1 != 0.99...

Consider the floor function. It ignores the decimal part (everything right of the decimal point) of a number.

Thus, floor(0.999...) = 0, and if 0.999... = 1, floor(1) = 0.
However, floor(1) = 1, therefore we must conclude that either 1 = 0, or the hypothesis was wrong.

>>1
You don't need to go through any clever nonsense.  All you have to do is remember that the stardard definitions used by mathemeticians now make it so that ununderstandable infinites are inherent in how they define their idea of numbers.  We get bogged in notational bullshit that doesn't make sense if we try to imagine it in the world of our experience of apples and kittens, which is not the same thing as the wonderland of conceived math.  So, fuck it.