Suppose $(a_n)$ and $(b_n)$ are two real monotonely increasing sequences with $a_n, b_n\to\infty$. Suppose further there is $c_0\ge 0$ such that $\frac{a_n}{b_n}\to c_0.$
Under which conditions is it then true that for any $c > c_0$ there is a subsequence $(a_{k_n})$ such that $\frac{a_{k_n}}{b_n}\to c \; ?$