Question 5
Given $n$ real numbers $\{a_1, a_2, \ldots, a_n\}$, prove that you can find $n$ integers $\{b_1, b_2, \ldots, b_n\}$, such that $|a_i - b_i| \lt 1$ and the sum of any subset of the original numbers differs from the sum of the corresponding $\{b_i\}$ by at most $(n + 1)/4$.