partition of unity \ multiplication in sobolev?

Question

*let $\Omega$=$\omega_1$ × $\omega_2$
q1: why for any open set $ω'_1$ satisfying $ω'_1$ ⊂⊂ $ω_1$ , we can find an other open set $ω''_1$ such that $ω'_1$ ⊂⊂ $ω''_1$ ⊂⊂ $ω_1$ and a smooth function ρ satisfies suppρ ⊂ $ω''_1$ , ρ = 1 on $ω'_1$
q2: if $u_\varepsilon$ and $u_0$ are in $H_0^1(\Omega )$ why $({u_\varepsilon } - {u_0}){\rho ^2} \in H_0^1(\Omega )$ ?