The question was taken from Apostol's Mathematical Analysis, question 3.7.
I have taken a look at the solution for this from this website: https://www.csie.ntu.edu.tw/~b89089/book/Apostol/ch3-all.pdf
Also posted here:
However, the notation makes absolutely no sense, and the argument is not at all satisfying to me. That, or perhaps I just don't understand it. For example, I do not accept the argument that $[inf S, sup S] - S$, for a closed and bounded set, is anything but $\phi$ (This explanation helped a little bit: Representation Theorem for Open Sets on The Real Line (Proof-explanation).).
I would like to prove this using only the machinery developed in prior chapters in the book, but I have absolutely no idea where to start. I will also post my solution when it is done. Thanks guys!
Edit: I have been spending time trying to work from Munkres's Topology in hopes of understanding the topic from a different perspective--do you guys think that this is the right way to go about solving this problem?