I have a question regarding the proof of Hilbert's Basis Theorem.
Say $I=(f_1,f_2,f_3,\dots)$ is an ideal in $A[x]$, where A is a Noetherian ring. Say we take the leading coefficients $a_i$ of all the polynomials generating $I$. Then the ideal $J$ generated by those coefficients will be finitely generated. We assume $J=(a_1,a_2,\dots,a_m)$. The proof says that $I=(f_1,f_2,\dots,f_m)$, where $f_i$ is the polynomial with leading coefficient $a_i$ for $i\in\{1,2,3,\dots,m\}$. How Hilbert goes about this (by successively selecting the polynomial of minimal degree) is well known, and can be seen in the link given.
My question is this:
Why does $f_i$ have to be the polynomial of minimal degree in $I$ for some $i\in\{1,2,3,\dots,m\}$? In fact, why do all $f_i$ have to be the minimal polynomials in the set $I\setminus (f_1,f_2,\dots,f_{i-1})$? I see no correlation between generating the ideal of leading coefficients and being the polynomial of lowest degree in $I$.