Prove that two sets A and B with $A \cap B=\emptyset$, $\sup A = \sup B$, $\sup A \notin A$ and $\sup B \notin B$ cannot exist.

Question

I have to show that it is either possible or impossible to have two such sets. I understand intuitively that they cannot exist (correct me if I'm wrong, please), but can't seem to figure out how to even put it in words, let alone prove it.

I was thinking of assuming all of these things and showing that it leads to a contradiction, but I got stuck several times. Could someone please give me some pointers as to how to begin?

Answer 1

Consider $\Bbb Q \cap (0,1)$ and $(\Bbb R \setminus \Bbb Q )\cap (0,1)$.

Answer 2

Let $A:=\left\{1-\frac{1}{2n-1}\mid n\in\left\{1,2,\dots\right\}\right\}.$
Let $B:=\left\{1-\frac{1}{2n}\mid n\in\left\{1,2,\dots\right\}\right\}.$