Analysis 2- upper/lower integral vs integral

[perlawin]

1. True or false (show work): For all bounded functions:
(L) ∫_a^b▒f(x)dx≤∫_a^b▒f(x)dx≤(U)∫_a^b▒f(x)dx



2. (L) ∫_a^b▒f(x)dx= sup{L(f,P) s.t P is a partition of [a,b]}
(U)∫_a^b▒f(x)dx= inf{U(f,P) s.t. P is a partition of [a,b]}




3. I am sure that this is true. What I want to do is prove it by induction. Specifically, prove that the first inequality holds and then show that the second one does. I have drawn pictures representing a base case (how the lower integral is less than the regular one), and I have pictures that illustrate how the amount of area increases but is never exact. How do I actually write it out?

 

[snipez90]

I'm not sure what you are inducting on, as I don't see any natural numbers in the problem. However, those inequalities should follow from the basic definitions that you've already listed in the relevant equations section.