Equivalence of Definitions of Distance to Nearest Integer Function

Theorem
The following definitions of the distance to nearest integer function $\norm \cdot: \R \to \closedint 0 {\dfrac 1 2}$ are equivalent:

Proof
Let $\alpha \in \R$, $n \in \Z$.

From Real Number is between Floor Functions:
 * $\floor \alpha \le \alpha < \floor \alpha + 1$

For any $n < \floor \alpha \le \alpha$:

For any $n > \floor \alpha + 1 > \alpha$:

Thus:

which shows that the definitions are indeed equivalent.