Question 161
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
$$
\frac{a_{1}}{b_{1}} + \frac{a_{2}}{b_{2}} + \cdots + \frac{a_{n}}{b_{n}} = \left\lceil \frac{1}{b_{1}} + \frac{1}{b_{2}} + \cdots + \frac{1}{b_{n}} \right\rceil.
$$
$$
\frac{a_{1}}{b_{1}} + \frac{a_{2}}{b_{2}} + \cdots + \frac{a_{n}}{b_{n}} = \left\lceil \frac{1}{b_{1}} + \frac{1}{b_{2}} + \cdots + \frac{1}{b_{n}} \right\rceil.
$$