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.

$\blacksquare$

Sources

• 1971: Wilfred Kaplan and Donald J. Lewis: Calculus and Linear Algebra ... (previous) ... (next): Introduction: Review of Algebra, Geometry, and Trigonometry: $\text{0-2}$: Inequalities
• 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 1$: Real Numbers: $\S 1.4$: Inequalities: $\text{(I)}$
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): inequality
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): inequality: 2.
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): order properties (of real numbers): $(2)$ Transitive law