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Thread: 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 $\displaystyle a_n=2^n$. Then

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

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

    Even if you meant $\displaystyle \liminf\left(\frac{a_n+1}{a_n}\right)$ instead of $\displaystyle \liminf\left(a_n+\frac{1}{a_n}\right)$, the inequality still doesn't hold (e.g. when $\displaystyle a_n=(1/2)^n$).
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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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