Let $(a_n)$ and $(b_n)$ be positive sequences with $a_n \sim b_n$. Suppose also that for some other positive sequence $(c_n)$, I have $a_n \leq c_n$ for all $n$. Also suppose that $a_n$ and $b_n$ alternate only finitely many times.
I would like to say that: $$b_n \leq c_n \textrm{ eventually }$$ i.e. there exists $N$ such that $b_n \leq c_n$ for all $n \geq N$. Clearly the statement isn't true in general, for example taking $a_n = n$, $b_n = n+1$, and $c_n = n$. So what is the weakest assumption I can impose to guarantee this holds in general?
The specific case I'm looking at is where all the sequences $a_n, b_n, c_n$ are $\in (0,1)$ and increasing to $1$. But I'm curious about the general case too.