prove that $supp(π^{∗} f) = (supp f)×N.$ Please can you check my answer? Also more explanation please.

Question

My question is that Let $f \colon M \to R$ be a $C^{\infty}$ function on a manifold $M$. If $N$ is another manifold and $π \colon M \times N \to M$ is the projection onto the first factor, prove that $\operatorname{supp}(\pi^{*} f) = (\operatorname{supp} f)\times N$.

And my answer is here. Please can you check my answer? Do you have any mistake or drawback? Also pleas more explain about what I write.Thank you.

Answer 1

It all seems correct. Can you expound why ${\rm cl}_{M\times N}(A\times N)={\rm cl}_M(A)\times N$?