# 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.

• The ti­tle men­tions Cauchy se­quences, but the body does not. Doesn’t this defi­ni­tion con­sider classes of non-con­verg­ing se­quences as real num­bers?

• You’re right; I was sloppy. I’ll fix it, thanks.