IR-TST 2007 Problem 2

Let $A$ be the largest subset of $\{1,\dots,n\}$ such that for each $x\in A$ , $x$ divides at most one other element in $A$ . Prove that $$\frac{2n}3\leq |A|\leq \left\lceil \frac{3n}4\right\rceil.$$