Let $\{f_n\}_{n=1}^{\infty},$ $\{\beta_{n}\}_{n=1}^{\infty}$ are two sequences where each $f_n,\beta_n:[a,b]\to\mathbb{R}$, and each $f_n$ and $\beta_n$ is of the class $C^2$ convex function. Moreover $f'_n(a)=A\leq0$, $f'_n(b)=B\geq0$ and $0\leq f_n(x)\leq\beta_n(x)$ on $[a,b]$ and for all $n\in\mathbb{N},$ where $\{\beta_{n}\}$ converges uniformly to $0$ on $[a,b]$ and the sequence of its first derivative $\{\beta'_{n}\}$ converges uniformly to $0$ on every compact subset $I\subset(a,b).$ I would need to prove that also $\{f'_n\}$ converges uniformly to $0$ on $I$ for $n\to\infty$, if this holds.
Can someone give me some hints on how to solve this problem please? Thank you.