\begin{align*} 0.999999... & = 0.9 + 0.09 + 0.009 + 0.0009 + ... = 0.9(1 + 0.1 + 0.1^2 + 0.1^3 + ...\\ & = 0.9 \frac{1}{1-0.1} = 0.9 \frac{1}{0.9} = 1 \end{align*}