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Let ${\delta }^{\prime }$ be any decision rule. For every $j$,because $r_j$ is the infimum of the integrated risk under ${\pi }_j$. Taking the lower limit givesThe given rule $\delta$ has constant risk $r$, and hence worst-case risk exactly $r$. It attains this universal lower bound and is therefore minimax. This is the constant-risk limit-of-Bayes-risk criterion.