Elementary initial segments form a club

Use the witness family $\mathcal{H}$. For $\beta <\kappa$, let $g(\beta )=\sup (\{h(\vec{a})+1:h\in \mathcal{H},\vec{a}\in (\beta +1{)}^{\mathrm{arity}(h)}\}\cup \{\beta +1\})$. There are fewer than kappa values: the finite-string cardinal count in the witness lemma bounds all tuples, and multiplying by $\mathrm{\mid }\mathcal{H}\mathrm{\mid }<\kappa$ still gives fewer than kappa. Regularity therefore gives $g(\beta )<\kappa$.

The nonzero closure points alpha of g form an unbounded set. Since $g(\beta )>\beta$, such alpha are limits. Every finite tuple below alpha is contained in some $\beta +1<\alpha$; hence every h value on it is below alpha. This includes the empty tuple for constants. The witness lemma gives $\alpha \in E_M$. Thus $E_M$ is unbounded.

If delta is a nonzero limit point of $E_M$, every finite tuple below delta lies below some $\alpha \in E_M\cap \delta$. Language-function closure at alpha makes the restriction to delta a substructure. Any existential formula true in M with such a tuple has a witness below alpha by elementarity there, hence below delta. The witness criterion proves elementarity at delta. Thus $E_M$ itself is closed, completing the club claim.