$\{ x : x,c \in \mathbb{R} \land c>0 \land \forall j,k \in \mathbb{Z} \ ( k \ge 0 \Rightarrow |x-j2^{-k}| \ge c2^{-k} ) \}$ dense in $\mathbb{R}$? We define the subset $A\subset \mathbb{R}$ as follows: $x\in A$ if and only if there exists $c>0$ so that
$$ |x-j2^{-k}|\geq c2^{-k} $$
holds for all $j\in \mathbb{Z}$ and integers $k\geq 0$. Prove that $A$ is dense

So I tried showing that for any interval $(a,b)\in \mathbb{R}, (a,b)\cap A \neq \emptyset$. But I am having quite a bit of trouble. Does anyone have a hint on how to proceed?

If $x $ is a rational number wchich can be represent as $\frac{s}{l} $ with $GCD(s,l)=1 $ and such that $2 $  not divide $l$ then $x\in A.$ This is because that $$|2^k x -j|\geq\frac{1}{|l|}$$ for all $j\in \mathbb{Z}.$ But set of such $x$ is dense in $\mathbb{R}.$
