I am reading the proof of Riemann-Roch theorem from Shafarevich's Basic Algebraic Geometry 1 (3rd edition), but I'm stuck on a Lemma on pg 215. It says
(II) Every divisor $D$ on $X$ is dominated by a divisor linearly equivalent to $mA$ for some integer $m$.
Here $X$ is a non-singular projective curve and $A$ is the divisor of poles of some $f \in k(X)$ (though I believe that's not important, it seems that the only important thing is that $A$ must be effective).
The book dismisses the proof as an easy verification but I have not been able to see it. Any help will be appreciated.