Also, if we assume that they are not equal and that it is "as close to one as possible without being one". Then we define must define a smallest number which represents their difference. If we assume a smallest possible number, bad things happen. Like the number line ceases to be continuous...
Ram a repeating infinite decimal is rational. It can be expressed as a fraction, and so is rational. The definition of a rational number includes repeating infinite decimals. On this point you are simply wrong.
Wow, took you a while to post this. Anyway, 0.9... is not a limit. It always has had and always will have an infinite number of nines. There are no more nines being added. 0.9... can be expressed as a sum of a series or as a limit, but this doesn't show that it is not equal to one. 0.9...
Since there are already an infinite number of zeroes before the one, adding one more will not make a difference (property of infinity). Thats part of the problem. Giving the 1 at the end of infinity a value means a) allowing the infinity to end and b) since the value of the one is equal to...
If we allow .0...1 (sorry not sure how to code for the bar notation) to have a non-zero value, then it must be the smallest positive number. If we allow for numbers to have a minimum value then the number line ceases to be continuous as any 2 numbers no longer have an infinite range of values...