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?