Solution
= 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 surjection
$$
s:\kappa\longrightarrow\alpha.
$$
All ordinal values of $s$ belong to the transitive model $M$, so closure gives $s\in M$. Therefore
$$
M\models|\alpha|\leq\kappa<\alpha,
$$
and $M$ does not regard $\alpha$ as a cardinal.