Is the following true about sups and infs?

  • Context: Graduate 
  • Thread starter Thread starter AxiomOfChoice
  • Start date Start date
Click For Summary

Discussion Overview

The discussion revolves around the properties of the supremum and infimum of a set of real numbers, specifically whether the statement regarding the supremum of the absolute values of a set is true. The scope includes mathematical reasoning and proof strategies related to this concept.

Discussion Character

  • Mathematical reasoning

Main Points Raised

  • One participant proposes that for a subset A of real numbers, the relationship between the supremum of the absolute values and the maximum of the absolute values of the infimum and supremum of A needs to be established.
  • Another participant agrees that the statement seems true but emphasizes the need for a proof.
  • A different participant suggests considering different cases based on the relationship between the absolute values of the infimum and supremum to demonstrate the claim.
  • One participant provides a reasoning approach, noting that the absolute values of elements in A are bounded by the maximum of the absolute values of the supremum and infimum, and suggests showing that there exists an element in A that is arbitrarily close to either of these bounds.

Areas of Agreement / Disagreement

Participants generally agree that the statement appears plausible but do not reach a consensus on its truth without a formal proof. Multiple approaches to proving the statement are discussed, indicating a lack of resolution.

Contextual Notes

The discussion does not resolve the assumptions required for the proof, nor does it clarify the definitions of supremum and infimum in this context. The mathematical steps to reach a conclusion remain unresolved.

AxiomOfChoice
Messages
531
Reaction score
1
I am trying to determine whether the following is true.

Let [itex]A\subset \mathbb R[/itex], and let [itex]|A| = \{|a|:a\in A\}[/itex]. Then [itex]\sup |A| = \max \{|\inf A|, |\sup A|\}[/itex].

I think I could figure this out on my own, but there is something else demanding my attention right now...so, hopefully this isn't too tough. Thanks!
 
Physics news on Phys.org
Seems true... But it still needs to be proven. How would you begin such a proof?
 
I'm on the phone now, but I can't help but read this :p

I think, show different cases: if |inf A| < |sup A|, show that that sup|A| = |sup A|; etc.
 
Last edited:
First note that, for any a in A, [itex]inf(A)\le a\le sup(A)[/itex] so that [itex]|a|\le max(|sup(A)|, |inf(A)|)[/itex] so that "max(|sup(A)|, |min(A)|)" is an upper bound. Now, to show it is the least upper bound, show that there must be a in A such that |a| is arbitrarily close to either |sup(A)| or |inf(A)|, whichever is larger.
 

Similar threads

High School Proving half of the Heine-Borel theorem
  • · Replies 3 ·
Replies
3
Views
2K
Graduate Unclear step in diam(S) = supS − inf S proof
  • · Replies 1 ·
Replies
1
Views
3K
High School Functions which relate to calculus: Questions about Notation
  • · Replies 36 ·
2
Replies
36
Views
7K
Graduate Question about proof of Great Picard Theorem
  • · Replies 3 ·
Replies
3
Views
4K
Undergrad Partial derivative of Dirac delta of a composite argument
  • · Replies 2 ·
Replies
2
Views
4K
Undergrad Correct Upper/Lower limits for continuation of solutions for ODE
  • · Replies 0 ·
Replies
0
Views
4K
High School How do I invert this exponential function?
  • · Replies 3 ·
Replies
3
Views
4K
Graduate Does the Sup/Inf Operator Apply in Vector/Matrix Conditions?
  • · Replies 2 ·
Replies
2
Views
2K
Graduate Yet another cross-product integral
  • · Replies 1 ·
Replies
1
Views
2K
Sup. and Lim. Sup. are Measurable Functions
  • · Replies 7 ·
Replies
7
Views
3K