OPEN

Let $A$ be a finite set and
\[B=\{ n \geq 1 : a\nmid n\textrm{ for all }a\in A\}.\]
Is it true that, for every $m>n$,
\[\frac{\lvert B\cap [1,m]\rvert }{m}< 2\frac{\lvert B\cap [1,n]\rvert}{n}?\]

The example $A=\{a\}$ and $n=2a-1$ and $m=2a$ shows that $2$ would be best possible.