Is there a name for the relationship between sequences $A_n$ and $B_n$ which means that the sequence $A_n  B_n$ converges to zero? I want to say something like "sequence $A$ converges to sequence $B$" which might not mean the right thing, or something like "sequences $A$ and $B$ converge" which certainly doesn't mean what I want it to. Sorry if this question is too noob.

Note that this is an equivalence relation. In fact, if you start with just the rational numbers, then this relation for Cauchy sequences of rationals is usually called "equivalence", and the equivalence classes can then be identified with precisely the field of real numbers. (This is my personal favorite way for constructing the reals from the rationals.) So I think that in any reasonable context it would be quite reasonable to just call such sequences equivalent. 


In one context they are called equivalent. If $A_n$ and $B_n$ are Cauchy sequences in a metric space $X$, they are called equivalent when $d(A_n,B_n)$ converges to 0, and the completion of $X$ consists of equivalence classes of Cauchy sequences. This can match your $A_nB_n$ in cases where $X$ is a subset of a normed linear space. 


For sequences of real numbers there is a French term: "suites adjacentes" (may be translated as adjacent sequences) which means that the two sequences satisfy $\lim_{k\to\infty}(A_kB_k)=0$, but with $A_k$ decreasing and $B_k$ increasing. 


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 $\limsup_{n>\infty}A_nB_n$. 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 

