11.3.3 The algebraic structure of Cauchy reals

11.3.3 The algebraic structure of Cauchy reals

We first define the

additive

structure

(

ℝ

𝖼

,

0

,

+

,

-

)

. Clearly, the additive unit element

0

is just

𝗋𝖺𝗍

⁢

(

0

)

, while the

additive inverse

-

:

ℝ

𝖼

→

ℝ

𝖼

is obtained as the

extension

of the additive inverse

-

:

ℚ

→

ℚ

, using

\autoref

RC-extend-Q-Lipschitz
with

Lipschitz constant

1

. We have to work a bit harder for

addition

.

Lemma 11.3.1

.

Suppose

f

:

Q

×

Q

→

Q

satisfies

, for all

q

,

r

,

s

:

Q

,

|

f

⁢

(

q

,

s

)

-

f

⁢

(

r

,

s

)

|

≤

|

q

-

r

|

𝑎𝑛𝑑

|

f

⁢

(

q

,

r

)

-

f

⁢

(

q

,

s

)

|

≤

|

r

-

s

|

.

Then there is a

function

f

¯

:

R

c

×

R

c

→

R

c

such that

f

¯

⁢

(

rat

⁢

(

q

)

,

rat

⁢

(

r

)

)

=

f

⁢

(

q

,

r

)

for all

q

,

r

:

Q

. Furthermore,
for all

u

,

v

,

w

:

R

c

and

q

:

Q

+

,

u

∼

ϵ

v

⇒

f

¯

⁢

(

u

,

w

)

∼

ϵ

f

¯

⁢

(

v

,

w

)

𝑎𝑛𝑑

v

∼

ϵ

w

⇒

f

¯

⁢

(

u

,

v

)

∼

ϵ

f

¯

⁢

(

u

,

w

)

.

Proof.

We use

(

ℝ

𝖼

,

∼

)

-

recursion

to construct the curried form of

f

¯

as a map

ℝ

𝖼

→

A

where

A

is the space of non-expanding real-valued
functions:

A

:

≡

\setof

h

:

ℝ

𝖼

→

ℝ

𝖼

|

∀

(

ϵ

:

ℚ

+

)

.

∀

(

u

,

v

:

ℝ

𝖼

)

.

u

∼

ϵ

v

⇒

h

(

u

)

∼

ϵ

h

(

v

)

.

We shall also need a suitable

⌢

ϵ

on

A

, which we define as

(

h

⌢

ϵ

k

)

:

≡

∀

(

u

:

ℝ

𝖼

)

.

h

(

u

)

∼

ϵ

k

(

u

)

.

Clearly, if

∀

(

ϵ

:

ℚ

+

)

.

h

⌢

ϵ

k

then

h

⁢

(

u

)

=

k

⁢

(

u

)

for all

u

:

ℝ

𝖼

, so

⌢

is separated.

For the

base case

we define

f

¯

⁢

(

𝗋𝖺𝗍

⁢

(

q

)

)

:

A

, where

q

:

ℚ

, as the
extension of the Lipschitz map

λ

⁢

r

.

f

⁢

(

q

,

r

)

from

ℚ

→

ℚ

to

ℝ

𝖼

→

ℝ

𝖼

, as
constructed in

\autoref

RC-extend-Q-Lipschitz with Lipschitz constant

1

. Next, for a
Cauchy approximation

x

, we define

f

¯

⁢

(

𝗅𝗂𝗆

⁢

(

x

)

)

:

ℝ

𝖼

→

ℝ

𝖼

as

f

¯

(

𝗅𝗂𝗆

(

x

)

)

(

v

)

:

≡

𝗅𝗂𝗆

(

λ

ϵ

.

f

¯

(

x

ϵ

)

(

v

)

)

.

For this to be a valid definition,

λ

⁢

ϵ

.

f

¯

⁢

(

x

ϵ

)

⁢

(

v

)

should be a
Cauchy approximation, so consider any

δ

,

ϵ

:

ℚ

. Then by

assumption

f

¯

⁢

(

x

δ

)

⌢

δ

+

ϵ

f

¯

⁢

(

x

ϵ

)

, hence

f

¯

⁢

(

x

δ

)

⁢

(

v

)

∼

δ

+

ϵ

f

¯

⁢

(

x

ϵ

)

⁢

(

v

)

. Furthermore,

f

¯

⁢

(

𝗅𝗂𝗆

⁢

(

x

)

)

is non-expanding because

f

¯

⁢

(

x

ϵ

)

is such by induction
hypothesis. Indeed, if

u

∼

ϵ

v

then, for all

ϵ

:

ℚ

,

f

¯

⁢

(

x

ϵ

/

3

)

⁢

(

u

)

∼

ϵ

/

3

f

¯

⁢

(

x

ϵ

/

3

)

⁢

(

v

)

,

therefore

f

¯

⁢

(

𝗅𝗂𝗆

⁢

(

x

)

)

⁢

(

u

)

∼

ϵ

f

¯

⁢

(

𝗅𝗂𝗆

⁢

(

x

)

)

⁢

(

v

)

by the fourth constructor of

∼

.

We still have to check four more conditions, let us illustrate just one. Suppose

ϵ

:

ℚ

+

and for some

δ

:

ℚ

+

we have

𝗋𝖺𝗍

⁢

(

q

)

∼

ϵ

-

δ

y

δ

and

f

¯

⁢

(

𝗋𝖺𝗍

⁢

(

q

)

)

⌢

ϵ

-

δ

f

¯

⁢

(

y

δ

)

. To show

f

¯

⁢

(

𝗋𝖺𝗍

⁢

(

q

)

)

⌢

ϵ

f

¯

⁢

(

𝗅𝗂𝗆

⁢

(

y

)

)

, consider any

v

:

ℝ

𝖼

and observe that

f

¯

⁢

(

𝗋𝖺𝗍

⁢

(

q

)

)

⁢

(

v

)

∼

ϵ

-

δ

f

¯

⁢

(

y

δ

)

⁢

(

v

)

.

Therefore, by the second constructor of

∼

, we have

f

¯

⁢

(

𝗋𝖺𝗍

⁢

(

q

)

)

⁢

(

v

)

∼

ϵ

f

¯

⁢

(

𝗅𝗂𝗆

⁢

(

y

)

)

⁢

(

v

)

as required.
∎

We may apply

\autoref

RC-binary-nonexpanding-extension to any bivariate

rational function

which is non-expanding separately in each

variable

. Addition is such a function, therefore
we get

+

:

ℝ

𝖼

×

ℝ

𝖼

→

ℝ

𝖼

.
Furthermore, the extension is unique as long as we
require it to be non-expanding in each variable, and just as in the univariate case,

identities

on

rationals

extend to identities on reals. Since composition of non-expanding
maps is again non-expanding, we may conclude that addition satisfies the usual properties,
such as

commutativity

and

associativity

.
Therefore,

(

ℝ

𝖼

,

0

,

+

,

-

)

is a

commutative
group

.

We may also apply

\autoref

RC-binary-nonexpanding-extension to the functions

min

:

ℚ

×

ℚ

→

ℚ

and

max

:

ℚ

×

ℚ

→

ℚ

, which turns

ℝ

𝖼

into a

lattice

. The

partial
order

≤

on

ℝ

𝖼

is defined in terms of

max

as

(

u

≤

v

)

:

≡

(

max

(

u

,

v

)

=

v

)

.

The

relation

≤

is a partial order because it is such on

ℚ

, and the axioms of a
partial order are

expressible

as

equations

in terms of

min

and

max

, so they transfer
to

ℝ

𝖼

.

Another function which extends to

ℝ

𝖼

by the same method is the

absolute value

|

–

|

.
Again, it has the expected properties because they transfer from

ℚ

to

ℝ

𝖼

.

From

≤

we get the

strict order

<

by

(

u

<

v

)

:

≡

∃

(

q

,

r

:

ℚ

)

.

(

u

≤

𝗋𝖺𝗍

(

q

)

)

∧

(

q

<

r

)

∧

(

𝗋𝖺𝗍

(

r

)

≤

v

)

.

That is,

u

<

v

holds when there merely exists a pair of rational numbers

q

<

r

such that

x

≤

𝗋𝖺𝗍

⁢

(

q

)

and

𝗋𝖺𝗍

⁢

(

r

)

≤

v

. It is not hard to check that

<

is

irreflexive

and

transitive

, and has other properties that are expected for an

ordered field

.
The

archimedean principle

follows directly from the definition of

<

.

Theorem 11.3.2

(Archimedean principle for

R

c

).

For every

u

,

v

:

R

c

such that

u

<

v

there merely exists

q

:

Q

such that

u

<

q

<

v

.

Proof.

From

u

<

v

we merely get

r

,

s

:

ℚ

such that

u

≤

r

<

s

≤

v

, and we may take

q

:

≡

(

r

+

s

)

/

2

.
∎

We now have enough structure on

ℝ

𝖼

to express

u

∼

ϵ

v

with standard

concepts

.

Lemma 11.3.3

.

If

q

:

Q

and

u

:

R

c

satisfy

u

≤

rat

⁢

(

q

)

, then for any

v

:

R

c

and

ϵ

:

Q

+

, if

u

∼

ϵ

v

then

v

≤

rat

⁢

(

q

+

ϵ

)

.

Proof.

Note that the function

max

⁡

(

𝗋𝖺𝗍

⁢

(

q

)

,

–

)

:

ℝ

𝖼

→

ℝ

𝖼

is Lipschitz with

constant

1

.
First consider the case when

u

=

𝗋𝖺𝗍

⁢

(

r

)

is rational.
For this we use

induction

on

v

.
If

v

is rational, then the statement is obvious.
If

v

is

𝗅𝗂𝗆

⁢

(

y

)

, we assume inductively that for any

ϵ

,

δ

, if

𝗋𝖺𝗍

⁢

(

r

)

∼

ϵ

y

δ

then

y

δ

≤

𝗋𝖺𝗍

⁢

(

q

+

ϵ

)

, i.e.

max

⁡

(

𝗋𝖺𝗍

⁢

(

q

+

ϵ

)

,

y

δ

)

=

𝗋𝖺𝗍

⁢

(

q

+

ϵ

)

.

Now assuming

ϵ

and

𝗋𝖺𝗍

⁢

(

r

)

∼

ϵ

𝗅𝗂𝗆

⁢

(

y

)

, we have

θ

such that

𝗋𝖺𝗍

⁢

(

r

)

∼

ϵ

-

θ

𝗅𝗂𝗆

⁢

(

y

)

, hence

𝗋𝖺𝗍

⁢

(

r

)

∼

ϵ

y

δ

whenever

δ

<

θ

.
Thus, the inductive hypothesis gives

max

⁡

(

𝗋𝖺𝗍

⁢

(

q

+

ϵ

)

,

y

δ

)

=

𝗋𝖺𝗍

⁢

(

q

+

ϵ

)

for such

δ

.
But by definition,

max

(

𝗋𝖺𝗍

(

q

+

ϵ

)

,

𝗅𝗂𝗆

(

y

)

)

≡

𝗅𝗂𝗆

(

λ

δ

.

max

(

𝗋𝖺𝗍

(

q

+

ϵ

)

,

y

δ

)

)

.

Since the limit of an

eventually constant

Cauchy approximation is that constant, we have

max

⁡

(

𝗋𝖺𝗍

⁢

(

q

+

ϵ

)

,

𝗅𝗂𝗆

⁢

(

y

)

)

=

𝗋𝖺𝗍

⁢

(

q

+

ϵ

)

,

hence

𝗅𝗂𝗆

⁢

(

y

)

≤

𝗋𝖺𝗍

⁢

(

q

+

ϵ

)

.

Now consider a general

u

:

ℝ

𝖼

.
Since

u

≤

𝗋𝖺𝗍

⁢

(

q

)

means

max

⁡

(

𝗋𝖺𝗍

⁢

(

q

)

,

u

)

=

𝗋𝖺𝗍

⁢

(

q

)

, the assumption

u

∼

ϵ

v

and the Lipschitz property of

max

⁡

(

𝗋𝖺𝗍

⁢

(

q

)

,

-

)

imply

max

⁡

(

𝗋𝖺𝗍

⁢

(

q

)

,

v

)

∼

ϵ

𝗋𝖺𝗍

⁢

(

q

)

.
Thus, since

𝗋𝖺𝗍

⁢

(

q

)

≤

𝗋𝖺𝗍

⁢

(

q

)

, the first case implies

max

⁡

(

𝗋𝖺𝗍

⁢

(

q

)

,

v

)

≤

𝗋𝖺𝗍

⁢

(

q

+

ϵ

)

, and hence

v

≤

𝗋𝖺𝗍

⁢

(

q

+

ϵ

)

by

transitivity

of

≤

.
∎

Lemma 11.3.4

.

Suppose

q

:

Q

and

u

:

R

c

satisfy

u

<

rat

⁢

(

q

)

. Then:

1.

For any

v

:

ℝ

𝖼

and

ϵ

:

ℚ

+

, if

u

∼

ϵ

v

then

v

<

𝗋𝖺𝗍

⁢

(

q

+

ϵ

)

.

2.

There exists

ϵ

:

ℚ

+

such that for any

v

:

ℝ

𝖼

, if

u

∼

ϵ

v

we have

v

<

𝗋𝖺𝗍

⁢

(

q

)

.

Proof.

By definition,

u

<

𝗋𝖺𝗍

⁢

(

q

)

means there is

r

:

ℚ

with

r

<

q

and

u

≤

𝗋𝖺𝗍

⁢

(

r

)

.
Then by

\autoref

thm:RC-le-grow, for any

ϵ

, if

u

∼

ϵ

v

then

v

≤

𝗋𝖺𝗍

⁢

(

r

+

ϵ

)

.

Conclusion

1

follows immediately since

r

+

ϵ

<

q

+

ϵ

, while for

2

we can take any

ϵ

<

q

-

r

.
∎

We are now able to show that the auxiliary relation

∼

is what we think it is.

Theorem 11.3.5

.

(

u

∼

ϵ

v

)

≃

(

|

u

-

v

|

<

𝗋𝖺𝗍

(

ϵ

)

)

for all

u

,

v

:

R

c

and

ϵ

:

Q

+

.

Proof.

The Lipschitz properties of

subtraction

and absolute value imply that if

u

∼

ϵ

v

, then

|

u

-

v

|

∼

ϵ

|

u

-

u

|

=

0

.
Thus, for the left-to-right direction, it will suffice to show that if

u

∼

ϵ

0

, then

|

u

|

<

𝗋𝖺𝗍

⁢

(

ϵ

)

.
We proceed by

ℝ

𝖼

-induction on

u

.

If

u

is rational, the statement follows immediately since absolute value and order extend the standard ones on

ℚ

+

.
If

u

is

𝗅𝗂𝗆

⁢

(

x

)

, then by roundedness we have

θ

:

ℚ

+

with

𝗅𝗂𝗆

⁢

(

x

)

∼

ϵ

-

θ

0

.
By the

triangle inequality

, therefore, we have

x

θ

/

3

∼

ϵ

-

2

⁢

θ

/

3

0

, so the inductive hypothesis yields

|

x

θ

/

3

|

<

𝗋𝖺𝗍

⁢

(

ϵ

-

2

⁢

θ

/

3

)

.
But

x

θ

/

3

∼

2

⁢

θ

/

3

𝗅𝗂𝗆

⁢

(

x

)

, hence

|

x

θ

/

3

|

∼

2

⁢

θ

/

3

|

𝗅𝗂𝗆

⁢

(

x

)

|

by the Lipschitz property, so

\autoref

thm:RC-lt-open

1

implies

|

𝗅𝗂𝗆

⁢

(

x

)

|

<

𝗋𝖺𝗍

⁢

(

ϵ

)

.

In the other direction, we use

ℝ

𝖼

-induction on

u

and

v

.
If both are rational, this is the first constructor of

∼

.

If

u

is

𝗋𝖺𝗍

⁢

(

q

)

and

v

is

𝗅𝗂𝗆

⁢

(

y

)

, we assume inductively that for any

ϵ

,

δ

, if

|

𝗋𝖺𝗍

⁢

(

q

)

-

y

δ

|

<

𝗋𝖺𝗍

⁢

(

ϵ

)

then

𝗋𝖺𝗍

⁢

(

q

)

∼

ϵ

y

δ

.
Fix an

ϵ

such that

|

𝗋𝖺𝗍

⁢

(

q

)

-

𝗅𝗂𝗆

⁢

(

y

)

|

<

𝗋𝖺𝗍

⁢

(

ϵ

)

.
Since

ℚ

is order-dense in

ℝ

𝖼

, there exists

θ

<

ϵ

with

|

𝗋𝖺𝗍

⁢

(

q

)

-

𝗅𝗂𝗆

⁢

(

y

)

|

<

𝗋𝖺𝗍

⁢

(

θ

)

.
Now for any

δ

,

η

we have

𝗅𝗂𝗆

⁢

(

y

)

∼

2

⁢

δ

y

δ

, hence by the Lipschitz property

|

𝗋𝖺𝗍

⁢

(

q

)

-

𝗅𝗂𝗆

⁢

(

y

)

|

∼

δ

+

η

|

𝗋𝖺𝗍

⁢

(

q

)

-

y

δ

|

.

Thus, by

\autoref

thm:RC-lt-open

1

, we have

|

𝗋𝖺𝗍

⁢

(

q

)

-

y

δ

|

<

𝗋𝖺𝗍

⁢

(

θ

+

2

⁢

δ

)

.
So by the inductive hypothesis,

𝗋𝖺𝗍

⁢

(

q

)

∼

θ

+

2

⁢

δ

y

δ

, and thus

𝗋𝖺𝗍

⁢

(

q

)

∼

θ

+

4

⁢

δ

𝗅𝗂𝗆

⁢

(

y

)

by the triangle inequality.
Thus, it suffices to choose

δ

:

≡

(

ϵ

-

θ

)

/

4

.

The remaining two cases are entirely analogous.
∎

Next, we would like to equip

ℝ

𝖼

with

multiplicative

structure. For each

q

:

ℚ

the
map

r

↦

q

⋅

r

is Lipschitz with constant

1

1

We defined Lipschitz
constants as

positive

rational numbers.

|

q

|

+

1

, and so we can extend it to

multiplication

by

q

on the

real numbers

. Therefore

ℝ

𝖼

is a

vector space

over

ℚ

.
In general, we can define multiplication of real numbers as

u

⋅

v

:

≡

1

2

⋅

(

(

u

+

v

)

2

-

u

2

-

v

2

)

,

(11.3.6)

so we just need squaring

u

↦

u

2

as a map

ℝ

𝖼

→

ℝ

𝖼

. Squaring is not a
Lipschitz map, but it is Lipschitz on every

bounded

domain, which allows us to patch it
together. Define the open and

closed intervals

[

u

,

v

]

:

≡

\setof

x

:

ℝ

𝖼

|

u

≤

x

≤

v

and

(

u

,

v

)

:

≡

\setof

x

:

ℝ

𝖼

|

u

<

x

<

v

.

Although technically an element of

[

u

,

v

]

or

(

u

,

v

)

is a Cauchy real number together with a proof, since the latter inhabits a mere proposition it is uninteresting.
Thus, as is common with subset types, we generally write simply

x

:

[

u

,

v

]

whenever

x

:

ℝ

𝖼

is such that

u

≤

x

≤

v

, and similarly.

Theorem 11.3.7

.

There exists a unique function

(

–

)

2

:

R

c

→

R

c

which extends squaring

q

↦

q

2

of rational numbers and satisfies

∀

(

n

:

ℕ

)

.

∀

(

u

,

v

:

[

-

n

,

n

]

)

.

|

u

2

-

v

2

|

≤

2

⋅

n

⋅

|

u

-

v

|

.

Proof.

We first observe that for every

u

:

ℝ

𝖼

there merely exists

n

:

ℕ

such that

-

n

≤

u

≤

n

, see

\autoref

ex:traditional-archimedean, so the map

e

:

(

∑

n

:

ℕ

[

-

n

,

n

]

)

→

ℝ

𝖼

defined by

e

(

n

,

x

)

:

≡

x

is

surjective

. Next, for each

n

:

ℕ

, the squaring map

s

n

:

\setof

q

:

ℚ

|

-

n

≤

q

≤

n

→

ℚ

defined by

s

n

(

q

)

:

≡

q

2

is Lipschitz with constant

2

⁢

n

, so we can use

\autoref

RC-extend-Q-Lipschitz to
extend it to a map

s

¯

n

:

[

-

n

,

n

]

→

ℝ

𝖼

with Lipschitz constant

2

⁢

n

, see

\autoref

RC-Lipschitz-on-interval for details. The maps

s

¯

n

are

compatible

: if

m

<

n

for some

m

,

n

:

ℕ

then

s

n

restricted to

[

-

m

,

m

]

must agree with

s

m

because both are Lipschitz, and therefore

continuous

in the sense
of

\autoref

RC-continuous-eq. Therefore, by

\autoref

lem:images_are_coequalizers the map

(

∑

n

:

ℕ

[

-

n

,

n

]

)

→

ℝ

𝖼

,

given by

(

n

,

x

)

↦

s

n

⁢

(

x

)

factors uniquely through

ℝ

𝖼

to give us the desired function.
∎

At this point we have the ring structure of the reals and the

archimedean

order. To
establish

ℝ

𝖼

as an archimedean ordered field, we still need

inverses

.

Theorem 11.3.8

.

A Cauchy real is

invertible

if, and only if, it is apart from zero.

Proof.

First, suppose

u

:

ℝ

𝖼

has an inverse

v

:

ℝ

𝖼

By the archimedean principle there is

q

:

ℚ

such that

|

v

|

<

q

. Then

1

=

|

u

⁢

v

|

<

|

u

|

⋅

v

<

|

u

|

⋅

q

and hence

|

u

|

>

1

/

q

, which is to say that

u

#

0

.

For the

converse

we construct the inverse map

(

–

)

-

1

:

\setof

u

:

ℝ

𝖼

|

u

#

0

→

ℝ

𝖼

by patching together functions, similarly to the construction of squaring in

\autoref

RC-squaring. We only outline the main steps. For every

q

:

ℚ

let

[

q

,

∞

)

:

≡

\setof

u

:

ℝ

𝖼

|

q

≤

u

and

(

-

∞

,

q

]

:

≡

\setof

u

:

ℝ

𝖼

|

u

≤

-

q

.

Then, as

q

ranges over

ℚ

+

, the types

(

-

∞

,

q

]

and

[

q

,

∞

)

jointly cover

\setof

u

:

ℝ

𝖼

|

u

#

0

. On each such

[

q

,

∞

)

and

(

-

∞

,

q

]

the
inverse function is obtained by an application of

\autoref

RC-extend-Q-Lipschitz
with Lipschitz constant

1

/

q

2

. Finally,

\autoref

lem:images_are_coequalizers
guarantees that the inverse function factors uniquely through

\setof

u

:

ℝ

𝖼

|

u

#

0

.
∎

We summarize the

algebraic structure

of

ℝ

𝖼

with a

theorem

.

Theorem 11.3.9

.

The Cauchy reals form an archimedean ordered field.

Title

11.3.3 The algebraic structure of Cauchy reals

\metatable

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