If your nines can go on forever then our zeroes can go on forever, too. Just because their is an infinite number of zeroes or nines does not mean that the integer at the infinite position must be the same.
The point is, when we specify an infinte number of nines that is exactly what we mean, that EVERY digit is a nine and continues on that way with no possibility of variation.
When you say an infinite number of zeros then a 1 you have contradicted yourself because there can be no end to infinity.
I have already attempted to convey this information, perhaps if I keep trying certian individuals may start to catch on.
Let D(n) be the digit in the nth position of a Real number. So, for example,
2.000... means that D(n) = 0 for ALL n
This is what is implied when we write 2
.999.... means that D(n) = 9 for All n
I can completly specify the digits of those numbers within the context of real numbers therefore they are valid real numbers.
If I write .00...001 This implies that for some N>0 (pick N to be any integer you choose)
D(n) = 0 for n = 1 to N-1
D(N) = 1
Then
D(n) = 0 for all n > N
So formally for any decimal numbers , x, it is written like this
x = Sum(n=1 to infinity) D(n)* 10^-n
To specifiy x you must specify the D(n), these are the digits of the number.
if x = .000...00100...
Then
x = Sum( n= 1 to N-1)( 0 * 10^-n) + 10^-N + sum (n= N+1 to infinity)( 0 * 10^-n)
All real numbers can be written as this type of sum, if you cannot express the sum then it is not a real number. Therefore it is not a valid real number if you attempt to write
x = Sum(n= 1 to infinity) 0* 10^-n + 10^ - ??? (what do I put in the exponent? we have already committed all integers in the first sum, there are none left for use, we cannot recycle them, a sum to infinity means exactly that, not a sum to some unspecified number. That is how Madrat is treating infinity, he seems to think infinity is the same as unspecified large, this is not the case. A sum to infinity means that ALL integers are used in the sum indexes, none are free to be reused.