Non-Zero Real Numbers under Multiplication form Abelian Group

Theorem
Let $$\R^*$$ be the set of real numbers without zero, i.e. $$\R^* = \R - \left\{{0}\right\}$$.

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

Proof
Taking the group axioms in turn:

G0: Closure
Real Multiplication is Closed.

G1: Associativity
Real Multiplication is Associative.

G2: Identity
The identity element of $$\left({\R^*, \times}\right)$$ is the real number $$1$$:

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

Let $$\left \langle {1_n} \right \rangle$$ be such that $$\forall i: 1_n = 1$$.

Then we have:

$$ $$ $$

Similarly for $$\left[\!\left[{\left \langle {x_n} \right \rangle}\right]\!\right] \times \left[\!\left[{\left \langle {1_n} \right \rangle}\right]\!\right]$$.

G3: Inverses
The inverse of $$x \in \left({\R^*, \times}\right)$$ is $$x^{-1} = \frac 1 x$$:

We have:

$$ $$

Similarly for $$\frac 1 {\left[\!\left[{\left \langle {x_n} \right \rangle}\right]\!\right]} \times \left[\!\left[{\left \langle {x_n} \right \rangle}\right]\!\right]$$.

C: Commutativity
Real Multiplication is Commutative.

Infinite
Real Numbers are Uncountably Infinite.