Equivalence of Definitions of Countably Infinite Set

Theorem
Let $S$ be a set.

The following definitions for $S$ to be countably infinite are equivalent:

Proof
From Integers are Countably Infinite there is a bijection between $\Z$, the set of integers, and $\N$, the set of natural numbers.

Let $h: \N \to \Z$ be such a bijection.

Let $f: S \to \N$ be a bijection.

From Composite of Bijections is Bijection:
 * $h \circ f: S \to \Z$ is a bijection.

Similarly, let $g: S \to \Z$ be a bijection.

By Inverse of Bijection is Bijection, $h^{-1}: \Z \to \N$ is a bijection.

Again from Composite of Bijections is Bijection:
 * $h^{-1} \circ g: S \to \N$ is a bijection.

Hence the result.