A tower of irrationals?

Question

Set $x_0:=1$ and for every $n\in\mathbb{N}$ set $x_n:=2^{\frac{1}{2}x_{n-1}}$, so $$x_1=\sqrt{2},\; x_2=\sqrt{2}^{\sqrt{2}}, x_3=\sqrt{2}^{\sqrt{2}^{\sqrt{2}}},\ldots$$ Undoubtedly the $x_n$ are irrational for all $n\in\mathbb{N}=\{1, 2,\ldots\}$. (How) can we prove that?