Suppose $A$ and $B$ are nonempty bounded subsets of $\mathbb R$ with $A \cap B = \emptyset$. Prove $\sup \, (A \cup B) \geq \max \, \{ \sup A,\sup B\}$.
So I thought for this proof it was enough to show that the $\sup \, (A \cup B)$ is an upper bound for the maximum of the two individual $\sup$'s.
But my professor's work starts off as "It suffices to show that, for each $\epsilon>0$ there exists an $x$ in $A \cup B$ such that $x> \max \, \{\sup A,\sup B \} - \epsilon$."
Why do we need to choose an arbitrary epsilon?