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This is the definition of integrable we have :

A bounded function $f : [a,b] \to \mathbb {R} $ is said to be integrable if $\sup\, L(f,P)= \inf\, U(f,P)$.

I've looked through the book but it involves the 'norm' of a partition, which we sort of don't have..

My lecturer gave an example which involved ( I think ) the uniform partition and creating a 'sequence' of partitions. I don't remember much more, so could someone help me out?

Edited the title as everywhere I look for Riemann integrals, the norm shows up, and the Darboux one seems to be the one we have, but my lecturer has always called it Riemann integral, so I'm not sure

Paramanand Singh
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John
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    You have the $\sup, \inf$ mixed up there. – zhw. Jan 14 '18 at 08:44
  • A Riemann integral can be defined as $\sup/\inf$ of Darboux sums as well as a limit of Riemann sums as norm of partition tends to $0$. That these two definitions are equivalent is a non-obvious fact which is somewhat difficult to prove. Moreover none of these definitions are used in evaluating integrals. Integrals are almost always evaluated using theorems meant to evaluate them and these theorems in turn are proved using these definitions. – Paramanand Singh Jan 14 '18 at 09:15
  • Also fixed minor typo in the use of $\sup, \inf$ in the definition. – Paramanand Singh Jan 14 '18 at 09:19
  • But how do we evaluate them using definitions? For example, the integral of $x^3$ from 0 to 1? – John Jan 14 '18 at 09:24
  • Ok form a partition $P$ of $[0,1]$ into $n$ intervals via points $x_i=i/n$ and find $L(f, P), U(f, P) $ and use the non-trivial theorem that sup, inf of these numbers is same as the limit of these numbers when norm of partition (here $1/n$) tends to $0$. – Paramanand Singh Jan 14 '18 at 09:41
  • Do we need the norm to tend to 0? I ask because I don't think my lecturer mentioned anything about that at all.. Does it not suffice to have $L(f,P)$ and $U(f,P)$ converge to the same limit if we already know that the function is integrable ( since it's continuous )? – John Jan 14 '18 at 09:49
  • What do you mean by converge? The word converge always involves a limit operation. On the other hand sup, inf are not defined via limit and they are not supposed to be limits. So how do you actually find the inf sup of Darboux sums? You use the theorem which says that inf, sup of Darboux sums is same as the limit of these sums when norm tends to $0$. Again this theorem has a complicated proof. – Paramanand Singh Jan 14 '18 at 09:54
  • Sorry, what I meant to say is that whether it suffices to find $L(f,P_n)$ and $U(f,P_n)$, then let $n-->$ infinity, so that the integral is 'squeezed' between them, regardless of whether the norm goes to 0 or not – John Jan 14 '18 at 10:04
  • When $n\to\infty $ then the norm $1/n$ obviously tends to $0$. – Paramanand Singh Jan 14 '18 at 11:51
  • Could you link me to a proof of this? Thanks – John Jan 14 '18 at 22:52

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