Suppose that $\{f_n\}_{n = 1}^\infty$ is a sequence of functions in $BV[0, 1]$ that converges pointwise to a function $f$ on $[a, b]$. Show that $V_a^b f \leq \liminf_{n \to \infty} V_a^b f_n$.
I know that $\forall \varepsilon > 0$, $\exists N > 0$ such that $V_a^b f_n > \liminf_{n \to \infty} V_a^b f_n - \varepsilon$.
I was hoping to get a hint.