The chapter began with a question: the fractions look complete and are not. It ends with the answer stated precisely. This lesson mixes the ideas without saying which is which, because recognising which notion a question is about is most of the skill.

The pieces have been laid out one at a time. Here they arrive shuffled, as they do in any argument written by somebody who is not teaching you.

No fraction squares to two. An upper bound need not belong to the set, and the least one is the supremum. The reals are complete and the rationals are not, and from completeness follow the Archimedean property and density.

• A supremum need not be a member; a maximum is a supremum that happens to be one.
• Dense and complete are different properties, and $\mathbb{Q}$ separates them.
• Completeness is assumed, not proved, and nearly everything ahead traces back to it.
• Choosing $n$ with $\tfrac{1}{n}$ below a given tolerance is the move that begins most proofs to come.
• Completeness is the one axiom; the Archimedean property, density and nested intervals are its consequences.
• An interval is a set with no holes, and a closed one keeps its endpoints.

Great work! You now know how to:

• Move between an absolute value, a pair of inequalities and an interval
• Compute a supremum and justify both of its conditions
• Tell the axiom of this chapter from its consequences