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0531 sequence 5th edition Round 3 p1

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May 6, 2021
algebra5th edition

Problem Statement

Let {xn}n\{x_n\}_n be a sequence of positive rational numbers, such that x1x_1 is a positive integer, and for all positive integers nn. xn=2(n−1)nxn−1x_n = \frac{2(n - 1)}{n} x_{n-1}, if xn1≤1x_{n_1} \le 1 xn=(n−1)xn−1−1nx_n = \frac{(n - 1)x_{n-1} - 1}{n} , if xn1>1x_{n_1} > 1. Prove that there exists a constant subsequence of {xn}n\{x_n\}_n.
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