This is a question on a proof of Theorem 1.2.7 in Sturmfel's Algorithms in Invariant Theory. I will restate the whole proof up to where I am confused.
First some notation. Let $\sigma_i(x_1, \ldots, x_n)$ be the $i$-th elementary symmetric polynomial. Let $h_i(x_1, \ldots,x_n)$ be the $i$-th complete symmetric polynomial. Let $I$ be the ideal of $\mathbb{C}[\overline{x}, \overline{y}] = \mathbb{C}[x_1, \ldots, x_n, y_1, \ldots, y_n]$ generated by $\sigma_i(x_1, \ldots, x_n) - y_i$.
Next I will state the following useful theorem without proof:
$\textbf{Theorem 1.2.6:}$ Let $I$ be any ideal of $\mathbb{C}[\overline{x}]$ and fix any monomial order. Then the residue classes of standard monomials (monomials that are not in the initial ideal of $I$, $\text{init}(I)$) form a $\mathbb{C}$ vector space basis of $\mathbb{C}[\overline{c}] / I$.
$\textbf{Theorem 1.2.7:}$ The unique reduced Groebner basis of $I$ with respect to the lexicographic monoimial order induced by $x_1 > x_2 > \cdots > x_n > y_1 > y_2 > \cdots > y_n$ is $$G = \left\{ h_k(x_k, \ldots, x_n) + \sum_{i = 1}^l (-1)^i h_{k - i}(x_k, \ldots, x_n) \cdot y_i \mid k = 1, \ldots, n \right\}.$$
$\textbf{Proof}$: First, note that for all $k = 1, \ldots, n$ that $$h_k(x_k, \ldots, x_n) + \sum_{i = 1}^l (-1)^i h_{k - i}(x_k, \ldots, x_n) \cdot \sigma_i(x_1, \ldots, x_n) = 0.$$ This, $G \subseteq I$.
Now grade $\mathbb{C}[\overline{x}, \overline{y}]$ by $\deg(x_i) = 1$ and $\deg(y_i) = i)$. Thus, $I$ is homogenous and $R = \mathbb{C}[\overline{x}, \overline{y}] / I$ is isomorphic as a graded algebra to $\mathbb{C}[\overline{x}]$. Thus, the Hilbert Series of $R$ is $\frac{1}{(1 - z)^{-n}}$. By $\textbf{Theorem 1.2.6}$ we have that the Hilbert series of $\mathbb{C}[\overline{x}, \overline{y}] / \text{init}(I)$ is also $\frac{1}{(1 - z)^{-n}}$.
Now consider the ideal $J = \langle x_1, x_2^2, \ldots, x_n^n\rangle$. This is the initial ideal of $G$. Clearly $J \subseteq \text{init}(I)$. (And now this is where I start getting confused) Our assertion states that these two ideals are equal. For the proof, it is sufficient to verify that the Hilbert series of $R' = \mathbb{C}[\overline{x}, \overline{y}] / J$ is equal to the Hilbert series of $R$.
So here are my questions
1: When the proof says "Our assertion states that these two ideals are equal" what assertion are they referring to?
2: More importantly, why is it sufficient to verify that these Hilbert series are equivalent?