Is proper morphism from affine scheme affine?

Question

I'm reading Mumford-Oda's lecture notes http://www.math.upenn.edu/~chai/624_08/mumford-oda_chap1-6.pdf. And they use the fact:"Let $f:U \to Y$ be a proper morphism of noetherian schemes and $U$ is affine, then $f$ is affine" to prove Chow's lemma(p.73). In the errata they say that there was assumed in this chapter that scheme $Y$ is separated over $\mathbb Z$, in these assumptions the fact is true and I'm able to prove it(even if $f$ is not proper). But I don't know is this fact true if $Y$ is not separated.