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I am studying basic topology so I came across a set named as set of Dyadic rationals. I want to know that how can I prove their density in R?

Antonio Maria Di Mauro
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prakash nainwal
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  • https://math.stackexchange.com/questions/263128/how-can-we-show-that-the-dyadic-rationals-are-dense-in-mathbb-r –  Oct 12 '17 at 05:24

1 Answers1

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Let the set of dyadic rationals be $$A = \left\{ \frac{m}{2^n}: m\in\mathbb{Z}, n\in\mathbb{N} \right\}.$$

We want to show that $A$ is dense in $\mathbb{R},$ that is, for every nonempty open interval $(a,b),$ there exists $m\in\mathbb{Z}$ and $n\in\mathbb{N}$ such that $$a <\frac{m}{2^n}<b.$$

(Remark: The proof is very similar to showing set of rationals $\mathbb{Q}$ is dense in $\mathbb{R}$)

Since $(a,b)$ is nonempty, we have $b-a>0.$ Choose large $n\in\mathbb{N}$ such that $$b-a>\frac{1}{2^n}$$ (This can be done due to the Archimedean Property). Then $$2^nb-2^na>1.$$ Observe that the interval $(2^na,2^nb)$ has length more than $1.$ Therefore, it contains an integer $m,$ that is, $$2^na<m<2^nb.$$ Therefore, $$a<\frac{m}{2^n}<b.$$ Hence, $A$ is dense in $\mathbb{R}.$

Idonknow
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