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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.

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    Do you mean $F_{+}(i)=F(i)$ if F(i)>0 and $0$ otherwise?2012-12-06

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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 $F(i) \geq kG(i)$.

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    That's great! Thank you very much!2012-12-06