I have just encountered the following question: Let $C_n$ be a sequence of real numbers with the following three properties: 1) $C_n$ is subadditive, such that $C_{m+n} \leq C_m +C_n$
2) $C_n=O(\sqrt{n})$
3) $C_{q+1} \leq 2 \sqrt{q}$ for every prime power $q=p^m$
But I have no idea what's the use of the first condition in order to prove that: $\lim_n \sup \frac{C_n}{2\sqrt{n}} \leq 1$