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:''(Could someone who understands explain why the set of rational numbers does not have property 4?)''
Property 4 says that if you partition the set into two halves, then there must be a boundary point ''in the set''. This is not true for the rationals: take as ''A'' the set of all rationals smaller than √2 and as ''B'' the set of all rational above √2. Then all rationals are covered, since √2 is irrational, so this is a valid partition. There is no boundary point ''in the set of rational numbers'' that separates ''A'' from ''B'' however. [[User:AxelBoldt|AxelBoldt]] 02:09, 23 May 2006 (UTC)
==Complete is the wrong word==
Technically, the real numbers are not [[complete]]. Of course, the [[extended reals]] are complete. It may be better to remove the completeness requirement and leave the "i.e.". Really, all we need is what Rudin calls the "least-upper-bound property." That is, the least upper bound of any set that is bounded from above exists (and the similar claim about sets bounded from below and infimum). --[[User:TedPavlic|TedPavlic]] 15:32, 7 March 2007 (UTC)
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