If a nonempty set, S, of real numbers has an upper bound M ([itex] x \leq M [/itex] for all x in S), then S has a least upper bound b. (This means that b is an upper bound for S, but if M is any other upper bound, then [itex] b \leq M [/itex].) The Completeness Axiom is an expression of the fact that there is no gap or hole in the real number line. I am confused about this theorem. I apologize, for I have not been formally taught any number theory, or whatever branch of math into which this falls. I only came across this in my calc text when reviewing sequences and series. It was used to prove the Monotonic Sequence Theorem. However, I do not understand why the condition that the set S must have a least upper bound guarantees that the real number line has no holes or gaps. What about this situation (hypothetical "broken" real number line): <----------------Smax_______________M----------------------> So, at the end of the left section of the real line, we have Smax, the highest number in the set S. At the beginning of the right section, we have the upper bound M. Is M not, by necessity, a least upper bound? What part of the completeness axiom is not satisfied by the broken number line? Thank you.