Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 4 / 2G / c / Solution

Letand suppose $x=p/q$. Choose $n\ge \max (q,2)$ such that some ${\theta }_j=1$ for $j>n$. Then $n!x$ is an integer, while its first $n$ terms also sum to an integer. Their difference is strictly positive and satisfiesThis cannot be the difference of two integers, so $x$ is irrational.