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Test Review 1 - lim sups and lim infs

  1. Oct 15, 2005 #1

    I am taking Adv. Calc, and we have a test next week. I am going to post a few questions that I have from the review where I got stuck. If you have any help, please steer me in the right direction!

    Question 1: Suppose sn <= 0 <= tn for n in N. Prove
    (lim inf sn)(lim sup tn) <= lim inf (sntn), provided none of these products is of the form 0 * infinity.

    Here is what I have so far:
    Since sn <= 0, we must have lim inf sn <= 0.
    Also, since tn >= 0, we must have lim sup tn >= 0.
    Thus (lim inf sn)(lim sup tn) <= 0.

    We also know that (sntn) <= 0.
    This means that lim inf (sntn) <= 0.

    I am having difficulties at this point, because the two things that I want to compare are both <= 0, so I don't have a way of comparing them.

    I'm not sure where to go with this. Any suggestions?
  2. jcsd
  3. Oct 16, 2005 #2
    Would it be at all helpful to look at the fact that lim inf sn = -lim sup -sn, or is that just complicating matters?

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