Let $\{q_n\}$ and $\{b_n\}$ be two positive integer sequences with $\{b_n\}$ is bounded, $q_n\ge 2$ and $q_n| b_n$ (it means $q_n$ divides $b_n$) for any $n\ge 1.$ For any $0<t<\limsup\limits_{n\to \infty}\frac{\log(q_1\cdots q_n)}{\log(b_1\cdots b_n)}$, can we find integers $q_n'\in \{0,1,\ldots, q_n\}, n=1,2,\ldots$ such that $$t=\limsup\limits_{n\to \infty}\frac{\log(q_1'\cdots q_n')}{\log(b_1\cdots b_n)}?$$
It seems that we can do it by properly choosing $q_n'$ from $\{0,1,\ldots, q_n\}$ for each $n$.
Thanks!