Are two languages regular if the union is regular? Statement:


  For any two languages $L_1$ and $L_2$ if $L_1 \cup L_2$ is regular, then $L_1$ and $L_2$ are regular.


Why is this statement false? Could somebody give me an example.
Pick $\Sigma = \{a,b\}$, $L_1 = \{ a^n b^n : n \in \mathbb{N}\}$ and $L_2 = \Sigma^* \setminus L_1$.

The union $L_1 \cup L_2$ is $\Sigma^*$ and hence is regular, but none of $L_1$ and $L_2$ is.