?. Max, According to Lemma 5.5, ?t ? [t 1 , t 2 ] U f +1 (t) ? max(S(t)) So ?t ? [t 1 , t 2 ] U f +1 (t) ? S max . By Lemma 5.4, for each correct robot i and for each t ? [t 1 , t 2 ], D i (t) ? (U m (t) + U f +1 (t))/2. So for each correct robot i and for each t ? [t 1, D i (t) ? (U m (t) + S max ) . Since the algorithm is cautious, ?t ? [t 1 , t 2 ] U m (t) ? U m (t 1 ) and the lemma follows

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