0
$\begingroup$

I want to prove this:

If $P^{*}$ is a finer partition than $P$, then show that $L(f,P, \alpha) \leq L(f,P^{*}, \alpha)$ and $U(f,P^{*}, \alpha) \leq U(f,P, \alpha)$.

If you have a set $S = \{1.2.3 \}$ then adding an element can change the infimum. If $S' = \{\frac{1}{2},1.2.3 \}$ then $\inf S' \leq \inf S$. I don't get how the above holds then.

  • 1
    smaller the interval bigger the infimum and smaller the supremum. Now apply this fact to your example.2011-08-25
  • 0
    What's $\alpha$? (Edit: Probably a Stieltjes integral)2011-08-25
  • 1
    When we talk about partitions we are usually fixing an inverval $[a, b]$, and our partitions then start at $a$ and end at $b$, so usually it isn't correct to introduce points outside of that interval and call the result a refinement. If this were allowed, then the statement of your problem wouldn't be true.2011-08-25

2 Answers 2