Past exam of the mathematics course of the University of Cambridge / 2015 / ii / Paper 4 / 13I / iii / Solution

Represent a function by its graph of Kuratowski ordered pairs. For $f:\omega \to {\omega }^2$,The bound is attained by $f(n)=\omega n$, which is a cofinal function into ${\omega }^2$. Taking the rank of the set of all such functions adds one at this attained maximum. Thus $\boxed{\mathrm{rk}(({\omega }^2{)}^{\omega })={\omega }^2+1}$. The extra successor is essential: unlike the previous product, this set actually contains a member of rank ${\omega }^2$.