I am reading Chapter 10 of Albiac and Kalton's book $\textit{Topics in Banach Space Theory}$, and am trying to understand the proof of Theorem 10.1.3, namely that subsets of $\mathcal{P}_{\infty}\mathbb{N}$ that are Borel for the Ellentuck topology are completely Ramsey.
In particular, Step 1 of the proof is to show that if $\mathscr{U}$ is an open and dense set for the Ellentuck topology, then for any pair $(A,E)$ and finite subset $B \subset E$, there is some $M \in \mathcal{P}_{\infty}(B,E)$ such that $\mathcal{P}_{\infty}(A,M) \subset \mathscr{U}$. Their argument is that if we denote by $(B_{j})_{j=1}^{N}$ a list of all subsets of $B$, then since $\mathscr{U}$ is both open and dense there is some $H_{1} \in \mathcal{P}_{\infty}(E)$ such that $\mathcal{P}_{\infty}(A \cup B_{1},H_{1}) \subset \mathscr{U}$ (using the pair $(A \cup B_{1},E)$). The process is then repeated where we look at the pair $(A \cup B_{2},H_{1})$ and obtain an $H_{2} \in \mathcal{P}_{\infty}(H_{1})$ such that $\mathcal{P}_{\infty}(A \cup B_{2},H_{2}) \subset \mathscr{U}$. This is then repeated for all $2 \le j \le N$, and letting $M = H_{N}$ we get that $\mathcal{P}_{\infty}(A,M) \subset \mathscr{U}$.
I understand why $\mathcal{P}_{\infty}(A,M) \subset \mathscr{U}$, but I don't understand however why $M=H_{N}$ must contain $B$. Indeed, if it did contain $B$, then we would get that $H_{1}$ contains $B$ which seems a bit strange to me (as $B_{1}$ could be the empty set for example). I've been trying to show that $B \subset M$ using that $\mathcal{P}_{\infty}(A \cup B_{j},M) \subset \mathscr{U}$ for every $1 \le j \le N$ (and that $\mathscr{U}$ is Ellentuck-open), but I haven't gotten very far. Any help would be greatly appreciated.
We have constructed an $M \in \mathcal{P}_{\infty}(E)$ such that $\mathcal{P}_{\infty}(A \cup B_{j},M) \subset \mathscr{U}$ for every $1 \le j \le N$. So, if we let $M' = M \cup B$, then $M' \in \mathcal{P}_{\infty}(B,E)$ and any $N \in \mathcal{P}_{\infty}(A,M')$ is in $\mathcal{P}_{\infty}(A \cup B_{j},M \cup B_{j}) = \mathcal{P}_{\infty}(A \cup B_{j},M)$ for some $1 \le j \le N$, which is a subset of $\mathscr{U}$.