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good sequences, when each derivative consists of positive numbers

Source: Tournament of Towns 2020 oral p4 (15 March 2020)

May 18, 2020
Sequencerecurrence relationpositive realalgebra

Problem Statement

For an infinite sequence a1,a2,...a_1, a_2,. . . denote as it's first derivative is the sequence an′=an+1−ana'_n= a_{n + 1} - a_n (where n=1,2,..n = 1, 2,...), and her kk- th derivative as the first derivative of its (k−1)(k-1)-th derivative (k=2,3,...k = 2, 3,...). We call a sequence good if it and all its derivatives consist of positive numbers. Prove that if a1,a2,...a_1, a_2,. . . and b1,b2,...b_1, b_2,. . . are good sequences, then sequence a1⋅b1,a2⋅b2,..a_1\cdot b_1, a_2 \cdot b_2,.. is also a good one.
R. Salimov
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