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.