Let $\omega:[0,\infty)\to[-\infty,\infty)$ be subadditive. How can we show that $$[0,\infty)\ni t\mapsto\frac{\omega(t)}t\tag 1$$ is monotonically decreasing?
Let $t>0$. Then, $$t=kq$$ for some $k\in\mathbb N$ and $q\in(0,1]$. Since $\omega$ is subadditive, $$\frac{\omega(t)}t\le\frac{\omega(q)}q\tag 2$$ as pointed out by Oles Wohnzimmer.
Can we conclude that $(1)$ is monotonically decreasing? If $0<s\le t$, then $$t=\underbrace{\left\lfloor\frac ts\right\rfloor}_{=:\:n}s+\underbrace{\left(t-\left\lfloor\frac ts\right\rfloor s\right)}_{=:\:r}\;.\tag 3$$ If $r=0$, then $$\frac{\omega(t)}t\le\frac{\omega(s)}s\tag 4$$ follows as in $(2)$. How can we prove $(4)$, if $r\in(0,s)$?