Real Addition is Closed

Theorem
The set of real numbers $$\R$$ is closed under addition.

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