Some ideas about $H=W$ 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 . 
 A: It does hold for $W$.
When $\Omega$ has certain smoothness, Sobolev embedding theorem reads that $u\in W^{1,p}(\Omega)$ can be compactly embedded into Hölder continuous function space when the integrability $p>n$. The intuition is that, when the integrability is large enough, we can "trade" the integrability for certain differentiability.
In formal language it should be:

If $n<p<\infty$, for all $0<\alpha< 1-n/p$, the embedding
  $$
W^{1,p}(\Omega)\hookrightarrow C^{0,\alpha}(\Omega)
$$
  is compact.

Compact embeddings are continuous, basically it means the function is being mapped to itself, but now measured under a different norm, such that this identity map is bounded: For any $u\in W^{1,p}(\Omega)$
$$
\|u\|_{C^{0,\alpha}(\Omega)} \leq C\|u\|_{W^{1,p}(\Omega)},
$$
which is to say $u$ is also in $C^{0,\alpha}(\Omega)$, i.e.,$W^{1,p}(\Omega)\subset C^{0,\alpha}(\Omega)$ . You can refer to Theorem 3.36 in this note by John K. Hunter (I like it a lot), it phrases the Sobolev embedding in a different way, but essentially says the same thing.
