Positive Real Number Inequalities can be Multiplied/Disproof for Negative Parameters
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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
Let $a = c = -1, b = d = -2$.
Then $a c = 1$ but $b d = 2$.
$\blacksquare$
Sources
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 1$: Real Numbers: Exercise $\S 1.8 \ (4) \ \text{(iii)}$