Proof. (1)$\implies$(2) Let $\curl{K_n}_{1}^{\infty}\suf\cali{P}(X)$ be a covering of $X$ by compact sets and $U_{0}=\emp$. Suppose inducitvely we have constructed $\curl{U_i}_{0}^{n}$ satisfying:

1. (a)

   For each $0\leq j\leq n,U_{j}$ is precompact open set.

2. (b)

   For each $0\leq j<n,\cl{U_j}\suf U_{j+1}$.

3. (c)

   For each $1\leq j\leq n, U_{j}\supf \cups{j}{i=1}K_{i}$.

Pick $U_{n+1}\in \cali{N}^{o}(\cl{U}_{n}\cup K_{n+1})$ precompact by Lemma 3.13.3 then by (c)

Thus $\curl{U_i}_{0}^{n+1}$ satisfies (a,b,c) and $\curl{U_n}_{1}^{\infty}$ is an exhaustion of $X$ by compact sets. (2)$\implies$(1) follows by definition.$\square$