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
From Real Numbers under Addition form Group, $\struct {\R, +}$ is a group.

Then:
 * Real Addition is Commutative

and:
 * Real Numbers are Uncountably Infinite.