Show that a sequence admits converging subsequence

Question

I don't know how to solve the following exercise. I think I should use Ascoli-Arzelà's theorem, but I don't know how.

Let $\{ u_n\}_n$ be a sequence of functions in $C^1[0,1]$ with $u_n(0)=0$ for every $n \in \mathbb{N}$, and $\sup \{ \int_0^1 u_n'(d)\varphi(x)dx: \varphi \in L^p(0,1), ||\varphi||_{L^{p}} \leq 1 \} \leq C$ for some $p \in (0,1)$ and $C >0$. Then $\{ u_n \}$ admits a converging subsequence in $C^0[0,1]$.

I'd like to show that such a sequence is equicontinuous and equibounded, but don't know how to use the property in the text. Moreover, that integral makes me think about some weak convergence properties, but I'm still stuck.

Any hint is really appreciated

Answer 1

Big hint: The hypothesis shows that $$|u_n(x)-u_n(y)|=\left|\int_x^yu_n'(t)\right|\le C||\chi_{[x,y]}||_p.$$