I was reading Sacks forcing from Jech's Multiple Forcing and reached the definition of a fusion sequence which said:
Definition. $p \le_n q$ means $p \le q$ and every nth branching point of $q$ is a branching point of $p$. And we call a sequence $\{p_n\}_{n \in \omega}$ a fusion sequence if: $p_0 \ge_0 p_1 \ge_1 \dots \ge_{n-1} p_n \ge_n \dots$
And then i tried to prove the fusion lemma and i came up short. So naturally i checked Jech's Set Theory but there he defined a fusion sequence like this:
Definition. $p \le_n q$ means $p \le q$ and every nth branching point of $q$ is an nth branching point of $p$. And we call a sequence $\{p_n\}_{n \in \omega}$ a fusion sequence if: $p_0 \ge_0 p_1 \ge_1 \dots \ge_{n-1} p_n \ge_n \dots$
So which definition is correct? And for the fusion lemma the only part which i can prove is that the intersection is downward closed... I would really appreciate any hints that would help me prove the intersection is a perfect tree. Thanks for your patience.
Edit I: I forgot the statement of the fusion lemma:
Lemma. If $\{p_n\}_{n \in \omega}$ is a fusion sequence, then $p = \cap_{n \in \omega} p_n$ is a perfect tree. Moreover, $p \le_n p_n$ for all $n$.
The two are actually equivalent. Every branching point of $p$ is certainly a branching point of $q$ and the first version says that every $n$th branching point of $q$ is a branching point of $p$, in particular it is in $p$. This means that all points of $q$ below its $n$th splitting level are in $p$ and because $p$ is a subset of $q$ this means that $p$ and $q$ are exactly the same below the $n$th splitting level of $q$. Therefore the $n$th splitting level of $q$ is also the $n$th splitting level of $p$.
As to the proof: prove that the $n$th splitting level of $p_n$ survives to become the $n$th splitting level of $p$.