Is the sequence $x_n$ convergent, where $x_n= a_n - \sqrt{2n}$ and $a_n=a_{n-1}+\frac{1}{a_{n-1}}$

Question

A sequence $\{x_n\}$ is given by $x_n= a_n-\sqrt{2n}$ and $a_n$ is given by $a_1=1$ and $a_n=a_{n-1}+\frac{1}{a_{n-1}}$.

Please help. I could not find a way out.