Definition of supremum and infimum using epsilons ?

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Discussion Overview

The discussion revolves around the definitions of supremum and infimum in the context of mathematical analysis, specifically focusing on the use of epsilon in these definitions. Participants explore the nuances of these definitions and the implications of the conditions involved.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant proposes a definition of infimum involving epsilon, stating that for every \(\epsilon > 0\), there exists \(x \in A\) such that \(x \leq \inf A + \epsilon\).
  • Another participant challenges this definition, arguing that any number greater than all numbers in \(A\) would also satisfy the proposed condition.
  • A different participant suggests that the original definition's phrasing of "there exists" diminishes the importance of upper bounds, indicating that the inequality might not need to be strict.
  • One participant seeks clarification on what is meant by "first definition," reiterating the original statement about the infimum and questioning its sufficiency.
  • Another participant emphasizes that the infimum must also be recognized as a lower bound for the set \(A\).
  • A later reply acknowledges a misunderstanding, clarifying that a theorem relates to lower bounds and the infimum, rather than being a definition itself.

Areas of Agreement / Disagreement

Participants express disagreement regarding the sufficiency and correctness of the proposed definitions, with multiple competing views on the matter. The discussion remains unresolved as participants refine their positions.

Contextual Notes

There are limitations in the discussion regarding the assumptions made about the definitions and the implications of the conditions stated. The distinction between definitions and theorems is also not fully resolved.

AxiomOfChoice
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Is this what it is:

"For every [itex]\epsilon > 0[/itex] there exists [itex]x\in A[/itex] such that [itex]x \leq \inf A + \epsilon[/itex]."

...and similarly for the supremum?
 
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No, that's not quite it. Any number y which is greater than all numbers in A would satisfy your definition.
 
Well in the first definition the OP said "there exists", so really the concern of upper bounds isn't really important. This is a useful proposition, and sometimes the inequality is strict, though that doesn't matter all that much.
 
What "first definition"? Are you talking about something other than:

"For every [tex]\epsilon > 0[/tex] there exists [tex]x\in A[/tex] such that [tex]x \leq \inf A + \epsilon .[/tex]"?

While that is true about the infinimum, it won't do for the definition for the reason I gave.
 
LCKurtz said:
What "first definition"? Are you talking about something other than:

"For every [tex]\epsilon > 0[/tex] there exists [tex]x\in A[/tex] such that [tex]x \leq \inf A + \epsilon .[/tex]"?

While that is true about the infinimum, it won't do for the definition for the reason I gave.

That's correct. We must also state that inf A is a lower bound.
 
Yeah I realized I was thinking of the theorem that states that if L is a lower bound for a set A in R, then L = inf A iff for every epsilon > 0, there is an x in A with x - L < epsilon. My apologies, of course it's not a definition.
 

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