>>68
Your proof is essentially circular. You start by assuming 1/9 = 0.111..., and then show 0.999... = 1. While it's certainly true that 0.111... = 1/9, people who don't believe 0.999... = 1 obviously are not going to accept your proof anymore than they would accept "0.999... = 1 because I said so." They will simply resort to claiming 1/9 != 0.111...
The better algebraic proofs (or better still, analytic proofs) do not rely on the decimal expansions of other fractions.