I've got $\sum_{i} F_+(i) \ge k \sum_{i} G(i)$ and it says that implies there's an $i$ such that $F(i) \ge k G(i)$, $F_+$ is the positive part of $F$ and $\sum_{i} F(i) = 0$. How does it follow?
From Thomas Andrews I realized $\sum_{i} F_+(i) = \sum_{i,F(i)\ge 0} F(i) \ge k \sum_{i} G(i).$ This seems like it could be useful.