suppose ${x_n}$ is a numerical sequence such that $$0<x_{n+m}\leq x_n+x_m$$ for all $n$ and $m\in \mathbb{N}$ and $x_1>0$. Prove that $lim_{n\rightarrow \infty }\frac{x_n}{n}$ exists.
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2I don't know how to prove the limit exists, I only know that $\frac{x_n}{n}$ has a upper bound $x_1$ – Shine Jun 23 '14 at 10:38