I have been looking for various proofs on why the infinite repeating decimal .999....=1 and I came across an explanation using Dedekind cuts on Wikipedia's website: https://en.wikipedia.org/wiki/0.999...#cite_note-13
But the definition presented brought up other questions for me that I had some difficulty answering.
1) My understanding from the Wikipedia page is that real numbers are defined by Dedekind cuts, where a real number is then equal to the infinite set of rationals less than it. If real numbers are sets then how do the usual operations on real numbers translate? For example, is addition set union? What about division?
2) I would like to present the proof to some high school senior students, but would like to read a "friendly" introduction of Dedekind cuts first. Do you know of any good resources?