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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: $\lim_n \sup \frac{C_n}{2\sqrt{n}} \leq 1$

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    @Thijs: Ah, right. Not that it matters much, but it is interesting to note that the question was deleted by the OP himself -- and it was a list of four homework problems on which the OP worked hard according to a comment in the now deleted question and the question is now sold as "just encountered"... Anyway, the post I linked to should lead to a proof in a relatively straightforward way. Also, the $\limsup$ used to be $\leq 1$ instead of $= 1$ (the $\leq$ makes more sense).2011-09-11

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