Sobolev Embedding says that $W^{1,p}_0\rightarrow L^q$ continuously for $1<q<p^{*}$ and $1<p<N$. For the case $p=N$, $W^{1,N}_0\rightarrow L^{\infty}$ is not true.
My confusion is the following, at the limit $p=N$, $W^{1,N}$ should be continuously embedded in $(\cap L^p:=L^{\infty})$, for a bounded and smooth domain $\Omega$.
So where am I going wrong? Any help is appreciated.
The identity $\bigcap_{q<\infty}L^q=L^\infty$ is simply not true. For example, $\log \in \bigcap_{q<\infty}L^q(0,1)$, but clearly $\log\notin L^\infty(0,1)$. What is true is that $f\in \bigcap_{q<\infty}L^q$ with $\sup_{q<\infty}\|f\|_q<\infty$ implies $f\in L^\infty$, but the constant of the Sobolev embedding explodes as $p\nearrow N$, so you can't use that either.