Proving LUB and GLB Properties in Ordered Fields

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Homework Help Overview

The discussion revolves around proving properties related to the least upper bound (LUB) and greatest lower bound (GLB) in the context of ordered fields. The original poster seeks assistance in establishing a proof that connects these two properties.

Discussion Character

  • Exploratory, Assumption checking, Conceptual clarification

Approaches and Questions Raised

  • The original poster attempts to clarify the relationship between the LUB and GLB properties and questions whether the proof requires assuming the ordered field is bounded. Participants explore the implications of lower bounds and their relation to upper bounds through hints and follow-up questions.

Discussion Status

Participants are engaging in a back-and-forth exchange, providing hints and prompting further exploration of the concepts involved. There is an emphasis on understanding the definitions and relationships between bounds in ordered fields, but no consensus or resolution has been reached yet.

Contextual Notes

There is a focus on the properties of ordered fields, and participants are reminded that the discussion is not limited to the field of real numbers. The nature of the elements and operations within the ordered field is under consideration.

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1. Prove that an ordered field has the LUB property iff it has the GLB property.


I know that I need to prove that if the ordered field has the GLB property, then it has the LUB property, and that if the ordered field does NOT has the GLB property, then it also does not have the LUB property. I'm just really stuck on how to start the proof - do you assume the ordered field is bounded?

Any help would be appreciated!
 
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Hint: if a set S has a lower bound, what can you say about the set (-1)*S?
 
That it has an upper bound?...
 
ragnes said:
That it has an upper bound?...

Yes. And how is the lower bound of S related to the upper bound of (-1)*S?
 
ragnes said:
That it has an upper bound?...
Don't guess! If U is the LUB, it is, first, of all, an upper bound. In other words, for any x in the field, x\le U. Multiplying both sides by -1, -x\ge -U. But if y is any member of the field, x= -y is also in the field and so y= -x\ge -U. That is, -U is a lower bound. Now you need to show it is the greatest lower bound.

Don't forget that this is an "ordered field", not necessarily the field of real numbers. What is meant by "-1"? Have you proven or can you prove that "if a< b, then -a> -b"?
 

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