Additive monotone function $g: (0, \infty) \to \mathbb{R}$ satisfies $g(s) = sg(1)$

Question

I'm struggling to prove the following assertion:

Let $g: (0, \infty) \to \mathbb{R}$ be a monotone function satisfying $g(s + t) = g(s) + g(t)$ for all $s,t > 0$. Then $g$ satisfies $g(s) = sg(1)$ for all $s > 0$.

I don't really know where to start, so any help is greatly appreciated.