Rudin exercise 1.5.considers a nonempty subset of reals, $A$, and the set $-A = \{-x : x \in A\}$, and asks for a proof that $\inf A = - \sup(-A)$.

My question on this proof is unrelated to the actual proof. It seems there are three different ways to go about the proof:

(a) Assume that $\inf A$ and $-\sup(-A)$ exist, and if they do, this identity holds.

(b) Extend our notion of infimum and supremum to the extended real line, in which case we can always talk about the $\inf$ and $\sup$, even if a set is not actually bounded.

(c) Prove that $\inf A$ and $-\sup(-A)$ exist so that we are allowed to talk about them and then prove this equality.

I usually opt for (c) when writing a proof of this kind, and it isn't particularly difficult in this case. My question is: would this be standard? I cannot tell which of these techniques Rudin was intending when writing this problem.

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    $\begingroup$ How do you intend to prove "The Axiom of Completeness?" $\endgroup$ – Mark Viola Aug 9 '19 at 17:10
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    $\begingroup$ This is usually taken as an axiom as Mark Viola suggests. What definition of real numbers do you use? And which axioms? $\endgroup$ – Jakobian Aug 9 '19 at 17:17
  • $\begingroup$ At the moment I'm only using the properties of the reals given in chapter 1 of Rudin. So I'm taking the least upper bound and greatest lower bound property for granted, from which existence follows. If I am not mistaken, though, his definition of supremum and infimum doesn't include the extended real line. $\endgroup$ – user465188 Aug 9 '19 at 17:28
  • $\begingroup$ @Matt.P Not everyone has Rudin's book or read it, please include the axioms in your question. $\endgroup$ – Jakobian Aug 9 '19 at 17:44

My reasoning when writing a proof is as follows: I should include enough details so that someone with a similar level of mathematical knowledge could follow my proof and understand it. If it is obvious to you that $\inf A$ and $\sup (-A)$ exist, and you believe your peers would see immediately that it exists as well, you can mention it without going into the details of the proof.


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