Let $\sup(A) = \max\{A\}$ and $\sup(B) = \max\{B\}$ hold,
that is, that the suprema are contained in the respective amount and thus maxima.
Must then $A \cap B$ also have a maximum?
If so, prove it. If not, give a counterexample.
I literally have no clue what to do. Please help, I'd appreciate it.
Assuming further that $A\cap B \neq \emptyset$ otherwise this could already be a good case for a counterexample.
Take $A= (0,1)\cup \{2\}$ and $B=(0,1)\cup\{3\}$ then $A\cap B = (0,1)$ has no maximum.