Real Numbers under Addition form Infinite Abelian Group

Theorem
Let $$\R$$ be the set of real numbers.

The structure $$\left({\R, +}\right)$$ is an infinite abelian group.

Proof
Taking the group axioms in turn:

G0: Closure
Real Addition is Closed.

G1: Associativity
Real Addition is Associative.

G2: Identity
The identity element of $$\left({\R, +}\right)$$ is the real number $$0$$:

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

G3: Inverses
The inverse of $$x \in \left({\R, +}\right)$$ is $$-x$$:

We have:

$$ $$

Similarly for $$\left({-\left[\!\left[{\left \langle {x_n} \right \rangle}\right]\!\right]}\right) + \left[\!\left[{\left \langle {x_n} \right \rangle}\right]\!\right]$$.

C: Commutativity
Real Addition is Commutative.

Infinite
Real Numbers are Uncountably Infinite.