[banach2]

Okay, this is a little late but when you say 10X = 9.999... then X = 0.999... not because you have removed the 9 but because that's what X is; a tenth of 10X! So when you say in your argument 10X = 9.9 and therefore 10X - X = 9.9 - 0.9, that's wrong. 10X = 9.9 implies X = 0.99. No wiggle room. No need to consider infinites, just one movement of a decimal point.

[And hence 10X - X = 9.9 - 0.99 = 8.91 = 9X implying that X = 0.99]

Following, as delineated above, from there, you'll see there's no contradiction. It's not so easy to break arithmetic that easily without dividing by zero :)

Sorry for dragging on this meaningless thread.