Name: .999...9 =/= 1 2007-05-12 6:26 ID:/qcJW6wi
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
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