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Multiply infinity by a positive number

  1. Oct 31, 2011 #1
    1. The problem statement, all variables and given/known data
    Prove that if limXn = +∞ and limYn>0 then limXnYn=+∞


    3. The attempt at a solution
    limXnYn = limXnlimYn = (c)(+∞) where c is a positive real number

    I know in my head that a positive number multiplied by infinity is positive, but I am unsure how to prove this and we have not yet done this particular example in class.
     
  2. jcsd
  3. Oct 31, 2011 #2

    Deveno

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    Science Advisor

    instead of thinking of ∞ as a number, think of:

    [tex]\lim_{n \to \infty} x_n = \infty[/tex]

    meaning, no matter how large a positive real number N we choose, for all large enough n, xn > N.

    now, suppose

    [tex]\lim_{n \to \infty} y_n = L > 0[/tex]

    for large enough n, we can ensure that yn > L/2 > 0.

    can we make xnyn larger than any positive real number M?

    (what happens if we pick n so that xnis larger than 2M/L, and yn is larger than _____?)
     
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