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