# Real Number Ordering is Transitive

## Theorem

The usual ordering on the real numbers is a transitive relation.

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

Then:

$a > c$

## Proof

From Real Numbers form Ordered Integral Domain, $\struct {\R, +, \times, \le}$ forms an ordered integral domain.

From Ordered Integral Domain is Totally Ordered Ring, the usual ordering $\le$ is a total ordering.

From Relation Induced by Strict Positivity Property is Transitive it follows that $<$ is transitive.

$\blacksquare$