# Question about the proof that $e^{\sqrt 2}$ is irrational

Show that $e^{\sqrt 2}$ is irrational
$0<p\frac{(2n)!}{2^n}-qs_n\frac{(2n)!}{2^n}<\frac{2}{(2n+1)(2n+2)}\frac{(2n+3)^2}{(2n+3)^2-2}$
"But $\left(p\frac{(2n)!}{2^n}-qs_n\frac{(2n)!}{2^n}\right)\in\mathbb{N}$ which is a contradiction for large n. Thus s is irrational."