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.
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3$\begingroup$ Perhaps that the two sequences are asymptotic? $\endgroup$– Willie WongCommented Oct 27, 2010 at 18:58
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10$\begingroup$ How about $A_n-B_n\to0$? And what is "noob"? $\endgroup$– Robin ChapmanCommented Oct 27, 2010 at 20:23
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2$\begingroup$ $A_n$ and $B_n$ are "asymptotically equal" - proofwiki.org/wiki/Definition:Asymptotically_Equal $\endgroup$– James DavidoffCommented Oct 27, 2010 at 20:27
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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 ChoiCommented Oct 27, 2010 at 20:36
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3$\begingroup$ What's wrong with "The sequences differ by a null sequence"? $\endgroup$– Franz LemmermeyerCommented Oct 28, 2010 at 8:54
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4 Answers
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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.
answered Oct 27, 2010 at 19:13
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$\begingroup$ So making "equivalent" an even more overloaded term than it already is :-) $\endgroup$ Commented Oct 27, 2010 at 20:28
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$\begingroup$ 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 context---such as the one I mentioned, the construction o the reals from the rationals. $\endgroup$ Commented Oct 27, 2010 at 20:31
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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 $A_n-B_n$ in cases where $X$ is a subset of a normed linear space.
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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 $\lim_{k\to\infty}(A_k-B_k)=0$, but with $A_k$ decreasing and $B_k$ increasing.
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2$\begingroup$ Indeed, but the non-increasing/non-decreasing condition makes it a lot more restrictive than the OP's question. $\endgroup$ Commented Oct 27, 2010 at 19:47
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$\begingroup$ Yes. I thought it may suggest a suitable terminology. $\endgroup$– HanyCommented Oct 27, 2010 at 21:56
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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 eventually converge to B or}, as per Willie Wong's suggestion A is asymptotically equivalent to B.
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$\begingroup$ how can we talk about convergence without a notion of distance ? $\endgroup$– VagabondCommented Oct 27, 2010 at 20:42
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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. $\endgroup$– GMRACommented Oct 27, 2010 at 21:06
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$\begingroup$ 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" $\endgroup$– VagabondCommented Oct 27, 2010 at 21:27
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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 $\endgroup$ Commented Oct 28, 2010 at 1:46