I consider $f$ a continuous function on $[a,b]\subset\mathbb{R}$ and $F$ a primitive of $f$. I define the sequence of functions
$$ u_n(x) = \frac{F(x+1/n) - F(x)}{1/n}\mathbb{1}_{[a,b]},\quad \forall n\geq 1 $$
I want to show that :
$$ \exists M>0, \forall n\geq 1, \forall x\in\mathbb{R} : \lvert u_n(x)\rvert\leq M\mathbb{1}_{[a,b]} $$
My attempt : since $F$ is a primitive of $f$, it is differentiable on [a,b]. It follows from the mean value theorem that I can find for all $n\geq 1$ an $a_n\in[x,x+1/n]$ such that
$$ u_n(x) = \frac{F(x+1/n) - F(x)}{1/n}\mathbb{1}_{[a,b]} = F'(a_n)\mathbb{1}_{[a,b]} = f(a_n)\mathbb{1}_{[a,b]} $$
And conclude by the fact that $f$ being continuous on a compact set, it is bounded.
However as you can see, there is a problem since for all $n\geq1$ we can find a $x\in[a,b]$ such that the segment $[x, x+1/n]$ does not belong in $[a,b]$ but the differentiability assumption holds only $[a,b]$. So I don't know if this proof can be improved or I need to use a totally different argument ?
Thank you a lot