DE-MO 2013 Problem 2

Let $\alpha$ be a real number with $\alpha>1$ . Let the sequence $(a_n)$ be defined as $$a_n=1+\sqrt[\alpha]{2+\sqrt[\alpha]{3+\ldots+\sqrt[\alpha]{n+\sqrt[\alpha]{n+1}}}}$$ for all positive integers $n$ . Show that there exists a positive real constant $C$ such that $a_n<C$ for all positive integers $n$ .