Using the dense subset property and darboux's definition

Question

first off I would like to say that the solution in the picture is most likely incorrect since I didn't get full marks.

When the question refers to part a.) it's just referring to the darboux's def'n in the prev. question that I did not include in the picture.

Anyway, I need help understanding when I can use a dense subset. I feel like I don't really understand the concept and I'm just using it randomly. I know that my prof said that a dense subset can be used for reasoning when the range alternates from rational to irrational. Can anybody help? Thanks!

I also apologize for the ugly hand writing.

Answer 1

The issue is that $m_{i}=\ln 1=0$, $M_{i}=\ln\sqrt{3}$ so $L(f,P)=0$, $U(f,P)=\ln\sqrt{3}$.

As @Berci has noted, there is also an issue on the $\sup,\inf$, they are numbers, not sets.

Answer 2

It is simple the fact that in every interval of a partition you will find an irrational number and a rational number (density), later you know, the upper sum for a partition P is always the same and not equal to the sup(lower sums)