Set of Odd Integers is Countably Infinite

Theorem
Let $\Bbb O$ be the set of odd integers.

Then $\Bbb O$ is countably infinite.

Proof
Let $f: \Bbb O \to \Z$ be the mapping defined as:
 * $\forall x \in \Bbb O: f \left({x}\right) = \dfrac {x + 1} 2$