Again let \(X\) be a curve, and let \(P_1, \ldots, P_r \in X\) be points. Then there is a rational function \(f \in K(X)\) having poles (of some order) at each of the \(P_i\), and regular elsewhere.
Solution
Let \(D= P_1 + \cdots + P_r\), we then want \(f\in \globsec{X; \mcl(nD)}\) for \(n\gg 0\). By RR, \[ \chi(\mcl(nD)) = \deg D + 1-g \implies h^0(\mcl(nD)) = rn + 1 - g ,\] which is non-negative for \(n\gg 0\).