Real Addition is Closed

Theorem
The set of real numbers $\R$ is closed under addition:
 * $\forall x, y \in \R: x + y \in \R$

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 $\left[\!\left[{\left \langle {x_n} \right \rangle}\right]\!\right] + \left[\!\left[{\left \langle {y_n} \right \rangle}\right]\!\right] = \left[\!\left[{\left \langle {x_n + y_n} \right \rangle}\right]\!\right]$.

We have that $\forall i \in \N: x_i \in \Q, y_i \in \Q$, therefore $x_i + y_i \in \Q$.

So it follows that $\left[\!\left[{\left \langle {x_n + y_n} \right \rangle}\right]\!\right] \in \R$.