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Math Help - Monotone Sequence

  1. #1
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    Monotone Sequence

    Hi. I was wondering if someone could show me how to prove this statement,
    let P > 0, prove that (P^(1/n) is a monotone sequence and deduce by using a subsequence that the limit must be 1.

    I understand that if 0 < P < 1, then the sequence is increasing and if P > 1, the sequence is decreasing, but how do you show it? Also, how I do use the subsequence to prove its limit?

    Thanks for your help.
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  2. #2
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    There are many ways to prove the proposition. I suspect that whoever set this for you to do has one particular way in mind. However, here is one way that I like.
    Given p > 1\quad  \Rightarrow \quad \ln (p) > 0.
    Then \begin{array}{rcl}<br />
 0 < a < b\quad  & \Rightarrow & \quad a\ln (p) < b\ln (p) \\ <br />
 \quad  & \Rightarrow & \quad \ln \left( {p^a } \right) < \ln \left( {p^b } \right) \\ <br />
 \quad  & \Rightarrow & \quad p^a  < p^b  \\ <br />
 \end{array}.

    This shows that the sequence p > 1,\quad \left( {p^{\frac{1}{n}} } \right) is decreasing; it is also bounded below by 1. I would use a subsequence converging to 1.
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  3. #3
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    What's an appropriate subsequence?
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  4. #4
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    I would just use the factor that (say a\geq 1) 1\leq a\leq n for sufficiently large n it means 1\leq a^{1/n} \leq n^{1/n} now use the squeeze theorem.
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  5. #5
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    Honestly though, what's an easy subsequence of this? We don't know what P is, so I'm not even sure what the terms in this sequence look like. Can you just pick a P and make that a subsequence? Somehow I don't think so.
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  6. #6
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    Don't worry about it. I figured something out.
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