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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.

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1 Answer

Fix an integer $N$ and consider $a=t_0<t_1<\dots<t_N=b$. Then $$\sum_{j=1}^N|f(x_j)-f(x_{j-1})|=\liminf_{n\to +\infty}\sum_{j=1}^N|f_n(x_j)-f_n(x_{j-1})|\leqslant \liminf_{n\to +\infty}V_a^b(f_n).$$

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