Prove $\lim\limits_{n\to\infty}f_n'(x)=0$ for a.e. $x\in[a,b]$.

Question

For all $n\in \Bbb N$, let $f_n\colon[a,b]\to\Bbb R$ an increasing differentiable function with $\int_a^bf_n'(x)dx\le\frac{1}{3^n}$. Prove that $\lim\limits_{n\to\infty}f_n'(x)=0$ for a.e. $x\in[a,b]$.

I want to prove $m\{x:\exists \varepsilon\gt0,\forall n,|f_n'(x)|\ge0\}=0$, but it is hard to extend.

Answer 1

The assumptions on the functions are not at all clear. Without extra assumptions a simple counterexample is the following: let $f_n(x)=-x$ for all $n$ and all $x$ with $a=0,b=1$.

Answer 2

By assumption, $f_n'$ is non-negative and hence $$ 0\le\int_a^b \sum_{n=1}^\infty f_n'(x)dx=\sum_{n=1}^\infty \int_a^b f_n'(x)dx \le \sum_{n=1}^\infty 3^{-n}<\infty.$$ Therefore, $$ \sum_{n=1}^\infty f_n'(x) < \infty \quad \text{for almost all $x\in[a,b]$,}$$ so in particular $$ f_n'(x) \xrightarrow{n\to \infty}0 \quad \text{for almost all $x\in[a,b]$.}$$