TUYMAADA 2023 S8

Given is a positive integer $n$ . Let $A$ be the set of points $x \in (0;1)$ such that $|x-\frac{p} {q}|>\frac{1}{n^3}$ for each rational fraction $\frac{p} {q}$ with denominator $q \leq n^2$ . Prove that $A$ is a union of intervals with total length not exceeding $\frac{100}{n}$ .

Proposed by Fedor Petrov