About smooth approximation in a Sobolev space I want to prove the following fact :
Consider $\Omega \subset R^n$ a bounded and open set. Let $v \in H^{1}_{0}(\Omega)$ a nonnegative function. Then exists a sequence $v_m$ in $C^{\infty}_{0}(\Omega)$ of nongative functions converging to $v$ in $H^{1}( \Omega)$.
I know how to prove this fact (I think this fact can help): If $u_m$ is a sequence converging to $u$ in $H^1(\Omega)$ , then ${u^{+}_m} \rightarrow  u^+$.
Someone can give me a hint ?
Thanks in advance

Since $v+1/n\to v$, it suffices to approximate $v+1/n$. As in your previous question Sobolev spaces - about smooth aproximation you can use a smooth threshhold function $\phi:\mathbb R\to\mathbb R$ such that 


*

*$\phi(x)=x$ when $x\ge 1/n$

*$\phi\ge 0$ everywhere

*$0\le \phi'\le 1$ everywhere


The proof stays the same: pick a smooth approximation $u_m\to v+1/n$, compose with $\phi$, argue that a subsequence of $\phi\circ u_m$ converges to $v+1/n$.
