Prove/Disprove: $L_1, L_2 \in RE-R \implies L_1 \cup L_2 \notin R$
Hint: Let $L$ be any language in $\mathsf{RE} - \mathsf{R}$, and consider
$$ L_1 = 0\Sigma^* \cup 1L, \quad L_2 = 1\Sigma^* \cup 0L. $$