$x_1=\sqrt{2}$ and we have $x_{n+1}=(\sqrt{2})^{x_n}$. Find out the limit.

Suppose we have $ x_1=\sqrt{2}$ and we have $ x_{n+1}=(\sqrt{2})^{x_n}$ . We have to find out the limit of the sequence.

I was observing that $ x_2=(\sqrt{2})^\sqrt{2}$ and $ x_3=(\sqrt{2})^{{\sqrt{2}}^{\sqrt{2}}}$ and so on now it is clear that the function is increasing. Now how to prove that the sequence is convergent and then how to find the limit? The calculative value is going towards $ 2$ . But how to prove it!!