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 $$1$$:

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

C: Commutativity
Real Multiplication is Commutative.

Infinite
Real Numbers are Uncountably Infinite.