I'm trying to prove that the collection of function $\{F^*\rho_\alpha\}$ is a partition of unity on $N$ subordinate to the open cover $\{F^{-1}(U_\alpha)\}$ of $N$. Here $\{\rho_\alpha\}$ is a partition of unity on a manifold $M$ subordinate to an open cover $\{U_\alpha\}$ of $M$ and $F:N\rightarrow M$ is a $C^\infty$ map.
For the partition $\{\rho_\alpha\}$, $\sum\rho_\alpha=1$ implies that $$\sum F^*\rho_\alpha=\sum\rho_\alpha\circ F=1$$ on $N$. We are reminded that we need to demonstrate is that $\{\mathrm{supp}\ F^*\rho_\alpha\}$ is locally finite. I know that it is easy to verify that the pullback $F^*$ is a linear map and hence we have $F^*(C^\infty(N))\subset C^\infty(M)$, but how can one conclude that $\{\mathrm{supp}\ F^*\rho_\alpha\}$ is locally finite from the fact that $\{\mathrm{supp}\ \rho_\alpha\}$ is locally finite?
At present, I have one idea. By the linearity of $F^*$, it follows from $\rho_\alpha\neq 0$ that $F^*\rho_\alpha\neq 0$. Then we might find some relation between $\{\mathrm{supp}\ \rho_\alpha\}$ and $\{\mathrm{supp}\ F^*\rho_\alpha\}$, but I don't know what to do next.
Could you give me some hints or a solution?
For each $\alpha$, let's define $V_\alpha := \{ x \in N : F^\star \rho_\alpha (x) > 0 \}$ and $W_\alpha := \{ y \in M : \rho_\alpha (y) > 0 \}$.
Now, let's make some observations by unpacking the definitions.
Firstly, from the definitions of $V_\alpha$ and $W_\alpha$, we can see that $V_\alpha = F^{-1}(W_\alpha)$, for each $\alpha$.
Also, from the definition of a support, it's clear that $\text{supp}(F^\star \rho_\alpha) = \overline{V_\alpha}$ and $\text{supp}(\rho_\alpha) = \overline{W_\alpha}$, for each $\alpha$.
By assumption, $\{ \text{supp}(\rho_\alpha) \}$ is locally finite on $M$. So for any $y \in M$, there exists an open set $S_y \subset M$ such that $y \in S_y$ and such that $S_y$ intersects with only finitely many of the $\text{supp}(\rho_\alpha)$'s.
And finally, because $F: N \to M$ is continuous, $F^{-1}(S_y)$ is an open subset of $N$, for each $y \in M$.
I hope you can assemble an argument from these observations.