global regular functions projective variety

Put $A=S(X)$. Irreducibility makes $A$ a graded domain, so $f$ is a [given, algebra] degree-zero element of $\mathrm{Frac}(A)$. If $x_i=0$ in $A$, take $N_i=1$, and then $x_i^{N_i}f=0\in A_{N_i}$. Otherwise $U_i=X\cap D_{+}(x_i)$ is nonempty. Normalizing $x_i=1$ identifies its defining ideal with the dehomogenizations of the homogeneous elements of $I_{+}(X)$; hence its affine coordinate ring is canonically the degree-zero localization $A_{(x_i)}$. The affine global-functions theorem places $f{\mathrm{\mid }}_{U_i}$ in $A_{(x_i)}$, so it has the form $a\mathrm{/}{x}_i^{N_i}$ with $a\in A_{N_i}$. Therefore, for every $i$, there is $N_i\ge 0$ such that $x_i^{N_i}f\in A_{N_i}$.

Choose $N\ge N_i$ for all $i$. If $d>(n+1)(N-1)$, every degree-$d$ monomial is divisible by some $x_i^N$, so multiplication by $f$ sends the finite-dimensional space $A_d$ into itself. This space is nonzero: choose $p\in X$ and a coordinate $x_i$ nonzero at $p$; then $x_i^d$ is nonzero in $A_d$.

Cayley--Hamilton applied to the $k$-linear endomorphism $a\mapsto fa$ of $A_d$ gives a nonzero polynomial $P\in k[T]$ with $P(f)a=0$ for every $a\in A_d$. Taking $0\mathrm{\ne }a\in A_d$ and working in the field $\mathrm{Frac}(A)$ gives $P(f)=0$. Since $k$ is algebraically closed, $P$ splits into linear factors, and the domain property forces $f=c$ for some $c\in k$. Thus every global regular function is constant.