| 1. | In the real numbers every Cauchy sequence converges to some limit.
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| 2. | In these cases, the concept of a Cauchy sequence is useful.
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| 3. | In metric spaces, one can define Cauchy sequences.
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| 4. | In a similar way one can define Cauchy sequences of rational or complex numbers.
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| 5. | Clearly, any sequence with a modulus of Cauchy convergence is a Cauchy sequence.
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| 6. | In a general metric space, however, a Cauchy sequence need not converge.
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| 7. | By definition, in a Hilbert space any Cauchy sequence converges to a limit.
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| 8. | The usual decimal notation can be translated to Cauchy sequences in a natural way.
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| 9. | Thus one may speak of uniform continuity, uniform convergence, and Cauchy sequences.
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| 10. | Cauchy completeness is related to the construction of the real numbers using Cauchy sequences.
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