For d = 1 and nonnested partitions where each partition consists of nonaccumulating intervals, prove the assertions of Lemma 24.9 with factor 2 on the right-hand side of the inequalities, and then...

For d = 1 and nonnested partitions where each partition consists of nonaccumulating intervals, prove the assertions of Lemma 24.9 with factor 2 on the right-hand side of the inequalities, and then prove Theorem 24.3 for d = 1 without condition (24.24). Hint: Use arguments in the proof of Lemma 24.4.