5 added 13 characters in body

To define convergence one needs a "metric" or a concept of "distance", and there can be many different notion of "distances". For example one can consider $\lim_{n -> \infty} \frac {1}{n} \sum_{k=1}^n |(A_k - B_k)|^p$. Or alternatively $\lim_{n -> \infty} \frac {1}{n} \sum_{k=n}^{2n} |(A_k - B_k)|^p$.

Though your notion of distance is much stronger than the above, to be precise its $\limsup_{n->\infty}|A_n-B_n|$. So if the "distance" between two sequences is zero one can define an equivalence relation in a natural way and then you do actually get a proper metric. As everyone has mentioned this is how we go about constructing the real number system using Cauchy sequences.

So, I would suggest the sequence \sout{A {A eventually converge to B}, B or}, as per Willie Wong's suggestion A is asymptotically equivalent to B.

4 added 7 characters in body

To define convergence one needs a "metric" or a concept of "distance", and there can be many different notion of "distances". For example one can consider $\lim_{n -> \infty} \frac {1}{n} \sum_{k=1}^n |(A_k - B_k)|^p$. Or alternatively $\lim_{n -> \infty} \frac {1}{n} \sum_{k=n}^{2n} |(A_k - B_k)|^p$.

Though your notion of distance is much stronger than the above, to be precise its $\limsup_{n->\infty}|A_n-B_n|$. So if the "distance" between two sequences is zero one can define an equivalence relation in a natural way and then you do actually get a proper metric. As everyone has mentioned this is how we go about constructing the real number system using Cauchy sequences.

So, I would suggest the sequence A \sout{A eventually converge to BB}, or as per Willie Wong's suggestion A is asymptotically equivalent to B.

3 added 55 characters in body

To define convergence one needs a "metric" or a concept of "distance". distance", and there can be many different notion of "distances". For example one can consider $\lim_{n -> \infty} \frac {1}{n} \sum_{k=1}^n |(A_k - B_k)|^p$. Or alternatively $\lim_{n -> \infty} \frac {1}{n} \sum_{k=n}^{2n} |(A_k - B_k)|^p$.

Though your notion of distance is much stronger than the above, to be precise its $\limsup_{n->\infty}|A_n-B_n|$. So if the "distance" between two sequences is zero one can define an equivalence relation in a natural way and then you do actually get a proper metric. As everyone has mentioned this is how we go about constructing the real number system using Cauchy sequences.

So, I would suggest the sequence A eventually converge to B, or as per Willie Wong's suggestion A is asymptotically equivalent to B.

2 added 137 characters in body
1