jmc

First, we convert $0.\overline{9}$ to fraction form. Let $a=0.\overline{9}.$ Multiply both sides of this equation by $10$ to get $10a=9.\overline{9}.$ Subtracting the left-hand sides $10a$ and $a$ as well as the right-hand sides $9.\overline{9}$ and $0.\overline{9}$ yields $9a=9$, which implies that $a=1.$ So, $$\begin{array}{rl}1-0.\overline{9}& =1-a\\ & =1-1\\ & =\boxed{0}.\end{array}$$