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.

1$\begingroup$ I'm tempted to write $A_n \sim B_n$ for this, but I would actually use that notation for $\lim_{n \to \infty} A_n/B_n = 1$. $\endgroup$ – Michael Lugo Oct 27 '10 at 18:52

3$\begingroup$ Perhaps that the two sequences are asymptotic? $\endgroup$ – Willie Wong Oct 27 '10 at 18:58

10$\begingroup$ How about $A_nB_n\to0$? And what is "noob"? $\endgroup$ – Robin Chapman Oct 27 '10 at 20:23

5$\begingroup$ Saying sequence A converges to sequence B seems like saying that $1/n$ converges to $1/n^2$ as $n\to\infty$, a statement which makes me feel queasy and shouty at the same time $\endgroup$ – Yemon Choi Oct 27 '10 at 20:36

3$\begingroup$ What's wrong with "The sequences differ by a null sequence"? $\endgroup$ – Franz Lemmermeyer Oct 28 '10 at 8:54
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.

$\begingroup$ So making "equivalent" an even more overloaded term than it already is :) $\endgroup$ – Robin Chapman Oct 27 '10 at 20:28

$\begingroup$ Well, yes, but notice I said "in any reasonable context". This is not at all likely to be a universal definition, but only one used in some limited contextsuch as the one I mentioned, the construction o the reals from the rationals. $\endgroup$ – Dick Palais Oct 27 '10 at 20:31
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.

1$\begingroup$ Indeed, but the nonincreasing/nondecreasing condition makes it a lot more restrictive than the OP's question. $\endgroup$ – Thierry Zell Oct 27 '10 at 19:47

$\begingroup$ Yes. I thought it may suggest a suitable terminology. $\endgroup$ – Hany Oct 27 '10 at 21:56
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 {A eventually converge to B or}, as per Willie Wong's suggestion A is asymptotically equivalent to B.

2

$\begingroup$ how can we talk about convergence without a notion of distance ? $\endgroup$ – Vagabond Oct 27 '10 at 20:42

4$\begingroup$ You only need a topology, not a metric. You probably want a Hausdorff topology, because otherwise the limit will not necessarily be unique. $\endgroup$ – GMRA Oct 27 '10 at 21:06

$\begingroup$ point taken, I made that statement to emphasize that it matters a lot what notion of distance (if you like topolgy) we are considering when we are talking about convergence... so just saying two sequence converges without clarifying what the notion of convergence is inadequate. The title now edited by Yemon Choi is in fact the perfect answer to the original question which incidentally was "two sequences converge to each other" $\endgroup$ – Vagabond Oct 27 '10 at 21:27

2$\begingroup$ In my book, saying two sequences converge to each other is like saying you can cancel the d's in dy/dx, only even more like someone scraping nails down a blackboard $\endgroup$ – Yemon Choi Oct 28 '10 at 1:46