Given two real sequences that go to infinity, is it possible to select two subsequences that grow at the same rate asympotically?

Question

Given two positive real sequences $a_n$ and $b_n$ that both diverge to infinity, is it possible to choose two subsequences $a_{s_n}$ and $b_{t_n}$ such that $a_{s_n}/b_{t_n}\rightarrow1$?

Answer 1

Not necessarily. If $$a_n=2^{2n}\quad\hbox{and}\quad b_n=2^{2n+1}$$ then any $a_{s_n}$ and any $b_{t_n}$ have a ratio of at least $2$ or at most $\frac12$.