Most of the explanations I've seen of why 0.999...repeating equals 1.0 have seemed a little esoteric, but I just thought of a really simple demonstration showing that they're equal
Start with the number 1.1111....repeating
Subtract 1 (leaving you with 0.1111...repeating)
add 0.9999....repeating
and you see immediately that it gives you back the original number