Operator in R^n

Let $A\in{B(R^n)}$ be linear invertible operator. Prove that $\forall\ x\in {R^n},\ \|x\|=1[$ ,

$<br /> \exists i\in\{\pm1,\ldots,\pm n\}$ , such that $\|A^i(x)\|\geq 1/n$ .

Prove that constant $1/n$ can't be increased. Prove also that if $S\subset\{\pm1,\ldots,\pm n\}$ is proper subset, then for
arbitrary $\varepsilon>0$ exists $A$ and $x$ so that $\|A^i(x)\|<\varepsilon,\ \forall\ i\in S$ .