Polish spaces are continuous images of the Baire space I'm having some troubles understanding the proof of Theorem 7.9 (pag. 39) in Kechris' "Classical Descriptive Set Theory":

There are two points of the proof proposed that I don't quite understand.
First of all, when we define the Lusin scheme, why do we need to specify that every $F_s$ is going to be a  $F_\sigma$ if right afterwards we set $F_s = \bigcup_i \overline{F_{s^\smallfrown i}}$, making it a $F_\sigma$ set by definition. Moreover I truly don't get how does he manage, at the end of the proof, to write $C_{i+1} \setminus C_i = \bigcup_j E_j^{(i)}$ with $E_j^{(i)}$ being pairwise disjoint $F_\sigma$ sets of diameter $< \epsilon$. Where do $(E_j^{(i)})_j$ come from? How do we know that  $C_{i+1} \setminus C_i$ can be covered by pairwise disjoint $F_\sigma$ sets of diameter $< \epsilon$?

(ii) is somewhat superfluous. (iii) is already the intended construction: each $F_s$ is partitioned by its successors $F_{s\smallfrown i}$ and these sets are all $F_\sigma$ too. (ii) is just to reinforce and anticipate (iii), I think. (iii) is a statement of intent, not the definition of $F_s$.
The final point is more subtle: $C_{i+1}\setminus C_i$ is a relatively open set of the Polish space $C_{i+1}$. Every open subset of a Polish space can be written as a pairwise disjoint countable union of small diameter $F_\sigma$ sets. This follows as we can take a cover by small (diameter) open sets and as the set is hereditarily Lindelöf (being second countable) we can find a countable subcover  of it, which we enumerate. After that, taking the standard trick of subtracting all previous sets, we get a countable disjoint family of sets that are all $F_\sigma$ (the difference of open sets in a metric space, using open sets are $F_\sigma$) and still small.
