 Rational-point projection need not be Zariski closed

Over the real numbers, the closed equation $y_0^2=t{y}_1^2$ in ${\mathbb{A}}^1\times {\mathbb{P}}^1$ projects on real points to $\{t\ge 0\}$, which is not Zariski closed. Projective properness concerns scheme or geometric images, not the image of rational points over an arbitrary field. Similarly, $y_0^2+y_1^2$ has no real projective zero, but its homogeneous ideal contains no power of the irrelevant ideal of projective space, since it has a complex projective zero.