Throughout, 'proper' means 'pulls back compacts to compacts'.
I've read here and there some claims about properness implying closedness. I want to check whether my attempt at a generalization is correct.
Proposition. Suppose $Y$ is compactly generated Hausdorff. Let $f:X\rightarrow Y$ be a continuous function. Then $f$ is proper $\implies f$ is closed.
Proof. Since $Y$ is compactly generated $f(X)$ is closed in $Y$ iff $f_\ast f^\ast (K)$ is closed in $K$ for all compacts $K$ in $Y$. Since $Y$ is Hausdorff, $K$ is compact Hausdorff and hence $f_\ast f^\ast (K)$ is closed in $K$ iff its compact in $K$. $f^\ast(K)$ is compact since $f$ is proper and its image is compact as a continuous image of a compact set. Since $K$ is arbitrary this proves $f(X)$ is closed in $Y$ as desired.
Is my proof correct?