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Math Help - Proving limit

  1. #1
    Junior Member mremwo's Avatar
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    Proving limit

    Show that if lim(\frac{a_{n}}{n}) = L for L>0, then lim(a_{n}) = + \infty

    Here's what I did:

    lim(\frac{a_{n}}{n}) = \frac{lim(a_{n})}{lim(n)} = L

    Then \\ lim(a_{n}) = L \cdot  lim(n)

    And since  \\ lim(n) goes to +\infty and L is a constant greater than 0,
    the lim(a_{n}) also goes to + \infty

    However, I'm pretty sure I need to use the definition of convergence to proves this, but I'm not sure how. Thank you.
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  2. #2
    Senior Member roninpro's Avatar
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    What you have might serve as some intuition, but it does not work as a proof. You used the limit laws to rearrange the expressions, but they only work when the corresponding limits exist. Here is a slightly more formal argument:

    Since \lim_{n\to \infty} a_n/n = L, we know that given any \varepsilon>0, when n is sufficiently large, we have |a_n/n-L|<\varepsilon. Multiplying through by n gives |a_n-nL|<n\varepsilon. Breaking the absolute values and rearranging, n(L-\varepsilon)<a_n.

    This shows that as long as we pick \varepsilon<L and take n\to \infty, we have n(L-\varepsilon)\to \infty, so a_n\to \infty.
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  3. #3
    Junior Member mremwo's Avatar
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    Great, thanks! I had this all the way down to |a_n - nL| < ne but I always seem to forget that I am able to pick e to fit what I need so long as it is greater than 0.
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