Proof of Density: D = dyadic rationals set is dense on [0,1] | Homework Help

[aortizmena]

Homework Statement

Prove that D={\frac{m}{2^{n}} : n\in N , m=0,1,2,...,2^{n}} (dyatic rationals set) is dense on [0,1] , i.e. if (a,b) \subset [0,1] then (a,b) \bigcap D \neq emptyset

Homework Equations

The Attempt at a Solution

Is it wrong if I just state that because a,b\in\Re we know that there exists m,n\inN, l=\frac{m}{2^{n+1}}, u=\frac{m}{2^{n-1}} and r=\frac{m}{2^{n}} such that l \leq a \leq r \leq b \leq u?

 

[Gib Z]

Why do you need the extremal inequalities? You should only need a \leq r \leq b. Also it seems like you are trying to say "Because a and b are real, its is obvious there are naturals m,n that allow us to conclude the result", which really is not substantial enough.

Instead, I would use the Archimedian property to argue where there exists n \in \mathbb{N} such that b-a < \frac{1}{2^n}, then you a Pigenhole Principle type argument to see why D should intersect with (a,b) .