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**Completeness axiom as having "no holes" in the set**

My textbook describes the completeness axiom as essential to showing that there are no "holes or gaps" in the real numbers. That is, for any two reals A and B, there exists a real C such that A<C<B.

Of course, we all know that the actual statement of the completeness axiom is that any bounded set of real numbers has a least upper bound.

I was wondering, how can we use the explicit statement of the completeness axiom to show that there are no "holes or gaps" in the reals? Is it possible or did my textbook just use this as an intuitive explanation for the completeness axiom?

Thanks all!

BiP

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