Let languages $A, B$ be co-recursively enumerable and disjoint. Prove the existence of a decidable language $L$ such that $A \subset L$ and $B \cap L = \emptyset$. That is, such that $A \subset L \subset B^c$.
The question should not be too hard (was on a rather easy exam, but I did not know how to do this one. Exam is over but still would be cool to know how to do this one!). I have the feeling it has something to do with the fact that the intersection of recursively and co-recursively enumerable languages is the set of decidable languages. Then starting with $A$, a co-recursively enumerable language, we add more and more words, and finally end up with $B^c$, a recursively enumerable language. I have some intuition that somewhere along the way we must have necessarily passed through some decidable languages, and cannot flip from co-recursively enumerable to recursively enumerable in an "instant" when adding words in such a way. But I am not sure how to put this intuition into a valid proof.