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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 $\frac{a_{k_n}}{b_n}\to c \; ?$

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    You can reduce the problem to the two cases $c_0=0$ and $c_0=1$. An interesting base case is $a_n=b_n$ with $c_0=1$. Try to answer for that case, first.2012-07-12

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Take $a_n = b_n = 2^n$, and $c$ not a power of two. You will need some condition limiting the growth rate of $a_n$. I wonder if perhaps concavity is sufficient?

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    It feels like you can suffice to say $b_n<$\alpha$ n$ for some $\alpha$, which is a bit weaker than concavity.2012-07-12