Real Number Ordering is Compatible with Multiplication/Positive Factor/Corollary

Theorem

$\forall a, b, c, d \in \R: 0 < a < b \land 0 < c < d \implies a c < b d$

where $\R$ is the set of real numbers.

Proof

\(\ds a < b\)

\(\implies\)

\(\ds a \times c < b \times c\)

Real Number Ordering is Compatible with Multiplication: Positive Factor as $c > 0$

\(\ds c < d\)

\(\implies\)

\(\ds b \times c < b \times d\)

Real Number Ordering is Compatible with Multiplication: Positive Factor as $b > 0$

The result follows by Real Number Ordering is Transitive.

$\blacksquare$

Sources

• 1957: Tom M. Apostol: Mathematical Analysis ... (previous) ... (next): Chapter $1$: The Real and Complex Number Systems: $\text{1-3}$ Order properties of real numbers