Rationals are Dense in the Reals

Last updated Nov 1, 2022

# Statement 1

$\mathbb{Q}$ is a Dense

subset of

\mathbb{R}

# Proof 1

Recalling the Sequence Completion Construction of the Reals, every $x \in \mathbb{R}$ is defined as a an Equivalence Relation

over Cauchy Sequences in

\mathbb{Q}

, where elements of

\mathbb{Q}

are identified with constant sequences. The space is endowed with the Limsup Norm. Let

x \in \mathbb{R}

be identified by the Sequence $({x}{n}){n=1}^{\infty} \subset \mathbb{Q}$. This Sequence precisely gives us a Sequence of

\mathbb{Q} \subset \mathbb{R}

that converges to

x

. Thus, because Closure of a Set in a Metric Space is all its Sequential Limits,

\text{cl}\mathbb{Q} = \mathbb{R}

and

\mathbb{Q}

is Dense.

TODO

Complete Sequence Completion Construction of the Reals

# Proof 2

TODO

by Dedekind Cut Construction of the Reals.

# Statement 2

$\mathbb{Q}^{n}$ is a Dense

subset of

\mathbb{R}^{n}

for

n \in \mathbb{N}

# Proof

Use Statement 1 and apply Product of Dense Sets is Dense. $\blacksquare$

# Statement 3

$\mathbb{Q}^{\kappa}$ is a Dense

subset of

\mathbb{R}^{\kappa}

for any cardinal

\kappa

# Proof

Use Statement 1 and apply Product of Dense Sets is Dense. $\blacksquare$

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Rationals are Dense in the Reals

# Statement 1

Dense

# Proof 1

Equivalence Relation

Cauchy Sequence

Sequence

Sequence

Sequence

Closure of a Set in a Metric Space is all its Sequential Limits

Dense

TODO

# Proof 2

TODO

# Statement 2

Dense

# Proof

# Statement 3

Dense

# Proof

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