Question 18
Given any set $S$ of positive integers, show that at least one of the following two assertions holds:
(1) There exist distinct finite subsets $F$ and $G$ of $S$ such that $\sum_{x \in F} 1 / x = \sum_{x \in G} 1 / x$;
(2) There exists a positive rational number $r \lt 1$ such that $\sum_{x \in F} 1 / x \neq r$ for all finite subsets $F$ of $S$.
(1) There exist distinct finite subsets $F$ and $G$ of $S$ such that $\sum_{x \in F} 1 / x = \sum_{x \in G} 1 / x$;
(2) There exists a positive rational number $r \lt 1$ such that $\sum_{x \in F} 1 / x \neq r$ for all finite subsets $F$ of $S$.