$\Box_{\kappa}$ principle, unknown derivation of its property

Question

In Set Theory: The Third Millennium Edition, revised and expanded, page $443$, Jech states the $\Box_{\kappa}$ principle as follows:

$(\Box_\kappa)$ There exists a sequence $\langle C_\alpha:\alpha\in\operatorname{Lim}(\kappa^+)\rangle$ such that

1. $C_\alpha$ is a closed unbounded subset of $\alpha$;
2. if $\beta\in\operatorname{Lim}(C_\alpha)$ then $C_\beta=C_\alpha\cap\beta$;
3. if $\operatorname{cf}\alpha<\kappa$ then $|C_\alpha|<\kappa$.

He then says that it follows from 2. and 3. that the order-type of every $C_{\alpha}$ is at most $\kappa$. I would like to know how this follows.

Answer 1

If $\alpha$ has cofinality $\kappa$, fix some closed and unbounded sequence of length $\kappa$, now for every $\beta<\alpha$ which is a limit point of the sequence $C_\alpha\cap\beta=C_\beta$, but since $\beta$ must have cofinality less than $\kappa$, $|C_\beta|<\kappa$.

Therefore every initial segment of $C_\alpha$ has size less than $\kappa$, the order type of $C_\alpha$ is $\kappa$.