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Math Help - cauchy

  1. #1
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    cauchy

    Suppose that {xm} is a Cauchy sequence, and that 􏰫xm(k)􏰬 is a subsequence of {xm} which converges to some point p. Prove that {xm} itself must also converge to p.
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  2. #2
    MHF Contributor chiph588@'s Avatar
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    Let  \epsilon>0

    Since  \{x_n\} is Cauchy, we know for some  N>0 \;\; \forall\; n,m>N, \; d(x_n,x_m)<\tfrac{\epsilon}{2} .

    Since  \{x_{n_k}\} converges to  p we also know for some  M>0 \;\; \forall n_i,m>M, \; d(x_{n_i},p)<\tfrac{\epsilon}{2} \text{ and } d(x_{n_i},x_m)<\tfrac{\epsilon}{2} .

    By the triangle inequality  d(x_m,p)<\epsilon .
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