Uniqueness of the True cofinality

Question

Why is the true cofinality here on the page 220 in the definition 5.7(5) unique ?

Answer 1

Suppose that a partial order $\mathcal{P}$ have two $\leq_{\mathcal{P}}$-increasing sequences $A:=\{p_i:i<\lambda\}$ and $B:=\{q_i:i<\kappa\}$ such that that $\lambda<\kappa$ and both of them are witness for $tcf(\mathcal{P})$. So, you can build an alternating sequence between the elements of $A$ and $B$, $p_0<q_{\alpha_0}<p_{\beta_1}<q_{\alpha_1}<...$ where $q_{\alpha_i}>p_i$. But, as $\kappa$ is regular and $\lambda<\kappa$ the sequence $\{q_{\alpha_i}:i\leq\lambda\}$ is bounded in $B$. Let $q\in B$ such that $q_{\alpha_i}<q$. So, for every $p_i\in A$ we have that $p_i<q$. This implies that $A$ fails to be a witness of $tcf(\mathcal{P})$.