I am reading Sharp's book "Steps in Commutative Algebra".
Let $R$ be a commutative Noetherian ring and let $M$ be a non-zero finitely generated $R$-module. Let $a_1,\ldots,a_n \in R$. Suppose that there exist positive integers $t_1,\ldots,t_n$ such that $(a_i^{t_i})_{i=1}^n$ is an $M$-sequence. Prove that $(a_i)_{i=1}^n$ is an $M$-sequence.
I know the definition of $M$-sequences. So, I try to show that $a_i$ is not a zero divisor on $R/(a_1,\ldots,a_{i-1})$ using $a_i^{t_i}$ is not a zero divisor on $R/ (a_1^{t_1}, \ldots, a_{i-1}^{t_{i-1}})$. But I didn't find a relation between them. So I need a hint for this.
