Question 9
Given a set $S$ of positive integers, prove that at least one of the following two statements holds:
(1) There exist distinct finite subsets $F$ and $G$ of $S$ such that
$$
\sum_{x \in F} \frac{1}{x} = \sum_{x \in G} \frac{1}{x};
$$
(2) There exists a positive rational number $r \lt 1$ such that for every finite subset $F$ of $S$,
$$
\sum_{x \in F} \frac{1}{x} \neq r.
$$
(1) There exist distinct finite subsets $F$ and $G$ of $S$ such that
$$
\sum_{x \in F} \frac{1}{x} = \sum_{x \in G} \frac{1}{x};
$$
(2) There exists a positive rational number $r \lt 1$ such that for every finite subset $F$ of $S$,
$$
\sum_{x \in F} \frac{1}{x} \neq r.
$$