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