Let $(f\_n)\_{n \in \mathbb{N}} \subset C([0,1], \mathbb{R})$ be a sequence of functions such that $\int\_0^1 f\_n (t) dt =0$ and $\vert f\_n'(x) \vert < x$ then there is a subsequence that converge uniformly. 

By Arzela-Ascoli theorem all I need to do is to prove that $(f\_n)\_{n \in \mathbb{N}}$ is uniformly bounded and equicontinuous, I already proved that is equicontinuous but I don't know how to use the hypothesis to prove that is uniformly bounded.