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.

>>13
While your conclusion is correct, the reasoning is not. The question is how to evaluate the expression. You did not show lim n->inf (sum(9*10^-n, n, 1, n)) = 1, which is the essence of the entire proof. I'm going to assume you do not understand how to rigorously use limits.