Inequality of Product of Unequal Numbers

Theorem

Let $a, b, c, d \in \R$.

Then:

$0 < a < b \land 0 < c < d \implies 0 < a c < b d$

$\text {(1)}: \quad$

$\ds $

$\ds 0 < a < b$

$\text {(2)}: \quad$

$\ds $

$\leadsto$

$\ds 0 < b$

Ordering is Transitive

$\text {(3)}: \quad$

$\ds $

$\ds 0 < c < d$

$\text {(4)}: \quad$

$\ds $

$\leadsto$

$\ds 0 < c$

Ordering is Transitive

$\ds $

$\leadsto$

$\ds 0 < a c < b c$

$(1)$ and $(4)$: Real Number Axiom $\R \text O2$: Usual Ordering is Compatible with Multiplication

$\ds $

$\ds 0 < b c < b d$

$(2)$ and $(3)$: Real Number Axiom $\R \text O2$: Usual Ordering is Compatible with Multiplication

$\ds $

$\leadsto$

$\ds 0 < a c < b d$

Ordering is Transitive

$\blacksquare$