Proof of measurability of set of all rationals

Question

How we can prove that set of all rational numbers is measurable by showing that inner and outer measures are equal?

Answer 1

If we arrange the set of rational numbers as $\{r_1,r_2,...\}$ and consider the intervals $(r_i-\frac {\epsilon} {2^{i}},r_i+\frac {\epsilon} {2^{i}})$ we see that outer measure at most $\epsilon$. Since $\epsilon$ is arbitrary this shows that the outer measure is $0$. But inner measure is $\leq$ outer measure so it is also $0$.