Archimedean property for unbounded sets

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.

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.).

The real numbers are unbounded and have the Archimedean property.

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.