I am stuck with this problem

Show that for real functions $\displaystyle f$ and $\displaystyle g$ such that $\displaystyle f+g<x$, there exists rationals $\displaystyle r$ and $\displaystyle s$ such that $\displaystyle r+s<x, f<r$ and $\displaystyle g<s$

Would using the Archimedean principle help?

Let $\displaystyle \alpha \in \mathbb{Q}$ and $\displaystyle \alpha>0$.

By the Archimedean principle, there exists $\displaystyle k \in \mathbb{Z}$ such that

$\displaystyle k\alpha \leq x < (k+1)\alpha$

Is this in the right direction?

Thanks in advance