This question has been bugging me for quite some time now.

Say we have some $\beta$ smaller than some $\gamma$ and a sequence

$\beta$_{$\epsilon$} : $\epsilon$ smaller than cf($\beta$) cofinal in $\beta$ and say

we have some sets $A$_{n}^{$\epsilon$} and each of these $A$_{n}^{$\epsilon$} has order type less than $\gamma$^{$n$}.

Now $\forall n$ in $\omega$ let $B$_{n}= $\cup$ $A$_{n}^{$\epsilon$} for all $\epsilon$ < $\gamma$ and suppose in the end I can write $\beta$ as the union of all the $B$_{n} (but that is not really my problem here)

Why can I deduce that $B$_{n} has order type less than $\gamma$^{$n+1$} ** only if** all my sets $A$

_{n}

^{$\epsilon$}are disjoint and do not overlap?

(since we have a union of less then $\gamma$ sets each of which is of order type less than $\gamma$^{$n$})

Why can't I still guarantee that the $B$_{n} will still have order type $\gamma$^{$n+1$} if all the $A$_{n}^{$\epsilon$} are not disjoint?

I know that I need to take the $A$_{n}^{$\epsilon$} to be [$\epsilon$,$\epsilon+1$) so that they are disjoint.

But why does everything in the union have to be in order?

I hope I conveyed my question clearly. Thanks in advance for any help.