A normal Moore space is completely normal

Hello;

I face difficulties in understanding the proof of this theorem: Every normal Moore space is completely normal.

First we suppose that $\displaystyle H$ and $\displaystyle K$ are two separated sets of a normal Moore space $\displaystyle X$. Let $\displaystyle g_1,g_2,...g_n$ be the sequence of open covers of $\displaystyle X$. For each $\displaystyle n$ let $\displaystyle H_n$ denote the set of all points $\displaystyle p$of closure of $\displaystyle H$ such that no open set of $\displaystyle g_n$ which cotains $\displaystyle p$ contains a point of closure of $\displaystyle K$. Honestly, I don't know why this is possible as well as how can I proceed.

Please guide me.

Thaank you in advance