Exercise 8.1.5

Question

Use the results of the previous exercise to finish the proof of Theorem 8.1.2.

Ulisse Mini · Accepted Answer

Given arbitrary $\epsilon > 0$, find a $\delta$ and a corresponding $\delta$-fine partition $P$ so that for any set of tags $\{c_k\}$, we have
$$\left|R(f,P) - A\right| < \frac{\epsilon}{4}$$
Then pick tags $c_1$ so that $R(f, (P,c_1)) - L(f,P) < \epsilon/4$ and tags $c_2$ so that $U(f,P) - R(f, (P,c_2)) < \epsilon/4$. For conciseness let $L = L(f,P)$, $U=U(f,P)$, $R_1 = R(f,(P,c_1))$, $R_2 = R(f, (P,c_2))$. Then
$$U - L \leq U - R_1 + \left|R_1 - A\right| + \left|A - R_2\right| + R_2 - L < \epsilon$$
showing $f$ is Riemann-integrable.

Exercise 8.1.5

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Exercise 8.1.5

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