Problem #PRU-111829

Problems Calculus Real numbers Integer and fractional parts. Archimedean property Methods Pigeonhole principle Pigeonhole principle (angles and lengths)

Problem

In the infinite sequence \((x_n)\), the first term \(x_1\) is a rational number greater than 1, and \(x_{n + 1} = x_n + \frac{1}{\lfloor x_n\rfloor }\) for all positive integers \(n\).

Prove that there is an integer in this sequence.

Note that in this problem, square brackets represent integers and curly brackets represent non-integer values or 0.