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Let $f:[a,b]\to\Bbb R$. These two Riemann integrability criterion are equivalent:

  1. $\forall \epsilon>0,~\exists \delta >0$, $\forall P$: partition of $[a,b]$, $\forall T$: sample points of $P$, ($\lVert P\rVert<\delta\Rightarrow S(f,P,T)<\epsilon$).

  2. $\forall \epsilon >0,~\exists P$: partition of $[a,b]$ such that $U(f,P)-L(f,P)<\epsilon$.

I can prove $1.\Rightarrow 2.$ And I also have the reference of the proof of converse direction $2.\Rightarrow 1.$ However, the proof of it seems too complicated, and not straightforward to me. Though I can finally understand it by reading it patiently, I'm not sure it is very pedagogical. Thus I want to see if there're another proof of $2.\Rightarrow 1.$

Update: My reference. (From Ethan D. Bloch "The Real Numbers and Real Analysis") The author first proved (a)$\Rightarrow$(b), then (b)$\Rightarrow$(a). Then (b)$\Rightarrow$(c), which is trivial. Then (c)$\Rightarrow $(b).

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PS: I omit the $(a)\Rightarrow (b)$ part here. enter image description here enter image description here enter image description here

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Eric
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  • Is that proof similar to this? – RRL Oct 15 '17 at 06:44
  • @PRL I saw your proof an hour ago, and it looks new to me. Moreover, at the first glance, it's very simple and straightforward than the one I have! However, I haven't checked the final step myself, since I'm at home(not so quite as library.) I'm going to think it deeply tomorrow. – Eric Oct 15 '17 at 09:36
  • For this to make sense condition (1) should be $|P| < \delta \implies \left|S(f,P,T) - \int_a^b f(x) , dx \right| < \epsilon$. – RRL Oct 15 '17 at 21:52
  • @PRL I have just studied it! The proof is very easy and clear. Thanks! Is it discovered by you? If one day I have chance to teach other students, I'll definitely told them this elegant proof. If this is yours, I'll say that I saw it at MSE. ;) By the way, I include the proof of my reference, see the edit. And I'm also curious about your another version of the related proofs! – Eric Oct 16 '17 at 15:25
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    The proof of b->a is basically that given any $(f,P,T_1)$ and $(f,P,T_2)$, $|S(f,P,T_1)-S(f,P,T_2)| \leq U(f,P)-L(f,P)$. – Ian Oct 16 '17 at 15:58
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    @Eric: You're welcome. Not my discovery. The equivalence of the two approaches to Riemann integration is important and most treatments follow one path or the other but never explain the connection. As you can see proving the equivalence takes some effort. The proof you are citing is similar to others in that $\delta = O(\epsilon/(Mn))$ where $M$ bounds $f$ and $n$ is the fixed number of intervals in the Darboux partition. Then it is a matter of relating sums and this is where some are clearer than others. – RRL Oct 16 '17 at 16:03

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