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:
 * $\forall x \in \Bbb O: \map f x = \dfrac {x + 1} 2$

$f$ is well-defined as $x + 1$ is even and so $\dfrac {x + 1} 2 \in \Z$.

Let $x, y \in \Bbb O$ such that $\map f x = \map f y$.

Then:

Thus $f$ is injective by definition.

Consider the inverse $f^{-1}$.

By inspection:
 * $\forall x \in \Z: \map {f^{-1} } x = 2 x - 1$

$f^{-1}$ is well-defined, and $2 x - 1$ is odd.

Thus $f^{-1}$ is a mapping from $\Z$ to $\Bbb O$.

Then:

Thus $f^{-1}$ is injective by definition.

It follows by the Cantor-Bernstein-Schröder Theorem that there exists a bijection between $\Z$ and $\Bbb O$.