Proof check for closed sub scheme of an affine scheme is affine

Question

Basically I was solving an exercise of Hartshorne. So let $X=Spec A$ be an affine scheme and $i : Y \rightarrow X$ is a closed immersion. I have to show that $Y$ is affine. This is how I proceed. Firstly $Y$ is quasicompact from elementary topology since $X$ is quasicompact. Now $Y$ can be covered by open affines of the form $i^{-1}(V)$ where $V$ is open in $X$. Now for $D(f) \subset V, f \in A$ we have $D(f)= V_{f|_V}$ and hence $i^{-1}(D(f)) = i^{-1}(V_{f|_V})=(i^{-1}V)_{i^*(f|_V)}$ which is affine since $i^{-1}V$ is affine. Thus we have shown Y can be covered by finitely many open affines of the form $i^{-1}(D(f))=Y_{i^*(f)}$. Thus to apply the criterion for affineness we have to show these $i^*(f)$’s generate the unit ideal in $\mathscr {O}_Y(Y)$. But since $X-i(Y)$ is open by putting in some additional finitely many $f$s we can assume that $D(f)$ cover $Spec A$ and hence they generate the unit ideal and hence $i^*(f)$ generate unit ideal as well. So we can conclude $Y$ is affine. I am new to scheme theory so kindly point out if there’s any flaw in the above arguments. Also other suggestions/comments/insights are most welcome. Also I can’t figure out where I used the surjectivity of the sheaf morphism.