Real Numbers under Addition Modulo 1 form Group

Theorem
Let $S = \left\{{x \in \R: 0 \le x < 1}\right\}$.

Let $\circ: S \times S \to S$ be the operation defined as:
 * $x \circ y = x + y - \left \lfloor {x + y} \right \rfloor$

That is, $\circ$ is defined as addition modulo $1$.

Then $\left({S, \circ}\right)$ is a group.

Proof
First note that Modulo Addition is Well-Defined.

Taking the group axioms in turn:

G0: Closure
Demonstrated in Real Number Minus Floor.

G1: Associativity

 * The commutativity and associativity of $\circ$ follows from those of the sum of real numbers.

G2: Identity

 * There is an identity element which is $0$.

G3: Inverses

 * The inverse of $x$ is $1-x-\operatorname{floor}(1-x)$, so $\operatorname{inv} x=1-x$ if $x>0$