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

  • #1
aortizmena
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Homework Statement


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

Homework Equations


The Attempt at a Solution


Is it wrong if I just state that because a,b[tex]\in[/tex][tex]\Re[/tex] we know that there exists m,n[tex]\in[/tex]N, l=[tex]\frac{m}{2^{n+1}}[/tex], u=[tex]\frac{m}{2^{n-1}}[/tex] and r=[tex]\frac{m}{2^{n}}[/tex] such that l [tex]\leq[/tex] a [tex]\leq[/tex] r [tex]\leq[/tex] b [tex]\leq[/tex] u?
 
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  • #2
Why do you need the extremal inequalities? You should only need [itex] a \leq r \leq b [/itex]. 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 [itex] n \in \mathbb{N} [/itex] such that [tex] b-a < \frac{1}{2^n} [/tex], then you a Pigenhole Principle type argument to see why D should intersect with (a,b) .
 
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