Question 29
Let $\mathcal{S}$ be a finite nonempty set of prime numbers. Let $1 = b_1 \lt b_2 \lt \cdots$ be the sequence of all positive integers whose prime divisors all belong to $\mathcal{S}$. Prove that, for all but finitely many positive integers $n$, there exist positive integers $a_1, a_2, \ldots, a_n$ such that
$$ \frac{a_{1}}{b_{1}}+\frac{a_{2}}{b_{2}}+\cdots+\frac{a_{n}}{b_{n}}=\left\lceil\frac{1}{b_{1}}+\frac{1}{b_{2}}+\cdots+\frac{1}{b_{n}}\right\rceil. $$
$$ \frac{a_{1}}{b_{1}}+\frac{a_{2}}{b_{2}}+\cdots+\frac{a_{n}}{b_{n}}=\left\lceil\frac{1}{b_{1}}+\frac{1}{b_{2}}+\cdots+\frac{1}{b_{n}}\right\rceil. $$