Putnam 2001 B4

Let $S$ denote the set of rational numbers different from $\{-1,0,1\}$. Define $f:S\rightarrow S$ by $f(x)=x-1/x$. Prove or disprove that $$\bigcap_{n=1}^\infty f^{(n)}(S) = \emptyset,$$ where $f^{(n)}$ denotes $f$ composed with itself $n$ times.