Infinite Set has Countably Infinite Subset

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.

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