Let $S$ be a $\mathbb Z^{\ge 0}$-graded ring over $A$.
Suppose $S_+= (a_1, \dots, a_n)$ where $a_i$ are homogeneous of positive degree.
I want to show that $S$ is a finitely generated graded $A$-algebra.
I believe we want to show that $S=A[a_1, \dots, a_n]$.
If we let $x=s_0 + s_1 + \dots + s_m \in S$ where the $s_i$ are homogeneous of degree $i$, then for $i \ge 1$, we can write $s_i = f_1a_1 + \cdots + f_na_n$ where $f_i \in S$. Writing each $f_i$ in terms of its homogeneous components, we can then write the $f_i$ as linear combinations of the $a_i$ with coefficients in $S$.
Now we just have this loop of writing the homogeneous elements of positive degree as linear combinations of the $a_i$ with cofficients in $S$, then rewriting those coefficients in terms of homogeneous components, then rewriting those homogeneous components of positive degree as linear combinations of the $a_i$ with cofficients in $S$...
How do we go about this?