Real Multiplication Identity is One

Theorem
The identity of real number multiplication is $$1$$:
 * $$\exists 1 \in \R: \forall a \in \R: a \times 1 = a = 1 \times a$$

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

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

Let $$\left \langle {1_n} \right \rangle$$ be such that $$\forall i: 1_n = 1$$.

Then we have:

$$ $$ $$

Similarly for $$\left[\!\left[{\left \langle {x_n} \right \rangle}\right]\!\right] \times \left[\!\left[{\left \langle {1_n} \right \rangle}\right]\!\right]$$.

So the identity element of $$\left({\R^*, \times}\right)$$ is the real number $$1$$.