Archimedean property for unbounded sets

  • #1

Main Question or Discussion Point

Does the Archimedean property work for unbounded sets? My book does a proof of the Archimedean property relying on the existence of sup which relies on the existence of a bound.
 

Answers and Replies

  • #2
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Does the Archimedean property work for unbounded sets? My book does a proof of the Archimedean property relying on the existence of sup which relies on the existence of a bound.
Unbounded sets of what type of elements? Real numbers? I think you should reword or provide more details of your question and maybe even cite the text you're referring to (including page number, theorem number, etc.).
 
  • #3
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The real numbers are unbounded and have the Archimedean property.
 
  • #4
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If I'm recalling the proof of the Archimedean property correctly, I wonder if the OP actually meant to ask if this property holds true for sets that do not satisfy the Axiom of Completeness (i.e., the existence of sups)...

I suppose the rational numbers are an example of a set that satisfies the Archimedean property without satisfying the Axiom of Completeness.
 

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