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