Is there a way to prove that the canonical divisor $W$ of an algebraic function field in one variable $F$ over a field $K$ (that is the function field of an algebraic curve) of genus $g>0$ is base point free, *without* using Clifford´s theorem?

Note that K is not necessarily algebraically closed! in that case I know how to solve the problem.

Equivalently, I have to show that for any place $P$ of $F/K$ there exists an holomorphic differential $\omega$ of $F/K$ such that $v_P(\omega)=0$, that is the support of the associated canonical divisor $(\omega)$ to $\omega$ does not contain the place $P$.