# Riemann integrability criteria

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?

• Are there any conditions on $P$? Jul 12, 2020 at 13:23
• 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 $\delta$ Jul 12, 2020 at 13:24
• Sorry, I meant is this theorem supposed to hold for any arbitrary function (whose domain we are creating P over) Jul 12, 2020 at 13:28
• 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. Jul 12, 2020 at 13:32
• @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 Jul 12, 2020 at 13:51