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Putnam 1964 A3

Source: Putnam 1964

March 5, 2022
PutnamConvergencetopology

Problem Statement

Let P1,P2,…P_1 , P_2 , \ldots be a sequence of distinct points which is dense in the interval (0,1)(0,1). The points P1,…,Pn−1P_1 , \ldots , P_{n-1} decompose the interval into nn parts, and PnP_n decomposes one of these into two parts. Let ana_n and bnb_n be the length of these two intervals. Prove that \sum_{n=1}^{\infty} a_n b_n (a_n +b_n) =1 \slash 3.
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