Bukovský-Hechler theorem ($2^{\kappa }={\lim }_{\mu <\kappa }{2}^{\mu }\text{ on an eventual plateau}$)

If the power-set function is eventually constant below a singular cardinal $\kappa$, its constant value persists at $\kappa$. Writing $\theta =\mathrm{cf}(\kappa )$ gives $2^{\kappa }=(2^{<\kappa }{)}^{\theta }$. On the plateau choose a cardinal $\mu \ge \theta$, so $(2^{\mu }{)}^{\theta }=2^{\mu \cdot \theta }=2^{\mu }$.