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

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

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