On changing cofinality of partially ordered sets

1. for every n, 0 < n < ω, [f p(n) ] U≤n = p(n),
2. for every m, 1 ≤ m < ω and every sequence ⃗ν = 〈ν 1 , ..., ν m 〉 the sequence
〈f p(m) (⃗ν), [f p(m+1),⃗ν ] Um+1 , ..., [f p(k),⃗ν ] Um+1 ×...×U k
, ... | m < k < ω〉
satisfies Definition 4.12 only the second member is in M m+1 instead of M 1 , etc.
Proof. Note that for every m, 1 ≤ m < ω, the sequence 〈p(m + 1), p(m + 2), ..., p(n), ... |
m + 1 ≤ n < ω〉 ∈ M ≤m . Just for each n, 1 ≤ n < ω we have p(n) ∈ M ≤n , and given
m, 1 ≤ m < n < ω, M ≤n ≃ M j ≤n(κ m+1 )×...×j ≤n (κ n)
≤m /j ≤n (U m+1 × ... × U n ).
If f p(n) represents p(n) in M ≤n , i.e. [f p(n) ] U1 ×...×U n
, or equivalently j ≤n (f p(n) )(κ 1 , ..., κ n ) =
p(n), then
j ≤m,≤n ((j ≤m (f p(n) )) 〈κ1 ,...,κ m〉)(κ m+1 , ..., κ n ) = p(n).
Now the elementarity of the embeddings provides the desired conclusion.
□
Given a good sequence ⃗p = 〈p(0), p(1), ..., p(n), ...〉 with p(0) ∈ G.
associate to it a tree T (⃗p) ⊆ [κ]