Maybe I've been doing this stuff for too long tonight, but let's consider the following... let $X_1, \dots, X_n$ be numbers such that $0 < X_i \leq 1$, and assume $X
= \sum_{i=1}^{n}X_i$ satisfies $0 < X \leq 1$ as well. How do I know that $X_j \leq \dfrac{1}{n}$ for some $j$?