I am trying to understand the following proof on page 25 of AG.
Let $Y$ to be a quasi-affine variety in $\mathbb{A}^n$, with $p\in Y$. Define $Z = \overline{Y} - Y$, this is a closed set in $\mathbb{A}^n$. Form the ideal $\mathfrak{a}\subseteq k[x_1,...,x_n]$ corresponding to $Z$. Choose a polynomial $f$such that $f \in \mathfrak{a}$ but $f(p) \not = 0$. Define $H$ to be the hypersurface of $f$. Hartshorne says (i) $Y - (Y\cap H)$ is a closed subset of $(\mathbb{A}^n - H)$ and (ii) therefore $Y-(Y\cap H)$ is affine since $(\mathbb{A}^n - H)$ is affine.
Where is (i) coming from? And how does (ii) follow, a closed subset of an affine set is not necessarily affine, is it?
(i) Check that $Y - H = \overline{Y} \cap (\mathbb{A}^n - H)$. Note that $H$ contains $Z$.
(ii) Closed subsets of closed subsets are closed! Maybe I'm misunderstanding you. Of course, when we say that $\mathbb{A}^n - H$ is affine we mean that it's isomorphic to a closed subset of $\mathbb{A}^{n+1}$.