(adsbygoogle = window.adsbygoogle || []).push({}); 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

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Completeness axiom as having no holes in the set

**Physics Forums | Science Articles, Homework Help, Discussion**