Real Numbers form Ordered Field

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

Proof

 * From Additive Group of Real Numbers, we have that $\left({\R, +}\right)$ forms an abelian group.


 * From Multiplicative Group of Real Numbers, we have that $\left({\R^*, \times}\right)$ forms an abelian group.


 * Next we have that Real Multiplication Distributes over Addition.


 * Finally we have that Real Numbers are Totally Ordered.

Thus all the criteria are fulfilled, and $\left({\R, +, \times, \le}\right)$ is a totally ordered field.