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This is problem II.5.14(d) in Hartshorne - I have done parts (a) - (c). The question is this.

A scheme is called normal if all its local rings are integrally closed integral domains, and a subscheme of $\mathbb{P}^r_A$ is projectively normal if its homogeneous coordinate ring is an integrally closed domain.

The question is to show that such a subscheme is projectively normal if and only if it is normal, and for every $n \geq 0$, the natural map $\Gamma(\mathbb{P}^r, \mathcal{O}_{\mathbb{P}^r}(n)) \to \Gamma(X, \mathcal{O}_X(n))$ is surjective. I'm having trouble specifically with the 'normal + surjection $\implies$ projectively normal' direction.

I am told this is a corollary to part (a) of the problem, which says that if $X$ is a connected normal closed subscheme of $\mathbb{P}^r_k$, then the ring $S' = \bigoplus_{n \geq 0} \Gamma(X, \mathcal{O}_X(n))$ is the integral closure of $S$. The only thing that stands out to me is that I can put these surjections together to get a surjection onto $S'$. But I need to show $S$ is integrally closed, and I appear to be stuck. Can anyone point me in the right direction?

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This wasn't hard - I'm just dumb. Sticking these maps together like I thought, we have a map $$\bigoplus_{n \geq 0} \Gamma (\mathbb{P}^r, \mathcal{O}_{\mathbb{P}^r}(n)) \to \bigoplus_{n \geq 0} \Gamma(X, \mathcal{O}_X(n))$$

And this map is surjective onto $S \subseteq S'$, from the definition of $S$. Yet this map is surjective, and the summand on the right is $S'$. So if the image is $S$ and the whole ring is $S'$, they're equal.

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