Show that $S_f^{\ge0}=\bigoplus_{d\ge0}(S_f)_d$ is a normal domain, where $S$ is an $\mathbf N$-graded domain, $S_{(f)}$ a normal domain $f\in S_1$

Question

Let $S$ be an $\mathbf N$-graded domain with $S_{(f)}$ a normal domain for some $f\in S_1$. Then $S_f^{\geq0}=\bigoplus_{d\geq0}(S_f)_d$ is a normal domain.

Answer 1

$S_f^{\geq0}=S_{(f)}[f]\cong S_{(f)}[x]$, and the polynomial ring over a normal domain is normal.

Answer 2

$S_f=S_{(f)}\left[\frac{1}{f}\right]$, which is a polynomial ring in $\dfrac{1}{f}$ over $S_{(f)}$ and hence integrally closed.