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