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.

Actually, the limit $\lim a_n$ does not exist. For, assume it does though, and let it equal $\ell$. Then, $a_{n + 1} = a_n + 1/a_n$ yields $\ell = \ell + 1 / \ell$, a contradiction.
We now show that $\lim a_n/n$ exists. Since for every $n$ we know that $0 < a_n < n$, we get that $0 < a_n/n < 1$. Observe also that $a_n/n$ is a decreasing sequence. Thus, $a_n/n$ is bounded and monotone, so it has a limit.
Also we can prove that for any $N$ there exists $\varepsilon_N > 0$ such that for all $n > N$ we have $a_n < n \varepsilon_N$. This would prove that $a_n/n \to 0$.
