Real Numbers form Ordered Field
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Theorem
The set of real numbers $\R$ forms an ordered field under addition and multiplication: $\struct {\R, +, \times, \le}$.
Proof
From Real Numbers form Field, we have that $\struct {\R, +, \times}$ forms a field.
From Ordering Properties of Real Numbers we have that $\struct {\R, +, \times, \le}$ is a ordered field.
$\blacksquare$
Sources
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $1$: Review of some real analysis: $\S 1.1$: Real Numbers
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): order properties (of real numbers)