Absolute Proof that 0.999... IS NOT EQUAL TO 1 
A math-wiki:

You are wrong about infinite representation. The ancient Greeks *never* represented 1/3 as 0.333... 

Now let me refute your argument regarding the transitive property:

A)    1=3(1/3)
B)    3(1/3) = 3(0.333...)
C)    3(0.333...) =0.999... 

The problem here occurs in B) where you assume that 3 * 0.333... = 3 * 1/3. You have 'not proved' anything. On the contrary, you have only proved that you are assuming 1/3 = 0.333... which it is obviously 'not equal'. 0.333... is an 'indeterminate' number because it cannot be represented finitely in base 10. 
0.333... is not equal to 1/3 so you cannot say that 1/3 is a finite representation. An indeterminate number is a number that is repeating in any
radix system.

You mention the division algorithm but do you know what it says? The division algorithm has 'nothing' to do with radix systems. Let me state it for you - If a and b are two given integers, there are unique integers q and r
such that a=bq + r and  0<= r < b. Now I will use the division algorithm to prove that you are WRONG once and for all. Suppose a = 1 and b = 0.999..., then 
what are the integers q and r such that the above is true?

Suppose that q = 0 and r = 1. Then the division algorithm fails because it makes a false statement i.e. q and r can both be 0 and 1 respectively.
For example: 1 = (0.999...)(1)+0 and 1 = (0.999...)(0)+1. So by default, the
division algorithm is proved false if 0.999... is equal to 1. And you can say a lot of things about the Greeks, but they would not make a stupid mistake like this. They were the smartest race in ancient times.

Now suppose that you try to tell me that either form is acceptable in the case of 0 and 1, you will still end up with a terrible contradiction:

1=(0.999...)(0)+1. The second form in this case implies that 1 < 0.999... since r < b. But how can 1 be less than 0.999... ??!!

I want the world to know what f..g idiots run this site. You are a bunch of retarded, conceited fools who I look upon with the utmost hatred and condemnation. Your site is absolute crap because it is run by fools like Michael
Hardy, Kmrsq, Melchoir, Meni (idiot) Rosenfeld, et al. DIE WIKIPEDIA!!!! HAIL KNOLL!!!! 98.195.24.26