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A6

Part of 1982 Putnam

Problems(1)

permutation of N, limit is 1

Source: Putnam 1982 A6

9/24/2021
Let σ\sigma be a bijection on the positive integers. Let x1,x2,x3,…x_1,x_2,x_3,\ldots be a sequence of real numbers with the following three properties:
(i)(\text i) ∣xn∣|x_n| is a strictly decreasing function of nn; (ii)(\text{ii}) ∣σ(n)−n∣⋅∣xn∣→0|\sigma(n)-n|\cdot|x_n|\to0 as n→∞n\to\infty; (iii)(\text{iii}) lim⁡n→∞∑k=1nxk=1\lim_{n\to\infty}\sum_{k=1}^nx_k=1.
Prove or disprove that these conditions imply that lim⁡n→∞∑k=1nxσ(k)=1.\lim_{n\to\infty}\sum_{k=1}^nx_{\sigma(k)}=1.
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