Finding equivalent of an iterative sequence

Question

Define $(u_n)_{n\in\mathbb{N}^*}$:

$$u_1 =1 , u_{n+1}=1+\dfrac{n}{u_n}$$

Find an asymptotic equivalent of $u_n$ when $n\to+\infty$.

I guess that the answer should be $\sqrt{n}$, but I couldn't prove it...

Answer 1

You could try to show by induction that $\sqrt{n}<u_{n}<1+\sqrt{n}$ or something like that.