Let $(f_{n})$ be a sequence of functions such that $f(x)=\lim f_{n}(x)$ for every $x\in[a,b]$. Show that $$V_{f}(a,b)\leq \liminf V_{f_{n}}(a,b). $$ I'll try to write what I have done so far...
As $V_{f}(a,b)=\sup\{\sum(f,P): P\in\mathscr{P}([a,b])\}$, then for all $\epsilon>0$, there is a partition $P=\{x_{0}, x_{1},...,x_{m}\}\in\mathscr{P}([a,b])$ such that $V_{f}(a,b)-\epsilon/2<\sum(f,P)$.
As $(f_{n})$ be a sequence of functions such that $f(x)=\lim f_{n}(x)$ for every $x\in[a,b]$, then there is $N\in\mathbb{N}$ such that if $N\leq n$, then $|f_{n}(x)-f(x)|<\epsilon/4m$ for every $x\in [a,b]$.
Thus if $N=\max\{ N_{0}, N_{1},...,N_{m}\}\leq n$ we have $$ \begin{split} V_{f}(a,b)-\epsilon/2&<\sum(f,P)=\sum_{k=1}^{m}|f(x_{k})-f(x_{k-1})|\\ &\leq \sum_{k=1}^{m}|f_{n}(x_{k})-f(x_{k})|+\sum_{k=1}^{m}|f_{n}(x_{k-1})-f(x_{k-1})|+\sum_{k=1}^{m}|f_{n}(x_{k})-f_{n}(x_{k-1})|\\ &\leq \sum_{k=1}^{m}\epsilon/4m + \sum_{k=1}^{m}\epsilon/4m + \sum_{k=1}^{m}|f_{n}(x_{k})-f_{n}(x_{k-1})|\\ &= m\epsilon/4m + m\epsilon/4m + \sum_{k=1}^{m}|f_{n}(x_{k})-f_{n}(x_{k-1})|\\ &= \epsilon/2 +\sum_{k=1}^{m}|f_{n}(x_{k})-f_{n}(x_{k-1})|\\ &\leq \epsilon/2 + V_{f_{n}}(a,b). \end{split} $$
Hence if $N\leq n$, we have $V_{f}(a,b)<\epsilon + V_{f_{n}}(a,b)$
I don't know what else to do ...
You have shown that for any $\epsilon > 0$, there exists $N$ (depending on $\epsilon$ and some fixed partition $P$) such that for all $n \geqslant N$ we have $V_f(a,b) - \epsilon < V_{f_n}(a,b)$.
Hence,
$$V_f(a,b) - \epsilon < \inf_{k \geqslant n} V_{f_k}(a,b) \leqslant \sup_{n \geqslant N}\inf_{k \geqslant n} V_{f_k}(a,b) \leqslant \sup_{n \in \mathbb{N}}\inf_{k \geqslant n} V_{f_k}(a,b) = \liminf_{n \to \infty}\, V_{f_n}(a,b)$$
Since $\epsilon > 0$ can be arbitrarily close to $0$ it follows that
$$V_f(a,b) \leqslant \liminf_{n \to \infty}\, V_{f_n}(a,b)$$
For a somewhat shorter proof note that if $\liminf_{n \to \infty}\, V_{f_n}(a,b) = \alpha < \infty$ then for any $\epsilon > 0$ there must be a subsequence $n_j$ such that $V_{f_{n_j}}(a,b) < \alpha + \epsilon$. For any partition we have
$$\sum_{k=1}^n|f_{n_j}(x_k)- f_{n_j}(x_{k-1})| \leqslant V_{f_{n_j}}(a,b) + \epsilon< \alpha + \epsilon$$
Taking the limit of the LHS as $j \to \infty$ we have
$$\sum_{k=1}^n|f_{n_j}(x_k)- f_{n_j}(x_{k-1})| \leqslant \alpha + \epsilon,$$
and again, since $\epsilon$ is arbitrary, the desired conclusion follows.
The case where $\liminf_{n \to \infty}\, V_{f_n}(a,b) = +\infty$ is trivial.