How to show that the coordinate ring of a finite set of points in projective space is Cohen-Macaulay?

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Could anyone give me a hint how to show this one:

Let $V$ be a finite set of points in projective space. How to show that the coordinate ring of $V$ is Cohen-Macaulay?

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Proof 1: Regular $\implies$ Cohen-Macaulay.

Proof 2: Artinian $\implies$ Cohen-Macaulay.