Rationals are Everywhere Dense in Reals/Normed Vector Space

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Theorem

Let $\struct {\R, \size {\, \cdot \,}}$ be the normed vector space of real numbers.

Let $\Q$ be the set of rational numbers.


Then $\Q$ are everywhere dense in $\struct {\R, \size {\, \cdot \,}}$

Proof

We have that Between two Real Numbers exists Rational Number:

$\forall a, b \in \R : a < b : \exists r \in \Q : a < r < b$

Let $a := x$ with $x \in \R$.

Let $\epsilon \in \R_{\mathop > 0} : r - a < \epsilon$.

Let $b := x + \epsilon$.

Then:

\(\displaystyle x - \epsilon\) \(<\) \(\displaystyle x\)
\(\displaystyle \) \(<\) \(\displaystyle r\)
\(\displaystyle \) \(<\) \(\displaystyle x + \epsilon\)

Since this holds for all $x$, we have that:

$\forall x \in \R : \exists \epsilon \in \R_{\mathop > 0} : \exists r \in \Q : \size {x - r} < \epsilon$

By definition, $\Q$ is dense in $\R$.


Sources