Convergence of arc lengths of a sequence

Question

I'm new to this community. I'm trying to solve the following problem. I have a sequence of differentiable functions $f_n: [a,b] \rightarrow [a,b]$ which convergences uniformly to a differentiable function $f: [a,b] \rightarrow [a,b]$. In general, this does not imply convergence of the arc lengths $L(f_n)= \int_a^b \sqrt{1+\left(f_n^\prime(x)\right)^2}\ dx \rightarrow L(f)= \int_a^b \sqrt{1+\left(f^\prime(x)\right)^2}\ dx.$ See for example

Does there exist a if and only if condition so that arc length of a convergent sequence of functions to converge to the arc length of the limit.

But my functions $f_n$ and $f$ are all convex. Following the link above, one user said that this would imply convergence of arc lengths.

Does anyone of you know a source in literature or does there exist an elementary proof? I haven't found anything in literature.

Thank you!