Real Numbers form Totally Ordered Field

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Theorem

The set of real numbers $\R$ forms a totally ordered field under addition and multiplication: $\left({\R, +, \times, \le}\right)$.


Proof

From Real Numbers form Field, we have that $\left({\R, +, \times}\right)$ forms a field.

Then we have that Ordering on Real Numbers is Total Ordering.

Hence $\left({\R, +, \times, \le}\right)$ is a totally ordered field.

$\blacksquare$


Sources