Can you tell me, why the following is true: Let $C$ be a smooth, complete curve over an algebraically closed field. Let $D=P_1+...+P_n$ be an effective divisor that is linear combination of (not necessarily different) points $P_1,..., P_n \in C$. Let $P_n$ be not a basepoint of the linear system of $D$. Can you tell me why the dimension of $H^0(O_C(D))$ is smaller by at least $1$ than the dimension of $H^0(O_C(DP_n))$, i.e. the dimension of the global sections drops by $1$ when removing $P_n$.

This is exactly Proposition 3.1(a) in Chapter IV of Hartshorne. The proof, in my opinion, provides a good explanation of why the result is true. 

