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:

S0: Closure
Real Addition is Closed.

S1: Associativity
Real Addition is Associative.

S2: Identity
From Real Addition Identity is Zero, we have that the identity element of $\struct {\R, +}$ is the real number $0$.

Hence the result.