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Math Help - Cauchy sequences

  1. #1
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    Cauchy sequences

    Suppose x_n is a Cauchy sequence in a metric spare X, and some subsequence x_k converges to a point p \in X. Show the full series converges to p.

    Is this correct?

    First prove convergence of x_n. Find an integer N such that \forall a,b \geq N, d(x_a, x_b) \leq \epsilon/2. Now set N so that  d(x_k,x) \geq \epsilon /2 for x_k \geq N. Now replace x_b with x_k, then \epsilon \geq d(x_a,x_k) +d(x_k,p) \geq d(x_a,x)

    Now prove that the sequence goes to p. Assume x _n \rightarrow q and d(p,q) \leq \epsilon for \epsilon \geq 0. Pick an integer N such that d(x_n,x_{n+1}) \geq \epsilon This cannot be possible since it would violate the fact that x_n is Cauchy, so d(p,q)=0 \,\,and \,\,p=q
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  2. #2
    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by terr13 View Post
    Suppose x_n is a Cauchy sequence in a metric spare X, and some subsequence x_k converges to a point p \in X. Show the full series converges to p.

    Is this correct?

    First prove convergence of x_n. Find an integer N such that \forall a,b \geq N, d(x_a, x_b) \leq \epsilon/2. Now set N so that  d(x_k,x) \geq \epsilon /2 for x_k \geq N. Now replace x_b with x_k, then \epsilon \geq d(x_a,x_k) +d(x_k,p) \geq d(x_a,x)

    Now prove that the sequence goes to p. Assume x _n \rightarrow q and d(p,q) \leq \epsilon for \epsilon \geq 0. Pick an integer N such that d(x_n,x_{n+1}) \geq \epsilon This cannot be possible since it would violate the fact that x_n is Cauchy, so d(p,q)=0 \,\,and \,\,p=q
    maybe this would help..
    d(x_n, p) \leq d(x_n, x_k) + d(x_k,p)
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