Let $C$ be an open convex set, $(f_n)_{n=1}^\infty$ be a sequence of continuous and convex functions on $C$ that converges pointwisely to $f$. Then, it is proved as the Theorem 10.8 in the book Convex Analysis by Rockafellar that $f_n$ converges to $f$ uniformly on any compact subset of $C$.
My question is, if convex and continuous $f_n$ converges pointwisely to $f$ at the rate of $a_n$, i.e. $a_n(f_n-f)(x)\to 0$ for any $x\in C$, is it possible to prove that $a_n (f_n-f)$ also converges uniformly to $0$ on any compact subset of $C$. Here $a_n$ is a sequence of positive and diverging sequence, for instance, $a_n=n$ or $a_n=n^2$.
I have tried to understand how this result proved in Rockafellar's book, but it seems that the proof cannot be transfered to my question directly, if $f_n-f$'s are not convex (or concave) functions.
Moreover, if there exist counterexamples that pointwise convergence with rate $a_n$ cannot implies uniform convergence with the same rate, is it possible to prove that $\{a_n(f_n-f)\}_n$ is a sequence of uniformly bounded functions w.r.t. $n$ and $x\in C$?