Talk:Least-upper-bound property

Dedekind completeness

Dedekind completeness is the property that every Dedekind cut of the real numbers is generated by a real number. I would have simply removed the comment that Dedekind completeness is sometimes used to refer to the LUB property, but I wanted to get a second opinion on whether this misnomer is actually in sufficiently common usage. Hurkyl (talk) 11:15, 2 July 2012 (

Willard describes a space as dedekind complete space as a space where any subset with an upper bound has a least upper bound, which is the LUB property. This is from page 124/125 from exercise 17E. 130.63.185.213 (talk) 18:36, 17 April 2013 (UTC)

Thanks for pointing that out. Citation and quote from Willard added here. --50.53.61.240 (talk) 17:38, 9 September 2014 (UTC)

Proof of Bolzano Weierstrass Theorem

The proof seems unconvincing. Someone better than me in mathematics please verify it, and perhaps replace it with the proof in the main page involving the monotone subsequence theorem followed by monotone convergence theorem. Venkatarun95 (talk) 04:06, 10 July 2014 (UTC)

Proof using Cauchy sequences

It says: "suppose that S has an upper bound B1". But S may contain no upper bound at all, all its upper bound being real numbers but not in S. For example, in the set ]0,1[ (or {0,1} if you prefer this notation), there is no upper bound belonging to S. Even its least one, 1, is not member of S. In that case, there is no "B1". and the proof does not work. wku2m5rr (talk) 07:17, 18 July 2023 (UTC)

Well, I answer myself, I am wrong because the sentence "suppose that S has an upper bound B1" does not imply that this upper bound belongs to S and, of course, it never does. Sorry for this message. wku2m5rr (talk) 07:33, 18 July 2023 (UTC)

Merge Completeness (order theory) into Least-upper-bound property

Should Completeness (order theory) be merged with Least-upper-bound property? They refer to precisely the same property, if I'm not wrong. Mechanikin (talk) 12:19, 10 June 2026 (UTC)

There is scope for two articles: least upper bound property is about the real case, while the order theory article is about its generalization to posets. Neither article is in very good shape, but I think having two distinct (but bad) clearly-scoped articles is better than having one (worse and imbalanced) article with a schizophrenic scope. The concept in order theory is almost never treated alongside that of the reals, because the latter is too special. The order theory concept is more often applied to genuine posets which are not totally ordered, like Heyting algebras, the algebra of open sets or Borel sets in topological spaces, etc. Sławomir Biały (talk) 13:25, 10 June 2026 (UTC)

There already is an article about that for the real numbers, titled Completeness of the real numbers. Mechanikin (talk) 13:33, 10 June 2026 (UTC)

I didn't know about it, but that article discusses at least three notions of completeness, only one of which is the least upper bound. I still think the final arrangement of content would be better if completeness (order theory) and least upper bound property were improved prior than being merged into a single bad article. Sławomir Biały (talk) 13:51, 10 June 2026 (UTC)

Reading Completeness (order theory) I am unable to decipher the true subject of this article. The title suggests that this subject is the same as the one of Complete partial order. So, I suggest to redirect Completeness (order theory) to Complete partial order. If something in the first article deserves to be kept, a merge could be considered, but I doubt that it is the case.

It is only after such an edit that we could discuss further reorganization of the articles on these subjects. Personally, I find that the structure resulting from the redirect is fine, D.Lazard (talk) 15:14, 10 June 2026 (UTC)

It's sort of a mess, but I think complete partial order is about a different notion of completeness than the usual one. A merge might be appropriate, but we should at least have a clear article somewhere that covers the basic "partial order is complete if every bounded subset has a least upper bound" notion (and the corresponding dual statement). I don't think dcpo is what most people mean be, e.g., a complete lattice. But maybe I'm wrong. (Edit: oh no, complete lattice, good grief!) Sławomir Biały (talk) 15:26, 10 June 2026 (UTC)