Infinite Set has Countably Infinite Subset

Theorem
Every infinite set has a countably infinite subset.

Proof 2
This proof follows the same steps as the intuitive one, but with more formality.

Note
Although this is intuitively an "obvious" theorem, it is a theorem about cardinalities. The theory of cardinal numbers depends strongly on the axiom of choice. Without AoC one cannot guarantee "elementary" and "intuitive" results, like Surjection iff Right Inverse: if $f: X \to Y$ is surjective, then $\exists g: Y \to X$ injective. It is not true that only "complex" or "strange" results (like the Banach-Tarski Paradox, or the existence of a basis for $\R$ as a vector space over $\Q$) depend on the AoC.

Comment
What this in effect shows is that countably infinite sets are the smallest possible infinite sets.