Positive Real Number Inequalities can be Multiplied/Disproof for Negative Parameters

Theorem

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

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

From Positive Real Number Inequalities can be Multiplied, $a c > b d$ holds.

However, if $b < 0$ or $d < 0$ the inequality does not hold.

Proof

Proof by Counterexample:

Let $a = c = -1, b = d = -2$.

Then $ac = 1$ but $bd = 2$.

$\blacksquare$

Sources

• K.G. Binmore: Mathematical Analysis: A Straightforward Approach (1977)... (previous)... (next): $\S 1.8 \ (4) \ \text{(iii)}$