Talk:Supremum

Difference between sup and max

What's the difference between 'sup' and 'max'? It seems like 'sup' is 'max', plus definitions about what the value is for an empty set, as well as an unbounded set. I think it'd be nice to compare sup to max for other naive readers like me (and I'd like to know). Zashaw 21:15, 28 Aug 2003 (UTC)

given a set S, max S is a member of that set. sup S might not be. Also, max S might not exist. Eg, suppose S = { -1, -1/2, -1/3, -1/4 , ........ }. Then there is no max S -- any element of S you select as a candidate for max, I can pick one that beats it. But there IS a sup S, it's zero. -- Tarquin 21:19, 28 Aug 2003 (UTC)

I'm new to this, but the line "An important property of the real numbers is that every set of real numbers has a supremum." appears completely false to me. Isn't there only a supremum when there's an upper bound? (ie The set of all real numbers doesn't have a supremum since it's not bounded) -- Muso 19:02, 6 Sep 2003 (UTC)

What is meant is that there is one, perhaps + or - infinity. See also infimum, where I added a phrase to clarify, we can do that here also. - Patrick 19:41, 6 Sep 2003 (UTC)

I've reworded things in an attempt to make it clearer. --Zundark 19:54, 6 Sep 2003 (UTC)

Small correction to last section

In the example: "Let $S$ be the set of all rational numbers $q$ such that $q^{2}<2$. Then $S$ has an upper bound (1000, for example, or 6) but no least upper bound. For suppose $p$ is an upper bound for $S$, so $p^{2}>2$."

shouldn't the last expression be closed i.e. $p^{2}\geq 2$ since $q^{2}<2$ is open.

p is supposed to be rational, so equality is impossible. I've modified the paragraph to make this a bit clearer. --Zundark 14:41, 11 Mar 2004 (UTC)

Questions for clarification

I want to make the definition more clear, but since I don't know much about supremum, I need to ask a couple things:

• Is it true that every finite set has a supremum and infimum that is in that set?
  • True in a total order (or indeed any join lattice), not true in general. -lethe ^{talk +}
• Would it be correct to say that the supremum of a set with no upper bound is the limit as n->∞ of the set element a_n ?
  • This would be fairly inaccurate. If the set is countable then there exists such a sequence, but there exist many other sequences as well which do not have that limit. And if the set is not first countable, then sequences can't get you there. -lethe ^{talk +}
• What does the set being a subset of something else have anything to do with?
  • The most interesting examples are the ones where the supremum is not in the set, otherwise our supremum is actually a greatest element. This can only occur when we consider the supremum of a subset of some other set. -lethe ^{talk +} 12:38, 29 March 2006 (UTC)

Thanks, Fresheneesz 11:52, 29 March 2006 (UTC)

So what finite sets don't have a supremum and infimum inside itself? Fresheneesz 05:48, 31 March 2006 (UTC)

The set {{1},{2}}, ordered by containment, has supremum {1,2} (the smallest set which contains both {1} and {2} as subsets), and infimum ∅ (the largest set which is contained in both {1} and {2} as a subset). Neither of these is an element of the set. -lethe ^{talk +} 12:59, 31 March 2006 (UTC)

Slight clarification in introduction

I clarified in the introduction that S is a subset of an ordered set T, and that the supremum of S is an element of T. Previously the language was slightly confused and referred to the supremum being an element of S, then switching and saying it isn't always an element of S. Clarifying that S is a subset of a possibly larger ordered set T makes the introduction (hopefully) more accurate. Dugwiki 21:13, 9 November 2006 (UTC)

Illogical proof and possible merge with infimum

I just made a bold move to remove a proof (of the Approximation Property) that did not seem correct, as well as subsequent proofs. Perhaps it was just worded poorly, and someone can present sound reasoning for it.

Also, this article should be merged with infimum so they're not essentially mirror articles with opposite definitions, properties, examples, etc. Least upper bound has already been redirecting to Supremum since 2002. –Pomte 05:15, 15 March 2007 (UTC)