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Putnam 1969 B5

Source: Putnam 1969

March 31, 2022
PutnamSequences

Problem Statement

Let a1<a2<…a_1 <a_2 < \ldots be an increasing sequence of positive integers. Let the series ∑i=1∞1ai\sum_{i=1}^{\infty} \frac{1}{a_i } be convergent. For any real number xx, let k(x)k(x) be the number of the aia_i which do not exceed xx. Show that lim⁡x→∞k(x)x=0.\lim_{x\to \infty} \frac{k(x)}{x}=0.
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