Talk:Nested intervals

Merge or not?

Don't merge. There is a particular and useful and basic result here. Charles Matthews 19:49, 31 May 2007 (UTC)[reply]

Agreed. Nested intervals play an important role in real one variable analysis. Geometry guy 00:48, 1 June 2007 (UTC)[reply]

Do you ever disagree with him? Also, a single "useful and basic result" does not justify the existence of this article. The article on intervals could quite easily incorporate the notions in this article in a section. Myrkkyhammas 09:33, 3 June 2007 (UTC)[reply]

Well, could and should are different. I looked round the site at where the nested intervals theorem is actually used, and discovered it cited, for example, on the page for 0.9999.... Which is a classic source of confusion for people who do not understand real analysis.

By the way, snarky remarks such as Do you ever disagree with him? are not well taken. Charles Matthews 11:26, 3 June 2007 (UTC)[reply]

A link

I came to Nested intervals with link from Cardinal number, and the content of this article looks unrelated to the link. I would expect Cantor's schema there. Mashiah (talk) 20:49, 17 November 2007 (UTC)[reply]

For what it's worth, in Cantor's proof a real number is constructed by intersecting a sequence of nested intervals, in exactly the same way as it is described in this article. -- EJ (talk) 12:23, 30 November 2007 (UTC)[reply]

Problems with this article

Problem 1. The condition that the length of the intervals get arbitrarily small should not be part of the definition (in the intro or in the "construction of real numbers section").

A sequence $(I_{n})$ of intervals is said to be nested if $I_{n+1}\subseteq I_{n}$ for all </math> n </math>. (More generally, a sequence of sets $(C_{n})$ is said to be nested if $C_{n+1}\subseteq C_{n}$ for all </math> n </math>.) If $a_{n}$ and $b_{n}$ denote the left and right endpoints of the interval $I_{n}$ in the sequence, saying $(I_{n})$ is nested means that $(a_{n})$ is monotonically increasing ( $a_{n+1}\geq a_{n}$ ) and the sequence $(b_{n})$ is monotonically decreasing ( $b_{n+1}\leq b_{n}$ ). It may happen that the lengths of the intervals $|I_n| = b_n - a_n$ approach $0$ as $n$ approaches infinity, but this is not necessary.

Problem 2. The section "The construction of the real numbers" is a mess.

It is helpful to distinguish between two versions of the "axiom of completeness":

Nested Intervals Property (NIP): The intersection of every nested sequence of closed intervals is non-empty.

Shrinking Nested Intervals Property (SNIP): The intersection of every nested sequence of closed intervals whose lengths shrink to zero is non-empty.

Contrary to what the article asserts, neither NIP or SNIP is equivalent to the Least Upper Bound Property (every non-empty set having an upper bound has a least upper bound) or the Bolzano–Weierstrass Property (every bounded sequence has a convergent subsequence), unless we also assume the Archimedean Property. The Archimedean Property is used in the "proof" in the article (see section "Direct consequences of the axiom") when asserting that the lengths of the constructed intervals get arbitrarily small.

For an arbitrary ordered field, Least Upper Bound Property -> NIP -> SNIP -> Cauchy Completeness, and none of these implications reverse. For an Archimedean ordered field, these four properties are equivalent.

Remark: The Theorem in the section "The construction of the real numbers": It says that in SNIP, the words "is non-empty" can be replaced by the words "contains exactly one point."

Remark: This article should link to Cantor's Intersection Theorem. NIP and SNIP are theorems in R. They are, respectively, special cases of the Cantor's Intersection Theorem in topological spaces and in complete metric spaces.

Response to Problems with this article

Thanks for the remarks, I will get back to updating the article, when I've got some more time on my hands.

Problem 1. The condition that the length of the intervals get arbitrarily small should not be part of the definition (in the intro or in the "construction of real numbers section").

This might vary from author to author. Many textbooks I'm familiar with include the property of arbitrarily small intervals in their definition (e.g. the books Otto Forster and by Konrad Königsberger, which are standard works in the German speaking world). In the context of the completeness axiom it makes sense to do so, but I'll make a remark in the Wiki-page. The monotony of the bounding sequences however should not be in the definition, since it is a equivalent to the nesting property. I might add that as a remark though.

Problem 2. The section "The construction of the real numbers" is a mess. No, it is consistent with the rest of the article.

It is helpful to distinguish between two versions of the "axiom of completeness": Nested Intervals Property (NIP): The intersection of every nested sequence of closed intervals is non-empty. Shrinking Nested Intervals Property (SNIP): The intersection of every nested sequence of closed intervals whose lengths shrink to zero is non-empty.

It's hard to see, why the distinguation should be helpful in the context of the whole article.

Contrary to what the article asserts, neither NIP or SNIP is equivalent to the Least Upper Bound Property (every non-empty set having an upper bound has a least upper bound) or the Bolzano–Weierstrass Property (every bounded sequence has a convergent subsequence), unless we also assume the Archimedean Property. The Archimedean Property is used in the "proof" in the article (see section "Direct consequences of the axiom") when asserting that the lengths of the constructed intervals get arbitrarily small.

Yes, the Archimedean property is needed, but again, in the context of the whole article it's a given, we're talking about the real numbers with + and \cdot as an Archimedean ordered field.

For an arbitrary ordered field, Least Upper Bound Property -> NIP -> SNIP -> Cauchy Completeness, and none of these implications reverse. For an Archimedean ordered field, these four properties are equivalent.

Remark: The Theorem in the section "The construction of the real numbers": It says that in SNIP, the words "is non-empty" can be replaced by the words "contains exactly one point."

Remark: This article should link to Cantor's Intersection Theorem. NIP and SNIP are theorems in R. They are, respectively, special cases of the Cantor's Intersection Theorem in topological spaces and in complete metric spaces. — Preceding unsigned comment added by OmniPraesent (talk • contribs) 18:09, 20 February 2022 (UTC)[reply]

> Yes, the Archimedean property is needed, but again, in the context of the whole article it's a given, we're talking about the real numbers with + and \cdot as an Archimedean ordered field.

I disagree with this. I reached this article via a link from the article on Completeness of the real numbers. In this article it states:

"Nested intervals theorem shares the same logical status as Cauchy completeness in this spectrum of expressions of completeness. In other words, nested intervals theorem by itself is weaker than other forms of completeness, although taken together with Archimedean property, it is equivalent to the others."

I was surprised when I clicked on the article, and saw that "Existence of infimum and supremum in bounded sets" was listed under "Direct consequences of the axiom". It seems clear to me that this does not refer to the axiom of completeness, since the heading "axiom of completeness" comes under a completely different subheading. Hence, this section seemed to be about the direct consequences of nested intervals theorem, as an axiom on its own (i.e. without Archimedean property). This is obviously not what this section is about, so either the heading should be changed or the section should be changed.

Furthermore, that much of the article implicitly uses the Archimedean property is hidden inside the subheading for the construction of the reals. I didn't even notice this until I read your comment; this should really be more front and center.

Alternatively, (and I think this is the best approach) it would be good to make explicit the usage of the Archimedean property in the proofs under "direct consequences". Not only would this make the proof more interesting by highlighting where it goes wrong in non-Archimedean settings, but it would clear up a lot of the potential for misunderstanding.

> Least Upper Bound Property -> NIP -> SNIP -> Cauchy Completeness

I would have liked to see this in the article, along with the distinction between NIP and SNIP. IToastPotatoes (talk) 18:52, 25 April 2025 (UTC)[reply]