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.