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?
Dedekind cuts are somewhat elementary from an analysis point of view, but understanding exactly why they give the (complete) real numbers with all arithmetic operations defined as expected with all the desired properties is a bit harder. For addition, if you have two upper bounded sets of rational numbers then you can form the set of all pairwise sums of those rational numbers, and then take the least upper bound and show that this satisfies axioms of addition. Multiplication is similar for non-negative upper bounded sets of rationals, and care has to be taken when signs between two reals are potentially one or both negative. Division is accomplished if you can establish that multiplicative inverses of positive reals exist.
I've seen a few presentations of Dedekind cuts and they're all basically the same. I don't think there's any "silver bullet" that would allow you to significantly more easily explain the concept (complete with operations defined on reals) to high school students.