11. Weakly Cauchy sequences in are weakly convergent, since-spaces are weakly sequentially complete. 12. Completeness can be proved in a similar way to the construction from the Cauchy sequences . 13. Every Cauchy sequence is bounded, although Cauchy nets or Cauchy filters may not be bounded. 14. This obviously defines two Cauchy sequences of rationals, and so we have real numbers and. 15. Instead of working with Cauchy sequences , one works with Cauchy filters ( or Cauchy nets ). 16. The truncations of the decimal expansion give just one choice of Cauchy sequence in the relevant equivalence class. 17. The concept of completeness of metric spaces, and its generalizations is defined in terms of Cauchy sequences . 18. The existence of limits of Cauchy sequences is what makes calculus work and is of great practical use. 19. Since the metric space is complete this Cauchy sequence converges to some point " x ". 20. A definite meaning is given to these sums based on Cauchy sequences , using the absolute value as metric.
