Past exam of the mathematics course of the University of Oxford / 2015 / Prelims / Mathematics II / 2 / b / iv

True. Choose $N$ such that $\ell /2<a_n<3\ell /2$ for every $n\ge N$. All the finitely many preceding terms are also positive. Define the positive minimum and finite maximumEvery term lies between $\eta$ and $B$, so every ratio is at most $B/\eta$. Consequently the required supremum satisfies