Meyers and Serrins theorem says that $H=W$. ie $H^j_p(\Omega) = W_p^j(\Omega)$ .
Here the norm of $$\|u\|_{H^j_p(\Omega)} = (\int_\Omega\sum_{|\alpha|\le j}|D^\alpha u|^pdx )^{1/p}$$ where $D^\alpha$ denotes the usual derivative .
and similarly for $W$ we define the norm $$\|u\|_{W_p^j(\Omega)} = (\int_\Omega\sum_{|\alpha|\le j}|D^\alpha u|^pdx )^{1/p}$$ where $D^\alpha$ denotes the $WEAK$ derivative .
here after following the proof i understood that if $u\in W$ then it can be approximated with the sequence $u_n \in H$ . The other way around is trivial . Now the question is that My lecture notes say that if $u\in H^1_p(\Omega)$ with $p>n$ then $u\in C^{0+\alpha} (\bar\Omega)$. This sounds very trivial , because if $u\in H^1_p(\Omega)$ then its not just hölder continuous but its continuous (ie the regularity is more than what the theorem says ) . What i strongly believe is that here it even holds for funtions from $W$ which can be approximated with the functions from $H$ .
Can someone enlighten me with some good ideas and my misunderstanding . Thanks .