Let $X$ be a connected proper smooth curve over an algebraically closed field $k$. Given a divisor on $X$ i.e. a finite linear combination of closed points $$D=\sum n_i x_i,\enspace\ n_i \in \mathbb{Z}, i \in I \text{ finite}$$ and recalling that the structure sheaf of $X$ at every point is has as stalks DVR's we can construct the sheaves,
$$\mathcal{O}_X(D)(U)=\{ f \in K | v_x(f) \geq -n_x \text{ for all } x \in U\}$$ where $v_x$ denotes the valuation of $\mathcal{O}_{X,x}$. We say that $D$ is an effective divisor if $n_i >0$ for all $i \in I$. Then it is clear that $D$ is effective we have $1 \in \Gamma(X,\mathcal{O}_X(D))$ as a global section. I am trying to find a proof for the converse. I would like to show that only effective divisors have non trivial global sections.
As a partial result I know that if $D=-\sum_i n_ix_i$ with $n_i \geq 0$ then $\mathcal{O}(D)$ is a subsheaf of the structure sheaf which has global sections equal to $k$. Clearly no element in $f \in k$ will satisfy $v_x(f)>0$ since after mapping $f$ to a stalk ring it will be a unit and cannot be contained in the maximal ideal. How can I finish with the general case?