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Sequence limits?

  1. Sep 9, 2009 #1
    Let lim n →∞ XnYn = 0. Is it true that Limn →∞ Xn= 0
    or Limn →∞Yn = 0 (or both)?
  2. jcsd
  3. Sep 9, 2009 #2
    lim n →∞ XnYn = lim n →∞ Xn * lim n →∞ Yn

    One or even both limits need to be 0 so that lim n →∞ Xn * Yn =0

    I guess you are right. :smile:
  4. Sep 9, 2009 #3


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    Well, this is only true when both xn, and yn have limits. So, what if they don't? :wink: Say, what if they're oscillating?
  5. Sep 9, 2009 #4

    This equation is only correct if the individual limits on the right-hand side exist. The proposition in the original post is false and can be disproven with a counterexample.
  6. Sep 10, 2009 #5
    Yup, sorry I forgot to mention that the series must converge, in other way they would be divergent and we could not separate them.
  7. Sep 10, 2009 #6
    Simple counterexample. Clearly, [tex]\lim 1/n = \lim \(n \cdot 1/n^2 \) = 0[/tex] but we have that [tex]\lim n = +\infty[/tex] and [tex]\lim 1/n^2 = 0[/tex].
  8. Sep 10, 2009 #7
    This example satisfies the original claim that at least one of the sequences converges to 0. A counter example for this must consist of two sequences whose product vanishes but both sequence do not converge to 0.

  9. Sep 11, 2009 #8


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    You've misquoted yet again, Elucidus.. :cry: :cry:

    You shouldn't forget to wear your glasses, then.. :wink:
  10. Sep 11, 2009 #9
    Indeed. I'm surprised. I was trying to quote this post:

    To which my original comment makes more sense. Sorry for any confusion.

  11. Sep 11, 2009 #10
    Let [itex]x_n [/itex] be alternately 1 and 0, let [itex]y_n [/itex] be alternately 0 and 1.
    Then [itex]x_n y_n = 0 [/itex] for all n, but neither individual limit exists. In particular, neither individual limit is zero.
  12. Sep 11, 2009 #11


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    Excellent example. However, neither [itex]x_n[/itex] nor [itex]y_n[/itex] converges and the O.P. finally told us.
  13. Sep 11, 2009 #12
    HallsofIvy I am not the O.P, mate. :biggrin: I was just adding, additional explanation of my first post. :cool:
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