I'm learning Real Analysis by working through, on my own*, Bartle & Sherbert's "Introduction to Real Analysis"; it seems to lend itself well to self-study, and has answers -- some of them even right! -- on Chegg. Occasionally, the text will suggest an exercise for the reader, which -- since it is not part of a problem set -- has no answer. These are good and profitable, but I've often no idea whether my answer is correct, or at least on the right track.
One such is from Section 10.1 ["The Generalized Riemann Integral: Definitions and Properties"], and it asks the reader why the proof for necessary-boundedness for the Riemann Integral fails for the Generalized Riemann Integral. And I believe that the answer (though this is probably primitive and/or naive) is that the use of the norm (or mesh) to create the partitions in Riemann integration always necessarily leaves at least one partition non-integrable (and thus the whole of that interval is not integrable). Whereas the use of the gauge to define Generalized Riemann integration allows one to reduce the non-integrable portions to a finite (countable?) set which can be excluded from the total value of the integral. I'd appreciate feedback on whether I'm on the right track here.
Also, if I am, is it correct -- in a crude, conceptual-only sense -- to distinguish between the processes of Riemann vs. Generalized Riemann integrability by noting that the latter can deal with a finite (countable?) number of locations, in the interval under question, where the function is unbounded, whereas strict Riemann integration cannot?
Thanks in advance for your thoughts and input (ie - contribution to my post-age-65 education!).
- I live in a small town without a university, so no (easy, at least) access to taking classes in -- or getting tutoring on -- subjects in the upper-division (and beyond) math zone.