# 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:

• 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.