Bounded subsequence in Sobolev Space

Question

The following is an exercise.

Let $I=(0,1)$ and let $(u_n)$ be a bounded sequence in Sobolev space $W^{1,p}$,

First question: does "bounded" here means that (for a suitable $M$) $$ \| u_n \|_p + \| u^\prime_n \|_p < M $$ for all $x \in I$ and all $n$ ?

I must prove that there exist a subsequence $(u_{n_k})$ and some $u \in W^{1,p}$ such that $\| u_{n_k} - u \|_\infty$ goes to $0$.

Second question: I know that there exist a subsequence $(u_{n_k})$ that goes to $u$, but this in $L^p$. Is it useful to use this information to prove the statement above? If not, how to get a bound on the expression $|u_{n_k}-u|$ ?

Answer 1

$\bf{\mbox{Answer to the first question}:}$ In general, when $(X,\|\|)$ is a normed space and $A\subset X$, we say that $A$ is bounded if there is $M>0$ such that $$\|a\|\le M,\ \forall a\in A$$

$\bf{\mbox{Answer to the second question}:}$ Here, the first thing you have to prove is that for $p\in (1,\infty]$, $W^{1,p}(0,1)$ is compactly embedded in $C(\overline{I})$. For a reference, see Brezis chapter 8.

If you have proved the first thing, now you are able to try and prove that if $u_n\to u$ weakly (in the weak topology) in $W^{1,p}(I)$ then, $u_n\to u$ strongly (in the norm topology) in $C(\overline{I})$.