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Subsequence Convergence

  1. Jan 15, 2008 #1
    1. The problem statement, all variables and given/known data

    Suppose that {Xn} is a sequence in R. Prove that Xn converges to a if and only if every subsequence of Xn converges to a.

    2. Relevant equations



    3. The attempt at a solution

    Let e>0, choose N in N st n >=N implies |Xn-a| <e. Since a subsequence, nk, is in N and n1<n2<n3...., then nk>=k for all k in N. So, k>= implies |Xnk-a|<e.

    That's the first part, but I can't figure out how to start the proof the other way around. i.e. how do you prove that a convergent subsequence implies a convergent sequence?
     
  2. jcsd
  3. Jan 15, 2008 #2

    EnumaElish

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    It seems to me the sequence must be Cauchy for it to work.
     
  4. Jan 16, 2008 #3

    morphism

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    Any sequence is a subsequence of itself. So it's a tautology really.
     
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