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.
I do know how to solve it with the pigeonhole principle.
Just use contraddiction: say that for each $i$ $F(i) < kG(i)$, take the sum: $\sum F(i) < \sum {kG(i)}$. Now $\sum_{F(i)\geq 0}{F(i)} \leq \sum F(i) < \sum {kG(i)}$, but this is a contraddiction of your initial hypothesis, so should exist at least an index $i$ such that $F(i) \geq kG(i)$.