Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 116 / 2 / b / i / Solution

The standard closure lemma for the ultrapower embedding says that every $\kappa$-sequence of members of $M$ which belongs to $V_{\lambda }$ is itself in $M$. If $\kappa <\alpha <{\kappa }^{+}$, choose in $V_{\lambda }$ a surjectionAll ordinal values of $s$ belong to the transitive model $M$, so closure gives $s\in M$. Thereforeand $M$ does not regard $\alpha$ as a cardinal.