The reals (constructed as classes of Cauchy sequences of rationals) form a field

https://arbital.com/p/reals_as_classes_of_cauchy_sequences_form_a_field

by Patrick Stevens Jul 4 2016 updated Jul 5 2016

The reals are an archetypal example of a field, but if we are to construct them from simpler objects, we need to show that our construction does indeed have the right properties.


The real numbers, when constructed as equivalence classes of Cauchy sequences of rationals, form a totally ordered field, with the inherited field structure given by

Proof

Firstly, we need to show that those operations are even well-defined: that is, if we pick two different representatives and of the same equivalence class , we don't somehow get different answers.

Well-definedness of

We wish to show that whenever and ; this is an exercise. %%hidden(Show solution): Since , it must be the case that both and are Cauchy sequences such that as . Similarly, as .

We require ; that is, we require as .

But this is true: if we fix rational , we can find such that for all , we have ; and we can find such that for all , we have . Letting be the maximum of the two , we have that for all , by the [-triangle_inequality], and hence . %%

Well-definedness of

We wish to show that whenever and ; this is also an exercise. %%hidden(Show solution): We require ; that is, as .

Let be rational. Then using the very handy trick of adding the expression .

By the triangle inequality, this is .

We now use the fact that [cauchy_sequences_are_bounded], to extract some such that and for all ; then our expression is less than .

Finally, for sufficiently large we have , and similarly for and , so the result follows that . %%

Well-definedness of

We wish to show that if and , then implies .

Suppose , but suppose for contradiction that is not : that is, and there are arbitrarily large such that . Then there are two cases.

Additive commutative group structure on

The additive identity is (formally, the equivalence class of the sequence ). Indeed, .

The additive inverse of the element is , because .

The operation is commutative: .

The operation is closed, because the sum of two Cauchy sequences is a Cauchy sequence (exercise). %%hidden(Show solution): If and are Cauchy sequences, then let . We wish to show that there is such that for all , we have .

But by the triangle inequality; so picking so that and for all , the result follows. %%

The operation is associative:

Ring structure

The multiplicative identity is (formally, the equivalence class of the sequence ). Indeed, .

is closed, because the product of two Cauchy sequences is a Cauchy sequence (exercise). %%hidden(Show solution): If and are Cauchy sequences, then let . We wish to show that there is such that for all , we have .

But by the triangle inequality.

Cauchy sequences are bounded, so there is such that and are both less than for all and .

So picking so that and for all , the result follows. %%

is clearly commutative: .

is associative:

distributes over : we need to show that . But this is true:

Field structure

To get from a ring to a field, it is necessary and sufficient to find a multiplicative inverse for any not equal to .

Since , there is some such that for all , . Then defining the sequence for , and for , we obtain a sequence which induces an element of ; and it is easy to check that . %%hidden(Show solution): ; but the sequence is for all , and so it lies in the same equivalence class as the sequence . %%

Ordering on the field

We need to show that:

We may assume that the inequalities are strict, because if equality holds in the assumption then everything is obvious. %%hidden(Show obvious bits): If , then for every we have by well-definedness of addition. Therefore .

If and , then , so it is certainly true that . %%

For the former: suppose , and let be an arbitrary equivalence class. Then ; ; but we have for all sufficiently large , because for sufficiently large . Therefore , as required.

For the latter: suppose and . Then for sufficiently large , we have both and are positive; so for sufficiently large , we have . But that is just saying that , as required.