Find $\lim\limits_{n\to+\infty}(u_n\sqrt{n})$

A solution from a friend of mine:

(i) Show $u_{n} > -1$ for all $n$. (Easy)
(ii) If $u_{0} = 2 - 2t$, where $0 \le t \le 1$ then $u_{n} < (n+2)(1-t)$ for all $n > 0$. (Induction)
(iii) There exists integer $K > 0$ s.t. $-1 < u_K < 1$. From (ii) we get that eventually $u_{n-1} < n$, whence $u_n < n$, and $u_{n+1} < n-1 $, etc.
$\text{(iv) } |u_n| \leqslant 1/n\text{ for all } n < K\text{.}\\\text{Therefore the limit is 0.}\\\text{I let the OP to complete the details. (to prove (i) and (ii)).}\\\text{Q.E.D. (Chris)}$