Proper integrally closed subring has infinite index

Statement
Suppose $$S$$ is a commutative unital ring and $$R$$ is a proper fact about::integrally closed subring of $$S$$. Then, $$R$$ has infinite index in $$S$$: in other words, the quotient group $$S/R$$ is infinite.

Proof
Given: A ring $$S$$, a proper integrally closed subring $$R$$ of $$S$$.

To prove: $$R$$ has infinite index in $$S$$.

Proof: Suppose $$R$$ has finite index, say $$r$$, in $$S$$. Let $$x$$ be an element in $$S \setminus R$$. Consider the elements $$x,x^2,x^3, \dots, $$. Since there are only $$r$$ cosets of $$R$$ in $$S$$, there must exist $$m > n \ge 1$$ such that $$x^m, x^n$$ are in the same coset. Let $$a = x^m - x^n$$. Then, $$a \in R$$, so $$x$$ satisfies the monic polynomial in $$R[x]$$:

$$x^m - x^n - a = 0$$.