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increasing sequence of sums of multiplicative inverses

Source: 2020 Simon Marais Mathematics Competition B2

November 17, 2020
number theory

Problem Statement

For each positive integer kk, let SkS_k be the set of real numbers that can be expressed in the form 1n1+1n2+⋯+1nk,\frac{1}{n_1}+\frac{1}{n_2}+\dots+\frac{1}{n_k}, where n1,n2…,nkn_1,n_2\dots,n_k are positive integers.
Prove that SkS_k does not contain an infinite strictly increasing sequence.
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