Non-Zero Real Numbers under Multiplication form Abelian Group/Proof 1

Proof
Taking the group axioms in turn:

$\text G 0$: Closure
From Non-Zero Real Numbers Closed under Multiplication: Proof 2, $\R_{\ne 0}$ is closed under multiplication.

Note that proof 2 needs to be used specifically here, as proof 1 rests on this result.

$\text G 1$: Associativity
Real Multiplication is Associative.

$\text C$: Commutativity
Real Multiplication is Commutative.

Infinite
Real Numbers are Uncountably Infinite.