Family of functions over compact have derivative with the same Lipschitz constant. Do the functions themselves have the same Lipschitz constant?

Question

Suppose a family of functions $\{f_n\}$ defined over a compact, with the $q$th derivatives ${f^{q}_n}$ Lipschitz continuous with the same Lipschitz constant for all $n$. Do the lower order derivatives and $\{f_n\}$ each have the same Lipschitz constant for all $n$?

Answer 1

A counter-exemple ($q=1$)

Define $f_n(x) = nx$ over $[0,1]$

then $f'_n(x) = n$ so they are 1-Lipschitz for all $n$ (you can choose whatever Lipschitz constant you prefer), but there is no uniform Lipschitz constant for the $f_n$ (no matter what $M$ you choose, $f_{M+1}$ is not $M$ Lipschitz)