Let $\{x_n\}$ be a sequence such that $\forall m,n\in\mathbb{N}$:
$$0\leqslant x_{m+n}\leqslant x_m+x_n$$
Prove that $\lim_{n\to\infty}\frac{x_n}{n}$ exists.
From the relation, we can show that
$$0\leq x_n\leq x_{n-1}+x_1\leq x_{n-2}+2x_1\leq\dots\leq nx_1$$
Hence
$$0\leq\frac{x_n}{n}\leq x_1$$ for all $n$. This means this sequence is bounded, so it has a convergent subsequence $\left\{\frac{x_{n_k}}{n_k}\right\}$ with
$$\lim_{k\to\infty}\frac{x_{n_k}}{n_k}=c$$
for some $c$. I want to show that the $\lim_{n\to\infty}\frac{x_n}{n}=c$, and I was stuck.
