I'm solving the following exercise:
Let $B = \{u \in C^1([0,1]): ||u'||_2 \leq 1\}$. Given a sequence $(u_n) \subseteq A = \{ u \in B : u(0)=0, u(1)=1\}$, show that there exists a subsequence ${u_n}_k$ which converges uniformly to some $u \in C^0([0,1])$.
My attempt: I've managed to show that $B$ is equicontinuous. This implies that $A \subseteq B$ is equicontinuous. If we prove that $A$ is also bounded with respect to the norm $||.||_{\infty}$, by Ascoli-Arzelà theorem we conclude that $A$ is relatively compact, so we are done. Now, let $u \in A$. Then $$|u(x)|=|u(x) - u(0)|=|\int_{0}^{x} u'(t) dt| \leq \int_{0}^{x} |u'(t)| dt \leq \int_{0}^{1} u'(t) dt = (||u'||_1)^2 \leq (||u'||_2)^2 \leq 1$$ where $ (||u'||_1)^2 \leq (||u'||_2)^2 $ holds since $[0,1] $ has measure 1. Hence $|u(x)| \leq 1$ for all $u$ and $x \in [0,1] $, so we are done. Is my solution correct? I have some doubts because I did not use the fact that $u(1)=1$ when $u \in A$.