I am trying to prove the following;
I am doing it by induction and the case $k=1$ is already done. So suppose the above is true for all integers less than or equal to k, first we want to show that if $p\in[1,n/(k+1))$ then
$$u\in W_0^{k+1,p} \implies u\in L^{\frac{np}{n-(k+1)p}}$$
I thought this could be done using some form of Holders inequality since $W_0^{m,p}(\mathbb{R}^n)\subset W_0^{k,p}(\mathbb{R}^n)$ for $k\leq m$ and the intervals in which $p$ can lie are also nested in the same way. However I don't see how it is possible to break $\frac{n-(k+1)p}{np}$ up into two fractions with which we can apply Holders (I mean a slight generalisation of Holders)
I think once I have the first bit the second should be ok so if someone could give me a little help that would be good because I've spent a while and can't see it.