Let $a_n$,$b_n$ with $b_n>0$ be two bounded sequences which are eventually close to, respectively, two other sequences $\bar a_n$,$\bar b_n$ with $\bar b_n>0$, that is, for every $\epsilon >0$ there exists an integer $N$ such that $|a_n-\bar a_n|<\epsilon$ and $|b_n-\bar b_n|<\epsilon$ for all $n>N$.

Is it possible to prove that the sequence $\frac{a_n}{b_n}$ will be eventually close to the sequence $\frac{\bar a_n}{\bar b_n}$? No assumptions are made on the sequences apart from boundedness and positivity of the denominators, but I'm particularly interested in the case in which all of them converge to zero.