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Math Help - Lim Sup and Lim Inf Proof

  1. #1
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    Lim Sup and Lim Inf Proof

    Prove that for any positive sequence a_n of real numbers
    lim inf (a _n+1/a_n) <= lim inf (a_n)^(1/n) <= lim sup (a_n)^(1/n)
    <= lim sup(a_n+1/a_n).
    Give examples where equality does not hold.
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  2. #2
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    Re: Lim Sup and Lim Inf Proof

    Quote Originally Posted by veronicak5678 View Post
    Prove that for any positive sequence a_n of real numbers
    lim inf (a _n+1/a_n) <= lim inf (a_n)^(1/n) <= lim sup (a_n)^(1/n)
    <= lim sup(a_n+1/a_n).
    Give examples where equality does not hold.
    The first inequality is invalid. For instance, let a_n=2^n. Then

    \liminf\left(a_n+\frac{1}{a_n}\right)=\liminf\left  (2^n+\frac{1}{2^n}\right)=\infty

    >2=\liminf\left((2^n)^{1/n}\right)=\liminf\left((a_n)^{1/n}\right).

    Even if you meant \liminf\left(\frac{a_n+1}{a_n}\right) instead of \liminf\left(a_n+\frac{1}{a_n}\right), the inequality still doesn't hold (e.g. when a_n=(1/2)^n).
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  3. #3
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    Re: Lim Sup and Lim Inf Proof

    I meant a_(n+1). The (n+1)th element. Sorry it's so unclear.
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  4. #4
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    Re: Lim Sup and Lim Inf Proof

    I think the sequence must be bounded too, right? If so, I know how to do it.

    veronicak5678, can you confirm that?
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