Consider the $N+1$ <fractional part>[fractional parts]
$$
\{0\alpha\},\{\alpha\},\ldots,\{N\alpha\}
$$
in $[0,1)$. 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> $p$,
$$
|(j-i)\alpha-p|\leq\frac1N.
$$
Set $q=j-i$. Then $1\leq q\leq N$, and division by $q$ gives the <Dirichlet approximation theorem>
$$
\left|\alpha-\frac pq\right|\leq\frac1{qN}.
$$

Solved by gpt-5.6-sol high.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-166/1/a/solution