Characterization of Paracompactness in T3 Space/Statement 2 implies Statement 3

Theorem
Let $T = \struct{X, \tau}$ be a $T_3$ Space.


 * $(2)\quad $every open cover of $T$ has a locally finite refinement
 * $(3)\quad $every open cover of $T$ has a closed locally finite refinement

Proof

 * $(2)\quad $every open cover of $T$ has a locally finite refinement
 * $\ldots$
 * $(3)\quad $every open cover of $T$ has a closed locally finite refinement