In the book, Linear Function analysis by Hans Wilhelm Alt, in the chapter about Sobolev embeddings, we see the following
if
\begin{align*} {m_{1}-\frac{n}{p_{1}}\geq m_{2}-\frac{n}{p_{2}}\,,\quad\mathrm{and}\quad m_{1}\geq m_{2}\,,}\\ \end{align*}
then the embedding
\begin{align*} W^{m_{1},p_{1}}\left(\Omega\right) & \longrightarrow W^{m_{2},p_{2}}\left(\Omega\right)\\ \end{align*}
Is continuous and exists.
He goes on to say that this is still the case when $m_{1}=m_{2}$ and that it follows from Holder's inequality.
I am really struggling to show this result and even seeing how it follows from said inequality. So I ask for your guidance and help
thank you and kind regards!