Question 167
Let $u_n$ be the least common multiple of the first $n$ terms of a strictly increasing sequence of positive integers $a_1, a_2, a_3, \dots, a_{1000}$. Prove that
$$
\sum_{k=1}^{1000} \frac{1}{u_k} \le 2.
$$
$$
\sum_{k=1}^{1000} \frac{1}{u_k} \le 2.
$$