Suppose $f,g:[0,1]\rightarrow[0,1]$, $g>0$, and $\frac{f}{g}$ is of bounded variation. If $f_n, f \in C^1[0,1]$ and $f_n \rightarrow f$ in $C^1$. Does it follow that $\exists N$ such that $\frac{f_n}{g}$ is of uniform variation for $n>N$?
Here by "$f_n\to f$ in $C^1$" we mean that $f_n\to f$ and $f'_n\to f'$ uniformly on the unit interval.