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.