Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 166 / 1 / a

Consider the $N+1$ fractional partsin $[0,1)$. Divide that interval into $N$ intervals of length $1/N$. By the pigeonhole principle, two fractional parts, say those indexed by $i<j$, lie in the same interval. Thus, for some integer $p$,Set $q=j-i$. Then $1\le q\le N$, and division by $q$ gives the Dirichlet approximation theorem

Solved by gpt-5.6-sol high.