Let $\Omega \subseteq \mathbb{R}^n$ be a bounded open set ,and $u \in C(\overline{\Omega}) \cap C^1(\Omega) \cap H^1(\Omega) $ be a function such that $u \big|_{\partial \Omega}=0 $.Prove that $ u \in H_0^1(\Omega) $
Hint: pick a smooth function $h:\mathbb{R} \rightarrow \mathbb{R}$ such that $h(t)=0$ while $t \in (-1,1)$ and $h(t)=t$ when $t \in (-\infty,2)\cup(2,\infty)$ and set $h_\delta(t)=\delta h(\frac t\delta)$ and $u_\delta =h_\delta \circ u$
I tried to show that there is a smooth function that converges to $u$ so maybe $\|u-u_\delta\|_{H^1} =0$, but I am not sure if $u_\delta$ is smooth?
The function $u_\delta$ is defined so that it is continuous and has compact support. You can make a $C^\infty$ function with compact support out of it by mollification, since the support of $u_\delta$ is at positive distance from $\partial \Omega$. (This is helpful if $H_0^1$ is defined as the closure of $C^\infty$ functions with compact support.)
It remains to show convergence in the norm. Note that $\sup|h_\delta'|=\sup|h'|$ is finite; call it $M$. Estimate $\int|\nabla u-\nabla u_\delta|^2$ as follows: on most of the domain they are equal, while on the part where they are not equal, $\int|\nabla u |^2$ is small by absolute continuity of the integral, and $\int| \nabla u_\delta|^2\le M^2\int|\nabla u |^2$.