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 $\{b, 2b,\dots, nb\}\subseteq A-A$.

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 $\{b, 2b,\dots, nb\}\subseteq A-A$, where $A-A=\{a_1-a_2: a_1, a_2\in A\}$.