# Rationals, irrational and dyadic numbers with archimedean propierty

i need help to test the following:

“prove that every open interval in real numbers contains rational, irrational and dyadic numbers.”

I had tried defining an isometry on a set where the property was fulfilled but my teacher told me that it must be through the Archimedean property, I would be very grateful if you could help me.

Let $$a be any real numbers. By Archimedean property there exists a natural number $$n$$ such that $$n(b-a)>1$$. It follows that $$2^n(b-a)>1$$, so the interval $$(a,b)$$ has length bigger that $$\tfrac {1}{2^n}$$ and so it contains a dyadic rational points of the form $$\tfrac {k}{2^n}$$ for some integer $$k$$, because such points are placed at the real line with a distance $$\tfrac {1}{2^n}<|b-a|$$ between consecutive points. Similarly we can show that the interval $$(a,b)$$ contains an rational point of the form $$\tfrac {k}{2^n}\cdot \tfrac{1}{\sqrt 2}$$ (with a little care assuring that $$k\ne 0$$).