Real numbers/Ordering axioms/Archimedean/Implications/Fact

1. For ${}x,y\in \mathbb {R}$ with ${}x>0$ there exists ${}n\in \mathbb {N}$ such that ${}nx\geq y$.
2. For ${}x>0$ there exists a natural number ${}n$ such that ${}{\frac {1}{n}}<x$.
3. For two real numbers ${}x<y$ there exists a rational number ${}n/k$ (with ${}n\in \mathbb {Z}$, ${}k\in \mathbb {N} _{+}$) such that

   ${}x<{\frac {n}{k}}<y\,.$

Proof 1, 2, Write another proof