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.



Ratio of eventually close sequences
If the sequences are bounded, and the b's are bounded away from 0, there is a simple proof by the triangle equality. If the a's are both 1/n, and the b's are 1/n and -1/n, this clearly fails. 


**Edit**: As far as I can see, this can't be rescued by insisting on positive sequences. Starting with the first counterexample, keep the a's equal to each other, but tending to zero a bit slower than 1/n. Take the b's to be 1/n, and 1/n slightly perturbed, by a factor 1 + o(1). Working out the difference of ratios, it can be made large.


