Zaremba's conjecture on bounded partial quotients

Every rational number $a/q\in(0,1)$ has a continued-fraction expansion

$$\frac{a}{q}=[a_1,a_2,\ldots,a_s],$$

where the $a_i\geq 1$ are its partial quotients. Zaremba's conjecture. There exists a constant $\mathcal M>0$ such that, for every $q\in\mathbb N$, there is a natural number $a<q$ with $\operatorname{gcd}(a,q)=1$ and a continued-fraction expansion $a/q=[a_1,a_2,\ldots,a_s]$ satisfying $a_i\leq\mathcal M$ for every $1\leq i\leq s$. Zaremba additionally conjectured that one may take $\mathcal M=5$. The paper states that it proves the conjecture, so the claim is resolved; the stronger specific value $\mathcal M=5$ is not asserted here to have been proved.