= Solution
Let $\widehat\kappa=\sup_{n<\omega}j^n(\kappa)$. Every term of the critical sequence is below $\delta$. If $\operatorname{cf}(\delta)>\aleph_0$, then its countable supremum also satisfies $\widehat\kappa<\delta$, and because $\delta$ is a limit ordinal,
$$
\widehat\kappa+2<\delta.
$$
Apply the <Kunen lemma> to the restriction available inside $V_\delta$. It gives
$$
j^{\prime\prime}\widehat\kappa\in
V_{\widehat\kappa+1}\setminus V_\delta,
$$
which is impossible because $V_{\widehat\kappa+1}\subseteq V_\delta$. Hence $\operatorname{cf}(\delta)\leq\aleph_0$. A nonzero limit ordinal has infinite cofinality, so
$$
\boxed{\operatorname{cf}(\delta)=\aleph_0.}
$$
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