Let $(g_n)$ be a sequence of twice differentiable functions defined on $[0,1]$, and assume that g_n(0)=g_n'(0) for all $n$. Suppose also that |g_n'(x)|\leq 1 for all $n\in\mathbb{N}$ and all $x\in[0,1]$. Prove that there is a subsequence of $(g_n)$ converging uniformly on $[0,1]$.
This problem is screaming at me Arzelà-Ascoli Theorem. However, I am neither sure how to show that $(g_n)$ is uniformly bounded nor how to show that $(g_n)$ is equicontinuous.
Any help would be appreciated.
Edit: $(g_n)$ is uniformly bounded since |g_n(x)|=\left|g(0)+\int_0^xg_n'(x)dx\right| \leq |g_n(0)|+\int_0^x|g_n'(x)|dx \leq |g_n(0)|+ \int_0^x dx=1+x\leq2.