 Solution

Use the usual algebraically closed field convention for classical algebraic sets, or equivalently interpret their points geometrically. The proof also gives the scheme-theoretic statement over any field; the difference from an arbitrary-field rational-point image is explained below.

Choose an affine open cover $X={\bigcup }_i{U}_i$. On each $U_i$, the subset $Y_i=Y\cap (U_i\times {\mathbb{P}}^n)$ is closed relative to $U_i\times {\mathbb{P}}^n$. If the affine-base assertion holds, its projected image is closed in $U_i$. ButThus the complement of $\pi (Y)$ is open on each member of the open cover and hence open in $X$. Therefore it suffices to prove closedness for affine $X$. No finite-cover assumption is needed for this localization argument.