I could give you the real reason, but would require the use of real analysis which is usually a 4th year mathematics course. As I stated earlier, it has to deal with the fact that there is no nonzero infinitesimal between .999... and 1.0. In other words, there is no number between .999... and 1.
For the purpose of a proof, lets assume they are distinct numbers. For any number not ending in a infinite sequence of 9's, you can always find another number in between two different numbers by simply finding the midpoint (x1+x2)/2. This doesn't work with numbers ending in the infinite sequence of 9's because the midpoint between .9 and 1 is .95. so if you could take the midpoint of the two, following that idea, the last number would have to be a 5, which contradicts the whole notion because .999 is an infinite sequence of 9's and would actually be greater than our midpoints. This creates an absurdity, which many mathematicians call a proof by contradiction. So since our original idea about them being two different numbers let to a contradiction, that means that assumption must be false, and thus means they have to be the same number.
