Real Numbers form Totally Ordered Field

Theorem

The set of real numbers $\R$ forms a totally 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 totally 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
  • 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): order properties (of real numbers)
  • 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): order properties (of real numbers)
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