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lim b_n = 0 and sum a_ib_i$ diverges when sum a_i diverges

Source: 1967 Swedish Mathematical Competition p4

March 21, 2021
algebraSumlimit

Problem Statement

The sequence a1,a2,a3,...a_1, a_2, a_3, ... of positive reals is such that ∑ai\sum a_i diverges. Show that there is a sequence b1,b2,b3,...b_1, b_2, b_3, ... of positive reals such that lim⁡bn=0\lim b_n = 0 and ∑aibi\sum a_ib_i diverges.
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