Let there be defined an order $\langle X, \leq \rangle$ and two sets $A, B \subseteq X$. Does existence of $\sup (A \cup B)$ imply that both $\sup A$ and $\sup B$ exist?
Here is my reasoning so far: Let $x \in X$, $x = \sup(A\cup B)$, then $x$ is an upper bound for both $A$ and $B$. I cannot see however if that means that both $A$ and $B$ have least upper bounds.