Hartshorne Lemma ( I 6.4 )

Question

I have difficulty to understand the proof of this lemma :

Lemma 6.4 Hartshorne Let $Y$ be a qausi-projective variety, let $P,Q\in Y,$ and suppose that $\mathcal{O}_P\subset\mathcal{O}_Q$ as subrings of $K(Y).$ Then $ P= Q.$

Embed $Y$ in $\mathbb{P}^n$ for some $n.$ Replacing $Y$ by its closure, we may assume $Y$ is projective. After a suitable linear change of coordinates in $\mathbb{P}^n$ we may assume that neither $Ρ$ nor $Q$ is in the hyperplane $H_0$ defined by $x_0 = 0.$ Thus $P,Q \in Y- (\mathbb{P}^n- H_0)$ which is affine, so we may assume that $Y$ is an affine variety.

I can not understand two points in the proof :

1. why we may replace $Y$ by its closure ?
2. what is the suitable linear change of coordinates in $\mathbb{P}^n$ ?

Any help would be appreciated.