Let $(u_n)$ be a sequence defined by:
$\begin{equation} \left\{ u_0 \geq 0 \\ \forall n \in \mathbb{N}^*, u_n = \sqrt{n+u_{n-1}} \right. \end{equation}$
I'd like to prove that when $n \rightarrow +\infty$ :
$u_n \sim \sqrt n$
This would basically mean that : $\lim_{n\rightarrow\infty}\frac{u_n}{\sqrt{n}} = 1$ That's to say : $\lim_{n\rightarrow\infty}\sqrt{\frac{n+u_{n-1}}{n}} = 1$
Well, we can't replace $u_{n-1}$ and go on down to $u_0$... The result seems quite logic though I have no idea how I can really prove that.