If $R(\delta_\lambda, \theta) = c$ for all $\theta$, then \begin{align*} R_\lambda(\delta_\lambda) = \mathbb{E}_\lambda[R(\delta_\lambda, \theta)] = \mathbb{E}_\lambda[c] = c = \sup_{\theta \in \Theta} R(\delta_\lambda, \theta). \end{align*} The hypothesis of the previous theorem is satisfied, so the conclusion follows.