Dominant map from affine space implies unirationality

The proof in the case when $k$ is a finite field is given in Lemma 11 in Unirationality of Cubic Hypersurfaces by Kollár, published in Journal of the Institute of Mathematics of Jussieu, Volume 1, Issue 3 (2002). But you can also find the article in

https://arxiv.org/abs/math/0005146

and Lemma 11 is given in page 5.

It seems to me that the following arguement doses not depend on characteristic. Consider a general fiber of $\varphi:\mathbb{A}^N\dashrightarrow X$. Take its closure in $\mathbb{P}^N$. It is a subvariety of dimension $N-n$. Then a general $\mathbb{P}^n$ intersects it in a finite subscheme, so restriction of $f$ to this plane is still dominant.