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I know that $L^p$ convergence in general does not tell anything about sobolev convergence, but now what if we know that the $L^p$-convergent sequence consists of only sobolev functions and its $L^p$ limit is again in the sobolev space. Or in other words:

We have a sequence $(u_n)_{n>0}$ in $W^{k,p}$ and a function $u \in W^{k,p}$ such that

$||u_n - u||_{L^p} \to 0 \quad \text{as } n\to \infty$.

Then can we say that

$||u_n - u||_{W^{k,p}} \to 0 \quad \text{as } n \to \infty$ ?

I have been trying for some time to wrap my head around this and I feel that it must be true but I can't seem to neatly write down why. Any clues would be more than welcome.

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    Hint: Try to find a counterexample. You should even be able to get $u_n \to 0$ in $L^p$, but $\|u_n\|_{W^{k,p}} \not \to 0$. – 2017-02-22

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A typical counterexample is $n^{-1}\sin nx$, which converges in any $L^p[a,b]$ to zero but does not converge in any $W^{k,p}$.