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:

Real Addition is Closed.

Real Addition is Associative.

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

From Inverse 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.