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