Real numbers/Ordering axioms/Archimedean/Implications/Fact/Proof/Exercise

Show that in ${}\mathbb {R}$ the following properties hold.

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

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

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