Because adding 9 to any other digit will produce 1 and the other digit - 1. Add these together and you have the original other digit. Because your rule necessitates repeating until you only have 1 digit it doesn't matter when this resolves itself.

For example: 9 6=> 1 5 => 6

Put another way, a nine will bump up the 10s place by one and lower the ones place by 1. These cancel out.