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How might someone construct a proof that for a, b [tex]\in[/tex] R such that a < b, there exists a real number c such that a < c < b, without using the Archimedean principle?

I know if we can assume the Archimedean principle that we can easily prove this. But is it possible to follow logic that just uses order axioms to prove it?

Just as a reminder, the Archimedean principle says that if I have reals a, b such that a > 0 and b> 0, then there exists some positive integer n such that an > b.

Thanks.

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# Way to prove inequality without Archimedean?

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