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Convergence of a series

  1. Sep 11, 2012 #1
    1. The problem statement, all variables and given/known data
    Consider sequences {an} and {bn}, where sequence {bn} converges to 0.
    Is it true that sequence {anbn} converges to 0?



    3. The attempt at a solution

    Proof.
    First I assumed (an) is bounded, and so there exists M > 0 such that |an| < M for all n 2
    {1, 2, 3, . . .}. Moreover, since lim bn = 0, there exists N such that, for all n≥ N,
    |bn − 0| <ε /M
    or, equivalently, |bn| < ε/M.
    Therefore, whenever n ≥ N, we have that
    |anbn − 0| = |anbn| = |an||bn| ≤M|bn| < M(ε/M)=ε

    Since the choice of  ε> 0 was arbitrary, this implies that lim(anbn) = 0.

    However, I know that {anbn} can also not converge if {an} is not bounded. Can someone help with how to go about that part of hte proof?
    Thank you!
     
  2. jcsd
  3. Sep 11, 2012 #2
    It obviously doesn't work when an is not bounded. I am understanding correctly that an does not converge to 0, right?
    if so, just consider bn=1/n and an=n^2. an*bn=n does not convergen. Bam!
     
  4. Sep 11, 2012 #3
    There isn't given any information on if {an} converges to 0 or not. Thanks
     
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