## Puzzle | Pell numbers – inequality

To prove the inequality:

Hint :

For all real numbers   $x$   and   $y$   with   $0 < y < \pi/2$,

we have

$\sin x + \cos x \; = \; \sqrt{2} \: \sin (x + \pi/4) \: \leq \: \sqrt{2} \: < \: 3/2 \: \leq \: 3/2 \: \sec y$