Do you mean:
Remember that: Every monotonic, bounded sequence is convergent.
Use induction to show that (i.e. it's bounded).
And you can use induction again to show that is true for all n (i.e. it's monotonic).
Now this proves convergence. So, let (we can assume this because we know converges to some limit which we call L).
Now take the limit from both sides of the recursive sequence:
and solve the quadratic, picking the appropriate solution.