Conditions: $A,B \subset \mathbb Q$ such that $A,B \neq \emptyset$ moreover $A\cup B = \mathbb Q$ with the condition $\forall a \in A$ and $ \forall b \in B$ we have $a\leq b$.
I reckon I can describe the situation more plainly as; show that there is a unique real number between any two rational numbers.
My work: I have been bouncing the idea of using the fact that there exists a supremum for $A$ which lies in $B$, although that's rather obvious and I've found no prevail from there. From my experience a problem like this that involves proving uniqueness can be usually solved easily using a proof by contradiction. But again... I'm not progressing. Any help/pointers will be much appreciated, many thanks.