32. There are computer applications of the Cauchy sequence , in which an iterative process may be set up to create such sequences. 33. Uniform spaces do not introduce distances, but still allow one to use uniform continuity, Cauchy sequences , completeness and completion. 34. Prove that in a normed field the following assertion holds : Let be a Cauchy sequence , but not a null sequence. 35. For 2 ), take a real number r and show that f _ i ( r ) is a Cauchy sequence . 36. Every Cauchy sequence of real numbers is bounded, hence by Bolzano-Weierstrass has a convergent subsequence, hence is itself convergent. 37. In constructive mathematics, Cauchy sequences often must be given with a " modulus of Cauchy convergence " to be useful. 38. We define \ overline { K } to be the set of Cauchy sequences in K modulo Cauchy sequences that converge to zero. 39. We define \ overline { K } to be the set of Cauchy sequences in K modulo Cauchy sequences that converge to zero. 40. Another approach is to define a real number as the "'limit of a Cauchy sequence of rational numbers " '.
