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Thinking back about limits and the original definition of the limit I thought that the Reimann integral (for some bound function $f$ in $[a,b]$) could be defined using limit-like definition. I found one definition and proved the equivalence of the two:

for any $\epsilon >0$ exists a partition $P$ for which $U(P)-L(P)<\epsilon$. where $L$ and $P$ are the lower and upper Darboux sums.

My Question

then I found this theorem for equivalence:

for every $\epsilon >0$ exists some $\delta >0$ such that $U(P)-L(P)<\epsilon$ for any partition $P$, $||P||<\delta$.

I was not able to prove this one, how can on prove this theorem?

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  • $\begingroup$ Are there any conditions on $P$? $\endgroup$
    – Annika
    Jul 12, 2020 at 13:23
  • $\begingroup$ In the theorem I want to prove? yes, $||P||<\delta$ as stated (in other words $\lambda (P)<\delta$) meaning the size of largest $\Delta x$ in $P$ is smaller than $\delta$ $\endgroup$
    – snatchysquid
    Jul 12, 2020 at 13:24
  • $\begingroup$ Sorry, I meant is this theorem supposed to hold for any arbitrary function (whose domain we are creating P over) $\endgroup$
    – Annika
    Jul 12, 2020 at 13:28
  • $\begingroup$ The second equivalence seems weaker than the one you already proved, no? The conditions needed for the first equivalence are a strict subset of the first. $\endgroup$
    – Annika
    Jul 12, 2020 at 13:32
  • $\begingroup$ @Bey oh sorry, $f$ is bound in $[a,b]$ and this is where we are asking about it's integrability. I agree they are similar (they are equivilant as stated) but I did find it useful in many cases $\endgroup$
    – snatchysquid
    Jul 12, 2020 at 13:51

1 Answer 1

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I believe the proof can be found here in theorem 2.4 (2.4').

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