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for any positive integer k exist indices i,jsuch that k =k =x_i -x_j

Source: Switzerland - Swiss TST 2002 p6

February 18, 2020
Sequenceinequalitiesalgebra

Problem Statement

A sequence x1,x2,x3,...x_1,x_2,x_3,... has the following properties: (a) 1=x1<x2<x3<...1 = x_1 < x_2 < x_3 < ... (b) xn+1≤2nx_{n+1} \le 2n for all n∈Nn \in N. Prove that for each positive integer kk there exist indices ii and jj such that k=xi−xjk =x_i -x_j.
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