I am trying to solve this exercise from Eisenbud's Commutative Algebra with a View Toward Algebraic Geometry. There is a hint or possibly a solution in the back, but I want to try to get some more organic feedback before peeking.
3.11(d): Let $k$ be a field, $R = k[x_1, x_2, ..., x_r]$, and $I \subset R$ a homogeneous ideal. Suppose $P = (x_1, \ldots, x_r)$ is an associated prime of $R/I$. Show that there exists a positive integer $N$ such that if $I = J_1 \cap J_2 \cap \ldots \cap J_n$ is a primary decomposition and $J_1$ is $P$-primary and maximal among all ideals that can serve as the $P$-primary entry of the intersection, then the length of $R/J_1$ is smaller than $N$ for all choices of $J_1$.
My approach has been to try to show that there exists $B$ such that $(x_1^B, \ldots, x_r^B) \subset J_1$ for all choices of $J_1$. If I had this, then we would have a surjection $R/(x_1^B, \ldots, x_r^B) \to R/J_1$. The first ring is a finite dimensional vector space and hence Artinian. Moreover, its dimension is uniformly bounded for all choices of $J_1$. Hence, the length of $R/J_1$ is also bounded by the same implied constant.
What do you suggest on how to proceed from here? Is my approach off track?