Question: If $A$and $B$are nonempty, disjoint sets with $A\cup B=\mathbb{R}$and $a<b$for all $a∈A$and $b∈B$. Show that there is a real number $c$ such that $x < c $ implies $x ∈ A$, and $x > c$ implies $x ∈ B$.
How do I prove this?
My professor hasn't covered this topic in class yet, so I am very confused on how to prove this. She gave us a hint: Set $A$ is non-empty and bounded above. (To show that $A$ is bounded above, pick any number from set B.)
Can anyone please explain this conceptually and show me how to write a proof for a question like this so I understand?