omg new proof

Assumptions
If A = B, then
a)  A-B = 0
b) A = (A+B)/2 = B

If A > B, then
a)  A-B > 0
b) A > (A+B)/2 > B


Let A = 1, B = .999...9
a) A-B = 1-.999...9 = .000...1 =/= 0
b) (A+B)/2 = (1+.999...9)/2 = 1.999...9/2 = 1.999...95 < A

Therefore, .999...9 =/= 1

>>102
Yes, that's a very good definition of finite vs infinite (thanks), but it doesn't apply.  You provided a detailed and thorough definition of finite vs infinite for sets, but we're not dealing with sets.  We're arguing over a single element (that may or may not belong to some set).  The original argument was whether .999... was finite or not.  I claim it is.  Random person denies that.