Regular cofinality strata are stationary

Given a club $C\subseteq \theta$, define a strictly increasing sequence $(c_{\xi }{)}_{\xi <\lambda }$ in $C$: start at $\min C$, take the least greater C point at successors, and at nonzero limits take the supremum. Every such supremum is below theta since its index is below $\mathrm{cf}(\theta )$, and closure places it in C. Likewise $\delta ={\sup }_{\xi <\lambda }{c}_{\xi }<\theta$ belongs to C.

This sequence gives $\mathrm{cf}(\delta )\le \lambda$. If a cofinal subset $B\subseteq \delta$ had size $\mu <\lambda$, assign to each $b\in B$ the least $\xi <\lambda$ with $b<c_{\xi }$. These indices would be unbounded in lambda: a bound would bound B below delta. Regularity of lambda rules this out. Therefore $\mathrm{cf}(\delta )=\lambda$, so C meets the stratum. Since C was arbitrary the stratum is stationary.