# Thread: Lim Sup and Lim Inf Proof

1. ## 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.

2. ## Re: Lim Sup and Lim Inf Proof

Originally Posted by veronicak5678
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 $\displaystyle a_n=(1/2)^n$).

3. ## Re: Lim Sup and Lim Inf Proof

I meant a_(n+1). The (n+1)th element. Sorry it's so unclear.

4. ## 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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