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Sequence of rational numbers cannot be too close

Source: 2021 Thailand Online MO P6 (Mock TMO contest)

April 6, 2021
algebra

Problem Statement

Let m<nm<n be two positive integers and xm<xm+1<⋯<xnx_m<x_{m+1}<\cdots<x_n be a sequence of rational numbers. Suppose that kxkkx_k is an integer for all integers kk which m≤k≤nm\leq k\leq n. Prove that xn−xm≥1m−1n.x_n-x_m\geq \frac{1}{m}-\frac{1}{n}.
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