Equivalence of Definitions of Distance to Nearest Integer Function

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Theorem

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

Definition 1

$\norm \alpha:= \min \set {\size {n - \alpha}: n \in \Z}$

Definition 2

$\norm \alpha:= \min \set {\set \alpha, 1 - \set \alpha}$

where $\set \alpha$ is the fractional part of $\alpha$.


Proof

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

From Real Number is between Floor Functions:

$\floor \alpha \le \alpha < \floor \alpha + 1$


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

\(\displaystyle \size {n - \alpha}\) \(=\) \(\displaystyle \alpha - n\) Definition of Absolute Value
\(\displaystyle \) \(>\) \(\displaystyle \alpha - \floor \alpha\)
\(\displaystyle \) \(=\) \(\displaystyle \size {\alpha - \floor \alpha}\) Definition of Absolute Value

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

\(\displaystyle \size {n - \alpha}\) \(=\) \(\displaystyle n - \alpha\) Definition of Absolute Value
\(\displaystyle \) \(>\) \(\displaystyle \floor \alpha + 1 - \alpha\)
\(\displaystyle \) \(=\) \(\displaystyle \size {\floor \alpha + 1 - \alpha}\) Definition of Absolute Value

Thus:

\(\displaystyle \min \set {\size {n - \alpha}: n \in \Z}\) \(=\) \(\displaystyle \min \set {\size {\floor \alpha - \alpha}, \size {\floor \alpha + 1 - \alpha} }\) Other $n$'s are disregarded by above
\(\displaystyle \) \(=\) \(\displaystyle \min \set {\alpha - \floor \alpha, \floor \alpha + 1 - \alpha}\)
\(\displaystyle \) \(=\) \(\displaystyle \min \set {\set \alpha, 1 - \set \alpha}\) Definition of Fractional Part

which shows that the definitions are indeed equivalent.

$\blacksquare$