Real number (as Cauchy sequence)

Con­sider the set of all Cauchy se­quences of ra­tio­nal num­bers: con­cretely, the set

$$X = \{ (a_n)_{n=1}^{\infty} : a_n \in \mathbb{Q}, (\forall \epsilon \in \mathbb{Q}^{>0}) (\exists N \in \mathbb{N})(\forall n, m \in \mathbb{N}^{>N})(|a_n - a_m| < \epsilon) \}$$

Define an equiv­alence re­la­tion on this set, by \((a_n) \sim (b_n)\) if and only if, for ev­ery ra­tio­nal \(\epsilon > 0\), there is a nat­u­ral num­ber \(N\) such that for all \(n \in \mathbb{N}\) big­ger than \(N\), we have \(|a_n - b_n| < \epsilon\). This is an equiv­alence re­la­tion (ex­er­cise).

  • It is sym­met­ric, be­cause \(|a_n - b_n| = |b_n - a_n|\).

  • It is re­flex­ive, be­cause \(|a_n - a_n| = 0\) for ev­ery \(n\), and this is \(< \epsilon\).

  • It is tran­si­tive, be­cause if \(|a_n - b_n| < \frac{\epsilon}{2}\) for suffi­ciently large \(n\), and \(|b_n - c_n| < \frac{\epsilon}{2}\) for suffi­ciently large \(n\), then \(|a_n - b_n| + |b_n - c_n| < \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon\) for suffi­ciently large \(n\); so by the tri­an­gle in­equal­ity, \(|a_n - c_n| < \epsilon\) for suffi­ciently large \(n\). <div><div>

Write \([a_n]\) for the equiv­alence class of \((a_n)_{n=1}^{\infty}\). (This is a slight abuse of no­ta­tion, omit­ting the brack­ets that in­di­cate that \(a_n\) is ac­tu­ally a se­quence rather than a ra­tio­nal num­ber.)

The set of real num­bers is the set of equiv­alence classes of \(X\) un­der this equiv­alence re­la­tion, en­dowed with the fol­low­ing to­tally or­dered field struc­ture:

  • \([a_n] + [b_n] := [a_n + b_n]\)

  • \([a_n] \times [b_n] := [a_n \times b_n]\)

  • \([a_n] \leq [b_n]\) if and only if \([a_n] = [b_n]\) or there is some \(N\) such that for all \(n > N\), \(a_n \leq b_n\).

This field struc­ture is well-defined (proof).

Examples

  • Any ra­tio­nal num­ber \(r\) may be viewed as a real num­ber, be­ing the class \([r]\) (for­mally, the equiv­alence class of the se­quence \((r, r, \dots)\)).

  • The real num­ber \(\pi\) is in­deed a real num­ber un­der this defi­ni­tion; it is rep­re­sented by, for in­stance, \((3, 3.1, 3.14, 3.141, \dots)\). It is also rep­re­sented as \((100, 3, 3.1, 3.14, \dots)\), along with many other pos­si­bil­ities.

Children:

Parents:

  • Real number
    • Mathematics

      Math­e­mat­ics is the study of num­bers and other ideal ob­jects that can be de­scribed by ax­ioms.