(Dis)prove that $\sup(A \cap B) = \min\{\sup A, \sup B\}$

Question

Just beginning real analysis so I'm having some trouble with disproving this statement:

$$\sup(A \cap B) = \min\{\sup A, \sup B\}$$

Initially it asks whether it's true or false and to provide a counterexample if false, which by basic intuition to me it is. However I'm not sure where to start, as my book is rather poor in its explanations of concepts and I'm just starting out. Thanks for any help

Answer 1

Hint: Take $A=\{1,2,3\}$ and $B=\{2,4,5\}$.

Answer 2

Let $A = \{1\}$, $B = \{2\}$. Note that $\sup A \cap B = -\infty$.

If you want to avoid infinities, try $A = \{1, 2\}$, $B = \{1, 3\}$.