Real Addition Identity is Zero

Theorem
The identity of real number addition is $$0$$:
 * $$\exists 0 \in \R: \forall a \in \R: a + 0 = a = 0 + a$$

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 addition, $$x + y$$ is defined as $$\left[\!\left[{\left \langle {x_n} \right \rangle}\right]\!\right] + \left[\!\left[{\left \langle {y_n} \right \rangle}\right]\!\right] = \left[\!\left[{\left \langle {x_n + y_n} \right \rangle}\right]\!\right]$$.

Let $$\left \langle {0_n} \right \rangle$$ be such that $$\forall i: 0_n = 0$$.

Then we have:

$$ $$ $$

Similarly for $$\left[\!\left[{\left \langle {x_n} \right \rangle}\right]\!\right] + \left[\!\left[{\left \langle {0_n} \right \rangle}\right]\!\right]$$.

Thus the identity element of $$\left({\R, +}\right)$$ is the real number $$0$$.