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On this page, a proof of the equivalence of two definitions of Riemann integrals is given by the user Pedro using Apostol's Hint for Exercise 7.4, Mathematical Analysis. However, I still find this claim unclear (particularly, the first inequality):

$$S_1\leq U(P_\epsilon,f)<I+\frac \epsilon 2$$

To me, it seems that this is true only if f(x)>=0 on [a,b]. I'm sure I must be missing a subtle point here, so I would really appreciate it if anyone could clarify this for me.

Gravitational Singularity
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  • Put your LaTeX between $$...$$ in order to get it to render. – JonathanZ Dec 28 '23 at 20:39
  • What is $S_1$? [didn't see on link] – coffeemath Dec 28 '23 at 20:44
  • It's in Pedro's answer, not in the original question. – Gravitational Singularity Dec 28 '23 at 20:48
  • The argument should be made for $U(P_\epsilon, f) - L(P_{\epsilon}, f) $ where the terms are non-negative. See details at https://math.stackexchange.com/a/1834341/72031 – Paramanand Singh Dec 30 '23 at 02:44
  • Dealing with just upper (or lower sum) is tricky because of sign of terms and I have discussed that in this answer: https://math.stackexchange.com/a/2047959/72031 – Paramanand Singh Dec 30 '23 at 02:46
  • @ParamanandSingh Thanks so much for your answers! They certainly clarify a lot. – Gravitational Singularity Dec 30 '23 at 17:42
  • @ParamanandSingh Also, one more minor point to clarify. You chose $\delta = \epsilon / (2MN).$ However, there are $N$ partition points in $P_\epsilon,$ and at most, 2 partition intervals of $P$ containing each point in $P_\epsilon$ (since partition points of $P_\epsilon$ can lie on the end points of the closed sub-intervals corresponding to partition $P$). Therefore, shouldn't we have chosen $\delta = \epsilon / (4MN),$ instead? – Gravitational Singularity Dec 30 '23 at 18:20

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