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 $0$.