Let $A$ and $B$ be sets of real numbers. My question is, under what circumstances is the supremum of $A$ equal to the infimum of $B$?
Now $x$ is the supremum of $A$ if and only if for any $\epsilon>0$ there exists an $a\in A$ such that $x\geq a>x-\epsilon$. And $y$ is the infimum of $B$ if and only if there exists a $b\in B$ such that $y\leq b<y+\epsilon$. Is there any way to combine these two conditions into a single condition for when the supremum of $A$ is equal to the infimum of $B$? Some kind of inequality involving elements of $A$ and elements of $B$?