On 31 May 2007 05:01:36 -0700, email@example.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. 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.
A sum of an infinite set is defined as a certain limit.
>> 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.
You have contradicted yourself. A sum of an infinite set is a limit. How else would you define a sum of an infinite set?
> 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.
You need to think again about what an infinite sum means.