What I'm asking has always been if there is a (proper) proof of the least upper bound axiom (or theorem). The discussion about the meaning of axiom is only a by-product. By saying "proof", the one that uses "Theorem 8" as the basis, I don't think that's a proper proof (or a proper proof I'm asking about); it's only a circular argument.
I'm unable to comment on the rigour of Euclidean geometry, or that of modern mathematics—I'm far too ignorant to be able to. I only refer to one of its "axioms" as an example of those for which there may not be a proof of whatever form, based on whatever other axioms of theorems. (I say "may", because I'm not sure. And as I already said, this looks like a philosophical question to me. Moreover, this is not at the centre of my question.)
In post #9, you quoted from Wikipedia, which gave a proof. So you have answered your own question. This is why I am baffled by your continuing to ask this. I think what you need to understand is why something can be both proved (in one book) and assumed in another.
Just above what you quoted, you will have seen
this:
The least-upper-bound property is equivalent to other forms of the completeness axiom, such as the convergence of Cauchy sequences or the nested intervals theorem. The logical status of the property depends on the construction of the real numbers used: in the synthetic approach, the property is usually taken as an axiom for the real numbers (see least upper bound axiom); in a constructive approach, the property must be proved as a theorem, either directly from the construction or as a consequence of some other form of completeness.
The source you originally asked about says the same thing:
The completeness axiom. There are various different logically equivalent statements that can be used as an axiom of the completeness of the real numbers. We’ll use one called the least upper bound axiom.
Axiom 7 (Least upper bound axiom). Each nonempty set of real numbers that has an upper bound has a least upper bound.
What these are both saying is that you can take this as an
axiom, and prove other things from it; or you can take some other statement as an axiom (as in the Wikipedia proof that you quoted), and prove this as a
theorem. These are all
equivalent.
This is what we have been saying: You can develop this topic in various ways, some of which take this axiom, some of which take that axiom, and you end up with the same results. It is not a circular argument; it is a choice of
one of several starting points for a "circle".
You insist that the meaning of "axiom" is a side issue; but that is exactly what you need to understand in order to answer your question. An axiom is not, as I suspect you assume, something that is
known to be true without proof; it is something that is
accepted as true, as a basis for a systematic development of a topic. If you don't want to believe the real numbers work a certain way, you can reject the associated axioms, and develop a different theory; if you consider some other axiom more natural than this one, you can use that as your starting point and be just as convinced of everything that follows. But you need to have
some starting point. And that is what an axiom is.
In order to consider something in mathematics to be "true", you have to start by accepting some set of facts as true. They are not necessarily
unprovable (because you could start with something else and prove them from that); they are simply
unproved in your system. In the page I referred you to previously, I said the link on the first line could also be relevant; that was
Why Does Geometry Start With Unproved Assumptions?, which is indeed something you need to know. I assume you haven't read either.
So the direct answer to your question is,
Yes, it can be proved, and you have seen such a proof. But,
no, you can't prove it
as part of the presentation in which it was taken as an axiom. In such a proof, you need to have something
else that you are using as an axiom.
No obligations are assumed.