Real numbers · Definition

Archimedean property

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Before you start you need to know

It ensures there are no infinitely large or infinitely small numbers in \(\mathbb R\), a silent requirement of every argument with limits.

Definition

The property of \(\mathbb R\) by which, however small a positive number \(x\) is, some multiple of it by a natural \(n\in\mathbb N\) exceeds any real \(y\).

\[x>0,\ y\in\mathbb R\ \Longrightarrow\ \exists\, n\in\mathbb N:\ nx>y\]

Taking \(x=1\) gives the best-known form: for every real \(y\) there is a natural \(n>y\), i.e., \(\mathbb N\) is not bounded above. Taking \(y=1\) gives the form used in limits: for every \(\varepsilon>0\) there is \(n\) with \(1/n<\varepsilon\).

It does not follow from the ordered field axioms: non-Archimedean ordered fields exist, such as rational functions ordered so that \(t\) exceeds every constant. It is proved with completeness: if \(\mathbb N\) were bounded it would have a supremum \(s\), but \(s-1\) would not be a bound, so some \(n>s-1\) and hence \(n+1>s\), absurd.

Immediate consequences: \(\inf\{1/n\}=0\); if \(0\le x<1/n\) for all \(n\), then \(x=0\); and every real is exceeded by some integer, which allows defining the floor function. It is also the ingredient that gives meaning to the first limit of the course, \(1/n\to0\).

A typical mistake is to think of “infinity” as a number larger than all others. The Archimedean property says the opposite: no real is larger than all naturals, which is why the symbol \(\infty\) is never an element of \(\mathbb R\).

Think about it

Is there a positive real smaller than \(1/n\) for every natural \(n\)?

No: if it existed, \(1/x\) would be an upper bound of \(\mathbb N\), against the Archimedean property.

Does the Archimedean property hold in \(\mathbb Q\)?

Yes: for rationals \(x>0\) and \(y\) it suffices to take a natural greater than \(y/x\). What fails in \(\mathbb Q\) is completeness, not this property.

Exercises 10

  1. What does the Archimedean property state?
  2. What is the smallest natural \(n\) with \(\tfrac1n<0.003\)?
  3. By the Archimedean property, what is the infimum of \(\{1/n: n\in\mathbb N\}\)?
  4. If \(x=0.002\) and \(y=5\), what is the smallest natural \(n\) with \(nx>y\)?
  5. Which statement is equivalent to the Archimedean property?
  6. What is the smallest natural \(n\) with \(\dfrac{n}{n+1}>0.999\)?
  7. In an ordered field there is \(\varepsilon>0\) with \(\varepsilon<\tfrac1n\) for every natural \(n\). Which property fails?
  8. For \(x=\tfrac37\) and \(y=100\), what is the smallest natural \(n\) with \(nx>y\)?
  9. Which is the correct idea for proving the Archimedean property from the completeness axiom?
  10. If \(x\ge0\) and \(x<\tfrac1n\) for every natural \(n\), what follows?

References

  • Stephen Abbott, Understanding Analysis, 2nd edition, Springer, 2015. §1.4
  • Walter Rudin, Principles of Mathematical Analysis, 3rd edition, McGraw-Hill, 1976. Cap. 1, 1.20
  • Robert G. Bartle, Donald R. Sherbert, Introduction to Real Analysis, 4th edition, Wiley, 2011. §2.4