Half-Integer is Half Odd Integer

Theorem
Let $r$ be a number.

Then $r$ is a half-integer iff $r = \dfrac n 2$ where $n$ is an odd integer.

Necessary Condition
Let $r$ be a half-integer.

Then by definition $r = n + \dfrac 1 2$ for some $n \in \Z$.

Thus:

thus showing that $r$ is half of $2 n + 1$ for some $n \in \Z$.

By Odd Integer 2n + 1 it follows that $r$ is half of an odd integer.

Sufficient Condition
Let $k$ be an odd integer.

Then by Odd Integer 2n + 1:
 * $k = 2 n + 1$

where $n \in \Z$.

Then:

thus showing that $\dfrac k 2$ is a half-integer.