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Putnam 1994 A5

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July 12, 2014
Putnamlimitcollege contests

Problem Statement

Let (rn)n≥0(r_n)_{n\ge 0} be a sequence of positive real numbers such that lim⁡n→∞rn=0\lim_{n\to \infty} r_n = 0. Let SS be the set of numbers representable as a sum ri1+ri2+⋯+ri1994, r_{i_1} + r_{i_2} +\cdots + r_{i_{1994}} , with i1<i2<⋯<i1994.i_1 < i_2 < \cdots< i_{1994}. Show that every nonempty interval (a,b)(a, b) contains a nonempty subinterval (c,d)(c, d) that does not intersect SS.
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