Let $(A, \le_A)$ and $(B, \le_B)$ be ordered sets.
(a) Prove or disprove: if both $A$ and $B$ are dense, then so is $A \times B$ with the product ordering
(b) Prove or disprove: if both $A$ and $B$ are dense, then so is $A \times B$ with the lexicographic ordering.