Darboux's integral [Zorich's book] 
Hello! This is an excerpt from Zorich's book (on page 345) and I solved parts a), b) and c). I would like to see some hint(s) on part d).
Let's try to prove that $$I^*=\lim \limits_{\lambda(P)\to 0}S(f;P).$$
It means that for any $\epsilon>0$ we can find $\delta>0$ such that for any partition $P$ with $\lambda(P)<\delta$ we have $|S(f;P)-I^*|<\epsilon.$
Since $I^*$ is defined as the supremum then it is enough to show that $S(f;P)-I^*<\epsilon.$
Let $\epsilon>0$ be given then by definition of $I^*$ we can find partition $P_{\epsilon}$ such that $S(f;P_{\epsilon})<I^*+\frac{\epsilon}{2}.$
Let's take any partition $P$ with $\lambda(P)<\delta$ where $\delta>0$ will be defined later. Then
$$S(f;P)-I^*=S(f;P_{\epsilon})-I^*+S(f;P)-S(f;P_{\epsilon})<\frac{\epsilon}{2}+S(f;P)-S(f;P_{\epsilon})\leq$$
$$\leq \frac{\epsilon}{2}+S(f;P)-S(f;P_{\epsilon}\cup P).$$ The last inequality follows from the fact that $S(f;P_{\epsilon})\geq S(f;P_{\epsilon}\cup P)$ since $P_{\epsilon}\subset P\cup P_{\epsilon}$.
And if we apply part b) to the difference $S(f;P)-S(f;P_{\epsilon}\cup P)$ we will get the following: $$S(f;P)-S(f;P_{\epsilon}\cup P)\leq \omega(f;[a,b])\cdot (|\Delta_{i_1}|+\dots+|\Delta_{i_k}|),$$ where $\omega(f;[a,b])=\sup \limits_{x',x''\in [a,b]} |f(x')-f(x'')|$ - oscillation of $f(x)$ on $[a,b]$ and $\Delta_{i_1},\dots, \Delta_{i_k}$ be the intervals of partition $P$ that contain points of $(P\cup P_{\epsilon})\setminus P.$
Since $f$ is bounded then $\omega(f;[a,b])\leq C$ and since $\lambda(P)<\delta$ then $|\Delta_{i_1}|,\dots, |\Delta_{i_k}|<\delta$. So we will get that $$S(f;P)-S(f;P_{\epsilon}\cup P)\leq C\delta k .$$
And I cannot finish my reasoning since this $k$ makes an issue.
Can anyone show to me how to handle it please?
 A: The idea is to first choose a partition $P_\varepsilon$ such that
$$  S(f;P_\varepsilon)< \overline{I}+\varepsilon/2$$
say $P=\{a=y_0<\ldots<y_\ell=b\}$.
Let   $\delta=\frac{\varepsilon}{2\omega(f;[a,b])\ell}$.
Suppose $P=\{a=x_0<\ldots x_n=b\}$ is a partition with $\lambda(P)<\delta$.
Divide the indices of the subintervals generated by $P$ in two set $I$ and $J$: $I$ is the indices of the $[x_{j-1},x_j]$ containing at least one point of $P_\varepsilon$; $I^c=\{0,\ldots,n\}\setminus I$.
It follows from part (a) and (b) of the problem that
$$\begin{align}
S(f,P)&=S(f,P)-S(f,P\cup P_\varepsilon)+S(f,P\cup P_\varepsilon)\\
&\leq \omega(f;[a,b])\sum_{j\in I}(x_j-x_{j-1})+ S(f,P\cup P_\varepsilon)\\
&\leq \omega(f;[a,b])\ell\delta+S(P_\varepsilon)<\frac{\varepsilon}{2}+\overline{I}+\frac{\varepsilon}{2}
\end{align}$$
where $\sum_{j\in J}(x_j-x_{j-1})\leq \delta\ell$ follows from the fact that there  are at most $\ell$ subintervals of the partition $P$ that contain points of $\mathcal{P}_\varepsilon$.
That shows that $\lim_{\lambda(P)\rightarrow0}S(f,P)=\overline{I}$.
