The proof of the lemma then the lemma is given below:
My questions are:
1- Why $V(f_{[a,b]}, P) = V(f_{[a,c]}, P_{1}) + V(f_{[c,b]}, P_{2})$ in equation 19?
2-In the line before inequality (22), why $TV(f_{[u,v]}) = TV(f_{[a,v]}) - TV(f_{[a,u]})$?
Could anyone explain those questions for me, please?
By hypothesis, partition $P = (x_0,x_1, \ldots, x_n)$ includes $c$ and thus $c = x_j$ for some $j$ with $0 < j < n$. The inducing partitions $P_1$ and $P_2$ are $P_1 = (x_0, \ldots,x_j)$ and $P_2 = (x_j, \ldots, n)$ and we have
$$V(f_{[a,b]},P) =\sum_{k=1}^n |f(x_k) - f(x_{k-1})| = \sum_{k=1}^j |f(x_k) - f(x_{k-1})|+ \sum_{k=j+1}^{n} |f(x_k) - f(x_{k-1})| \\ = V(f_{[a,c]},P_1)+ V(f_{[c,b]},P_2)$$
The result before (22) is a restatement of the previously proved result (21) which follows from (20) with $u = c$ and $v = b$.