convergence of a recursive sequence and calculate the limit

how to show that the sequence $x_{n+1}=\frac{x_n}{2}+\frac{5}{x_n}$, $x_1=2$ is convergent.
I tried to prove using induction that it is bounded but couldn't work it out.
Only thing i could figure out is that the limit of sequence is $\sqrt{10}$ so it is convergent.