In Evans' PDE book, Theorem 6 in Section 6.3.2, which proves infinite differentiability of weak solution, the author says:
According to Theorem 5 we have $u\in H^m(U)$ for each integer $m= 1,2,\cdots$ Thus Theorem 6 in Section 5.6.3 implies $u\in C^k(\bar{U})$ for each integer $k= 1,2,\cdots$
where Theorem 6 in Section 5.6.3 is the Sobolev embedding theorem, which states that either $W^{k,p}\hookrightarrow L^q$ when $k < \frac{n}{p}$ or $W^{k,p}\hookrightarrow C^{k-[\frac{n}{p}]-1, \gamma}$ when $k > \frac{n}{p}$. I can't see how these two spaces is embedded in $C^k$?
The key is that $u\in H^m$ for ALL $m$. For any $k$, let $m$ be large so that
$$m-[\frac{n}{2}]-1\ge k$$
Then since $u\in H^m$, $u\in C^k$ by the Sobolev embedding.