## Prove that there exists a nonempty subset $I$ of $\{1,2,…,n\}$ such that $\sum_{i\in I}{\frac {1}{b_i}}$ is an integer

Let $$a_1,a_2,…,a_n$$ and $$b_1,b_2,…,b_n$$ be positive integers such that any integer $$x$$ satisfies at least one congruence $$x\equiv a_i\pmod {b_i}$$ for some $$i$$. Prove that there exists a nonempty subset $$I$$ of $$\{1,2,…,n\}$$ such that $$\sum_{i\in I}{\frac {1}{b_i}}$$ is an integer. – (Problems from the book, chapter 17)

This is a solution I have found on AoPS:

Solution a la Vess: Consider $$\prod_j(e^{2\pi i\frac{x-a_j}{b_j}}-1)=\sum_I\pm e^{2\pi i (A_I x+B_I)}$$. Looking at the left hand side, we see that its average over $$\mathbb Z$$ (understood as $$\lim_{N\to\infty}\frac1{2N+1}\sum_{-N}^N$$) is $$0$$. Looking at the right hand side, we see that if no $$A_I$$ with $$I\ne\varnothing$$ is an integer, then it is $$(-1)^n$$.

Is this answer correct? If so, how can I understand this solution ? What is $$A_I$$ and $$B_I$$ ? Are there any other solutions for this problem ?