Let $I$ be a bounded interval and consider a sequence $(u_k)$ in $H^{1,2}(I)$ (usual Sobolev space). Suppose furthermore, that the sequence $(u_k)$ is bounded in $H^{1,2}(I)$. Then, by Rellich, we can extract a subsequence, still denoted by $u_k,$ s.t. $u_k$ converges to some $\bar{u}$ in $C^0(I)$. Furthermore, by weak compactness of bounded sets in $H^{1,2}(I)$ we can select a subsequence, s.t. $u_k$ converges to some $u$ weakly in $H^{1,2}(I)$. Thus, $u_k$ converges to $\bar{u}$ in $C^0$ and weakly to $u$ in $H^{1,2}$.

Do these limits coincide, i.e. is it true that $u=\bar{u}$?