Show that if $F(f(x))$ is AC on $[a,b]$, for all $f$ that are AC on $[a,b]$, then $F$ is lipshitz

Question

I have no clue how to prove this. I am not sure if duplicates are allowed but no one answered it here: $F$ is Lipschitzian if for every $f$ AC，$F◦f$ is AC for 3 years.

It is clear $F$ is AC by letting $f=x$ However, not sure how to conclude that $F$ is lipshitz. MY idea was by contradiction. Assume there exist sequences $x_n,y_n$ s.t $|F(x_n)-F(y_n)|>n|x_n-y_n|$ but i am not sure how to take it from here. Perhaps compactness of $[a,b]$ comes into play here.