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
From Real Addition Identity is Zero, we have that the identity element of $$\left({\R, +}\right)$$ is the real number $$0$$.

G3: Inverses
From Inverses for Real Addition, we have that the inverse of $$x \in \left({\R, +}\right)$$ is $$-x$$.

C: Commutativity
Real Addition is Commutative.

Infinite
Real Numbers are Uncountably Infinite.