Real Multiplication Identity is One

Theorem
The identity element of real number multiplication is the real number $1$:
 * $\exists 1 \in \R: \forall a \in \R_{\ne 0}: a \times 1 = a = 1 \times a$

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:
 * $\left[\!\left[{\left \langle {x_n} \right \rangle}\right]\!\right] \times \left[\!\left[{\left \langle {y_n} \right \rangle}\right]\!\right] = \left[\!\left[{\left \langle {x_n \times y_n} \right \rangle}\right]\!\right]$

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 $\left({\R_{\ne 0}, \times}\right)$ is the real number $1$.