# A (short) proof for the paracompactness of CW complexes

So, for a while now I've been looking for a short but concise proof of the fact that

Every CW complex is paracompact.

I finally found the following proof (shortest so far) of this theorem but couldn't understand a few things the author did here:

The lemma being referred to (4th line from below) is as follows:

Lemma. Let $$K$$ be compact, $$C\subseteq K$$ be closed, and $$\mathcal U=\{U_\alpha\}$$ be an open cover of $$K$$. If $$\{\psi_\alpha\}$$ is a partition of unity subordinate to $$\{C\cap U_\alpha\,:\,U_\alpha \in\mathcal U\}$$, then there exists a partition of unity $$\{\Psi_\alpha\}$$ subordinate to $$\mathcal U$$ s.t. $$\Psi_\alpha\big|_C=\psi_\alpha$$ for each $$\alpha$$.

Also, the carrier of $$x$$, written $$C(x)$$, is defined as follows:

Definition. If $$(X,\mathcal E)$$ is a cellcomplex, the carrier of $$A\subseteq X$$, $$C(A)$$, is the intersection of all subcomplexes of $$(X,\mathcal E)$$ containing $$A$$.

I have understood all the individual logical steps in the proof. The part that confuses me is the following:

The author proved in the end that $$(X_{\Gamma_0},U_{\Gamma_0},p_0)$$ cannot be maximal in $$T$$. How does this imply that there is a partition of unity subordinate to $$\mathcal U$$?

Maybe I'm just overlooking something very trivial here! The proof is taken from Lundell's The Topology of CW Complexes.

P.S.: I'm sorry for the image but typing the entire proof would have been too tedious.

The final step of the proof showed that $$(X_{\Gamma_0},U_{\Gamma_0},p_0)$$ cannot be maximal if $$X_{\Gamma_0}\neq X$$. So, the maximal element $$(X_{\Gamma_0},U_{\Gamma_0},p_0)$$ given by Zorn's lemma must satisfy $$X_{\Gamma_0}=X$$. This means exactly that $$p_0$$ is a partition of unity subordinate to $$\mathscr{U}$$ on all of $$X$$.