Prove there exists $\lim_\limits{n\to \infty}{a_n}$ and find $\lim_\limits{n\to \infty}{a_n\over n}$. Let $(a_n)$ be defined by $a_1=1$, $a_{n+1}=a_n+{1\over a_n}$. Prove there exists $\lim_\limits{n\to \infty}{a_n}$ and find $\lim_\limits{n\to \infty}{a_n\over n}$. 
I tried many things but they wouldn't work. I would appreciate your help.

Just for a reference, we can prove that
$$ a_n = \sqrt{2n} + \frac{\log n}{4\sqrt{2n}} + O \left( \frac{1}{\sqrt{n}} \right). $$
The idea is essentially the same as @LeGrandDODOM's proof, but in this case we successively improve the error term.
