On May 31, 4:02 am, chaja...@mail.com wrote: > On May 31, 12:33 am, William Hughes <wpihug...@hotmail.com> wrote:
> > I disagree with the definition given: > > let a = 0.999... be a real number. We do not need > > > to give a full definition at this point > > a<=1 > > and > > a>(1-(1/10^n) for any natural number n > > You have defined 0.999... to be a real number without jusitification. > I can make no such assumption.
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.
> > The fact that 0.999... is or is not a real number is of little concern > to my proof - it makes no assumption and has no need one way or the > other. What it demonstrates is that if x = 0.999... then 10x - x must > be less than 9 within the confines of multiplication upon real > numbers.
No. If you use the standard limit definition, 10x-x = 9. You are only correct if you use some other definition for 0.999... in which case the "subtraction" is not in the real numbers. You need to define a new set of "numbers". When you do so you will not get a field.