I have read that for a bounded open set $U$ with $C^1$ boundary, $C^{0,1}(U)=W^{1,\infty}(U)$. But is it also true that convergence in one norm implies convergence in the other?
In particular, if $u_k\rightarrow u$ in $W^{1,\infty}$, and we know that $|\nabla u_k|=1$ a.e. we are able to say that the derivative $|\nabla u|=1$ almost everywhere. Can we say the same when $u_k\rightarrow u$ in $C^{0,1}$?