- #1

mcastillo356

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- TL;DR Summary
- I've got a thirty three pages notes on Riemann integrability criterion, and stucked on page four. Haven`t read it entirely, but my intention is to understand completely all the document (if I can).

Hi, PF

I've began to read the notes, and at the fourth page, there is

##\{P_1\}=\{a,b\}##

##\{P_2\}=\{a,\frac{a+b}{2},b\}##

##\{P_3\}=\{a,\frac{2a+b}{3},\frac{a+2b}{3},b\}##

and so on, just to display an easy one.

##\{P_n\}=\frac{a+b}{n}?##

Best wishes!

PD: The notes are a Google link.

I've began to read the notes, and at the fourth page, there is

**Theorem 2.4.**Quoting it:**Theorem 2.4'.***Let ##\{P_n\}## be a sequence of partitions satisfying ##\lim_{n\rightarrow\infty}\,||P_n||=0##***1st question**: What is the shape of a sequence of partitions?**Attempt**:##\{P_1\}=\{a,b\}##

##\{P_2\}=\{a,\frac{a+b}{2},b\}##

##\{P_3\}=\{a,\frac{2a+b}{3},\frac{a+2b}{3},b\}##

and so on, just to display an easy one.

##\{P_n\}=\frac{a+b}{n}?##

**2nd question**It's about the expression ##0\leq\overline{S}(f,\,P)-\overline{S}(f,\,R)\leq\sum_{j\in{J}}2M\Delta{x_j}\leq{2M\times{m||P||}}##. What is the reason for introducing the number two?**Attempt:**There are two partitions going on: ##P##, and ##R=P\cup{Q}##, that is the reason for including number two.Best wishes!

PD: The notes are a Google link.