1/9 = .111...
2/9=.222...
3/9=.333...
...
7/9=.777...
8/9=.888...
9/9=?
Answer .999... and 1 are equally correct. While not a proof (I can provide one on request), it demonstrates that the sum of an infinite number of things can still equal a finite quantity, e.g. 0.9+0.09+0.009+0.0009+...=1