The following is from p.102 of Sobolev Spaces by R. Adams and J. Fournier.
Here $\|\cdot\|_q$ is the $L^q$ norm, $C_0^\infty$ means compactly supported and $C^\infty$, and $$|\phi|_{m,p}:=\bigg(\sum_{|\alpha|=m}\int|D^\alpha\phi(x)|^p\,\mathrm{d}x\bigg)^{1/p}.$$
I know the proof for $m=1$, which is standard and can be found in many books. However don't know how to do the induction on $m$. In the process we must use $mp<n$, but I don't see how... Any hint will be appreciated!
Here's how to do the induction step. Suppose that if $1\le m$ is an integer and $1 \le p$ is such that $m p < n$, then there exists $K = K(m,p) >0$ such that if $q = np/(n-mp)$ then $$ \Vert u \Vert_{L^q} \le K \vert u \vert_{m,p} \text{ for all } u \in C^\infty_0. $$ Further suppose that $p \ge 1$ is such that $(m+1) p < n$ and note in particular that this implies two important bounds: first, $m p < (m+1) p < n$, and second $$ p < n-mp \Rightarrow q_0 = \frac{np}{n-mp} < n. $$ Fix $u \in C^\infty_0$. We then apply the induction hypothesis for each $1 \le j \le n$ to see that $$ \Vert \partial_j u \Vert_{L^{q_0}} \le K(m,p) \vert \partial_j u \vert_{m,p} \le K(m,p) \vert u \vert_{m+1,p}. $$ Summing over $j$, we find that $$ \vert u \vert_{1,q_0} \le n K \vert u \vert_{m+1,p}. $$ In turn, the bound on $q_0$ allows us to apply the base case analysis with $m=1$ to see that $$ \Vert u \Vert_{L^{q}} \le K(1,q_0) \vert u \vert_{1,q_0} \le n K(1,q_0) K(m,p) \vert u \vert_{m+1,p} =: K(m+1,p) \vert u \vert_{m+1,p} $$ for $$ q = \frac{n q_0}{n- q_0} = \frac{n^2 p}{n-mp} \frac{n-mp}{n^2 -np(m+1)} = \frac{np}{n - (m+1)p}, $$ which is what we were after.