Little help showing strong convergence in $H^1$

Let $$u_n$$ be a sequence in $$H^1 (\Omega,\mathbb{C})$$ where $$\Omega \subset \mathbb{R} ^N$$ is bounded. Assume that we know that all the functions $$u_n$$ are smooth and $$\Vert u_n \Vert _{C^m} \leq K(m,N)$$ and moreover $$\vert u_n \vert \leq K_1$$ and $$\vert \nabla u_n \vert\leq K_2$$ where $$K_i>0$$ is a constant (not depending on $$n$$). I wonder if $$\lbrace u_n\rbrace$$ is strongly convergent in $$H^1 (\Omega)$$, up to a subsequence.

Since I have the uniform bounds $$\vert u_n \vert \leq K_1$$ and $$\vert \nabla u_n \vert\leq K_2$$, a good attempt is to use Dominated Convergence Theorem. But then I also need convergence pointwise of $$u_n$$ to some $$u\in H^1(\Omega,\mathbb{C})$$. If I show that $$u_n$$ converges pointwise to $$u$$ then, by DCT in $$L^p$$ spaces, $$u_n \rightarrow u$$ in $$L^2$$. Similarly, $$\nabla u_n \rightarrow \nabla u$$ in $$L^2$$. So, therefore $$u_n \rightarrow u$$ in $$H^1$$. I think that this reasoning is right... so I only have to prove that $$u_n$$ and $$\nabla u_n$$ converge pointwise. Any help to show that?

• What rules out an "oscillating" sequence like $u_n = (-1)^n$? Or do you mean to ask for convergence of a subsequence? Nov 1 '19 at 18:55
• Okay, so with your $C^m$ bounds, Arzela-Ascoli should give you that a subsequence converges uniformly on compact sets, along with its gradients. Nov 1 '19 at 18:57
• So I will only have the pointwise a.e. convergence on compact subsets of $\Omega$? What about if $\Omega$ is compact? Nov 1 '19 at 19:00