Question 163
Let $m$ and $n$ be positive integers with $m \gt n \ge 2$. Set $S = \{1, 2, \dots, m\}$, and $T = \{a_1, a_2, \dots, a_n\}$ is a subset of $S$ such that every element in $S$ is divisible by at most one element in $T$. Prove that
$$
\frac{1}{a_1} + \frac{1}{a_2} + \dots + \frac{1}{a_n} \lt \frac{m+n}{m}.
$$
$$
\frac{1}{a_1} + \frac{1}{a_2} + \dots + \frac{1}{a_n} \lt \frac{m+n}{m}.
$$