Least Upper Bound: What Is It & How to Prove It

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Discussion Overview

The discussion revolves around the concept of "least upper bound" in the context of sequences and sets. Participants explore its definition, implications, and how to prove its properties, with a focus on mathematical reasoning and conceptual clarification.

Discussion Character

  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant expresses confusion about whether the least upper bound is the limit of a sequence and seeks clarification on how to prove it.
  • Another participant defines the least upper bound of a set E as a number greater than every element of E, with the condition that any number less than this bound is included in E.
  • A different participant elaborates that the least upper bound is the smallest upper bound, stating that if A is a least upper bound for a sequence, then any other upper bound B must be greater than A.
  • One participant challenges the previous definition by providing a counterexample involving the set (0,1), arguing that while 1 is the least upper bound, 0 is not an element of the set despite being less than 1.
  • Another participant indicates that they have gained understanding through the discussion and external research.
  • One participant provides an example involving negative real numbers, noting that while 0 is an upper bound, it is the least upper bound.

Areas of Agreement / Disagreement

Participants express differing views on the definition and implications of the least upper bound, with some agreeing on its properties while others challenge specific interpretations. The discussion remains unresolved regarding the accuracy of certain definitions.

Contextual Notes

Some definitions presented may depend on specific interpretations of upper bounds and the nature of the sets involved. The discussion includes counterexamples that highlight potential limitations in the definitions provided.

soul
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I am confused about the concept of "least upper bound". Is this line the limt of the {an} sequence. If so, how can we prove it?
 
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the least upper bound of a set E is a number greater than every element of E s.t. if another number is less than the least upper bound, it is in E.
 
The term least upper bound is a loaded term.
First of all it is an upper bound of the sequence {an}, meaning it is greater than every term of the sequence. More generally the upper bound of some set is some number greater than any number in the set.
Secondly it is the smallest such upper bound. So if A is a least upper bound for {an} and B is some other upper bound then A<B
 
matticus said:
the least upper bound of a set E is a number greater than every element of E s.t. if another number is less than the least upper bound, it is in E.

This is false, consider the set (0,1) then 1 is obviously the least upper bound, and also 0 is less than 1, but 0 is not in the set.
 
With a search on google and with your help I understood the point. Thanks for your help guys.
 
Think of an upper bound of nevative real numbers. Obviously 0 and any number greater than 0 is an upper bound. But the least is 0.
 

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