 Solution

Write $F={\sum }_C{\min }_{G\in |H|}{\mathrm{mult}}_C(G)C$. This is the fixed part of a linear system. The movable part of a linear system is $|M|$, with $M=H-F$; multiplication by the defining section of $F$ identifies its sections with those of $H$. In particular $|H|=F+|M|$, and no curve occurs in every member of $|M|$. The occurrence of $|F|$ in the definition request is understood as $|H|$.

For any integral curve $C$, choose a member $G\in |M|$ not containing $C$. Their intersection number is a sum of nonnegative local intersection multiplicities, so $M\cdot C=G\cdot C\ge 0$. Therefore the movable divisor $M$ is nef. If $M=0$, the same assertion is immediate.