From the Geometric Invariant Theory book [Mumford - Fogarty - Kirwan], we have the following theorem

([MFK,Theorem 1.1)Let $X$ be an affine scheme over a field $k$, let $G$ be a reductive algebraic group over $k$, let $\sigma:G\times X \to X$ be an action of $G$ on X. Then, a universal categorical quotient $(Y,\Phi)$ of $X$ by $G$ exists.

To fix ideas let's suppose $X:=\operatorname{Spec} A$ for a finitely generated $k$-algebra $A$.

Since $G$ is reductive, we have that $A^G$ is a finitely generated $k$-algebra.

I don't see where the finite generation of $A^G$ is used in the proof of the theorem? For instance we can drop the assumption that $G$ is reductive but then we still need to require that : $A^G$ is finitely generated $k$-algebra in order to conclude that $\operatorname{Spec} A^G$ is the categorical quotient. Where this assumption is used in the proof of the theorem above?

Thank you for your help.