Let $C$ be a smooth curve of genus $g$, let $L$ be a line bundle on $C$ and let $V\subseteq H^0(X,L)$ be a linear subspace of dimension $r$. Fix $p\in C$. Prove that $$\{\text{ord}_p\sigma\}_{\sigma\in V\setminus\{0\}}$$ is a set of exactly $r$ nonnegative integers.
I managed to prove through a brute-force computation that for $\mathbb{P}^1$ and $L=\mathcal{O}(d)$ a generic choice of a basis of $V$ yields $r$ distinct orders.