Let $u \in H^1((0,1))$ ($H^1(\Omega)$ being the set of weakly differentiable functions over $\Omega \subset \mathbb{R}^d$, also known as Sobolev-space) which satisfies the following inequality: $$\vert u(x) - u(y)\vert \le \vert x-y \vert^{1/2} \Vert u^{\prime} \Vert_{L^2}\ \ , x,y \in [0,1]$$ where $\Vert u^{\prime} \Vert_{L^2} = \sqrt{\int_{0}^{1} (u^{\prime}(x))^2dx}$ is the norm of the square integrable functions.
Show that every $u\in H^1((0,1))$ has a representative in $C([0,1])$ (the space of continuous functions on $[0,1]$).
I really don't have any clue how to prove this. We are given the following hint:
Approximate $u$ by a sequence $(u_n) \subset C^\infty([0,1])$ (possible because $C^\infty(\overline{\Omega})$ is a dense subset of $H^k(\Omega)$), so $u_n \to u \in H^1((0,1))$. Then, show that $u_n \to \tilde{u}$ pointwise, even uniformly converges to a function $\tilde{u}$, such that $u = \tilde{u}$ almost everywhere.
I also have trouble following the steps of the hint, I can't put together why the proof follows from that. Help would be very much appreciated. Thanks very much in advance!