I have been reading the chapter of Sobolev Space in Partial Differential Equations by Lawrence C. Evan, and I came across the General Sobolev Inequality stated as follows:
Theorem (General Sobolev Inequality) Let $U\subset\mathbb{R}^n$ be a bounded open set, with $C^1$ boundary. Assume $u\in W^{k,p}(U)$.
(i) If $k<\frac{n}{p}$ then $\|u\|_{L^q\left(U\right)}\le C\|u\|_{W^{k,p}\left(U\right)}$ with $\frac{1}{q}=\frac{1}{p}-\frac{k}{n}$
(ii) If $k>\frac{n}{p}$ then$$\|u\|_{C^{k- \left\lfloor{\frac{n}{p}}\right\rfloor - 1,\gamma}\left(\bar{U}\right)}\le C\|u\|_{W^{k,p}\left(U\right)}\text{, with } \: \gamma=\begin{cases}\left\lfloor{\frac{n}{p}}\right\rfloor + 1 -\frac{n}{p},\;\text{if $\frac{n}{p}\notin \mathbb{Z}$}\\ \text{any positive number}<1,\;\text{if $\frac{n}{p}\in \mathbb{Z}$}\end{cases}$$
My question is about the case when $\frac{n}{p}\in\mathbb{Z}$ in (ii). In the book, Evan provided the proof for this case as follows:
Suppose $k>\frac{n}{p}$ and $\frac{n}{p}\in\mathbb{Z}$. Set $l=\left\lfloor{\frac{n}{p}}\right\rfloor-1=\frac{n}{p}-1$. Consequently, we have as above $u\in W^{k-l,r}\left(U\right)$ for $r=\frac{pn}{n-pl}=n$. Hence the Gagliardo-Nirenberg-Sobolev inequality shows $D^\alpha u\in L^q(U)$ for all $n\le q<\infty$ and all $\lvert \alpha\rvert \le k-l-1=k-\left\lfloor{\frac{n}{p}}\right\rfloor=k-\frac{n}{p}$. Therefore Morrey's inequality further implies $D^\alpha u\in C^{0,1-\frac{n}{q}}\left(\bar{U}\right)$ for all $n<q<\infty$ and all $\lvert \alpha\rvert \le k-\left\lfloor{\frac{n}{p}}\right\rfloor-1$. Consequently $u\in C^{k- \left\lfloor{\frac{n}{p}}\right\rfloor - 1,\gamma}\left(\bar{U}\right)$ for each $0<\gamma<1$. As before, the stated estimate follows as well.
I understand the way he gets $u\in W^{k-l,r}\left(U\right)$ for $r=n$ by iterating the Gagliardo-Nirenberg-Sobolev inequality. But what I don't understand is how he used the Gagliardo-Nirenberg-Sobolev inequality on $u\in W^{k-l,n}\left(U\right)$ to obtain that $D^\alpha u\in L^q(U)$ for all $n\le q<\infty$ and all $\lvert \alpha\rvert \le k-l-1=k-\frac{n}{p}$. Isn't the Gagliardo-Nirenberg-Sobolev inequality only valid when $1\le r<n$? Or am I missing some extra steps he skipped?
Any help would be very much appreciated!