Complex Numbers under Addition form Infinite Abelian Group

Theorem
Let $$\mathbb{C}$$ be the set of complex numbers.

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

Proof
Taking the group axioms in turn:

G0: Closure
Complex Addition is Closed.

G1: Associativity
Complex Addition is Associative.

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

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

C: Commutativity
Complex Addition is Commutative.

Infinite
Complex Numbers are Infinite.