rational numbers are dense in the reals

Rational numbers are dense in the reals.

a)

Archimedian Property

$$\forall x\in \mathbb{R}\exists n\in \mathbb{N}:n<x$$

… $\mathbb{N}$ has no upper bound in $\mathbb{R}$.

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b)

$$\forall ϵ>0,\exists n\in \mathbb{N}:\frac{1}{n}<ϵ$$

c)

$$\forall x,y\in \mathbb{R},x<y,\exists \frac{m}{n}\in \mathbb{Q}:x\le \frac{m}{n}\le y$$

→ If I pick two real numbers, there is always a rational number in between them.