I am reading Eisenbud's Commutative Algebra. The following is the proof I am trying to understand.
My question is the second sentence in the proof.
I understand that a power of $P_P$ annihilates $M_P$. However, to conclude $M_P$ is of finite length using Corollary 2.17, I need to know that $P_P$ is maximal in $R_P$. I really have no idea why $P_P$ is maximal in $R_P$. All I know is that it is a minimal prime containing $I_P$.
Do I miss something really obvious? Thanks!
EDIT: I found the answer, so I will put it below.