Exercise 7.2.1

Question

Let $f$ be a bounded function on $[a, b]$, and let $P$ be an arbitrary partition of $[a, b]$. First, explain why $U(f) \geq L(f, P)$. Now, prove Lemma 7.2.6.

Ulisse Mini · Accepted Answer

If $U(f) < L(f,P)$ then since $U(f) = \inf\{U(f,P) : P \in \mathcal{P}\}$ there must also be some $P_1$ with $U(f, P_1) < L(f,P)$ which contradicts Lemma 7.2.4.

Similarly, if $U(f) < L(f)$ then there must be some $P$ where $U(f) < L(f,P)$ we've just shown to be impossible.

Exercise 7.2.1

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Exercise 7.2.1

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