If 1 > 0.999..., then the difference between the two must be a positive,
real number, yes? Ok, let's assume that and see what happens.
First,
look at 0.999.... If you add anything to it, a nine somewhere will
carry, triggering a cascade of carries, and the result will either be 1.0 or
greater, right? So, if there's a difference, it must be the smallest
possible positive non-zero number, right? Otherwise, there would be a
smaller number that could be added to 0.999.. that would also
give a result equal to or greater than 1.0. So, we're dealing with the smallest
possible number. Let's call it x. x = 1 - 0.999....
With me so
far?
So what is x/2? That can't be a smaller number, because we've
already established that x is the smallest possible number. In fact, we've
defined it as the smallest possible number. So x/2 must be equal or
larger than x. If it's larger, then x < x/2. Multiply both sides by two, and
we get 2x < x. But we can't have a number smaller than x, so clearly x/2 is
not larger than x. It must be equal. Ok, x = x/2. Multiply both sides by two: 2x
= x. Subtract x from both sides: x = 0. But x is supposed to be larger than
zero. Contradiction, unless you admit that x is not greater than zero. Which, in
fact, is the case. 1 - 0.999... = 0.
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Do not meddle in the affairs
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