Am Donnerstag, 31. Mai 2007 16:07:56 UTC+2 schrieb William Hughes: > On May 31, 8:01 am, chaja...@mail.com wrote: > > > You do however have to give some sort of definition for 0.999... > > > Whatever you define it to be it will either be equal to 1 > > > or it will not be a real number. > > > > My definition of 0.999... is the sum of all elements of an infinite > > set defined by 0.9*(1/10)^n for all n in the set of wholes numbers > > including 0 and is included in my proof. > > You are defining 0.999... in terms of something else > you have not defined, the sum of all elements of an infinite > set. The standard definition uses a limit and produces > a real number.
The "standard definition" however is overridden by set theory. There we have as a limit the complete sequence of all indexed nines. We have a bijection between omega and the set of all indexed nines. Note that omega is not among the natural numbers. According to this fact 1 is not among all terms of the sequence.