Dtjytyk

Let $(f_{n})$ be a sequence of functions such that $f(x)=lim f_{n}(x)$ for every $xin[a,b]$. 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 $V_{f}(a,b)=sup{sum(f,P): Pinmathscr{P}([a,b])}$, then for all $epsilon>0$, there is a partition $P={x_{0}, x_{1},...,x_{m}}inmathscr{P}([a,b])$ 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 $xin[a,b]$, then there is $Ninmathbb{N}$ such that if $Nleq n$, then $|f_{n}(x)-f(x)|<epsilon/4m$ for every $xin [a,b]$.

Thus if $N=max{ N_{0}, N_{1},...,N_{m}}leq n$ we have
$$
begin{split}
V_{f}(a,b)-epsilon/2&<sum(f,P)=sum_{k=1}^{m}|f(x_{k})-f(x_{k-1})|\
&leq sum_{k=1}^{m}|f_{n}(x_{k})-f(x_{k})|+sum_{k=1}^{m}|f_{n}(x_{k-1})-f(x_{k-1})|+sum_{k=1}^{m}|f_{n}(x_{k})-f_{n}(x_{k-1})|\
&leq sum_{k=1}^{m}epsilon/4m + sum_{k=1}^{m}epsilon/4m + sum_{k=1}^{m}|f_{n}(x_{k})-f_{n}(x_{k-1})|\
&= mepsilon/4m + mepsilon/4m + sum_{k=1}^{m}|f_{n}(x_{k})-f_{n}(x_{k-1})|\
&= epsilon/2 +sum_{k=1}^{m}|f_{n}(x_{k})-f_{n}(x_{k-1})|\
&leq epsilon/2 + V_{f_{n}}(a,b).
end{split}
$$

Hence if $Nleq n$, 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.