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.