Since $R_\pi(\delta) = \mathbb{E}_\pi[R(\delta, \theta)]$ is an average of $R(\delta, \theta)$ over $\theta$, and every average is bounded above by the supremum of the quantity being averaged: \begin{align*} R_\pi(\delta) = \mathbb{E}_\pi[R(\delta, \theta)] \leq \sup_{\theta \in \Theta} R(\delta, \theta) = R_m(\delta, \Theta). \end{align*}