Below I include images of the proof provided in the book.
In the second case, the one where $A = A_1 \cup A_2 \cup \dots$ where the $A_i$'s are compact and $A_i \subseteq A_{i+1}^\circ$,the author defines partitions of unity $\Phi_i$ on $B_i = A_i - A_{i-1}^\circ$ subordinate to $\mathcal{O}_i$ using case 1, because $B_i$ is compact.
It seems to me however, that case 1 ensures the existence of such a partition only on an open set containing $B_i$.Also, following the proof, the author treats the partitions $\Phi_i$ as defined on the whole set $A$.
I know that $A$ is open, but to me it seems that case 1 does not ensure that the open set containing $B_i$ contains $A$.
It is not obvious to me how to extend (if it is possible) the domain of the partitions of unity to all of $A$, saving the smoothness of the functions in $\Phi_i$.
Sorry for my English, I hope the question is clear enough.