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Proving properties of lim sup. Good attempt at proof!

  1. Oct 13, 2012 #1
    [/itex]1. The problem statement, all variables and given/known data
    Is the following statement true or false: 'if (cn) and (dn) are bounded sequences of positive real numbers then:

    lim sup (cndn) = (lim sup cn)(lim sup dn)


    2. Relevant equations



    3. The attempt at a solution
    for all n in the positive reals. cn and dn are bounded.

    Since cn and dn are bounded we know they converge.

    Hence, by the theorem lim(cndn)=lim cnlim dn we can say that lim(cn)=lim sup (cn) and lim(dn)=lim sup (dn)

    Hence, lim sup (cndn) = (lim sup cn)(lim sup dn)

    HENCE, TRUE!!

    ******I am not sure about line 2 where I say bounded => converge. I know that you can say that converge means bounded but I don't know if I can do the reverse.*******
     
  2. jcsd
  3. Oct 13, 2012 #2
    You can't. You can say that a bounded monotone sequence converges and that a bounded sequence has a convergent subsequence. But a bounded sequence need not converge. You should be able to come up with a pretty simple example of a bounded sequence that does not converge (at least now that you know there is one). In the process of doing so, you may stumble across the answer to your problem.
     
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