jmc

We first convert $0.\overline{2}$ to fraction form. Let $u=0.\overline{2}$. Then $10u=2.\overline{2}$. Subtracting the left-hand sides and subtracting the right-hand sides yields $$\begin{array}{rl}10u-u& =2.\overline{2}-0.\overline{2}\\ \Rightarrow 9u& =2\\ \Rightarrow u& =\frac{2}{9}.\end{array}$$Thus, $$\begin{array}{rl}0.\overline{2}\cdot 6& =\frac{2}{9}\cdot 6\\ & =\frac{2\cdot 2}{3}\\ & =\boxed{\frac{4}{3}}.\end{array}$$