 Solution

Repeat the preceding construction starting at ${\alpha }_0=\gamma +1$, for any given ordinal $\gamma$. A stabilized value or the strictly increasing sequence's supremum is a fixed point at least $\gamma +1$. Thus fixed points occur above every ordinal.

If the fixed points formed a set $S$, their union would be an ordinal bounding them all. Applying the construction above that bound contradicts it. HenceThis is the unboundedness part of the fixed point of a normal ordinal function theorem.