There are 2 claims in this Proposition that I can't see. In the proof of part b) he writes $(f) \geq -D_0$ thus $f$ gives a global section of $\mathcal L(D_0)$ How does this follow? I think having something like $f|_{U_i} \cdot f_i \in \mathcal O_X(U_i)$ gives it but I don't know how that follows from $(f) \geq -D_0$.
Then in part c) he says that since $X$ is a projective variety over an algebraically closed field $k$ the global sections are isomorphic to $k$. How can you show this?
