Trouble understanding proof of Sobolev trace theorem

Question

I am trying to understand the proof of the Sobolev trace theorem. I am stuck at the bit where the boundary is flattend out using partitions of unity. See the following text (from the book of James C. Robinson):

What I don't understand is: the author says before the equation before (5.39) "using the previous result". Why does that result apply? I do not see why $(\Phi_j^{-1})^*(\Psi_j u)(x)$ is such that it tends to zero as $x_m \to \infty$ where $x=(x',x_m)$, as required. Why is it true??

Answer 1

It's because the function Ψj has compact support, since {Ψj} is a partition of unity.