Proof. Let $\cf \subset E^{*}$ be a $\sigma(E^{*}, E)$-compact set and $U \in \cn_{K}(0)$ be a barrel, then $V = \bigcap_{\phi \in \cf}\phi^{-1}(U)$ is convex, circled, and closed. For each $x \in E$, $\cf(x) = \bracs{\dpn{x, \phi}{E}|\phi \in \cf}$ is bounded. Thus $V$ is absorbing and hence a barrel. Since $E$ is barreled, $V \in \cn_{E}(0)$. Therefore the Mackey topology is contained in the topology of $E$, and $E$ is a Mackey space.$\square$