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Application of Bolanzo-Weierstrauss

  1. Aug 31, 2011 #1
    Let {xn}n = 1[itex]\infty[/itex] be a bounded sequence

    and

    {xnj} be a convergent subsequence, each one converging to L.

    Want to show that {xn[}[/itex]n = 1[itex]\infty[/itex] converges to L.

    My proof is as follows.

    Suppose that {xn}n = 1[itex]\infty[/itex] does not converge to L; this implies that there is a subsequence |{xnj}-L|[itex]\geq[/itex][itex]\epsilon[/itex]. However by B-W there exists a subsequence of that subsequence that converges, and it must converge to L. However this is a contradiction.

    Is this sufficient?
     
  2. jcsd
  3. Sep 1, 2011 #2

    micromass

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    What exactly is it that you need to prove??
     
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