Question 20
Let $A$ be a non-empty set of positive integers. Suppose that there are positive integers $b_1, \ldots, b_n$ and $c_1, \ldots, c_n$ such that
(i) for each $i$ the set $b_i A + c_i = \{b_i a + c_i : a \in A\}$ is a subset of $A$, and
(ii) the sets $b_{i}A + c_{i}$ and $b_{j}A + c_{j}$ are disjoint whenever $i \neq j$. Prove that
$$ \frac{1}{b_{1}}+\cdots+\frac{1}{b_{n}}\leq1. $$
(i) for each $i$ the set $b_i A + c_i = \{b_i a + c_i : a \in A\}$ is a subset of $A$, and
(ii) the sets $b_{i}A + c_{i}$ and $b_{j}A + c_{j}$ are disjoint whenever $i \neq j$. Prove that
$$ \frac{1}{b_{1}}+\cdots+\frac{1}{b_{n}}\leq1. $$