KZ-MO 2016 Problem 6

Given a strictly increasing infinite sequence $\{a_n\}$ of positive real numbers such that for any $n\in N$ : $$a_{n+2}=(a_{n+1}-a_{n})^{\sqrt{n}}+n^{-\sqrt{n}}$$ Prove that for any $C>0$ there exist a positive integer $m(C)$ (depended on $C$ ) such that $a_{m(C)}>C$ .