I'm stuck on a proof of Lemma 9.2 in Brézis's book (Functional analysis, Sobolev spaces and partial differential equations).
He starts by taking $\varphi\in C^1_c(Q)$, where $$Q=\{x=(x',x_N)\in \mathbb{R}^N;\ |x'|<1\}.$$ Then, he defines $\psi(x',x_N)=\varphi(x',x_N)+\varphi(x',-x_N)$, for $(x',x_N)\in Q_+=Q\cap \mathbb{R}^N_+$. Hence, he claims that $\eta_k(x_N)\varphi(x',x_N)\in C^1_c(Q_+)$ and that is my question. I was not able to prove that $\eta_k(x_N)\varphi(x',x_N)$ has compact support, althought I can believe in it. Could someone provide a proof or idea or some reference that provides the details?
It remains to define $\eta_k$. First take $\eta(t)\in C^\infty(\mathbb{R})$ such that $$\eta(t)=\begin{cases} 0, & t<1/2\\ 1, & t>1. \end{cases}$$ Thus $\eta_k(t)=\eta(kt)$.