Existence of function in subset of Sobolev space

Question

Let $\Omega$ be a bounded Lipschitz set in $\mathbb{R}^n$, $g\in W^{1.2}(\Omega)$ and $Tg\leq 0$ a.e., where $T$ is the trace operator. Let $X:=\{u\in W^{1,2}_0(\Omega)|u\geq g \text{ a.e.}\}$. We want to show that $X$ is non-empty. Can anyone offer some help? Thank you very much.