If we assume $\sup A ≤ \sup B$, why is there not necessarily an element of $B$ that is an upper bound to $A$. I know that it is the case for when $\sup A < \sup B$, because it implies that set $A$ is bounded above and the elements of $B$ are greater than $A$. Would it be because set $A$ could be the same as set $B$ or at least have the same supremum? 

What if $A=B=(0,1)$?${}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}$
I think your intuition is correct. If $\sup A\leq\sup B$, but strict inequality doesn't hold, then equality certainly holds. 
