Real Numbers form Ring

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

Proof
From Additive Group of Real Numbers, $\left({\R, +}\right)$ is an abelian group.

We also have that:
 * Real Multiplication is Closed:
 * $\forall x, y \in \R: x \times y \in \R$


 * Real Multiplication is Associative:
 * $\forall x, y, z \in \R: x \times \left({y \times z}\right) = \left({x \times y}\right) \times z$

Thus $\left({\R, +}\right)$ is a semigroup.

Finally we have that Real Multiplication Distributes over Addition:
 * $\forall x, y, z \in \R:$
 * $x \times \left({y + z}\right) = x \times y + x \times z$
 * $\left({y + z}\right) \times x = y \times x + z \times x$

Hence the result, by definition of ring.