PL-OM 2010 Problem 6

Real number $C > 1$ is given. Sequence of positive real numbers $a_1, a_2, a_3, \ldots$ , in which $a_1=1$ and $a_2=2$ , satisfy the conditions $$a_{mn}=a_ma_n,$$ $$a_{m+n} \leq C(a_m + a_n),$$ for $m, n = 1, 2, 3, \ldots$ . Prove that $a_n = n$ for $n=1, 2, 3, \ldots$ .