Real Numbers under Addition form Monoid

Theorem
The set of real numbers under addition $\struct {\R, +}$ forms a monoid.

Proof
Taking the monoid axioms in turn:

$\text S 0$: Closure
Real Addition is Closed.

$\text S 1$: Associativity
Real Addition is Associative.

$\text S 2$: Identity
From Real Addition Identity is Zero, we have that the identity element of $\struct {\R, +}$ is the real number $0$.

Hence the result.