Give an example where the inequalities are strict $\sup (\inf A,\inf B) \le \inf (A \cap B) \le \sup (A \cap B) \le \inf (\sup A, \sup B)$

Question

$A$ and $B$ are in the real set. $\sup (\inf A,\inf B) \le \inf (A \cap B) \le \sup (A \cap B) \le \inf (\sup A, \sup B)$ I proved the inequalties as I was asked to but couldn't find an example. Is it that I should take sets A and B as the inverse of some function or what?

Answer 1

Even simpler: take $A=\{0,2,3,5\}$, $B=\{1,2,3,4\}$.

Answer 2

Let $A=\{1,5,3,4\}$ and $B=\{2,6,3,4\}$. Then the first is $2$, the second is $3$, the third is $4$, and the fourth is $5$.