Past exam of the options II course of the University of Oxford / 2013 / D4 / b / i

Given $ϵ>0$, cover $E_1$ by open intervals of total length less than $ϵ/2$, and cover $E_2$ similarly. Combining the two countable covers gives a cover of their union of total length less than $ϵ$. Thus $E_1\cup E_2$ is a null set.