Proof that Darboux upper/lower sums always converge to the Darboux upper/lower integral as the partition gets thinner

Question

I'm going through my calculus lecture notes, and at some point the following claim is made: Let $f:\left[a,b\right]\to\mathbb{R}$ be a bound function. For any $\epsilon>0$ there exists some $\delta>0$ s.t. for any partition $P$ of $\left[a,b\right]$ for which $\Delta P<\delta$, it holds that

$$\left|U\left(f,P\right)-\bar{\int}f\left(x\right)dx\right|<\epsilon$$

Where $\Delta P$ is the size of the biggest segment in the partition (The text, which is not in English, refers to $\Delta P$ as the “partition parameter”, I don't think this is the correct translation into English since I did not find any reference to it). $U\left(f,P\right)$ is the upper Darboux sum for said partition and $\bar{\int}f\left(x\right)dx$ is the infimum for the upper Darboux sums of all possible partitions.

I have failed in proving this, and it seems important, could someone supply a source where this is proved?

Answer 1

The upper integral you mentioned is the infimum of all these upper sums, so by definition these sums have to be as close as possible to that integral....

Answer 2

For a continuous function on a closed interval $[a, b] $, it follows from uniform continuity that there is a union of partitions $\bigcup_i h_i= [a, b]$ such that $\sum_i 1_{h_i} |max${f}-$min${f}$|$ $<\epsilon $. Where $\epsilon > 0$ The inequalities remain true for any partition $ d$ such that $ length (d) \le length (h_i) $ for all $ i $ .

$ L(f, p) \le \int f dx \le U (f, p) $

But $ |L(f, p) - U (f, p) | <( b-a )\epsilon$