I have been banging my head against this proof for a few days now, as I can visualize why it is true in my head, but don't know how to prove it in words:
Let $A$ and $B$ be nonempty subsets of $\mathbb{R}$. Show that if there exist disjoint, open sets $U$ and $V$ with $A \subseteq U$ and $B \subseteq V$, then $A$ and $B$ are separated.
I've seen two answers to this proof on here, but I don't fully understand either one of them, and the question was asked so long ago that neither of those users are active on here anymore, so I can't even ask them specific questions to help my understanding. I have tried proving it directly, but immediately get bogged down in multiple cases of what $A$ looks like in $U$ while $B$ looks a certain way in $V$, and vice versa. I have also tried assuming that $A$ and $B$ aren't separated, and I can't find a way to reach a contradiction (or contrapositive) from that. I would greatly appreciate any assistance as it is the only proof from my homework that I haven't been able to figure out on my own.
I will prove that $\overline A\cap B=\emptyset$. Since $A\subset U$ and $U$ and $V$ are disjoint, then $A\subset V^\complement$. But $V$ is open and so $V^\complement$ is closed. So, and since $A\subset V^\complement$, $\overline A\subset V^\complement$. But then, since $B\subset V$, $\overline A\cap B=\emptyset$. For the same reason, $A\cap\overline B=\emptyset$.