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Suppose, from the rational numbers, we constructed real numbers using Dedekind cuts: the collection of pairs $(L,U)$, where $L,U$ are non-empty disjoint subsets of $\mathbb{Q}$ with union $\mathbb{Q}$, such that

(i) each member of $L$ is smaller than each member of $U$

(ii) $L$ contains no largest element (for every $x\in L$, there is $y\in L$ with $x<y$).

Then we proceed to define addition and multiplication of such cuts; multiplication is to be defined carefully.

I want to see, how can we prove the completeness property of real numbers then? (a non-empty subset of real numbers which is bounded above has a supremum) Can one suggest a reference for this if this is lengthy process?

Paramanand Singh
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Beginner
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    Just take the union of $L$'s for each cut in your set, it will be the $L$ of the supremum. – Conifold Sep 01 '20 at 06:57
  • https://proofwiki.org/wiki/Continuum_Property here they give two proof, look at the second which involve Dedekind cuts, I was searching for the same question this day :) – Predator Monarch Sep 01 '20 at 15:38
  • See chapter 1 of Rudin's Principles of Mathematical Analysis and also https://math.stackexchange.com/a/498660/589 and https://math.stackexchange.com/questions/987564/books-that-follow-axiomatic-approach – lhf Sep 02 '20 at 00:32

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My favourite introduction to (all of) the number system is Edmund Landau's Foundations of Analysis (Chelsea publishing, New York, 1966). You will find the proof there (Ch. III-IV).

Another option is The Number Systems by Solomon Feferman (Chelsea publishing, New York, 1989).

Erik D
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