>>299
Claim: \eqn{0.999...\neq 1}
Proof: We use induction. The base case is trivial \eqn{0.9 \neq 1}. Next we introduce notation that \[0.9_n = 0.9999... n times].
Now the inductive step: we assume \eqn{0.9_n \neq 1}. Then trivially \eqn{0.9_{n+1} \neq 1}. It might help to notice \eqn{1 - 0.9_{n+1} \neq 0 }
This implies that \eqn{0.9_n\neq 1 \forall n \in \mathbb{N}}
\eqn{\therefore 0.999... \neq 1}