[quote=mdk]
Another way to approach it is, 1 - 0.9999999..... = X. In this case, X must equal 0.0000.......1, where the 1 does not occur until after an inifinitely long sequence of zeroes. It's a null infinity, which can never be expressed; thus, there is no difference between 1 and 0.999... Where there is no difference, the values are equivalent.
[/quote]

But the easiest way to approach it is with some simple fractions. Start with 1 = 1, split one side into thirds, then turn them into decimals.

1 = 1
1 = (1/3) + (1/3) + (1/3)
1 = 0.3333... + 0.3333... + 0.3333...
1 = 0.9999...

Even a middle school kid can understand it with this explanation, no wall of text or even knowledge of algebra needed.