Are There Infinite Upper Bounds for a Given Set?

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Discussion Overview

The discussion centers around the concept of upper bounds for a given set, particularly in the context of real numbers. Participants explore the implications of having upper bounds, the uniqueness of the least upper bound, and the terminology used in mathematical analysis.

Discussion Character

  • Conceptual clarification, Technical explanation, Debate/contested

Main Points Raised

  • One participant questions the assertion that if a set has an upper bound, there are infinitely many upper bounds, suggesting that a finite set might only have one upper bound.
  • Another participant points out that if a is an upper bound, then a+1, a+2, and so on, are also upper bounds, implying the existence of infinitely many upper bounds.
  • A participant clarifies the definition of upper bounds after initially misreading it.
  • It is noted that for sets of real numbers, if a set has any upper bounds, it has a unique least upper bound, referred to as the supremum.
  • Participants discuss the terminology, confirming that "supreme" is typically referred to as "supremum" in English, and clarify that "supremum" is derived from Latin.

Areas of Agreement / Disagreement

Participants generally agree on the definitions of upper bounds and the concept of the least upper bound, but there is some initial confusion regarding the implications for finite sets. The discussion remains unresolved regarding the participant's initial assertion about finite sets having only one upper bound.

Contextual Notes

There is a potential limitation in understanding due to language translation issues, particularly concerning the terms "supremum" and "maximum." Additionally, the discussion does not resolve the implications of upper bounds for finite sets.

Who May Find This Useful

This discussion may be useful for students and practitioners of mathematical analysis, particularly those interested in the properties of sets and bounds in real numbers.

Max.Planck
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#Upper bounds of a given set

In my analysis book it says:
"If a set has a upper bound, there are infinitely many upper bounds".
Why is this correct? If you have some finite set, won't there be just one?
 
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If a is an upper bound, what about a+1, a+ 2, a+ 3, ...? Are you clear on what an "upper bound" is?
 


Ah yes of course, I misread the definition.
 


Note, that if you are talking about sets of real numbers, if a set has any upper bounds, then it has a unique least upper bound.
 


And that unique least upper bound is called supreme.

If the supreme belongs to the set, then it's called a maximum.

EDIT: I need confirmation on the terms "supreme" and "maximum" since I've learned Analysis in spanish and translations might not be accurate
 


In english it's typically called the supremum
 


Office_Shredder said:
In english it's typically called the supremum
Which is actually Latin!
 

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