If A is an infinite set, one should be careful when talking about its cardinality |A|: what kind of object is it ? Whatever it is, we know that if e.g. A is the set of natural numbers, and B is the reals, |A| ≠ |B|, so we cannot just say that |A| and |B| are ∞ (whatever that means). It turns out one can see |A|, |B| as some kind of infinitary numbers, called cardinals, that will not be defined in this course. Fortunately, you can still talk about what it means for A and B to have the same cardinality using bijections, but unless A is finite, it is best to avoid writing |A| entirely, in my opinion.
We might revisit this subject once we get to equivalence relations.
Your textbook defines a set as countable if it is in bijection with the natural numbers, thus if A is a countable set, one directly knows it is infinite (why ?).
When you want to prove a certain set A is countable (as in 4.49), it is enough to build a surjection from the natural into A, and prove that A is infinite.
The natural map many of you defined in 4.49 is not a bijection, as it could be that A1, A2, ... are not disjoint (e.g. they may all be equal), so the fact a surjection is enough comes in handy (but then you shouldn't forget to argue A is infinite).
A very serious mistake in 4.49 was to prove only that the union of A1, ..., Ak is countable for all k. This is not the same as showing the entire union is countable. For example, if you replace "countable" by "finite", it is true that a union of finitely many finite sets is finite, but a countable union of finite sets is not necessarily finite (why ?). This shows that even if a property is true of any "initial segment" of a countable union, it may not hold for the entire union.