Definition:Distance to Nearest Integer Function/Definition 1

Definition
The nearest integer function $\left\Vert{\cdot}\right\Vert : \R \to \left[{0 \,.\,.\, \dfrac 1 2}\right]$ is defined as:
 * $\left\Vert{\alpha}\right\Vert := \min \left\{{ \left\vert{n - \alpha}\right\vert : n \in \Z}\right\}$