If $S$ is countable then $S_0 \subset S$ is countable.

Define “countable” in the following way:

$ S$ is said to be countable if $ S$ is finite OR $ |\mathbb{N}| = |S|$ .

So my textbook proves the theorem by considering two cases.

Case 1: $ S$ is finite. Then any subset $ S_0$ is clearly finite as well. So by definition $ S_0$ is countable.

Case 2: $ S$ is infinite. Then the proof proceeds…

I don’t understand why we are considering these two cases, i.e. why do we consider $ S$ is finite or infinite…?

We have the hypothesis that $ S$ is countable MEANING $ S$ is finite or $ |\mathbb{N}| = |S|$ . In particular we have that $ S$ is finite why are we considering these two seeming weird cases?

Like aren’t we trying to show $ S_0$ is countable i.e. aren’t we trying to show that $ S_0$ is finite or its cardinality is the same as the natural numbers?