We denote by $\kappa = \lambda _ { N } / \lambda _ { 1 }$. Show that the real number $\alpha$ from question 22 equals $\alpha = \frac { \sqrt { \kappa } - 1 } { \sqrt { \kappa } + 1 }$ and deduce that
$$\left\| e _ { k } \right\| _ { A } \leq 2 \left\| e _ { 0 } \right\| _ { A } \left( \frac { \sqrt { \kappa } - 1 } { \sqrt { \kappa } + 1 } \right) ^ { k }$$