The theorem says if $Y$ is a subspace of a topological vector space $X$, and $Y$ is an $F$-space (in the topology inherited from $X$, then $Y$ is a closed subspace of $X$.
The proof can be found in Rudin (page 21). The author writes $E_n = (x + V_n) \cap Y$ for open sets in $Y$, then shows that $diam(E_n) \to 0 $ as $n \to \infty$. He then writes that because $Y$ is complete, it follows that the $Y$-closures of the sets $E_n$ have exactly one point $y_0$ in common.
I don't know how this is true. In addition to $Y$ being closed, wouldn't we actually need the diameter of the $Y$-closures of the sets $E_n$ to be approaching $0$ for the conclusion about the single point in the intersection to follow?