Let $A$ be a set of real numbers such that ⊆(1,∞) and dense in $(1,\infty)$, I would like to prove that $B= \left\{\frac{a}{(a+1)n^2}|a∈A,n∈N\right\}$ is not dense in $[1,0]$.
Well I'm having some trouble showing it after I said let $x,y\in[0,1]$ while $y>x$ that for every $b\in B$ we get $b\geq y$ or $b\leq x$ and then I get stuck in showing that it leads to $A$ and therefore $B$ is not dense in $[0,1]$.