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.
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 . Then
.
Even if you meant instead of , the inequality still doesn't hold (e.g. when ).