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

Show that in $ℝ$ 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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