Real numbers · Definition

Completeness axiom

real analysisreal numbers

Before you start you need to know

This is the axiom that turns the gap-filled rationals into the continuous real line on which all of analysis rests.

Definition

The property that sets \(\mathbb R\) apart from \(\mathbb Q\): every nonempty set \(S\) of reals with an upper bound \(M\) has a supremum.

\[S\ne\varnothing,\ \exists M:\ s\le M\ \forall s\in S\ \Longrightarrow\ \sup S\in\mathbb R\]

The field and order axioms are not enough for analysis: \(\mathbb Q\) satisfies them, yet the set \(\{x\in\mathbb Q: x^2<2\}\) is bounded above without having a least rational upper bound. Completeness demands that this gap not exist in \(\mathbb R\): its supremum is \(\sqrt2\).

It is an axiom, not a theorem, in the axiomatic presentation of \(\mathbb R\) (in Dedekind-cut or Cauchy-sequence constructions it is proved). It is also called the supremum property or the least upper bound property. Stated for suprema, it holds for infima too: if \(S\) is bounded below, \(\inf S=-\sup(-S)\) exists.

Both hypotheses are necessary. Without an upper bound there is no bound at all (\(\mathbb N\) has no supremum in \(\mathbb R\)), and for the empty set every number would be a bound, so there would be no least one. Also, the supremum is guaranteed, not the maximum: \((0,1)\) has supremum \(1\) and no maximum.

The central results of the course follow from this axiom: the Archimedean property, density of the rationals, existence of roots, the monotone bounded sequence theorem, Bolzano-Weierstrass and the intermediate value theorem. Whenever a proof says “let \(s=\sup S\)”, it is using completeness.

Think about it

Is \(\mathbb Q\) an ordered field? What does it lack to be \(\mathbb R\)?

Yes, it satisfies the field and order axioms; it lacks completeness, since there are bounded rational sets with no rational supremum.

If \(S\) has a maximum, how does it relate to its supremum?

They coincide: the maximum is an upper bound that belongs to \(S\), hence it is the least one.

Exercises 10

  1. What does the completeness axiom state?
  2. Which of these sets of rationals has no supremum in \(\mathbb Q\)?
  3. For which set does the hypothesis of the completeness axiom NOT hold?
  4. If \(S\) is nonempty and bounded below, what does the completeness axiom guarantee?
  5. Compute \(\sup\{x\in\mathbb R: x^3<8\}\).
  6. Let \(S=\{x\in\mathbb R: x^2+x<6\}\). What is \(\inf S+\sup S\)?
  7. Compute \(\sup\left\{\dfrac{2n}{n+1}: n\in\mathbb N\right\}\).
  8. Which structure satisfies the ordered field axioms but not completeness?
  9. Let \(S\) be nonempty and bounded above, and \(M=\sup S\). Which statement is true?
  10. Why does the completeness axiom require the set to be nonempty?

References

  • Stephen Abbott, Understanding Analysis, 2nd edition, Springer, 2015. §1.3
  • Walter Rudin, Principles of Mathematical Analysis, 3rd edition, McGraw-Hill, 1976. Cap. 1, 1.10–1.11
  • Robert G. Bartle, Donald R. Sherbert, Introduction to Real Analysis, 4th edition, Wiley, 2011. §2.3