Real Numbers form Field

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

Proof
From Real Numbers under Addition form Abelian Group, we have that $\left({\R, +}\right)$ forms an abelian group.

From Non-Zero Real Numbers under Multiplication form Abelian Group, we have that $\left({\R_{\ne 0}, \times}\right)$ forms an abelian group.

Next we have that Real Multiplication Distributes over Addition.

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

Also see

 * Real Numbers form Totally Ordered Field