Proof. By Proposition 3.13.7 and Lemma 3.13.3, there exists an exhaustion $\curl{U_n}_{1}^{\infty}\suf\cali{P}(X)$ of $X$ by precompact open sets. Let $U_{0}=\emp$ and $V_{n}=U_{n+1}\del \cl{U_{n-1}}$ for each $n\in\N^{+}$. For any $x\in X$, choose $n$ such that $x\in U_{n}\del U_{n-1}$. If $n>1$ then $x\in U_{n}\del\cl{U_{n-2}}=V_{n-1}$. If $n=1$ then $x\in U_{2}=V_{1}$ so $\curl{V_n}_{1}^{\infty}$ is an oepn cover of $X$. In addition, for any $m,n\in\N^{+}$, $V_{m}\cap V_{n}\neq\emp\implies \abs{n-m}\leq 1$ so $\curl{V_n}_{1}^{\infty}$ is locally finite. By Theorem 3.13.10, $X$ is paracompact.$\square$