Lim Sup and Lim Inf Proof

**veronicak5678** · August 19th 2011, 08:12 PM

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.

**hatsoff** · August 20th 2011, 03:19 AM

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 $a_n=2^n$ . 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 $a_n=(1/2)^n$ ).

**veronicak5678** · August 20th 2011, 08:34 AM

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

**dgomes** · August 22nd 2011, 03:08 PM

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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