The proof is by transfinite induction. Let be a limit ordinal (the induction is trivial for successor ordinals), and for each , let be a partition of satisfying the requirements of the theorem.
Fix an increasing sequence cofinal in with .
and so .
Let be the order type of . As for the order types, clearly .
Noting that the sets form a consecutive sequence of ordinal intervals, and that each is a tail segment of we get that: