Curves which are not universally closed are affine (Hartshorne IV.1.3) I’m working on exercise IV.1.3 in Hartshorne:

Let $X$ be an integral, separated, regular, one-dimensional scheme of finite type over $k$ which is not proper over $k$. Prove $X$ is affine.

In other words, $X$ satisfies all the hypotheses of an abstract curve except that it’s not universally closed over $k$.
Hartshorne’s hint is to embed $X$ in a projective curve $\overline{X}$. I’m assuming one should embed $X$ in projective space and then take the closure. But why should such an embedding exist?

There is a remark on page 105, number 4.10.2, which includes mention of a paper by Nagata proving that every variety $X$ can be embedded as an open and dense subset of a complete variety $\overline X$. Using the results of Chapter I.6, mostly Theorem 6.9, you can deduce that $\overline X$ is actually projective. 
