11.2.2 Dedekind reals are Cauchy complete

11.2.2 Dedekind reals are Cauchy complete

Recall that

x

:

ℕ

→

ℚ

is a

Cauchy sequence

when it satisfies

∏

(

ϵ

:

ℚ

+

)

∑

(

n

:

ℕ

)

∏

(

m

,

k

≥

n

)

|

x

m

-

x

k

|

<

ϵ

.

(11.2.1)

Note that we did

not

truncate the inner existential because we actually want to
compute rates of convergence—an approximation without an error estimate carries little
useful information. By

\autoref

thm:ttac, (

11.2.1

) yields a function

M

:

ℚ

+

→

ℕ

, called the

modulus of convergence

, such that

m

,

k

≥

M

⁢

(

ϵ

)

implies

|

x

m

-

x

k

|

<

ϵ

. From this we get

|

x

M

⁢

(

δ

/

2

)

-

x

M

⁢

(

ϵ

/

2

)

|

<

δ

+

ϵ

for all

ϵ

:

ℚ

+

. In fact, the map

(

ϵ

↦

x

M

⁢

(

ϵ

/

2

)

)

:

ℚ

+

→

ℚ

carries the same information about the limit as the
original Cauchy condition (

11.2.1

). We shall work with these
approximation functions rather than with Cauchy sequences.

Definition 11.2.2

.

A

Cauchy approximation

is a map

x

:

Q

+

→

R

d

which satisfies

∀

(

δ

,

ϵ

:

ℚ

+

)

.

|

x

δ

-

x

ϵ

|

<

δ

+

ϵ

.

(11.2.2)

The

limit

of a Cauchy approximation

x

:

Q

+

→

R

d

is a number

ℓ

:

R

d

such
that

∀

(

ϵ

,

θ

:

ℚ

+

)

.

|

x

ϵ

-

ℓ

|

<

ϵ

+

θ

.

Theorem 11.2.4

.

Every Cauchy approximation in

R

d

has a limit.

Proof.

Note that we are showing existence, not mere existence, of the limit.
Given a Cauchy approximation

x

:

ℚ

+

→

ℝ

𝖽

, define

L

y

⁢

(

q

)

:

≡

∃

(

ϵ

,

θ

:

ℚ

+

)

.

L

x

ϵ

(

q

+

ϵ

+

θ

)

,

U

y

⁢

(

q

)

:

≡

∃

(

ϵ

,

θ

:

ℚ

+

)

.

U

x

ϵ

(

q

-

ϵ

-

θ

)

.

It is clear that

L

y

and

U

y

are inhabited, rounded, and disjoint. To establish
locatedness, consider any

q

,

r

:

ℚ

such that

q

<

r

. There is

ϵ

:

ℚ

+

such
that

5

⁢

ϵ

<

r

-

q

. Since

q

+

2

⁢

ϵ

<

r

-

2

⁢

ϵ

merely

L

x

ϵ

⁢

(

q

+

2

⁢

ϵ

)

or

U

x

ϵ

⁢

(

r

-

2

⁢

ϵ

)

. In the first case
we have

L

y

⁢

(

q

)

and in the second

U

y

⁢

(

r

)

.

To show that

y

is the limit of

x

, consider any

ϵ

,

θ

:

ℚ

+

. Because

ℚ

is dense in

ℝ

𝖽

there merely exist

q

,

r

:

ℚ

such that

x

ϵ

-

ϵ

-

θ

/

2

<

q

<

x

ϵ

-

ϵ

-

θ

/

4

<

x

ϵ

<

x

ϵ

+

ϵ

+

θ

/

4

<

r

<

x

ϵ

+

ϵ

+

θ

/

2

,

and thus

q

<

y

<

r

. Now either

y

<

x

ϵ

+

θ

/

2

or

x

ϵ

-

θ

/

2

<

y

.
In the first case we have

x

ϵ

-

ϵ

-

θ

/

2

<

q

<

y

<

x

ϵ

+

θ

/

2

,

and in the second

x

ϵ

-

θ

/

2

<

y

<

r

<

x

ϵ

+

ϵ

+

θ

/

2

.

In either case it follows that

|

y

-

x

ϵ

|

<

ϵ

+

θ

.
∎

For sake of completeness we record the classic formulation as well.

Corollary 11.2.5

.

Suppose

x

:

N

→

R

d

satisfies the Cauchy condition (

11.2.1

). Then
there exists

y

:

R

d

such that

∏

(

ϵ

:

ℚ

+

)

∑

(

n

:

ℕ

)

∏

(

m

≥

n

)

|

x

m

-

y

|

<

ϵ

.

Proof.

By

\autoref

thm:ttac there is

M

:

ℚ

+

→

ℕ

such that

x

¯

(

ϵ

)

:

≡

x

M

⁢

(

ϵ

/

2

)

is a Cauchy approximation. Let

y

be its limit, which exists by

\autoref

RD-cauchy-complete. Given any

ϵ

:

ℚ

+

, let

n

:

≡

M

(

ϵ

/

4

)

and observe that, for any

m

≥

n

,

|

x

m

-

y

|

≤

|

x

m

-

x

n

|

+

|

x

n

-

y

|

=

|

x

m

-

x

n

|

+

|

x

¯

⁢

(

ϵ

/

2

)

-

y

|

<

ϵ

/

4

+

ϵ

/

2

+

ϵ

/

4

=

ϵ

.

∎

Title

11.2.2 Dedekind reals are Cauchy

complete

\metatable

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