Real Multiplication Distributes over Addition

Theorem
The operation of multiplication on the set of real numbers $$\R$$ is distributive over the operation of addition.

Proof
From the definition, the real numbers are the set of all equivalence classes $$\left[\!\left[{\left \langle {x_n} \right \rangle}\right]\!\right]$$ of Cauchy sequences of rational numbers.

Let $$x = \left[\!\left[{\left \langle {x_n} \right \rangle}\right]\!\right], y = \left[\!\left[{\left \langle {y_n} \right \rangle}\right]\!\right], z = \left[\!\left[{\left \langle {z_n} \right \rangle}\right]\!\right]$$, where $$\left[\!\left[{\left \langle {x_n} \right \rangle}\right]\!\right]$$, $$\left[\!\left[{\left \langle {y_n} \right \rangle}\right]\!\right]$$ and $$\left[\!\left[{\left \langle {z_n} \right \rangle}\right]\!\right]$$ are such equivalence classes.


 * From the definition of real multiplication, $$x \times y$$ is defined as $$\left[\!\left[{\left \langle {x_n} \right \rangle}\right]\!\right] \times \left[\!\left[{\left \langle {y_n} \right \rangle}\right]\!\right] = \left[\!\left[{\left \langle {x_n \times y_n} \right \rangle}\right]\!\right]$$;
 * From the definition of real addition, $$x + y$$ is defined as $$\left[\!\left[{\left \langle {x_n} \right \rangle}\right]\!\right] + \left[\!\left[{\left \langle {y_n} \right \rangle}\right]\!\right] = \left[\!\left[{\left \langle {x_n + y_n} \right \rangle}\right]\!\right]$$.

We note the fact that the rational numbers form a field, so we can use the fact of the distributivity of $$\times$$ over $$+$$ on $$\Q$$.

Thus we have:

$$ $$ $$ $$ $$ $$ $$ $$

By Real Addition is Commutative and Real Multiplication is Commutative, it follows that $$\left({y + z}\right) \times x = \left({y \times x}\right) + \left({z \times x}\right)$$