I am struggling to understand the proof of the Sobolev embedding theorem given in Sobolev Spaces by Adams. Specifically section 4.25 (2003 edition).
The aim is to prove $W^{m,1}(\Omega) \to L^{q}(\Omega_{k})$ for $1 \le q \le \frac{k}{n-m}$, where $k>n-m$, and $n \gt m$
$\Omega \subset \mathbb{R}^{n}$ is open and satisfies the cone condition and $\Omega_{k}$ is the intersection of $\Omega$ with a k-dimensional plane in $\mathbb{R}^{n}$.
Lemma 4.24 gives $W^{m,1}(\Omega) \to W^{m-1,p}(\Omega)$ for $1\le p \le \frac{n}{n-1}$
Since $k \gt n-m$, we have $k \ge n-m+1 \gt n-(m-1)r$ $\forall r \gt 1$
The proof (with some obvious misprints) now claims (from previous parts of the proof) that $W^{m-1,r}(\Omega) \to L^{q}(\Omega_{k})$ $\forall 1 \le q \le r^{*}$ But I can only see that this is true for $r \le q \le r^{*}$.
Hence finally I get $W^{m,1}(\Omega) \to L^{q}(\Omega_{k})$ for $1 \lt q \le \frac{k}{n-m}$.
Am I missing something very obvious?