Positive Real Number Inequalities can be Multiplied

Theorem
Let $a, b, c, d \in \R$ such that $a > b$ and $c > d$.

Let $b > 0$ and $d > 0$.

Then $a c > b d$.

If $b < 0$ or $d < 0$ the inequality does not hold.

Proof
Note that as $d > 0$ it follows that $c > d > 0$ and so $c > 0$.

Finally: