= Solution
Take the <projective line with a doubled point> over $\mathbb C$, obtained by gluing two copies of $\mathbb P_{\mathbb C}^1$ along the complement of one point. After every <base change of a morphism of schemes>, a closed subset has closed image from each of the two projective-line charts, so its total image, the union of those two images, is closed. The structure morphism is therefore <universally closed morphism>[universally closed]. The two doubled points have no disjoint neighborhoods, so the scheme is not <separated scheme>[separated] and hence is not <proper morphism>[proper].
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