Irrationals 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 $\R \setminus \Q$ be the set of irrational numbers.
Then $\R \setminus \Q$ is everywhere dense in $\struct {\R, \size {\, \cdot \,} }$
Proof
Let $x \in \R$.
Let $\epsilon \in \R_{\mathop > 0}$
Either $x \in \Q$ or $x \in \R \setminus \Q$.
Suppose $x \in \R \setminus \Q$.
Let $y := x$.
Then:
- $\size {x - y} < \epsilon$
Suppose $x \in \Q$.
Let $n \in \N : n > \dfrac {\sqrt 2} \epsilon$
Let $y := x + \dfrac {\sqrt 2} n$
Then $y \in \R \setminus \Q$.
Furthermore:
\(\ds \size {x - y}\) | \(=\) | \(\ds \size {\frac {\sqrt 2} n}\) | ||||||||||||
\(\ds \) | \(<\) | \(\ds \epsilon\) |
In both cases $x$ was arbitrary.
Hence:
- $\forall x \in \R : \exists \epsilon \in \R_{\mathop > 0} : \exists y \in \R \setminus \Q : \size {x - y} < \epsilon$
By definition, $\R \setminus \Q$ is dense in $\R$.
$\blacksquare$
Sources
- 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): $\S 1.3$: Normed and Banach spaces. Topology of normed spaces