Let $\left(u_{n}\right)_{n \geqslant 0}$ be a real sequence such that: $\forall (m,n) \in \mathbb{N}^{2}, \quad u_{m+n} \geqslant u_{m} + u_{n}$. We suppose that the set $\left\{\frac{u_{n}}{n}, n \in \mathbb{N}^{*}\right\}$ is bounded above and we denote by $s$ its supremum.
Let $m$, $q$ and $r$ be elements of $\mathbb{N}$. We set $n = mq + r$. Compare the two real numbers $u_{n}$ and $qu_{m} + u_{r}$ and show that $u_{n} - ns \geqslant q\left(u_{m} - ms\right) + u_{r} - rs$.