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Homework Help: Proof or Counterexaple that a_n and b_n converge

  1. Feb 10, 2014 #1
    1. The problem statement, all variables and given/known data

    If a_n+b_n and a_n-b_n converge then a_n and b_n must converge.
    This MAY be a counterexample problem.

    2. Relevant equations


    3. The attempt at a solution

    I have tried the following possibilities out of a_n and b_n and none of these give a counterexample:
    Let a_n and b_n= (-1)^n , 1/n , 1/n^2 , (-1)^2n , sqrt(n)
    I need to come up with something similar to this that holds true that a_n+b_n converge, a_n-b_n converge, but a_n and b_n alone do not converge.
    Last edited: Feb 10, 2014
  2. jcsd
  3. Feb 10, 2014 #2


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    How do you know there is a counterexample?
  4. Feb 10, 2014 #3


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    Why are you so sure there is a counterexample?
  5. Feb 10, 2014 #4
    We have been doing similar problems and they were all counterexamples, but there is a possibility that it is not.
  6. Feb 10, 2014 #5
    It may not actually be. Thanks, I have updated my question. We were doing lots of counterexamples and it led me to believe that was one of them.
  7. Feb 10, 2014 #6
    You should try proving it. Because it is true.
  8. Feb 10, 2014 #7
    Hint: the two rows converge for n>N and for n>M respectively, what happens if you look at n>(N+M) and add the two sequences?
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