CW implies paracompact Hausdorff

Statement
Every CW-space (viz., every space which can be given a CW-complex structure) is a paracompact Hausdorff space.

Goal of the proof
Let $$X$$ be a CW-space. Equip $$X$$ with a CW-complex structure and let $$X^n$$ denote the $$n$$-skeleton with respect to that structure.

Let $$\left \{ U_\alpha \right \}$$ be an open cover for $$X$$.

The goal is to constrict a partition of unity $$\phi_\beta$$ on $$X$$, subordinate to the open cover $$U_\alpha$$.

We shall do this inductively on the $$n$$-skeleton. In other words, we shall construct a partition of unity on $$X^n$$, and extend it to a partition of unity on $$X^{n+1}$$.

Proof details
Suppose we have constructed $$\phi_\beta$$ on the $$n$$-skeleton. The goal is to now extend it to the $$n+1$$-skeleton.

It suffices to describe how to extend $$\phi_\beta$$ for one specific $$(n+1)$$-cell. The question boils down to the following:

Given a partition of unity on the boundary of a disc, which is subordinate to the intersection of an open cover with the boundary, extend it to a partition of unity on the whole disc, which is subordinate to the open cover on the whole disc.

The idea in performing this extension is to observe that any open set of the boundary has some thickness about the boundary (using compactness of the boundary) and any function whose support is inside the open set can be extended so that the support of the function on the whole disc, still lies inside the open set.