No, you are forgetting the reoccuring values. 0.999 x 9 does indeed equal 8.991, but 0.999... (reoccuring) is the same value as 1, as my calculation showed.
You can do it the other way:
1/3 = 0.333... (reoccuring)
3 * 1/3 = 3 * 0.333...
1 = 0.999...
You can do it the other way:
1/3 = 0.333... (reoccuring)
3 * 1/3 = 3 * 0.333...
1 = 0.999...