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. So you are not using the standard definition. If you do not use the standard definition you need to supply a definition (and it would be good practice to use another name).
> I do not attribute to this > entity any other characteristics, need not for it to be real or non- > real. I allow the consequence of the infinite contibutive values alone > to dictate all else. > > > 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. > > > - William Hughes > > I avoid all limit definitions. To me limits are their own area of > study that tell you potentially more about what bounds an entity that > about the entity itself. > > The limit of 1/x as x approaches infinity is 0, but this is not > representative of the function at all which will never yield a zero > value. I do not reject ideas that discuss and define 0.999... as a > limit - I only reject ideas that attempt use a limit to ~equate~ to > the entity described by that limit when this is not justified. > I leave > this disclaimer only to take into account a constant limit, such as > the limit of 3 as x approaches infinity - where the limit is exactly > equal to the number that yields it. >
Unfortunately, there are sequences like (.9,.99,.999,...) that do not reach anything and do not have an element equal to the limit. You want to talk about such sequences but you are on the horns of a dilemma. You do not wish to use the limit, but you have nothing else to replace it with.