# Real Numbers under Addition form 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

$\Box$

### $\text G 1$: Associativity

$\Box$

### $\text G 2$: Identity

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

$\Box$

### $\text G 3$: Inverses

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

$\Box$

### $\text C$: Commutativity

$\Box$

### Infinite

$\blacksquare$