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Infinitely many close pairs of members of a sequence

Source: BMO SL 2023 A4

May 3, 2024
algebra

Problem Statement

Prove that there exists a real c<34c<\frac{3}{4}, such that for each sequence x1,x2,…x_1, x_2, \ldots satisfying 0≤xi≤10 \leq x_i \leq 1 for all ii, there exist infinitely many (m,n)(m, n) with m>nm>n, such that ∣xm−xn∣≤cm.|x_m-x_n|\leq \frac{c} {m}.
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