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What does it mean for a sequence to converge to an element if the limit might not necessarily be defined or known, or not necessarily in the universe under consideration (whatever this means)?

I am not just talking about real numbers; it can be more general. The definition of a sequence $(x_n)$ converging to $x$ seems to say: for any $\epsilon > 0$, there is $N \in \mathbb{N}$ such that $|x_n - x| < \epsilon$ for all $n > N$. But doesn’t this assume the existence of a point $x$ under consideration? For example, when we show that $\{p \in \mathbb{Q}: p^2 < 2\}$ “approaches” a number, e.g. by considering $1.4, 1.41, 1.414, ...$, what does that even mean, if we have not yet constructed the real numbers? What is meant by “number” in this case? Does it even make sense to say that?

Although we might not "know" what the limiting point $x$ is for $\{p \in \mathbb{Q}: p^2 < 2\}$, it seems to still make sense to speak of a limit. (Again, I'm assuming we do not as yet know what real numbers might be, or what completeness is, etc.). In such a case, how should we define convergence, if we cannot have a point to explicitly refer to?

In general, couldn’t we converge to “something”, but it is not at all clear what that “something” should or might be? If it isn’t clear what that something should be, then how can we even speak of converging to that something? Is this just a logical/semantic/notational thing in the definition of convergence?

twosigma
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    Have a look at the construction of real numbers by Cauchy sequences . – hamam_Abdallah Jul 08 '20 at 22:45
  • Unless I misunderstand your question, it is quite common to know that a sequence converges but to not know what value it converges to. You are right: knowing that a sequence converges does indeed imply the existence of an element which the sequence converges to (by definition) but in many cases in mathematics proving existence is not always the same as finding the thing we have proved to exist! In particular, see construction of the reals by Dedekind cuts/Cauchy sequences. – Benjamin Jul 08 '20 at 22:47
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    The idea behind the Cauchy completion construction, which can be used in particular to construct the reals from the rationals, is that there's a class of sequences that "look like they should converge to something, but it's not a priori clear that their limit actually exists". We call these sequences Cauchy. The Cauchy completion construction is basically a self-consistent way to assign all Cauchy sequences in a given metric space a unique limit. – Ian Jul 08 '20 at 23:16
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    (Cont.) It is a theorem (i.e. not an axiom) that doing the Cauchy completion a second time results in an isomorphic copy of the metric space you had after doing the Cauchy completion once, which is why it makes philosophical sense to only do it once. – Ian Jul 08 '20 at 23:18
  • To put it more briefly, your question makes sense, but it is one that is solved, and you can find one of the solutions in any introductory real analysis textbook. – Ian Jul 08 '20 at 23:20
  • Well, the definition of convergence / limits takes care of the issue you describe. Either explicitly or implicitly it is conveyed that the definition makes sense in some particular "universe under consideration" (eg a specific metric space). And depending on the metric space the same sequence may or may not be convergent. – Paramanand Singh Jul 09 '20 at 00:40
  • @Ian: you may write (at your wish and leisure) an answer based on your comments. – Paramanand Singh Jul 09 '20 at 00:42

2 Answers2

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Not sure where the confusion comes from but...

In any book of real analysis (e.g. in the simplest case - real analysis of functions of one variable), real numbers are introduced first, and only then the book starts talking about limits. So first the existence of real numbers is shown or at least assumed/postulated via a set of axioms (and rigorously introduced later via Dedekind cuts/Cauchy sequences). And only then limits are discussed.

So yes, the two things must come in the right order and they usually do.

If you stick to rationals but have no real numbers at your side,
there's no real analysis and hence no limits. Right?

Same for analysis of real multivariate functions, complex analysis etc. First you need to know or postulate (via some set of axioms) that "points" exist, and only then talk about sequences converging to points.

Maybe the confusion comes from calculus
(which is basically analysis without too much rigor as far as I know).

On a more philosophical (or funny) level... I've had a few discussions with a physicist who thinks that in a way real analysis is in general somewhat "flawed" but still yields useful results. Why flawed? Because in nature/physics infinitely small positive numbers simply do not exist. E.g. the smallest distance is the Planck length. So how come in analysis we have those $$\epsilon \gt 0$$ values as small as we want them to be?! :)

But of course

  1. As far as I know in math we are allowed to have absolutely abstract theories and concepts which don't map necessarily to natural phenomena.
  2. I don't know if the Planck length will still be the smallest known distance in several hundred years.

So at least to me this argument which refers to nature is not right here, and I am personally fine with real analysis.

peter.petrov
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  • You can't really define the real numbers without having some notion of limits, or at least the similar notion of a supremum and infimum. It is perfectly possible to talk about limits before introducing the real numbers, and in fact, the real numbers are often defined through objects (Cauchy sequences) which only make sense if limits are already known. Also, the Planck length is not the smallest possible physical length. Planck units are often useful order of magnitude estimates for scales at which new, interesting physics are expected to happen, but that doesn't make space discrete. – Vercassivelaunos Jul 08 '20 at 23:40
  • @Vercassivelaunos Thanks. If I could I would downvote my answer... but keep it in order to keep your comment. This is really interesting then... Does that mean we define concept B (reals) via concept A (limits) but A already assumes the existence of B. So there's some sort of loop here. I don't get it, I guess that's the essence of this question then. – peter.petrov Jul 08 '20 at 23:55
  • No, the definition of a limit doesn't assume the existence of the reals. We just have to specify where a sequence converges (as in, what's the set we're working with). A sequence might not converge in the rationals, but after defining the reals, we could see that the same sequence does converge in the reals. The concept of what convergence is doesn't assume the existence of certain limits. It just defines what it even means for such a limit to exist. – Vercassivelaunos Jul 09 '20 at 00:04
  • @Vercassivelaunos: the notion of a real number can be described independently of limit process via Dedekind cuts. The definition is just based on order relations and nothing more. Even terms like supremum and infimum aren't used. The definition by Cauchy sequences is inherent based on limits and requires lot of machinery and is suitable only when the student is already familiar with ideas of calculus. – Paramanand Singh Jul 09 '20 at 02:06
  • You can define Dedekind cuts without using the words supremum or infimum, but I doubt anyone would come up with that definition if they had no concept of a supremum, since the whole point of that construction is to give every set which is bounded above a supremum. – Vercassivelaunos Jul 09 '20 at 08:40
  • @Vercassivelaunos: Well Dedekind himself defined Dedekind cuts without any notion of supremum. Rather the motivation for the definition was to mimic the so called intuitive notion of continuity (unbroken nature) of a straight line. Hardy gave an exposition of Dedekind's approach without any notion of supremum in his A Course of Pure Mathematics. – Paramanand Singh Jul 09 '20 at 10:27
  • @ParamanandSingh: I believe you. But then I don't understand what made Dedekind think that the rationals weren't continuous, if he had no notion of a supremum or limit? How would this kind of continuity be characterized if not by the existence of least upper bounds or some kind of limit definition? – Vercassivelaunos Jul 09 '20 at 10:33
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    @Vercassivelaunos: Dedekind idea was far simpler. He thought and if a line is continuous and we cut into two two pieces at some point then the cutting point itself will be included in only one of the two parts. Based on this consider the set of rationals divided into two sets $A, B$ such that every member of $A$ is less than every member of $B$. If the rationals are supposed to describe each and every point of a line then there must be a rational $c$ which makes the division between $A$ and $B$ and if it lies in $A$ it is max of $A$ otherwise it is min of $B$. Cont'd... – Paramanand Singh Jul 09 '20 at 12:01
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    @Vercassivelaunos: however it is possible to divide rationals into two sets $A, B$ such that every member of $A$ is less than every member of $B$ and yet neither $A$ has a maximum nor $B$ has a minimum. For example $A$ contains rationals whose cube is less than $2$ and $B$ contains rationals whose cube is greater than $2$. Dedekind noted this problem of rationals and extended the system to create reals which don't have this problem. For Dedekind a real number number is nothing but division of rationals into two sets based on order. – Paramanand Singh Jul 09 '20 at 12:05
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    @Vercassivelaunos: you should read his original Continuity and irrational numbers. Also see my answer https://math.stackexchange.com/a/3651131/72031 – Paramanand Singh Jul 09 '20 at 12:06
  • @ParamanandSingh: My mind was kinda set on the open half of the cut, for which we would require a supremum or infimum to exist. I didn't think of switching to the closed half of the cut, where it's possible to talk about max and min instead of sup and inf. It's the same in practice, but yeah, I see how that doesn't require thinking about sup and inf. Thanks. – Vercassivelaunos Jul 09 '20 at 12:15
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This is why the definition of convergence assumes a metric space (or more generally a topological space, but let's stay in metric spaces):

Let $(X,d)$ be a metric space and $(x_n)_{n\in\mathbb N}$ a sequence in $X$. It is said to converge to $x\in X$ if for arbitrarily small $\varepsilon>0$ there exists a sufficiently large $N\in\mathbb N$ such that $d(x,x_n)<\varepsilon$ for all $n\geq N$.

$d(x,y)$ is essentially a generalization of the distance between $x$ and $y$. In the rational or real numbers, it's usually what we would intuitively think of as distance, namely $d(x,y)=\vert y-x\vert$. That's why $\vert x-x_n\vert$ comes up in your less general definition. Anyway, with this more rigorous definition, the answer to your question is: Convergence is very much defined by the underlying set. If there is some bigger set containing $X$ and $x_n$ "converges" to an element $y$ in that bigger set, but which is not in $X$, then we just don't say that $x_n$ converges. With this in mind, we have to make a careful distinction when determining the convergence of the sequence $1,1.4,1.41,1.415,\dots$: If we consider the rationals as the underlying set, then it does not converge, since the "limit" $\sqrt2$ is not in the underlying set. But if we consider the reals as the underlying set, it does converge.

There is also a certain notion of when a sequence "should" converge. If the members of the sequence get arbitrarily close to each other for large enough index $n$, then we would intuitively expect it to converge. Such a sequence is then called Cauchy sequence, named after French mathematician Augustin-Louis Cauchy, who just assumed it as obvious that a sequence converges if it is a "Cauchy sequence". But it turns out, we have to be more careful, because not every Cauchy sequence converges. At least not in every metric space, because the point to which such a sequence should converge might not be in the underlying set. Such a metric space is called incomplete, otherwise it is called complete. But for every incomplete metric space, there is a bigger, complete metric space in which every Cauchy sequence does converge. Such a metric space is then called the completion of the smaller space. For instance, the real numbers are the completion of the rational numbers: There are rational sequences converging to an irrational number because the rationals are incomplete, and there is no larger space than the reals containing additional points to which a real sequence might converge, because the reals are complete.

Vercassivelaunos
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    You have a bit of a typo; your example sequence converges to $\sqrt{2}$ not $\pi$. – Ian Jul 09 '20 at 01:17
  • @Ian, Huh, wonder how that happened. Thanks! – Vercassivelaunos Jul 09 '20 at 08:28
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    Interesting! This has helped shed light on the type of remark I have seen by some authors when, upon introducing the definition of Cauchy sequence, point out how it has the benefit of not referencing a limit. – twosigma Jul 11 '20 at 17:40