Putnam 1994 A5

Let $(r_n)_{n \geq 0}$ be a sequence of positive real numbers such that $\lim_{n \to \infty} r_n = 0$. Let $S$ be the set of numbers representable as a sum $$r_{i_1} + r_{i_2} + \cdots + r_{i_{1994}},$$ with $i_1 < i_2 < \cdots < i_{1994}$. Show that every nonempty interval $(a,b)$ contains a nonempty subinterval $(c,d)$ that does not intersect $S$.