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Convergence of a sequence.

  1. Feb 26, 2012 #1
    1. The problem statement, all variables and given/known data
    Assume [itex] a_n [/itex] is a bounded sequence with the property that every convergent sub sequence of [itex] a_n [/itex] converges to the same limit a. Show that
    [itex] a_n [/itex] must converge to a.
    3. The attempt at a solution
    Could I do a proof by contradiction. And assume that [itex] a_n [/itex] does not converge
    to a. but then this would imply that there would be a sub sequence that did not converge
    to a and this is a contradiction because I could pick a sub sequence that converged to the same thing that [itex] a_n [/itex] did
  2. jcsd
  3. Feb 26, 2012 #2


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    Science Advisor

    Yes, that will work. Just fill in the details- why does the fact that [itex]a_n[/itex] does not converge to a imply that there exist a subsequence that does not converge to a? You will need to look at several cases- the sequence does not converge or it converges to some number other than a.
  4. Feb 26, 2012 #3
    Could I say that eventually a sub sequence will have the same end behavior as
    [itex] a_n [/itex] Or I could take 2 sub sequences that when put together would equal
    [itex] a_n [/itex] Sub sequences aren't like subsets in the sense that a sub set could equal the set itself.
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