Talk:Bolzano–Weierstrass theorem

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[edit] Generalization to metric spaces

Shouldn't the Theorem's generalization to metric spaces be included in the article?

In a compact metric space (S,d), every infinite set included in S admits at least an accumulation (limit) point. Or in a compact metric space, every sequence admits a convergent subsequence. —Preceding unsigned comment added by Hearth (talk • contribs) 01:58, 24 April 2008 (UTC)

[edit] Spelling

Weierstrass/Weierstraß

It is not really my buisness, but I think in English it should be spelled Weierstrass, you know, all these books, and everything. Tosha 07:35, 23 Feb 2005 (UTC)

Agree. Oleg Alexandrov 21:16, 6 May 2005 (UTC)


What's up with the weird dash? Shouldn't it be a hyphen? 67.164.12.169 07:17, 22 October 2005 (UTC)

No, the en dash is correct. See Dash#En dash. —Caesura(t) 19:10, 6 December 2005 (UTC)

[edit] Equivalent theorems

Regarding this edit: What does it mean for theorems to be equivalent? I understand what it means that propositions A and B are equivalent (namely, A implies B and B implies A), but that does not make sense for theorems which are basically tautologies. -- Jitse Niesen (talk) 23:45, 8 November 2006 (UTC)

[edit] Alternative Theorems

In my real analysis class, I had two Bolzano-Weierstrass theorems:

one for sets, and one (as is here) for sequences

Is there somewhere where the sets one is taken care of?

Peter Stalin 18:21, 27 March 2007 (UTC)

I'm not exactly sure what you are asking (but I'm not that smart, so its probably my fault)? What book did you use (if you have a page reference, that would be awesome)? Are you thinking of this formulation? Smmurphy(Talk) 18:09, 29 May 2007 (UTC)

[edit] Applications to Economics

Hi. While the application to economics listed here is interesting, it makes use of technical language that seems out of place in this article. If we had a nice list of a few of the many applications of this theorem in Mathematics, it would be a lot better. We could then include this application as one example. Regardless, this little section needs rewritten to be more understandable by someone with more knowledge of economics than I. Grokmoo 18:24, 13 July 2007 (UTC)

[edit] Correctness of Theorem

I could be confused, but I think this page has a somewhat serious issue. The Heine-Borel theorem is that a set A\subset \Bbb{R}^n is compact if and only if it is closed and bounded. This is equivalent to what is written here. My understanding is that the Bolzano-Weierstrass theorem is much more general and deeper, that is, that a subset of a metric space is compact if and only if it is sequentially compact. Any thoughts here? If there are no arguments, I think I will update the page and try to include a proof. This could be very confusing for a beginning student in analysis. John 06:27, 23 October 2007 (UTC)

The Bolzano–Weierstrass theorem article states that:
A subset A of Rn is sequentially compact if and only if it is both closed and bounded.
So I guess you are referring to the theorem being generalized to an arbitrary metric space (with "bounded and closed" being replaced by "compact"). I would very much prefer that instead of rewriting the whole article from the more general view of metric spaces, you'd rather work on expanding the "Generalizations" section in the article. How would that sound? Oleg Alexandrov (talk) 03:17, 28 October 2007 (UTC)
Yeah, that sounds good. I will spend some time doing that when I have some time. John 04:27, 28 October 2007 (UTC)