Limiting total variation attached to sequence of uniformly vanishing functions of bounded variation

Question

Let $(f_n)_n$ be a sequence of real functions of a single real variable with compact support in $[0,1]$ and of bounded variation all of them. Let the sequence be uniformly convergent to $0$. Is it true that the sequence of their total variations (on $[0,1]$), say $(V_0^1(f_n))_n$, is convergent to $0$?

Answer 1

Definitely not. Consider the sequence $f_n(x)=\frac{1}{n}\cos(2n\pi x)$. This converges uniformly to $0$ on $[0,1]$, but $V_0^1(f_n)>1$ for all $n$.