Rationals are Dense in the Reals

Last updated Nov 1, 2022

$\mathbb{Q}$ is a Dense subset of $\mathbb{R}$.

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

TODO

• by Dedekind Cut Construction of the Reals.

$\mathbb{Q}^{n}$ is a Dense subset of $\mathbb{R}^{n}$ for $n \in \mathbb{N}$

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

$\mathbb{Q}^{\kappa}$ is a Dense subset of $\mathbb{R}^{\kappa}$ for any cardinal $\kappa$.

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