Infinite Set has Countably Infinite Subset/Intuitive Proof

Theorem
Every infinite set has a countably infinite subset.

Intuitive Proof
Let $S$ be an infinite set, and let $a_0 \in S$.

$S$ is infinite, so $\exists a_1 \in S, a_1 \ne a_0$, and $\exists a_2 \in S, a_2 \ne a_0, a_2 \ne a_1$, and so on.

That is, we can continue to pick elements out of $S$, and assign them the labels $a_0, a_1, a_2, \ldots$ and this procedure will never terminate as $S$ is infinite.

Each one of the elements is in one-to-one correspondence with the elements of $\N$, and therefore the set $\left\{{a_0, a_1, a_2, \ldots}\right\} \subseteq S$ is countably infinite.

Warning
The intuitive nature of this proof obscures the fact that it is not a trivial truth that one may choose elements of $S$ in this manner when $S$ is infinite.

In Zermelo–Fraenkel set theory, a rigorous application of the principle of mathematical induction would show that one can repeat the procedure any finite number of times to construct a finite set $\left\{{a_0, a_1, \ldots, a_n}\right\}$.

However, in general, one needs the axiom of dependent choice to justify repeating such a procedure indefinitely.

It should be noted that the weaker axiom of countable choice is sufficient to prove the stated theorem.