Let $f$ and $g$ be integrable functions on $[a, b]$. \begin{enumerate}[label=(\alph*)] \item Show that if $P$ is any partition of $[a, b]$, then $$ U(f+g, P) \leq U(f, P)+U(g, P) . $$ Provide a specific example where the inequality is strict. What does the corresponding inequality for lower sums look like? \item Review the proof of Theorem 7.4.2 (ii), and provide an argument for part (i) of this theorem. \end{enumerate}

Understanding Analysis

Exercise 7.4.5

Let $f$ and $g$ be integrable functions on $[a, b]$ .

(a): Show that if $P$ is any partition of $[a, b]$ , then $U (f + g, P) \leq U (f, P) + U (g, P) .$
Provide a specific example where the inequality is strict. What does the corresponding inequality for lower sums look like?
(b): Review the proof of Theorem 7.4.2 (ii), and provide an argument for part (i) of this theorem.

TODO

ulissemini

2022-01-27 00:00