The Stacks project

Proof. Pick $f \in I$, $f\not= 0$. It suffices to show the lemma for $k[x_1, \ldots , x_ n]/(f)$ since $S$ is a quotient of that ring. We will take $y_ i = x_ i - x_ n^{e_ i}$, $i = 1, \ldots , n-1$ for suitable integers $e_ i$. When does this work? It suffices to show that $\overline{x_ n} \in k[x_1, \ldots , x_ n]/(f)$ is integral over the ring $k[y_1, \ldots , y_{n-1}]$. The equation for $\overline{x_ n}$ over this ring is

Hence we are done if we can show there exists integers $e_ i$ such that the leading coefficient with respect to $x_ n$ of the equation above is a nonzero element of $k$. This can be achieved for example by choosing $e_1 \gg e_2 \gg \ldots \gg e_{n-1}$, see Lemma 10.115.2. $\square$