If $1 \leq p < n$. The Gagliardo-Nirenberg-Sobolev inequality states that there exists a constant C such that $||u||_{L^{p^{*}}(\mathbb{R}^{n})} \leq C||Du||_{L^{p}(\mathbb{R}^{n})}$ for all $u \in C_{c}^{1}(\mathbb{R}^{n})$.
The first part of the theorem of General Sobolev Inequalities states:
Let $U$ be a bounded open subset of $\mathbb{R}^{n}$, with a $C^{1}$ boundary. Assume $u \in W^{k,p}(U)$. If $k < \frac{n}{p}$ then $u \in L^{q}(U)$ where $\frac{1}{q} = \frac{1}{p} - \frac{k}{n}$.
At the start of proof they state:
Assume $k < \frac{n}{p}$ then since $D^{\alpha}u \in L^{p}(U)$ for all $|\alpha| = k$, the Sobolev-Nirenberg-Gagliardo inequality implies $||D^{\beta }u||_{L^{p^{*}}(U)} \leq C||u||_{W^{k,p}(U)}$ if $|\beta| = k-1$.
How do we use the Sobolev-Nirenberg-Gagliardo inequality without the assumption that $u \in C_{c}^{1}(\mathbb{R}^{n})$?