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We define the subset $Asubset mathbbR$ as follows: $xin A$ if and only if there exists $c>0$ so that
$$ |x-j2^-k|geq c2^-k $$
holds for all $jin mathbbZ$ and integers $kgeq 0$. Prove that $A$ is dense

So I tried showing that for any interval $(a,b)in mathbbR, (a,b)cap A neq emptyset$. 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 $fracsl $ with $GCD(s,l)=1 $ and such that $2 $ not divide $l$ then $xin A.$ This is because that $$|2^k x -j|geqfrac1$$ for all $jin mathbbZ.$ But set of such $x$ is dense in $mathbbR.$

$x in A Longleftrightarrow$

there exists $c > 0$ with $forall j in Z$, integer $n geq 0$, $c leq |2^nx - j|.$

$x notin A Longleftrightarrow$
$forall c > 0$, there exists $j in Z$, integer $n geq 0$ with $|2^nx - j| < c$
$Longleftrightarrow$

exists $j in Z$, integer $n geq 0$ with $x = j/2^n$
$Longleftrightarrow x$ is a dyadic rational.

Since $R - Q subset R - x : x text dyadic rational = A$

and $R - Q$ is dense, $A$ is dense.