Exercise 8.1.3

Question

\begin{enumerate}[label=(\alph*)] 
    \item In terms of $n$, what is the largest number of terms of the form $M_k (x_k - x_{k-1})$ that could appear in one of $U(f,P)$ or $U(f,P')$ but not the other?
    \item Finish the proof in this direction by arguing that
    $$U(f,P) - U(f,P') < \epsilon/3$$
 \end{enumerate}

Ulisse Mini · Accepted Answer

\begin{enumerate}[label=(\alph*)] 
\item In order to transform $P$ into $P'$, we add the $n-1$ points from $P_\epsilon$ which are not the endpoints $a$ or $b$. Each point added can increase the number of non-cancelled terms by at most three (by preventing an interval from $P$ being cancelled, and by creating two new intervals in $P'$). Therefore the maximum number of terms is $3n - 3$.
\item A triangle inequality gives that $U(f,P) - U(f,P') \leq \sum M_k (x_k - x_{k-1})$, where $k$ goes over all of the subintervals in both $P$ and $P'$ which weren't cancelled. Since the length of each subinterval in $P$ and $P'$ is no more than $\delta$,
$$\sum M_k (x_k - x_{k-1}) \leq \sum M \delta = (3n-3) \frac{\epsilon}{9n} < \frac{\epsilon}{3} $$
 \end{enumerate}

Exercise 8.1.3

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Exercise 8.1.3

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