Putnam 1982 A6

Let $\sigma$ be a bijection on the positive integers. Let $x_1,x_2,x_3,\ldots$ be a sequence of real numbers with the following three properties: $(\text i)$ $|x_n|$ is a strictly decreasing function of $n$ ; $(\text{ii})$ $|\sigma(n)-n|\cdot|x_n|\to0$ as $n\to\infty$ ; $(\text{iii})$ $\lim_{n\to\infty}\sum_{k=1}^nx_k=1$ .

Prove or disprove that these conditions imply that $$\lim_{n\to\infty}\sum_{k=1}^nx_{\sigma(k)}=1.$$