Question 189
Let $A$ be a set of positive integers and let $k$ be a positive integer. Set $A$ is said to be $k$-thin if there are $k$ positive integers $n_1, \dots, n_k$ so that for all $1 \le i \lt j \le k$ and all $a, a' \in A$, $n_i + a \ne n_j + a'$. Suppose that $A_r$ is a $k_r$-thin set for $1 \le r \le m$ and that $\cup_{r=1}^m A_r$ is the set of all positive integers. Show that
$$
\frac{1}{k_1} + \dots + \frac{1}{k_m} \ge 1.
$$
$$
\frac{1}{k_1} + \dots + \frac{1}{k_m} \ge 1.
$$