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Proof: Since ? ? 0, increasing ? results in a smaller utility of forwarding to neighbors in Eq. (5), and the optimal policy will not use more transmissions or higher transmit power, hence the optimal reliability and the optimal energy cost cannot become larger. Proof of Lemma 5.1: Proof: At each step of the DP, the number of choices are limited by the number of neighboring nodes and channel states. We have finite number policies that leads to a finite number of optimal reliabilities. Thus, R is a finite set. For any given ? 1 , ? 2 ? ? R , we have R (? 1 ) = R (? 2 ) with optimal polices ? (? 1 ) and ? (? 2 ), respectively, Suppose that C (? 1 ) = C (? 2 ). Without loss of generality, we let ? 1 < ? 2 . According to Lemma A.1, we have C (? 1 ) ? C (? 2 ) and since, by assumption, C (? 1 ) = C (? 2 ), we have C (? 1 ) > ,