Controlling supremum during weak convergence in Sobolev spaces

Question

Let $u_n \in H^1(0,1)$ with uniformly bounded derivatives, i.e. $|u'_n|\leq C $ almost everywhere, for all $n$. Suppose that $u_n$ converges weakly to $u$ in $H^1(0,1)$.

Can I conclude the same bound for $u'$?

I was thinking about using the compact embedding into $C[0,1]$ to conclude that the sequence above converges uniformly, to then apply a difference quotient argument I wasn't able to elaborate. Do you have any hint?

Answer 1

Ys. The set $\{ u\in H^1: \ |u'|\le C\}$ is closed in $H^1$ (prove using pointwise convergence) and convex, hence weakly closed.