I am reading Bredon's Introduction to compact transformation groups, and came across the following result and proof on page 34:

Even though he writes "Recall our standing assumption that $X$ is Hausdorff," I cannot see where any properties of Hausdorffness of $X$ are used in this proof. I suspect it would be used to ensure that convergent nets in $X$ converge uniquely, but I do not see why this fact would be needed for this proof.

Is it true that the continuous action of a compact topological group on a topological space $X$ is always closed, regardless of whether $X$ is Hausdorff? If not, then where in the above proof is Hausdorffness used?