on the immersion of schemes determined by an ample line bundle

Question

It is well known that for a Noetherian scheme $X$ and an ample line bundle $\mathcal{L}$ on $X$, there exists some integer $n\geq1$ such that the canonical map $X\to \mathbb{P}(\Gamma(\mathcal{L}^{\otimes n}))$ is an immersion.

Does $n$ exist such that the canonical map is a closed immersion?

Answer 1

This will be true if and only if $X$ is proper. If $X$ is proper, then in particular it is universally closed and so its image under the immersion is closed. If not, then its image cannot be closed (a closed subvariety of projective space is proper).