Are sections of vector bundles $E\to X$ over schemes necessarily closed embeddings? A brief question, if $\pi\colon E\to X$ is a vector bundle over a scheme $X$, is it automatic that any section $s\colon X\to E$ always a closed embedding?

First, a general observation:

If $p : Y \to X$ is a separated morphism of schemes and $s : X \to Y$ satisfies $p \circ s = \mathrm{id}_X$, then $s$ is a closed embedding.

Indeed, under the hypotheses, we have the following pullback diagram,
$$\require{AMScd}
\begin{CD}
X @>{s}>> Y \\
@V{s}VV @VV{\Delta}V \\
Y @>>{\langle s \circ p, \mathrm{id}_Y \rangle}> Y \times_X Y
\end{CD}$$
and $\Delta : Y \to Y \times_X Y$ is a closed embedding by definition, so $s : X \to Y$ is indeed a closed embedding.
Next, we should show that the projection of a vector bundle is a separated morphism. But vector bundle projections are affine morphisms, hence separated in particular.
