jmc

We have $$0.\overline{4}=\frac{4}{10}+\frac{4}{100}+\frac{4}{1000}+\cdots .$$This infinite geometric series has first term $4/10=2/5$ and common ratio $1/10$, so we have $$0.\overline{4}=\frac{2/5}{1-1/10}=\boxed{\frac{4}{9}}.$$