Real Numbers under Addition form Infinite Abelian Group

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

The structure $\struct {\R, +}$ is an infinite abelian group.

Proof
Taking the group axioms in turn:

$\text G 0$: Closure
Real Addition is Closed.

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

$\text G 2$: Identity
From Real Addition Identity is Zero, we have that the identity element of $\struct {\R, +}$ is the real number $0$.

$\text G 3$: Inverses
From Inverses for Real Addition, we have that the inverse of $x \in \struct {\R, +}$ is $-x$.

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

Infinite
Real Numbers are Uncountably Infinite.