Suppose $A\subseteq \mathbb{Z}$ has positive upper density, that is 
$$
\limsup_{n\to \infty} \frac{|A\cap\{1,2,\dots,n\}|}{n}>0.
$$
I would like to know whether the following statement for $A$ can be proven without applying Szemerédi's theorem on arithmetic progression:

For any integer $n$, there exists an integer $b$ such that $\{b, 2b,\dots, nb\}\subseteq A-A$.