11 - Archimedean Property of the Reals

Archimedean Property of the Reals corollary

Let $ϵ>0$ then

$$\exists n\in N\text{such that}0<\frac{1}{n}<ϵ$$

Proof

If not, then $\frac{1}{ϵ}$ would be an upper bound for $\mathbb{N}$
This would contradict with Archimedean Property of ℕ

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Properties of $\mathbb{R}$ and $\mathbb{R}\setminus \mathbb{Q}$

Take $a,b\in \mathbb{R}$ with $a<b$ then

1. There is $x\in \mathbb{Q}$ such that $a<x<b$
   (the rationals are dense in the reals)
2. There is $y\in \mathbb{R}\setminus \mathbb{Q}$ such that $a<y<b$
   The irrationals are dense in the reals