How can we identify a set is infinite when proving Bolzano–Weierstrass theorem?

Question

I'm proving the Bolzano–Weierstrass theorem. It says

$$ \text{If } A \text{ : bounded and infinite set}, \text{ then } A \text{ has at least one limit point.} $$

So the proof goes like the following.

Since $A \subset \mathbb{R}^N$ is bounded, it can be a subset of a box $B_1 = I_1 \times \cdots \times I_N$, where $I_i$ is an interval in $\mathbb{R}$. Let's divide the $B_1$ into $2^N$ sub-boxes. Then, at least one sub-box has the infinite elements of $A$. Let's say that sub-box as $B_2$. Let's say we can repeat this procedure for $B_3, B_4, \ldots$

Well, so proof ends as we prove some $x$ exists in $\cap_{n=1}^{\infty}B_n$ and it is a limit point of $A$.

But what I'm wondering is, how can we identify the $B_2$? (and $B_3$, and so on)

Yes, it's clear to me that such set exists, but the existence itself does not guarantee us to identify the set (that is, to select and label it as $B_2$), since we don't have any tools to discern if $B_2$ is infinite or not.

It sounds a little bit philosophical, but is there anyone to help?

Answer 1

It is not surprising that there is no constructive way to identify, at each stage, which of the $2^n$ boxes holds infinitely many points—because, from the outset, the premiss of the theorem assumes nothing about where the points are: we know only that there are infinitely many of them. There are thus at least continuum-many possibilities for how the points might be distributed. All we can do is make the logical observation that, at every stage, the boxes cannot all hold a finite number of points, and therefore at least one of them holds an infinite number. If we need to identify just which such a box might be, we might be able to do so if we were given a precise construction of $A$; but we are given no such information.

Answer 2

It is in fact a philosophical question. The "paradoxes of infinity" have puzzled people since ancient times. In 1851 Bolzano himself has written a book with the title "Paradoxien des Unendlichen" (in English "Paradoxes of the Infinite").

In your proof you start with an infinite set $A$. But how can you know that is infinite? Is it explicitly given by listing its elements? In broad terms let us understand by "listing" any rule that allows you to decide whether a given element $x$ belongs to $A$ or not.

If you have such a listing, then you will be able to identify $B_2, B_3, ...$. If not, you are not even able to identify $A$. In that case it is only possible to argue on an abstract level: If you put infinitely many things into finitely many boxes, than at least one box must contain infinitely many things. Imagine a "person" has put the things into the boxes, but you didn't watch the process. Then you will not know which box contains infinitely many things, but you will know that at least one box must contain infinitely many things. If you don't like to consider "infinitely many things", you can also consider $10$ things in $2$ boxes: You know that at least one must contain $5$ things or more, but you don't know which box. It is similar to the shell game: You know that the ball is placed beneath one the shells, but would you bet beneath which?

Answer 3

It seems that you understand why at least one box contains infinitely many elements of $A$, and your difficulty is only in finding which box that is (or finding one such box if there are several of them). That is, it's a question of existence versus finding. Notice, though, that the theorem you're proving says only that there exists a limit point. It doesn't claim anything about finding a limit point.

To prove the existence of a limit point, it's enough to have the existence of an appropriate box at every stage of the process.

If you wanted to prove a stronger theorem saying that one can find a limit point, then you'd have to worry about finding an appropriate box at every stage. Any alleged theorem of that sort would have to be preceded by information about (1) exactly how the set $A$ is presented and (2) exactly what it means to "find" a point. So such a theorem would not only be harder to prove but considerably more complicated to state than the Bolzano-Weierstrass theorem.