Rational Numbers under Addition form Infinite Abelian Group

Theorem
Let $$\mathbb{Q}$$ be the set of rational numbers.

The structure $$\left({\mathbb{Q}, +}\right)$$ is an infinite abelian group.

Proof
Taking the group axioms in turn:

G0: Closure
Rational Addition is Closed.

G1: Associativity
Rational Addition is Associative.

G2: Identity
The identity element of $$\left({\mathbb{Q}, +}\right)$$ is the rational number $$0$$:

G3: Inverses
The inverse of $$x \in \left({\mathbb{Q}, +}\right)$$ is $$-x$$:

C: Commutativity
Rational Addition is Commutative.

Infinite
Rational Numbers are Countable