In Gilbard & Trundinger at the end of the proof of theorem 7.10, regarding Gagliardo-Nirenberg-Sobolev inequality there is the fallowing
To extend to arbitrary $u \in W^{1,p}_0$, we let $\{ u_m \}$ be a sequence in $C^1_0$ functions tending to $u$ in $W^{1,p}$. Applying estimates (7.26) to differences $u_m - u_n$ we see that $\{ u_m \}$ will be a Cauchy sequence in $L^{p*}$ (...). Consequently the limit function $u$ will lie in the desired space
Ok, if $u_m \rightarrow u$ in $W^{1,p}$ we get that $\vert\vert Du_m -Du_n\vert\vert_{L^{p}} \rightarrow 0$ and aplaying GNS to the difference $u_m - u_n$ yields Cauchy. But, $u_m \rightarrow u$ in $W^{1,p}$ isn't sufficient to show that, from GNS, $\vert\vert u \vert\vert_{p*} \leq C \vert\vert u \vert\vert_{W^{1,p}}$?
Apply GNS to the sequence itself
$$\|u_m\|_{L^{p^*}}\leq C \|u_m\|_{W^{1,p}}.$$
Since $u_m \to u$ in both $W^{1,p}$ and $L^{p^*}$ you can pass to limits on both sides.